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S. Ishikawa
Mathematical Foundations of Measurement Theory
Keio University Press (2006)

Includes bibliographical references and index
AMS Subject Classiﬁcations (2000) Primary 62A01, Secondary 93Axx
1. applied mathematics, 2. statistics, 3. system theory
Keio University Press, Tokyo, JAPAN (URL: http://www.keio-up.co.jp/eng/)− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −−
The manuscript of this book was written with LaTeX.
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Published by the author,
distributed by Keio University Press Inc.
19-30, 2chome, Mita, Minato-ku
Tokyo 108-8346 Japan
Copyright © 2006 Shiro ISHIKAWA.
All rights reser ved.
No par t of thie book may be reproduced inany manner without the written consent of
the publisher, except in the case of brief
excerpts in critical ar ticles and reviews.
Printed in JapanFirst edition, 2006

Wn@MFMnooFMZtW@MF

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Contents
Preface
1 The philosophy of measurement theory
1
1.1 How to construct “measurement theory” . . . . . . . . . . . . . . . . . . . 1
1.2 What is measurement theory? . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Measurement theory in engineering . . . . . . . . . . . . . . . . . . . . . . 10
1.4 The spirit of “the mechanical world view” . . . . . . . . . . . . . . . . . . 12
2 Measurements (Axiom 1)
2.1 Mathematical preparations . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 The meanings of observables and crisp observables . . . . . . . . . . . . . .
2.4 Measurement (Axiom 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Operations of observables . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Frequency probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Appendix (Bell’s thought experiment) . . . . . . . . . . . . . . . . . . . .
2.9.1 EPR thought experiment . . . . . . . . . . . . . . . . . . . . . . . .
2.9.2 Bell’s thought experiment . . . . . . . . . . . . . . . . . . . . . . .

15
15
21
26
29
30
33
42
45
49
49
51

3

The relation among systems (Axiom 2)
3.1 Newton Equation and Schrödinger equation . . . . . . . . . . . . . . . . .
3.2 The relation among systems (Deﬁnition) . . . . . . . . . . . . . . . . . . .
3.3 Examples (Several tree structures) . . . . . . . . . . . . . . . . . . . . . .
3.4 The relation among systems (Axiom 2) . . . . . . . . . . . . . . . . . . .
3.5 Heisenberg picture and Schrödinger picture . . . . . . . . . . . . . . . . .
3.6 Measurability theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Appendix (Bell’s inequality) . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7.1 Deterministic evolution or Stochastic evolution? . . . . . . . . . . .
3.7.2 Generalized Bell’s inequality in mathematics . . . . . . . . . . . .
3.7.3 Generalized Bell’s inequality in Measurements . . . . . . . . . . . .

55
55
58
60
64
66
67
69
69
70
73

4 Boltzmann’s equilibrium statistical mechanics
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Dynamical aspects of equilibrium statistical mechanics . . . . . . . . . . .
4.3 Probabilistic aspects of equilibrium statistical mechanics . . . . . . . . . .
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Appendix (The law of large numbers) . . . . . . . . . . . . . . . . . . . . .

75
75
78
86
88
90

5 Fisher’s statistics I (under Axiom 1)
91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Fisher’s maximum likelihood method . . . . . . . . . . . . . . . . . . . . . 92
5.2.1 Fisher’s maximum likelihood method . . . . . . . . . . . . . . . . . 92
5.2.2 Monty Hall problem in PMT . . . . . . . . . . . . . . . . . . . . . . 105

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5.3
5.4
5.5
5.6

Inference interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Testing statistical hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Measurement error model in PMT . . . . . . . . . . . . . . . . . . . . . . 117
Appendix (Iterative likelihood function method) . . . . . . . . . . . . . . 121

6 Fisher’s statistics II (related to Axioms 1 and 2)
127
6.1 Regression analysis I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.1.2 Regression analysis I in measurements . . . . . . . . . . . . . . . . 130
6.2 Bayes operator, Schrödinger picture, and S-states . . . . . . . . . . . . . . 134
6.3 Regression analysis II in measurements . . . . . . . . . . . . . . . . . . . . 142
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7 Practical logic
149
7.1 Measurement, inference, control andpractical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.2 Quasi-product observables with dependence . . . . . . . . . . . . . . . . . 151
7.3 Consistency and syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.3.1 Consistent condition . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.3.2 Practical syllogism . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.5 Appendix (Zadeh’s fuzzy sets theory) . . . . . . . . . . . . . . . . . . . . 169
7.5.1 What is Zadeh’s fuzzy sets theory? . . . . . . . . . . . . . . . . . 169
7.5.2 Why is Zadeh’s paper cited frequently? . . . . . . . . . . . . . . . 172
8 Statistical measurements in C ∗ -algebraic formulation
175
8.1 Statistical measurements (C ∗ -algebraic formulation) . . . . . . . . . . . . . 176
8.1.1 General theory of statistical measurements . . . . . . . . . . . . . . 176
8.1.2 Examples of statistical measurements . . . . . . . . . . . . . . . . . 182
8.1.3 Problems (statistical measurements) . . . . . . . . . . . . . . . . . 188
8.2 General statistical system (Example) . . . . . . . . . . . . . . . . . . . . . 193
8.3 Bayes theorem in statistical MT . . . . . . . . . . . . . . . . . . . . . . . 195
8.4 Kalman ﬁlter in noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.4.1 The measurement theoretical formulation of Figure (8.45) . . . . . . 200
8.4.2 Kalman ﬁlter in Noise . . . . . . . . . . . . . . . . . . . . . . . . . 202
8.5 Information and entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
8.6 Belief measurement theory (=BMT) . . . . . . . . . . . . . . . . . . . . . 208
8.6.1 The general argument about BMT . . . . . . . . . . . . . . . . . . 209
8.6.2 The principle of equal weight . . . . . . . . . . . . . . . . . . . . . 212
8.6.3 Is BMT necessary? . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.7 Appendix (Bertrand’s paradox) . . . . . . . . . . . . . . . . . . . . . . . . 218
8.7.1 Review (Bertrand’s paradox) . . . . . . . . . . . . . . . . . . . . . . 218
8.7.2 Bertrand’s paradox in measurement theory . . . . . . . . . . . . . . 219
9 Statistical measurements in W ∗ -algebraic formulation
225
9.1 Statistical measurements (W ∗ -algebraic formulation) . . . . . . . . . . . . 225∗∗
9.2 The relation among systems (ProclaimW 2 in SMTW ) . . . . . . . . . . . 236
9.3 Quantum mechanics in B(L2 (R)) . . . . . . . . . . . . . . . . . . . . . . . 244
9.3.1 Schrödinger equation and Heisenberg kinetic equation . . . . . . . . 244
9.3.2 A simplest example of Schrödinger equation . . . . . . . . . . . . . 245
9.3.3 The de Broglie paradox . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.4 The method of moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
9.4.1 The moment method . . . . . . . . . . . . . . . . . . . . . . . . . . 248
9.4.2 Example 1 [Normal distribution (= Gaussian distribution)] . . . . 250
9.4.3 Example 2 (measurement error model in SMT) . . . . . . . . . . . 252
9.5 Principal components analysis in MT . . . . . . . . . . . . . . . . . . . . . 255

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10 Newtonian mechanics in measurement Theory261
10.1 Kolmogorov’s extension theorem in W ∗ -algebra . . . . . . . . . . . . . . . 261
10.2 The deﬁnition of “trajectories” . . . . . . . . . . . . . . . . . . . . . . . . 264
10.3 Trajectories and Newtonian mechanics . . . . . . . . . . . . . . . . . . . . 266
10.4 Brownian motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
10.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
11 Measurement error
275
11.1 Approximate measurements for quantities . . . . . . . . . . . . . . . . . . 275
11.2 The estimation under loss function in statistics . . . . . . . . . . . . . . . . 277
11.3 Random observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
11.4 The principle of equal probability . . . . . . . . . . . . . . . . . . . . . . . 288
12 Heisenberg’s uncertainty relation
293
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
12.2 Example due to Arthurs-Kelley . . . . . . . . . . . . . . . . . . . . . . . . 297
12.3 Approximate simultaneous measurement . . . . . . . . . . . . . . . . . . . 299
12.4 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
12.5 Existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
12.6 Uncertainty relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
12.7 EPR-experiment and Heisenberg’s uncertainty relation . . . . . . . . . . . 315
Postscript

319

Bibliography

325

Index

336

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PREFACEIt is well known that the dynamical system theory (DST) starts from the following
equations:

dx(t)

 dt = f (x(t), u1 (t), t), x(0) = x0 · · · (state equation),
DST

=


 y(t) = g(x(t), u2 (t), t)(D)(measurement equation)

where u1 and u2 are external forces (or noises). Also recall that quantum mechanics isformulated as the following form:

quantum mechanics= [the rule of time evolution](Schrödinger equation)+[measurement](Born’s quantum measurements)(Q)
The above two theories (D) and (Q) are, of course, fundamental and famous. Thus, aquarter of a century ago, I already knew them. However, about ﬁfteen years ago, I was
suddenly surprised by the similarity between (D) and (Q), particularly, the fact that:(F) the term “measurement” is common to both dynamical system theory (D) andquantum system theory (Q).
This surprise urged me to propose “measurement theory”.

I want to share my surprisewith all people .1 This is the reason for this book.
Shiro ISHIKAWA2
21st, October, 2006

1

Some sections of this book were lectured in the master-course program: “Advanced study of mathematics A” at Keio university (three-hour lecture every week from April to July in 2006).
2
For the further information of our theory, see “http://www.keio-up.co.jp/kup/mfomt/”

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It is recommended to read this book as follows:

Chap. 1yChap. 2yChap. 4←−−−

Chap. 3


y

−−−→

Chap. 7

Chap. 5yChap. 6yChap. 8yChap. 9yChap. 10yChap. 11yChap. 12

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Chapter 1
The philosophy of measurement
theoryThe purpose of this book is to propose “mathematical foundations of measurement theory”. The
statement:“There is no science without measurements”(1.1)

is an old famous saying, which of course emphasizes the importance of “measurement”. We believein the saying, i.e., the concept of “measurements” should be the most fundamental in science.
However, it is certain that we do not have an authorized “measurement theory” in science yet.
Thus, we think that it is worthwhile proposing the mathematical foundations of “measurement
theory” 1 :Chapters 2, 3, 8 · · · Mathematical foundations of measurement theory
Chapter 4· · · An application (of measurement theory) to statistical mechanics

Chapters 5∼12

· · · Several theories (e.g., statistics, classical and quantum system theories,etc.) in measurement theory

It should be noted that “measurement theory” and “theoretical physics” are diﬀerent. In particular, their philosophies are completely diﬀerent. Although it is a matter of course that it is
impossible to understand the philosophy of measurement theory without the complete knowledgeof measurements (i.e., the contents of Chapters 2 ∼ 12), the philosophy of measurement theory is
also indispensable for the understanding of measurement theory. Therefore, in this ﬁrst chapter,we devote ourselves to the philosophy of measurement theory.

1.1

How to construct “measurement theory”

It is well known that the dynamical system theory (DST, classical system theory)
1

The measurement theory is proposed in the references [41]∼[48],[55] in this book. We devote ourselvesto the mathematical aspect of “measurement theory”. For the other aspects (e.g., practical and general
aspects), see [30], which is educational and enlightening.

1
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starts from the following equations:DST = dx(t) dt = f (x(t), u1 (t), t), x(0) = x0

· · · ((stochastic) state equation) ,2(measurement equation)y(t) = g(x(t), u2 (t), t)(1.2a)

where u1 and u2 are external forces (or noises),or more precisely,= “Apply (1.2a) to every phenomenon by an analogy of Newtonian mechanics andthe coin-tossing problem” 3.

(1.2b)

That is, DST is usually believed to be a kind of epistemology called “the mechanicalworld view”, namely, an epistemology to understand and analyze (moreover, control)
every phenomenon — economics, psychology, engineering and so on — by an analogy of
Newtonian mechanics (and coin-tossing).
Also recall that quantum mechanics is formulated as the following form (cf. von
Neumann [84]):

quantum mechanics=[measurement](Born’s quantum measurement)+[the rule of time evolution](Schrödinger equation)(1.3)
which was discovered by W. Heisenberg, E. Schrödinger, M. Born in between 1924 and1926.
Here, it should be noted that the term “measurement” appears in both (1.2) and (1.3).
Thus, our proposal, i.e., “measurement theory (=MT)”, is constructed as follows:(I1 ) Quantum mechanics (1.3) is formulated in B(H), the algebra composed of all
bounded linear operators on a Hilbert space H (cf. von Neumann (1932: [84])).
Thus it is easy to generalize quantum mechanics in C ∗ -algebra A (⊆ B(H), cf.
Deﬁnition 2.1 in §2.1) such that it includes DST (1.2) as a special case. Namely,(1.2)+(1.3)⊂“MT”.
That is, as a kind of generalization of quantum mechanics (1.3), we can propose as follows:
2

A stochastic diﬀerential equation (or stochastic diﬀerence equation) in dynamical system theory isusually called a stochastic state equation.
3
That is, DST is, from the mathematical point of view, based on “the theory of diﬀerential
equations” and “probability theory”. Thus, I think that I.Newton (cf. [66]) and A.Kolmogorov (cf.
[56]) are greatest in DST.

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3

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/“measurement theory (or in short, MT)”=[measurement] + [“the rule of the relation among systems”]“Axiom 1 (2.37)”

in C ∗ -algebra A“Axiom 2 (3.26)”(1.4a)or more precisely,=“Apply (1.4a) to every phenomenon by an analogy of quantum mechanics”(1.4b)(For the details, see Chapter 2 [Axiom 1 (2.37)], and Chapter 3 [Axiom 2 (3.26)]). Hereit should be noted that MT (= Axiom 1 + Axiom 2) is composed of a few key-words i.e.,(I2 ) system, state, observable, measurement, measured-value, probability, Markov relation, sequential observable, Heisenberg picture, etc.
and Axioms 1 and 2 explain how to use these words. Roughly speaking, Axioms 1 and 2
say “Use these words by analogy of quantum mechanics”.4
We have the classiﬁcation of MT as follows:5
in a commutative C ∗ -algebra C0 (Ω)
 “classical MT”“MT” =
 “quantum MT” in a non-commutative C ∗ -algebra B(H)

(1.5)

where a C ∗ -algebra is either commutative or non-commutative. Also, as mentioned in(I1 ), we consider the following correspondence:



“MT”

=




“classical MT” in (1.5)↔

DST in (1.2)(1.6)“quantum MT” in (1.5)↔

quantum theory in (1.3)

4Thus, our approach is, from the philosophical point of view, characterized as so called foundationalism.
5
As seen later (i.e., Chapter 8), we also have the classiﬁcation of MT, i.e., “(pure) measurement theory(= PMT)” and “statistical measurement theory ( =SMT)”. That is, PMT (=“(pure) measurement theory”) in Chapters 2 ∼ 7
MT (=“measurement theory”)



SMT (=“statistical measurement theory”) in Chapters 8 ∼)PMT is essential. That is, we can say that there is no SMT without PMT. (Cf. Chapter 8.)

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1.2

What is measurement theory?

We think that the question “What is measurement theory?” is much more diﬃcult
than the question “How is measurement theory constructed?”.
As mentioned in (I1 ) in §1.1, MT is the mathematical generalization of quantum mechanics (1.3). That is, MT is not quantum mechanics but “something beyond mechanics”.
Thus, we can assert that(I3 ) MT is the mathematical representation of the epistemology called “the mechanicalworld view” (just like DST(1.2) is).
Also, it should be noted that MT is quite a wide theory, that is, we assert:(I4 ) MT is the most fundamental theory of so-called “theoretical informatics”, including
dynamical system theory, quantum system theory, practical logic, statistics, circuit
theory, control theory, chaotic system theory, multivariate analysis, information
theory, automata theory, OR, game theory, etc.
This will be discussed in Chapters 5 ∼ 12. Also, note that the above (I4 ) should be
regarded as the same as the following assertion:(I5 ) The term: “theoretical informatics” is deﬁned as the academic discipline that iscomposed of all theories understood in MT. That is, “theoretical informatics” =“MT”.
We assert that

the most fundamental theory of theoretical physics =⇒ ‘TOE (string theory(?))’6the most fundamental theory of theoretical informatics =⇒ MT

And therefore, we can present the following table, which indicates where MT is in science.
6

The string theory (cf. [28]) is not necessarily authorized yet. Thus, in this book, the term ‘TOE
(Theory of Everything)’ is used as the symbol of the most fundamental theory of theoretical physics.
As emphasized in this section, the philosophy of theoretical physics is diﬀerent from that of theoreticalinformatics. And thus, the meanings of “the most fundamental theory” are respectively diﬀerent intheoretical physics and in theoretical informatics.

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5

Table
(1.7)


Mathematics

mathematical logic, algebra, geometry, analysis, etc.
(C1 )


“Cantor’ssets theory”









Newtonian mechanics












quantum mechanics












 Theor. Physics  Maxwell’s electromagnetic theory






Einstein’s relativity theory
(C2 )







‘TOE’





 Weinberg-Salam theory











 quantum chromodynamics








etc.
























[My proposal in this book]












dynamical system theory












quantum system theoryMath. Sci.
(C3 )practical logic

Sci.


Theor.
 statistics, circuit theory
Theor. 










Theor.
Informatics

control theory








multivariate analysis
MT(=measurement
theory)











information theory












chaotic system theory










 automata theory












OR, game theory, etc.




























 The third mathematical scientiﬁc theory




(C4 )




(undiscovered)





8


>

>

< economics, chemistry, biology, medicine, psychology, Usual Sci. Theor.
statistical mechanics, ﬂuid mechanics,


>
>

:
(C5 )
engineering (also, see (I7 ) and (I8 ) later), etc.

That is, the mathematical structures of all theories in (C3 ) are common, and thus, they
are discussed in the framework of MT.
We add the following remark.
Remark 1.1. (About Table (1.7)).
(a). Note that the class (C3 ) (= (I4 )) is usually called “applied mathematics”.

In thissense, we think that MT is the main part of so-called applied mathematics.
(b). For example, if electromagnetic theory and relativity theory can not be uniﬁed, wemust consider two categories (e.g., “Theoretical physics (I)” and “Theoretical physics(II)”) in theoretical physics. However, most physicists believe that physics consists of only
one category, that is, the theories in (C2 ) must be uniﬁed in the most fundamental theory(= ‘TOE’). The purpose of this book is, of course, to show that the theories in (C3 ) are

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mathematically understood in MT. Also, in this book, “Newtonian mechanics” [resp.
“quantum mechanics” ]in MT is called “classical system theory (= dynamical sys³
tem theory)” [resp. “quantum system theory”] though the addition of “measurement
equation” to DTS(1.2a) should be regarded as the act of genius (since there is no concept´
of “measurement” in Newtonian mechanics) . That is, the two (i.e., Newtonian mechanics and quantum mechanics) are common in both “theoretical physics” and “theoreticalinformatics” (cf. §10.5).
(c). The purpose of theoretical physics is to represent “natural forces” in terms of mathematics. On the other hand, as mentioned in (I3 ), MT is a kind of epistemology called “themechanical world view”, namely, an epistemology to understand and analyze (moreover,control) every phenomenon — economics, psychology, engineering and so on — by ananalogy of mechanics. That is, MT is the mathematical representation of “the mechanical world view”. Or, precisely speaking, the deﬁnition of “the mechanical world view” isgiven by MT.
(d). From the mathematical point of view, the diﬀerence between “theoretical physics” and“theoretical informatics” is that of “diﬀerential geometry” and “the theory of Hilbert
spaces (or operator algebras)”. Cf. Remark 8.26.
(e). It is a matter of course that the theories in theoretical physics (= (C2 ) in (1.7))
should be tested by experiments. For example, the question: “Is electromagnetic theory
experimentally true or not?” is meaningful. In fact, serious experiments have been often
conducted as big projects (such as SERN, Kamioka Observatory (Japan), etc.) in theoretical physics. On the other hand, the experimental tests of the theories in theoreticalinformatics (= (C3 )) are nonsense. For example, the experimental test of statistics ismeaningless just like that of mathematics (e.g., linear algebra) is obviously meaningless.7
Thus, we think that the question: “Is statistics experimentally true or not?” is meaningless. However, it should be noted that the question: “Is statistics convenient (=useful)?” is meaningful.
(f). We hope that some will ﬁnd and propose “The third mathematical scientiﬁc theory
in (C4 )”.
¥7

There may be some truth in the assertion that statistics is a kind of mathematics. However, as
mentioned in Table (1.7), we think that “statistics” = “mathematics + something”.

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Summing up, we assert the following table:
Table(1). the theories
in this ﬁeld
(cf. Remark 1.1 (b))(2). the most fundamental theory(cf. Remark 1.1 (b))(3). the purpose(cf. Remark 1.1 (c))(4). mathematical language(cf. Remark 1.1 (d))(5). experimentallytrue or false(cf. Remark 1.1 (e))(1.8a)

Theoretical Physics

Theoretical Informatics

classical and quantum mechanics,
electromagnetic theory,
Weinberg-Salam theory, etc. (cf. (C2 ))‘TOE (Theory of Everything)’(will be proposed in the future)
the mathematical representationof “force”
diﬀerential geometry (gauge theory)

dynamical system theory, statistics,
logic, quantum system theory,
information theory, etc. (cf. (C3 ))MT (measurement theory)(proposed in this book cf. [41]∼[48],[55])the mathematical representationof “the mechanical world view”
operator algebra(functional analysis, real analysis)
meaninglessmeaningful

Next let us consider the following problem.
Problem 1.2. (“experimentally true or false” and “theoretically true or false”). Consider
the following problems (i) and (ii).
(i) Assume that someone proposes “psychokinetic theory” as a theory of theoreticalphysics. Determine whether his/her theory is true or false.
(ii) In [93], Zadeh proposed “the fuzzy sets theory” as a theory of theoretical informatics.
Determine whether his theory is true or false.8[Answer (i)]. The problem (i) is solved by two methods. One is the experimental test.
If it is OK (i.e., if it is experimentally true), “psychokinetic theory” should be accepted
as a physical theory. Also, if we have the most fundamental theory (= ‘TOE’), we candetermine whether “psychokinetic theory” is theoretically true in ‘TOE’. If it is OK (i.e.,
if it can be understood in ‘TOE’), the “psychokinetic theory” should be accepted as aphysical theory. Of course, it always holds that “experimentally true” = “theoreticallytrue”.
8

One of our motivations for this research may be inspired by the fashion of Zadeh’s fuzzy sets theory(cf. [93], which is the most cited paper in all ﬁelds of 20th century science) in 1980s ∼ 1990s. We had a lotof arguments about “Is Zadeh’s fuzzy sets theory true or false?” or “Can it be justiﬁed?” However, thesearguments may be fruitless. That is because all controversies were engaged without the understandingof the meaning of “true” (or “justiﬁcation”). It should be noted that we do not only have the answer
to the question: “Is Zadeh’s fuzzy³ sets theory (theoretical) true or false?” but also “Is Fisher’sstatistics´(theoretically) true or false?”.
These will be respectively answered in Chapter 5∼7. In this sense,
we can say that the purpose of this book is to introduce the criterion:“theoretically true or false” into
theoretical informatics. (Cf. Declaration (1.11) later). Here, two criterions of “theoretically true or
false (in theoretical informatics)” and “useful or not (in informatics-related engineering)” should not be
confused. Throughout this book we are not concerned with “useful or not” but “theoretically true or
false”, though we, of course, know that the criterion “useful or not” is also quite important.

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CHAPTER 1. THE PHILOSOPHY OF MEASUREMENT THEORYFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/[Answer (ii)]. On the other hand, the problem (ii) is solved by one method. If we have themost fundamental theory (=‘measurement theory’), we can determine whether “Zadeh’s
fuzzy sets theory” is theoretically true or false in the most fundamental theory. If it canbe understood in the most fundamental theory, “Zadeh’s fuzzy sets theory” should be
accepted as a theory of theoretical informatics. Our answer will be presented in Chapter
7. However, as mentioned in Remark 1.1 (e), it should be noted that the question: “Is
Zadeh’s fuzzy sets theory experimentally true or not?” is nonsense.
¥

Remark 1.3. (What should measurement theory be applied to?). Recall that MT isa kind of epistemology called “the mechanical world view”, namely, an epistemology tounderstand and analyze (moreover, control) every phenomenon by an analogy of mechanics. In this sense, MT may be applied to everything. However, it is certain that someproblems (or phenomena) are ﬁt for “the mechanical world view”, but others are not.
Thus, we have the following question.
(I6 ) What phenomenon should measurement theory be applied to?
The following ﬁelds are generally believed to be ﬁt for “the mechanical world view” tosome degree.
(I7 ) the ﬁelds in informatics-related engineering, e.g., information engineering, administration engineering, mathematical psychology, statistical medicine, mathematical
economics, ﬁnancial engineering, cognitive engineering, quality control engineering,
chaotic engineering, electrical circuit engineering 9, etc.
And further, we add(I8 ) statistical mechanics, ﬂuid mechanics, etc.10
though the two are usually believed to belong to theoretical physics. As mentioned later(i.e., the footnote under (I13 )), the theories in (C5 ) in Table (1.7) should be studied byseveral methods (and not only by “the mechanical world view” (= MT)). Also, we say
9For example, the distinction between “electrical circuit engineering” in (I7 ) and “circuit theory” in(I4 ) may be ambiguous. However, we want to say “MT itself is not engineering but the mathematicalrepresentation of “the mechanical world view”.
10
Boltzmann’s statistical mechanics will be discussed as one of applications (of MT) in Chapter 4.
Therefore, there is a reason to call “theoretical physics” [resp. “theoretical informatics” ] “the ﬁrst
physics” [resp.“the second physics”] .

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9(I9 ) It is too optimistic to consider that the completely precise theory exists in (I7 ) and(I8 ). However, the theories in (I7 ) and (I8 ) may be “almost experimentally true” tosuch a degree that they are assured to be “useful”. That is, every theory in (I7 ) and(I8 ) is, more or less, ambiguous. Although the challenge to make a precise theory
should be worthy of praise, what is most important is not “precise” but “useful” inengineering.
¥

Remark 1.4. (Aristotles and Plato). As mentioned before, theoretical physics must be
always checked by experimental tests. That is, it is based on realism (i.e., the Aristotles
spirit). On the other hand, recall that the experimental test for MT is nonsense. Therefore, we can not deny MT by any experimental tests.11

Thus, we may agree to theopinion that
“MT is self-righteous”.
In this sense, we cay say that MT is based on idealism (i.e., the Plato spirit).12 However,it does not imply “unfair”.

That is because, if some want to deny MT, it suﬃces topropose another fundamental theory better than MT. Here,(I10 ) the question:“Which is better?” is decided by majority (or popularity).
Here it should be noted that to win popularity is as diﬃcult as to ﬁnd the truth. Also, as
mentioned in (I9 ), we can expect that every theory in (I7 ) and (I8 ) is “almost experimentally true”. That is because, if it is not “almost experimentally true”, it can never win
popularity.
¥

11

Thus, I assume that MT itself is a kind of metaphysics (and not science in the sense of Popper [70],“falsiﬁability” ).
12
If I were familiar with the history of philosophy, I could stress the correspondences: “theoreticalphysics (realism) ↔ Aristotles” and “theoretical informatics (idealism)↔Plato”.

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Summing up, we assert the following table:
Table(6). important criterion(cf. Remark 1.1 (e), Remark 1.4)
(7). theoreticallytrue or false(cf. Problem 1.2 (i),(ii))(8). what to be applied to(cf. Remark 1.3)
(9). fundamental spirit(cf. Remark 1.4)

1.3

(1.8b)

Theoretical Physics

Theoretical Informatics

experimentally true or false
objective
meaningful in ‘TOE’

useful or not, likes or dislikes
popularity, economical, subjective
meaningful in MT

physical phenomena

all phenomena (particularly,
appearing in (I7 ) and (I8 ))
idealism (due to Plato)
Theory is free from experiment.

realism (due to Aristotles)
Theory is dominated by experiment.

Measurement theory in engineering

As mentioned in (I7 ) in §1.2, MT plays an important role in engineering. The theoretical physics (= ‘TOE’) itself may be worthy even if it has no applications. However,
MT is not so. Thus, in this section, we consider the relation between engineering and
MT. Here, engineering is usually considered to be composed of physics-related engineering(e.g., laser engineering, etc.), chemistry-related engineering (e.g., chemistry engineering,etc.), informatics-related engineering (e.g., ﬁnancial engineering, etc.), etc.
The area of physics-related engineering is clear. That is because the physics-related
engineering is generally believed to be supported by “physics” as the theoretical backbone.
We studied physics as one of the important subjects in high-school, and therefore, we
believe that theoretical physics is only one discipline, i.e., classical mechanics, relativity
theory, electromagnetic theory and so on that should be uniﬁed. That is, physics-related
engineering has the authorized root (= physics). Also, note that the circumstance ofchemistry-related engineering is similar to that of physics-related engineering.
On the other hand, the area of informatics-related engineering may be vague. Thisis due to the fact that we do not know the most fundamental root in informatics-related
engineering. Note that there is a possibility that informatics-related engineering has
two (or more than two) fundamental roots. If it is so, we must consider “informaticsrelated engineering (I)” and “informatics-related engineering (II)” (cf. Remark 1.1 (b)).
Therefore, we must answer the following question:³(I11 ) What subject is the most fundamental in informatics-related engineering?
´
theoretical informatics the only one?

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Of course, our answer is(I12 ) MT is the most fundamental theoretical backbone in informatics-related engineering.
MT (or, theoretical informatics) is not studied in high-school. However, statistics anddiﬀerential equations (which are closely related to MT (i.e., Axioms 1 and 2 in (1.4a)))are studied as mathematics in high-school. In this sense, theoretical informatics is notunderestimated in high-school education.
Thus we have the following table.
Table

(1.9)

fundamental subject

area (applications)

physics-related
engineering

physics (mathematical)
experimental test is possible

semiconductor engineering,laser engineering, etc.

chemistry-related
engineeringchemistry (non-mathematical)
experimental test is possible

chemical engineering

informatics-related
engineering

MT (mathematical)
experimental test is meaninglessCf. (I7 ) and (I8 )Thus we conclude that
(I13 ) The area of informatics-related engineering is roughly13 determined by MT (= “theoretical informatics” ), just like the area of physics-related engineering is roughly
determined by physics. Also, recall (I5 ).
That is, we say:
8
>
>
>
>
<
>
>
>
>
:

physical phenomena

M.R.
=⇒
ET

mechanical world view

M.R.
=⇒
PP

theoretical physics (=‘TOE’)

Appl
=⇒
AET

physics-related engineering(1.10)

theoretical informatics (= MT)

Appl
=⇒
AET

informatics-related engineering

where “M.R.” = “mathematical representation”, “ET” = “experimental test” (cf. Problem 1.2 (i)), “Appl” = “Applications” (cf. Table (1.9)), “PP” = “popularity” (cf. (I10 )),“AET” = “almost experimentally true” (cf. (I9 ) in §1.3).
13
For example, mechanical engineering is closely related to physics. However, control theory (in (C3 )of Table (1.7)) plays an important role in robot engineering (which is a kind of mechanical engineering).
Also, electrical circuit engineering may be close to electromagnetic theory as well as dynamical systemtheory. Thus, such a classiﬁcation of engineering (presented in Table (1.9)) is somewhat forcible. Thatis because “Use everything available” is the engineer’s spirit. Thus we must say that physics (as well as
measurement theory) is more or less inﬂuential to every ﬁeld in (I7 ) and (I8 ). However we can, at least,
assert that physics, chemistry and MT are the most fundamental subjects in the faculty of engineering.

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Here again note that

(I14 ) Theoretical physics has to be precise. On the other hand, engineering has to be
useful rather than precise. Since ambiguous statements can not be tested “exactly”,
we use the term : “AET (= almost experimentally true)” in the above (1.10).
Thus we see(I15 ) There is a possibility that a phenomenon has two (or, more than two) explanations in MT. And moreover, in this case, we may not choose one from the two by
experimental tests but a sense of beauty (≈ like or dislike).

1.4

The spirit of “the mechanical world view”

We think that “measurement”, “its philosophy” and “its applications (≈ informaticsrelated engineering)” should be regarded as “the Trinity”.

And we assert the following

declaration, which was essentially proposed in [Ishikawa, 2002, [48]].

Declaration(1.11)

We assert the following (i) ∼(iii), which should be understood as the diﬀerent representations of the same thing:(i) MT is the most fundamental theory of theoretical informatics, which is regarded as
the theoretical backbone of informatics-related engineering.
(ii) MT is the ultimately generalized form of the dynamical system theory (1.2). Thus,
MT is regarded as the mathematical representation of the epistemology called “themechanical world view”. And thus, MT is sometimes called the general dynamicalsystem theory (or in short, GDST).
(iii) MT is entitled to check all theories in theoretical informatics. In other words, we can,by using MT, introduce the criterion:“(theoretically) true or false” into theoreticalinformatics.
Here, note that:• in this book, “the mechanical world view” means “the quantum mechanical worldview” and not “the Newtonian mechanical world view”.

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13

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We might say too much in this chapter. It may suﬃce to say

The spirit of “the mechanical world view”(1.12)• Mind Declaration (1.11) and Tables (1.7) and (1.8). And further, at
any rate (= setting aside the reason), study every (physical or nonphysical) problem in the framework of MT.14

Summing up, we have “the Trinity” as follows:

the mechanical world view
(philosophy)]

the Trinitymeasurement theory(mathematical theory)·························



-

^

informatics-related engineering(applications)Here, again note that the philosophy of “theoretical informatics” is completely diﬀerent
from that of “theoretical physics”, Although it is a matter of course that it is impossible
to understand the philosophy of measurement theory without the complete knowledge ofmeasurements (i.e., the contents of Chapters 2 ∼ 12), the philosophy of measurement
theory is also indispensable for the understanding of measurement theory.

Remark 1.5 (Another important problem)

The problem:(I16 ) “Propose The third mathematical scientiﬁc theory in (C4 ) of Table (1.7)”
14

As mentioned in Remark 1.4, we do not necessarily need a perfect reason in theoretical informatics.
In this sense, the term: “extensive interpretation” is one of the most important terms in theoreticalinformatics.

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may be the most important. I think that the above problem (I16 ) is so diﬃcult. Thus I
may prefer waiting the appearance of a genius to doing it ourselves.15

15

As mentioned in this chapter, our purpose may be, brieﬂy speaking, to study all ﬁelds which can beunderstood in terms of “measurement (i.e., Axioms 1 and 2)”. In this sense, Frieden’s challenge [24] isalso interesting. His purpose seems to study all ﬁelds (of physics) which can be understood in terms of“Fisher information”. Although we do not completely understand his theory, we expect that his theory
may be one of the candidates of The third mathematical scientiﬁc theory . We never hope that MT is
the only one mathematical theory that belongs to the category of “idealism”.

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Chapter 2
Measurements (Axiom 1)
Measurement theory (MT) is classiﬁed two subjects, i.e., “(pure) measurement theory (PMT)” and“statistical measurement theory (SMT)”. That is, PMT (=“(pure) measurement theory”) in Chapters 2 ∼ 7
MT (=“measurement theory”)

SMT (=“statistical measurement theory” in Chapters 8 ∼)(2.1)
PMT is essential, and it should be noted that there is no SMT without PMT (cf. Chapter 8). In
Chapters 2 ∼ 7, we devote ourselves to PMT, which is formulated as follows:
PMT = measurement + the relation among systems[Axiom 1 (2.37)]

in C ∗ -algebra[Axiom 2 (3.26)]

.

(2.2)
(=(1.4))In this chapter we intend to explain “measurement (= Axiom 1)”. (In Chapter 3 we will devote
ourselves to Axiom 2 (i.e., “the relation among systems”).)

2.1

Mathematical preparations

The theory of operator algebras (i.e, C ∗ -algebra and W ∗ -algebra) is a convenientmathematical tool to describe both classical and quantum mechanics (cf. [76]). Thus
our theory is described in terms of C ∗ -algebras. Since our concern in this book is mainly
concentrated on classical systems and not quantum systems, it may suﬃce to deal with
only commutative C ∗ -algebras. In fact, most of our main results are related to classicalsystems. However, recall (1.4), that is:
PMT =“Apply Axioms 1 and 2 to every phenomenon by an analogy ofquantum mechanics”(2.3)(=(1.4))15
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Thus we think that the essence of measurements can not be appreciated deeply without
the knowledge of quantum measurements. In fact, the concept of measurements was ﬁrst
discovered and formulated by M. Born [13]1 as the most fundamental concept in quantum
mechanics. Thus, we begin with general C ∗ -algebras, in which both classical and quantum
systems are formulated.2
Let A be a linear associative algebra over the complex ﬁeld C. The algebra A is calleda Banach algebra if it is associated to each element T a real number ∥T ∥, called the normof T , with the properties:(i) ∥T ∥ ≥ 0,(ii) ∥T ∥ = 0 if and only if T = 0, (i.e., the 0-element in A),(iii) ∥T + S∥ ≤ ∥T ∥ + ∥S∥,(v) ∥T S∥ ≤ ∥T ∥ · ∥S∥,(iv) ∥λT ∥ = |λ| · ∥T ∥, λ ∈ C,(vi) A is complete with respect to the norm ∥ · ∥.

A mapping T 7→ T ∗ of A into itself is called an involution (and T ∗ is called the adjoint
element of T ) if it satisﬁes the following conditions:(i) (T ∗ )∗ = T ,(ii) (T + S)∗ = T ∗ + S ∗ ,(iii) (T S)∗ = S ∗ T ∗ ,(iv) (λT )∗ = λ̄T ∗ , λ ∈ C.
A Banach algebra with an involution ∗ is called a Banach ∗ -algebra.
Deﬁnition 2.1. [C ∗ -algebra, identity, commutative C ∗ -algebra]. A Banach ∗ -algebra A(with the norm ∥ · ∥A ) is called a C ∗ -algebra if it satisﬁes the C ∗ -condition, i.e., ∥T ∗ T ∥ =∥T ∥2 for any T ∈ A. A C ∗ -algebra A does not always have the identity element IA (i.e.,
IA T = T IA = T for all T ∈ A), though in this book we usually suppose that a C ∗ -algebra
A has the identity element IA . A C ∗ -algebra A is called unital, if it has the identity
element IA . Also, a C ∗ -algebra A is called commutative, if it holds that T1 T2 = T2 T1(∀T1 , T2 ∈ A).
¥
An element F in A is called self-adjoint if it holds that F = F ∗ . A self-adjoint element
F in A is called positive (and denoted by F ≥ 0) if there exists an element F0 in A such1

He proposed his theory in 1926, and he won the Nobel prize of physics in 1954.
I am afraid that the mathematical preparation (in this section) discourages readers to want to readthis book. Thus, it may be recommended to skip to Example 2.16 ﬁrstly. In order to read this book, it
suﬃces to understand Example 2.16.
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17

that F = F0∗ F0 where F0∗ is the adjoint element of F0 . Also, a positive element F is calleda projection if F = F 2 holds. Let A∗ be the dual Banach space of A. That is,A∗ = {ρ | ρ : A → C is continuous linear }³
with the norm ∥ρ∥A∗ ( ≡ sup{|ρ(F )| : ∥F ∥A ≤ 1}). The linear functional ρ(F ) isD
E ´
sometimes denoted by A∗ ρ, F A .
Deﬁne the mixed state space Sm (A∗ ) such that:
Sm (A∗ ) = {ρ ∈ A∗ | ∥ρ∥A∗ = 1 and ρ(F ) ≥ 0 for all F ≥ 0}.

(2.4)

A mixed state ρ ( ∈ Sm (A∗ )) is called a pure state if it satisﬁes that “ρ = θρ1 + (1 − θ)ρ2
for some ρ1 , ρ2 ∈ Sm (A∗ ) and 0 < θ < 1” implies “ρ = ρ1 = ρ2 ”. Deﬁne
Sp (A∗ ) ≡ {ρp ∈ Sm (A∗ ) | ρp is a pure state},

which is called a state space (or pure state space, phase space).

(2.5)

Note that Sm (A∗ ) isconvex and compact in the weak∗ topology σ(A∗ ; A). Also, Sp (A∗ ) is characterized as
the set of the extreme points of Sm (A∗ ). (Cf. [92, 76]). Since Sp (A∗ ) is the closed set of
Sm (A∗ ), the Sp (A∗ ) is also compact in the weak∗ topology.
The following Examples 2.2 and 2.3 will promote the understanding of Deﬁnition 2.1.
Example 2.2. [Commutative C ∗ -algebras; C(Ω), or generally, C0 (Ω)]. When A is acommutative C ∗ -algebra, that is, T1 · T2 = T2 · T1 holds for all T1 , T2 ∈ A, by Gelfand
theorem (cf. [74, 76]), we can put A = C(Ω), the algebra composed of all continuous³
complex-valued functions on a compact Hausdorﬀ space Ω. If the commutative C ∗ algebra A is not necessarily assumed to be unital, we can put A = C0 (Ω), the algebra
composed of all continuous complex-valued functions vanishing at inﬁnity on a locally´
compact Hausdorﬀ space Ω.
The norm ∥f ∥C(Ω) is, of course, deﬁned by ∥f ∥C(Ω) =max{|f (ω)| : ω ∈ Ω} (∀f ∈ C(Ω)). Also, we can easily see that it satisﬁes the C ∗ condition, i.e., ∥f · f ∗ ∥C(Ω) = ∥f ∥2C(Ω) where f ∗ (ω) ( ≡ f (ω)) is deﬁned by the conjugate“Re[f (ω)] − Im[f (ω)]i” (∀ω ∈ Ω) (where Re is “real part”, Im is “imaginary part”). It
is well known (i.e., Riese Theorem) that C(Ω)∗ = M(Ω), i.e., the Banach space composedof all regular complex-valued measures on Ω. And therefore,
Sm (M(Ω)) = {ρ ∈ M(Ω) | ρ ≥ 0, ∥ρ∥M(Ω) = 1},

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which is denoted by Mm+1 (Ω). Also, it is clear thatnoSp (M(Ω)) = δω ∈ M(Ω) | δω is a point measure at ω ∈ Ω
(2.7)
³
D
E
´
i.e., M(Ω) δω , f C(Ω) = f (ω) (∀f ∈ C(Ω), ∀ω ∈ Ω) , which is denoted by Mp+1 (Ω), andcalled a state space. And therefore, we have the identiﬁcation: Ω ≈ Mp+1 (Ω) in the senseofΩ ∋ ω ←→ δω ∈ Mp+1 (Ω).

(2.8)

Thus the compact Hausdorﬀ space Ω may be also called a state space.
¥
Example 2.3. [Non-commutative C ∗ -algebras; B(V ) and C(V )]. Let V be a (separable)
Hilbert space with the inner product ⟨·, ·⟩V . Here we always assume that ⟨v1 , αv2 ⟩V =α⟨v1 , v2 ⟩V (∀v1 , v2 ∈ V, α ∈ C). (Cf. [4, 71].) Put
B(V ) = {T : T is a bounded linear operator from a Hilbert space V into itself }.3
Deﬁne ∥T ∥B(V ) = sup{∥T v∥V : ∥v∥V = 1}, and (T1 T2 )(v) = T1 (T2 v) (∀v ∈ V ). AndT ∗ is deﬁned by the adjoint operator of T . Note that it holds that ∥T ∗ T ∥B(V ) = ∥T ∥2(∀T ∈ B(V )). Thus, we can easily see that the B(V ) is a non-commutative C ∗ -algebra.
Also note thatC(V ) ≡ {T ∈ B(V ) : T is a compact operator }(2.9)

is a C ∗ -subalgebra of B(V ). If the dimension of V is inﬁnite, this C ∗ -algebra C(V ) has
no identity I. We see that³´
C(V ) = T r(V ) ≡ {T ∈ B(V ) : ∥T ∥tr < ∞} .
∗(2.10)

Here T r(V ) is the class of trace operators with the trace norm ∥ · ∥tr such that:∥ρ∥tr =

∞
X

⟨en ,

√ ∗
ρ ρ en ⟩V

n=1

where {en }∞n=1 is the complete orthonormal system in V . It is well known that the value∥ρ∥tr is independent of the choice of a complete orthonormal basis {eλ |λ ∈ Λ} in V . And
we seem
Sm (C(V )∗ ) = T r+1(V ) ≡ {ρ ∈ T r(V ) : ρ ≥ 0, ∥ρ∥T r(V ) = 1}.
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And further,(T r(V ))∗ = B(V ).
Also, it is well known that“ρ ∈ Sp (C(V )∗ )” ⇔ “there exists ψ ∈ V (∥ψ∥V = 1) such that ρ = |ψ⟩⟨ψ|”,
where the Dirac notation “|ψ1 ⟩⟨ψ2 |”³(2.12)´∈ B(V ) , ψ1 , ψ2 ∈ V , is deﬁned by(|ψ1 ⟩⟨ψ2 |)ϕ = ⟨ψ2 , ϕ⟩V ψ1

for all ϕ ∈ V .

p
The state space Sp (C(V )∗ ) is denoted by T r+1(V ), that is,p
Sp (C(V )∗ ) ≡ T r+1(V ).

Also, it is well-known that “ρ ∈ Sm (C(V )∗ )” ⇔ “there exists an orthonormal systemP∞∞{ψn }∞
n=1 in V and non-negative real numbers {λn }n=1 (wheren=1 λn = 1) such thatP∞
ρ = n=1 λn |ψn ⟩⟨ψn |”.
¥The following theorem is one of the most important theorems in the theory of operator
algebras.
Theorem 2.4. [GNS-construction, Gelfand, Naimark, Siegel, cf. [50, 76]]. Let A be a
C ∗ -algebra. Then there exists a B(V ) such that:A ⊆ B(V ).

(2.13)

That is, A can be identiﬁed with the norm-closed C ∗ -subalgebra of a certain B(V ).
¥Example 2.5. [Commutative C ∗ -algebra MatD (n; C) as the subalgebra of B(Cn )]. Let
Cn be an n-dimensional Hilbert space with the inner product ⟨·, ·⟩Cn (that is, ∥z∥Cn =
pPn
n
∗
2
k=1 |zk | (∀z = (z1 , z2 , ..., zn ) ∈ C )). Consider the non-commutative C -algebra
B(Cn ) ≡ {T : T is a (bounded) linear operator from a Hilbert space Cn into itself },which is clearly equal toMat(n; C) ≡ {T : T is a complex (n × n)-matrix }.

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That is,
B(Cn ) = Mat(n; C).
Put
MatD (n; C) = {T : T is a complex (n × n)-diagonal matrix },(2.15)

which is clearly a commutative C ∗ -subalgebra of B(Cn ). Also, it is obvious that the
MatD (n; C) is equivalent to C(Ω), where Ω is the ﬁnite state space ({1, 2, ..., n}) with thediscrete topology.4 That is, we see the following identiﬁcations:
MatD (n; C) ≈ C({ω1 , ω2 , ..., ωn }) ≈ Cn³
nmax
where C is assumed to have the max-norm ∥z∥Cn = maxk=1,2,...,n |zk | (∀z = (z1 , z2 , ..., zn )´
∈ Cn ) . Also, the multiplication (z11 , z21 , ..., zn1 ) · (z12 , z22 , ..., zn2 ) is deﬁned by (z11 z12 , z21 z22 , ...,
zn1 zn2 ).
¥Remark 2.6. [(i): The identity]. Let A0 be a non-unital C ∗ -algebra. Theorem 2.4 (TheGNS-construction) says that there exists a B(V ) such that A0 ⊆ B(V ). That is, A0 canbe identiﬁed with the norm-closed subalgebra of B(V ). Thus we can deﬁne the C ∗ -algebra
AI such that it is the smallest algebra that includes {I} ∪ A0 ( ⊆ B(V )). Therefore, we
can always add the identity I to A0 , and construct the new unital C ∗ -algebra AI . This
argument implies that the “unital condition” is not so strict. Thus, throughout this book,
we usually deal with a unital C ∗ -algebra, though the C0 (Ω) is sometimes used.
[(ii): Minimal tensor C ∗ -algebras]. Here consider the minimal tensor C ∗ -algebra as follows:¡¢b = Nn Ak be the tensor product C ∗ -algebra of {Ak : k = 1, 2, ..., n}. This can
Let A
k=1

be easily constructed as follows: Since we can see, by Theorem 2.4 (GNS-construction),that
Ak ⊆ B(Vk )we can deﬁnethat contains(k = 1, 2, ..., n),(2.16)Nn

Nn

k=1 Ak such that the smallest norm-closed sub-algebra (of B(

k=1 Vk ))

n
o
³
´¯
O
¯
Fk ∈ B(
Vk ) ¯ Fk ∈ Ak , k = 1, 2, ..., n(2.17)

n
nO
k=1

k=1

Throughout this book, we assume that a ﬁnite state space Ω (≡ {ω1 , ω2 , ..., ωn }) has the discrete
metric dD (i.e., dD (ω1 , ω2 ) = 1 (ω1 ̸= ω2 ), = 0 (ω1 = ω2 )).
4

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Nn

k=1 Vk is the tensor Hilbert space of {Vk | k = 1, 2, ..., n}. Though the generalNntheory of tensor product C ∗ -algebras
k=1 Ak is not easy, we only use the following

whereproperties (i)∼(iii) of the tensor C ∗ -algebras:b for any Tk ∈ Ak , k = 1, 2, ..., n,(i) T1 ⊗ T2 ⊗ · · · ⊗ Tn ∈ Ab∗ ) for any ρk ∈ Sp (A∗ ), k = 1, 2, ..., n,(ii) ρ1 ⊗ ρ2 ⊗ · · · ⊗ ρn ∈ Sp (AkQn∗
k=1 ρk (Tk ) for any ρk ∈ Ak and any(iii) (ρ1 ⊗ ρ2 ⊗ · · · ⊗ ρn ) (T1 ⊗ T2 ⊗ · · · ⊗ Tn ) =Tk ∈ Ak , k = 1, 2, ..., n.

If we focus on only commutative cases, it is suﬃcient to know the fact thatnOnC(Ωk ) = C( × Ωk )

and

n
O

k=1

k=1

n

M(Ωk ) = M( × Ωk ),k=1(2.18)

k=1

where ×k=1 Ωk is the product topological space of Ω1 ,...,Ωn . Therefore, for example, thenabove (iii) implies the elementary property of product measure (Fubini’s theorem), i.e.,ZZZ
f1 (ω1 ) · f2 (ω2 )(ρ1 ⊗ ρ2 )(dω1 dω2 ) =f1 (ω1 )ρ1 (dω1 ) ·f2 (ω2 )ρ2 (dω2 )Ω1 ×Ω2Ω1(∀f1 ∈ C(Ω1 ),∀f2 ∈ C(Ω2 )).

Ω2(2.19)For the deep studies of “tensor C ∗ -algebra”, see [50].
¥2.2

Observables

Let X be a set. Let 2X (or, P(X)) be the power set of X. i.e., 2X = {Ξ | Ξ ⊆ X}.
A set F( ⊆ 2X ) is called a ﬁeld if the F is closed under the intersection (i.e., ∩) and thecompliment (i.e., [ · ]c ), that is, if “Ξ1 , Ξ2 ∈ F” implies “Ξ1 ∩ Ξ2 ∈ F” and “Ξc1 ∈ F”,/ Ξ1 }. Note that Ξ1 ∪ Ξ2 = (Ξc1 ∩ Ξc2 )c , Ξ1 \ Ξ2 =where Ξc1 = X \ Ξ1 = {x | x ∈ X ∧ x ∈Ξ1 ∩ Ξc2 and Ξ1 △ Ξ2 = (Ξ1 ∪ Ξ2 ) \ (Ξ1 ∩ Ξ2 ). Thus the ﬁeld F is also closed under theoperations ∪, \ and △.
Also, a set R( ⊆ 2X ) is called a ring if the R is closed under the intersection (i.e., ∩)and the symmetric diﬀerence (i.e., △), that is, if “Ξ1 , Ξ2 ∈ R” implies “Ξ1 ∩ Ξ2 ∈ R” andSample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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Thus the ring R is also closed under the operations ∪ and \ (cf. [29]).
Motivated by the Davies’ idea (in quantum mechanics, cf. [17]), we propose thefollowing deﬁnition.
Deﬁnition 2.7. [ C ∗ -observables in a unital A]. A C ∗ -observable ( or in short, observable,
fuzzy observable) O ≡ (X, F, F ) in a unital C ∗ -algebra A is deﬁned such that it satisﬁes
that(i) [ﬁeld]. X is a set (called a “measured value set” or “label set” ), and F is the subﬁeldof the power set P(X) ( ≡ {Ξ : Ξ ⊆ X}),(ii) for every Ξ ∈ F, F (Ξ) is a positive element in A such that F (∅) = 0 and F (X) = IA
(where 0 is the 0-element in A),³(iii) for any countable decomposition {Ξ1 , Ξ2 , ..., Ξn , ...} of Ξ, i.e., Ξ, Ξn ∈ F, ∪∞n=1 Ξn =
´
³
´
³P
´
N
Ξ, Ξn ∩ Ξm = ∅(if n ̸= m) , it holds that ρ F (Ξ) = limN →∞ ρ
F
(Ξ
)
n
n=1
(∀ρ ∈ Sm (A∗ )).
Also, if F (Ξ) is a projection for every Ξ ( ∈ F), a C ∗ -observable (X, F, F ) is called a crisp
C ∗ -observable (or, a crisp observable, an idea).
¥
Remark 2.8. [(1): The case that X is ﬁnite]. In chapters 2∼8, we will usually deal
with the case that X is ﬁnite. When we want to stress that X is ﬁnite, the (X, F, F ) isoften denoted by (X, 2X , F ) or (X, P(X), F ). Thus, in this case, the (iii) in Deﬁnition 2.7
means
F (Ξ1 ∪ Ξ2 ) = F (Ξ1 ) + F (Ξ2 ) (∀Ξ1 , ∀Ξ2 (∈ 2X ) such that Ξ1 ∩ Ξ2 = ∅)).
[(2): C ∗ -observables in general C ∗ -algebras]. Although we are usually concerned withunital C ∗ -algebras, we add the generalization of Deﬁnition 2.7 as follows: Let A be a
C ∗ -algebra, which does not necessarily have the identity I. A C ∗ -observable ( or in short,observable, fuzzy observable ) O ≡ (X, R, F ) in a C ∗ -algebra A is deﬁned such that it
satisﬁes that(i) X is a set, and R is the subring of the power set P(X) ( ≡ {Ξ : Ξ ⊆ X}), that is,“ Ξ1 , Ξ2 ∈ R” implies “ Ξ1 ∩ Ξ2 ∈ R” and “ Ξ1 △ Ξ2 ∈ R”,

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(ii) for every Ξ ∈ R, F (Ξ) is a positive element in A such that F (∅) = 0 (where 0 is the0-element in A),(iii) for any countable decomposition {Ξ1 , Ξ2 , ..., Ξn , ...} of Ξ, (Ξ, Ξn ∈ R), it holds that
³
´
³P
´
N
ρ F (Ξ) = limN →∞ ρ(∀ρ ∈ Sm (A∗ )),n=1 F (Ξn )00∞0(iv) there exists a sequence {Ξ0n }∞n=1 in R such that Ξ1 ⊆ Ξ2 ⊆ · · · and X = ∪n=1 Ξn³´
and limn→∞ ρ F (Ξ0n ) = 1 (∀ρ ∈ Sm (A∗ )).

Also, if F (Ξ) is a projection for every Ξ ( ∈ R), a C ∗ -observable (X, R, F ) is called a crisp
C ∗ -observable.
¥
Deﬁnition 2.9. [Image observable]. Let O ≡ (X, F, F ) be an observable in a C ∗ -algebra
A. Let G be a subﬁeld of 2Y . Let h : X → Y be a measurable map, i.e., h−1 (Γ) ∈ F(∀Γ ∈ G). Then, we can deﬁne the observable O[h] (≡ (Y, G, F ◦ h−1 )) in A such that:(F ◦ h−1 )(Γ) = F (h−1 (Γ))(Γ ∈ G).

(2.20)The O[h] ≡ (X, F, G◦h−1 ) is called the image observable of O ≡ (Y, G, G) (in a C ∗ -algebraA) concerning the map h : X → Y . The image observable O[h] is also denoted by h(O).
¥Deﬁnition 2.10. [Quasi-product observable]. For each k = 1, 2, ..., n, consider an obn
Nservable Ok ≡ (Xk , Fk , Fk ) in a C ∗ -algebra A. Deﬁne the ﬁeld nk=1 Fk ( ⊆ 2×k=1 Xk )such as the smallest ﬁeld (on ×k=1 Xk ) that contains ×k=1 Ξk , Ξk ∈ Fk . The prod³
Nnn
uct ﬁeldF
isusually
denotedby×F.
Throughout this book, the notak
k
k=1
k=1
´
n
n
b ≡
tion ×k=1 Fk does not mean the set { ×k=1 Ξk : Ξk ∈ Fk }.
An observable Onn( ×k=1 Xk , ×k=1 Fk , Fb) in A is called the quasi-product observable of {Ok : k = 1, 2, ..., n}³´
or, quasi-product observable with marginal observables {Ok : k = 1, 2, ..., n} if it holds
n

n

that
Fb(X1 × · · · × Xk−1 × Ξk × Xk+1 × · · · × Xn ) = Fk (Ξk ) (∀Ξk ∈ Fk , ∀k = 1, ..., n).
(2.21)b (of {Ok }n ) is denoted by
The quasi-product observable O
k=1
qp

×

k=1,2,...,nOk , or,

³ n

n

k=1

k=1

´

³ n

× Xk , × Fk , k=1,2,...,nFk , or × Xk ,
qp

×

k=1

n
O

Fk ,

k=1

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×

k=1,2,...,n´
Fk ,(2.22)

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qp

qp

qp

b = × k=1,2,...,n Ok , Fb = × k=1,2,...,n Fk . Also, × k=1,2,...,n Fk is sometimes written byi.e., ObO

× k=1,2,...,n Fk .
¥Note that the existence and the uniqueness of the quasi-product observable of {Ok :k = 1, 2, ..., n} are not guaranteed in general. However, when Ok , k = 1, 2, ..., n, commute,i.e.,
Fk (Ξk )Fk′ (Ξk′ ) = Fk′ (Ξk′ )Fk (Ξk )for all Ξk ∈ Fk , Ξk′ ∈ Fk′ such that k ̸= k ′ , (2.23)

we can construct the quasi-product observable ( ×k=1 Xk , ×k=1 Fk , Fe) in A such that:nnFe(Ξ1 × Ξ2 × · · · × Ξn ) = F1 (Ξ1 )F2 (Ξ2 ) · · · Fn (Ξn ).

(2.24)

This kind of quasi-product observable is called a product observable and denoted by
n

× Ok

k=1

³
=

×

k=1,2,...,n

n

n

n

n

k=1

k=1

k=1

k=1

Ok , or, ( × Xk , × Fk , × Fk ), or, ( × Xk ,

n
O

´
n
Fk , × Fk ), .

k=1

k=1

(2.25)

×nk=1 is sometimes written byQn

Q

×nk=1 Ok = nk=1 Ok , ×nk=1 Xk
Qn= nk=1 Xk , etc. Also, note that the product observable ×k=1 Ok always exists for any
k=1 , and thus, we write:

Ok in a commutative C ∗ -algebra C(Ω).
Summing up the above arguments, we can state the following theorem.
Theorem 2.11. For each k ∈ K ≡ {1, 2, ..., |K|}, consider an observable Ok ≡ (Xk , Fk , Fk )in a C ∗ -algebra A. If the commutativity condition:
Fk1 (Ξk1 )Fk2 (Ξk2 ) = Fk2 (Ξk2 )Fk1 (Ξk1 ) (∀Ξk1 ∈ Fk1 , ∀Ξk2 ∈ Fk2 , k1 ̸= k2 )(2.26)b ≡ ( ×k∈K Xk , ×k∈K Fk , Fe ≡holds, then we can construct a product observable O

×k∈K Fk ) such that:

Fe(Ξ1 × Ξ2 × · · · × Ξ|K| ) = F1 (Ξ1 )F2 (Ξ2 ) · · · F|K| (Ξ|K| ).

(2.27)

Note that the uniqueness (of quasi-product observables) is not guaranteed even under theabove commutativity condition. Also, note that the product observable ×k=1 Ok alwaysnexists for any Ok in a commutative C ∗ -algebra C(Ω).
¥

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Theorem 2.12. Let O ≡ (X, R, F ) be a C ∗ -observable in a general C ∗ -algebra A (i.e.,
it does not necessarily have the identity). Let A1 be a C ∗ -algebra with the identity I(generated by the A such as in Remark 2.6(i)). Then, there uniquely exists the observable(X, F, Fe) be a C ∗ -observable in A1 such that:F = R ∪ {X \ Γ | Γ ∈ R}½
F (Ξ)(Ξ ∈ R)e
F (Ξ) =c
I − F (Ξ ) (Ξc = (X \ Ξ) ∈ R).

(i)
(ii)

Proof. It suﬃces to show that F is the ﬁeld. Let Ξ1 ∈ R and Ξ2 ∈ {X \ Γ | Γ ∈ R}.
Thus Ξ2 = X \ Γ ( for some Γ ∈ R). Then, we see Ξ1 ∩ Ξ2 = Ξ1 ∩ (X \ Γ) = Ξ1 ∩ (Ξ1 \ Γ)∈ F. Also, Ξ1 ∪ Ξ2 = (Ξc1 ∩ Ξc2 )c = (Ξc1 ∩ Γ)c = (Γ \ Ξ1 )c ∈ F. Also, it is clear that“Ξ ∈ F” =⇒ “Ξc ∈ F”. Thus, we see that F is the ﬁeld.
The following theorem (and Theorem 9.8) will be often used throughout this book.
Theorem 2.13. [cf. [42]]. Let A be a C ∗ -algebra. Let O1 ≡ (X1 , F1 , F1 ) and O2 ≡³
(X2 , F2 , F2 ) be C ∗ -observables in A such that at least one of them is crisp. So, without´
loss of generality, we assume that O2 is crisp .
Then, the following statements areequivalent:(i) There exists a quasi-product observable O12 ≡ (X1 × X2 , F1 × F2 , F1 × F2 ) withqp

marginal observables O1 and O2 .
(ii) O1 and O2 commute, that is, F1 (Ξ1 )F2 (Ξ2 ) = F2 (Ξ2 )F1 (Ξ1 ) (∀Ξ1 ∈ F1 , ∀Ξ2 ∈ F2 ).
Furthermore, if the above statements (i) and (ii) hold, the uniqueness of the quasi-productobservable O12 of O1 and O2 is guaranteed.
Proof. It suﬃces to prove it in the case that A has the identity. When O1 ≡ (X1 , F1 , F1 )and O2 ≡ (X2 , F2 , F2 ) are both crisp observables, it is proved in [17]. By the same way, we
can prove this theorem. It is clear that (ii) =⇒ (i) since we can construct a C ∗ -observable(X1 × X2 , F1 × F2 , H) such that:
H(Ξ1 × Ξ2 ) = F1 (Ξ1 )F2 (Ξ2 ) (∀Ξ1 ∈ F1 , ∀Ξ2 ∈ F2 ).
Thus, it suﬃces to prove that (i) =⇒ (ii). Assume that (i) holds. Let Ξ1 and Ξ2 be any
element in F1 and F2 respectively. Put Ξ11 = Ξ1 , Ξ21 = X1 \Ξ1 , Ξ12 = Ξ2 and Ξ22 = X2 \Ξ2 .
qp

Put H = F1 × F2 . Note that:0 ≤ H(Ξi1 × Ξj2 ) ≤ H(X1 × Ξj2 ) ≡ F2 (Ξj2 ) ( = “projection”).

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This implies that H(Ξi1 × Ξj2 ) and F2 (Ξj2 ) commute, and so, H(Ξi1 × Ξj2 ) and I − F2 (Ξj2 )commute. Hence, F1 (Ξ1 ) ( = H(Ξ11 ×Ξ12 ) + H(Ξ11 ×Ξ22 )) and F2 (Ξ2 ) ( = F2 (Ξ12 )) commute.
Therefore, we get that (i) =⇒ (ii).
Next we prove the uniqueness of H under the assumption (i) (and so (ii)). Note that0 ≤ H(Ξi1 × Ξj2 ) ≤ H(Ξi1 × X2 ) ≡ F1 (Ξi1 ). This implies, by the commutativity condition(ii) and (2.28), that0 ≤ H(Ξi1 × Ξj2 ) ≤ F2 (Ξj2 )F1 (Ξi1 )F2 (Ξj2 ) = F1 (Ξi1 )F2 (Ξj2 ).
Therefore we see that I =

P

P
j
j
i
i
i,j=1,2 F1 (Ξ1 )F2 (Ξ2 ) = I.
i,j=1,2 H(Ξ1 × Ξ2 ) ≤(2.29)
Then, weobtain that H(Ξ1 × Ξ2 ) = F1 (Ξ1 )F2 (Ξ2 ), that is, H is unique. Therefore, we ﬁnish theproof.

2.3

The meanings of observables and crisp observables

In the conventional classical [resp. quantum] mechanics, the term “observable” usuallymeans a real valued continuous function on a state space Ω [resp. a self-adjoint operator in B(V )]. Thus, the “observable” (deﬁned in Deﬁnition 2.7) should be a kind ofgeneralization of the above conventional “observable”. In what follows we will see it.
Now we shall consider the several aspects (and properties) of the observable O ≡(X, F, F ) in a C ∗ -algebra A. Examining Deﬁnition 2.7, we can easily see(A1 ) An observable O¡≡ (X, F, F )¢in A can be regarded as the A-valued probability

space5 , i.e., the additive set-function:F ∋ Ξ 7→ F (Ξ) ∈ A.
Also, we may ﬁnd the similarity between an observable O and the resolution of the identity
I in what follows. Assume, for simplicity, that X is countable (i.e., X ≡ {x1 , x2 , ...}).
Then, it is clear that5

In this book, the term “probability space” is used as “a positive measure space whose total measureis equal to 1”. That is, the term “probability space” is used as the pure mathematical concept, and thus,
it is not always assured to be related to the concept of “probability”.

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27

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(i) F ({xk }) ≥ 0 for all k = 1, 2, ...
(ii)

P∞

k=1 F ({xk }) = IA in the sense of weak topology of A,

which imply that the [F ({xk }) : k = 1, 2, ..., n] can be regarded as the resolution of theidentity element IA . Thus we say that
(A2 ) An observable O¡≡ (X, F, F )¢in A can be regarded as“the fuzzy decomposition”(2.30)

that is, the resolution of the identity IA , i.e., [F ({xk }) : k = 1, 2, ..., n].
“The ﬁgure of O ≡ ({x1 , x2 , x3 }, 2{x1 ,x2 ,x3 } , F ) in C(Ω)”
1[F ({x1 })](ω)[F ({x3 })](ω)[F ({x2 })](ω)

0Ω

Also, we note that¡¢
(A3 ) An observable O ≡ (X, F, F ) in A can be characterized as a kind of generalizationof a self-adjoint element in A.
This is shown as follows: For simplicity, assume that A = B(CN ). And put
 
 
 
0
0
1
0
1
0
 
 
 
e1 =  ..  , e2 =  ..  , · · · , eN =  .. 
.
.
.
1
0
0
Thus we see that
1 0 ...
0 0 . . .

|e1 ⟩⟨e1 | =  .. .. . .
. .
.


0
0

..  ,.0 0 ... 000|e2 ⟩⟨e2 | =  ..
.

0 ...
1 ...
.. . .
.
.
0 0 ...



0
0


0
0
..  , · · · , |eN ⟩⟨eN | =  ..
.
.
0
0

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(2.31)0 ...
0 ...
.. . .
.
.
0 ...


0
0

..  .
.1

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The spectral theorem says that a self-adjoint matrix Fb ( ∈ B(CN )) can be representedby
λ1 0 . . . 0
 0 λ2 . . . 0  ∗b
F = U  ..
.. . .
..  U.
. . .
0 0 . . . λN³´
= U λ1 |e1 ⟩⟨e1 | + λ2 |e2 ⟩⟨e2 | + · · · + λN |eN ⟩⟨eN | U ∗=N
Xλn |U en ⟩⟨U en |(2.32)

n=1

where λn ∈ R (∀n = 1, 2, ..., N ) and U is a unitary matrix in B(CN ). For any Ξ( ∈ BR = “Borel ﬁeld” )6 , put

X

F (Ξ) =

|U en ⟩⟨U en |.

(2.33)λn ∈Ξ

Here it should be noted that F (Ξ) is a projection for all Ξ ( ∈ BR ). This implies the thefollowing identiﬁcation:
Fb←→ (R, BR , F ) in B(CN )(self-adjoint operator).

(2.34)(crisp observable)

That is because Fb is represented by (2.32), i.e.,ZFb =λF (dλ).
R

Next assume that A = C(Ω), where Ω is, for simplicity, assumed to be the ﬁnite set{ω1 , ω2 , ω3 , ..., ωN } with the discrete topology. Consider a real valued continuous function
Fb : Ω → R. Deﬁne the observable (R, BR , F ) in C(Ω) such that: 1 if ω ∈ Fb−1 (Ξ)[F (Ξ)](ω) =(∀ω ∈ Ω, ∀Ξ ∈ BR ).
(2.35)
−1b
0 if ω ∈/ F (Ξ)Note that
Z
N
³
´ X
³
´
X
Fb(ω) =
Fb(ωn ) [F ({ωn })](ω) =λ[F ({λ})](ω) = [ λF (dλ)](ω)
n=1(∀ω ∈ Ω).

R

λ∈RThis implies the the following identiﬁcation:
Fb←→ (R, BR , F ) in C(Ω)(real valued function on Ω).

(crisp observable)

Therefore, we say, by (2.34) and (2.36), that6

“Borel ﬁeld” = “the smallest σ-ﬁeld that contains all open sets”

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29←→(A4 ) “crisp observable (R, BR , F )” in A identiﬁcation“self-adjoint element” in A.
³´
where A = B(Cn ) or A = C({ω1 , ω2 , ..., ωN }) . Here, the “self-adjoint element” in A´¡i.e., “crisp observable (R, BR , F )” in A is sometimes called a “quantity (or, systemtheoretical quantity”) in A.
Remark 2.14. [OR (= operation research) and game theory].

In OR [resp. gametheory [85]], we are mainly concerned with the problem: “Study the maximal point [resp.
the saddle point] of Fb !”¥2.4

Measurement (Axiom 1)

Under the mathematical preparations in the previous sections, now we can describe
the fundamental concepts of measurement theory (2.2) (=(1.4)).
With any system S, a C ∗ -algebra A can be associated in which measurement theory ofthat system can be formulated. A state of the system S is represented by a pure state ρp( ∈ Sp (A∗ ), i.e., a state space ). Also, an observable is represented by a C ∗ -observable O≡ (X, F, F ) in the C ∗ -algebra A.7

The measurement of an observable O for the system¡¢
S with (or, in) the state ρp is represented by MA O, S[ρp ] in the C ∗ -algebra A. Also, we¡¢
can obtain a measured value x ( ∈ X) by the measurement MA O, S[ρp ] .
The axiom presented below is analogous to (or, a kind of generalizations of) Born’s
probabilistic interpretation of quantum mechanics [13]. We of course assert that the axiom
is a principle for all measurements, i.e., classical and quantum measurements. Cf. [41, 42].

¡
AXIOM 1. ¢[Measurement axiom]. Consider a measurement MA O ≡(X, F, F ), S[ρp ] formulated in a C ∗ -algebra A. Assume¡ that the¢ measured
value x ( ∈ X) is obtained by the measurement MA O, S[ρp ] . Then, theprobability
that Ethe x ( ∈ X) belongs to a set Ξ ( ∈ F) is given by ρp (F (Ξ))
D¡¢
≡ A∗ ρp , F (Ξ) A .
(2.37)
7

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We introduce the following classiﬁcation in measurement theory:classical measurement theory (for classical systems)measurement theoryquantum measurement theory (for quantum systems)(2.38)

where a C ∗ -algebra A is commutative or non-commutative.
Recall the (1.3), that is, quantum mechanics (cf. [71]) is formulated by“quantum mechanics”=

measurement+ the rule of time evolution(“Born’s quantum measurements”)(“Schrödinger equation”)(1.3)
Of course, Axiom 1 corresponds to “Born’s quantum measurements”. Note that quantum
measurement theory is well authorized as a principle of quantum mechanics (cf. [17, 34,84]). Our interest in this book is mainly concentrated on classical systems. Therefore, inmost cases, it suﬃces to assume that A = C(Ω).

2.5

Remarks

In this section we add some remarks concerning Axiom 1.
[(I): Probability]. It should be noted that the term “probability” appears in Axiom
1. Following the common knowledge of quantum mechanics (cf. [71, 84]), we believethat any scientiﬁc statement including the term “probability” is meaningless without theconcept of “measurement”. That is, we say that(♯) “There is no probability without measurements”.
Throughout this book, the above spirit (♯) is quite important.
[(II): It is prohibited to take measurements twice]. The quasi-product observable(or, the product observable) is used to represent “the measurement of (more than one )observables” as follows: For example, consider “the measurement of O1 and O2 for theqp

system with the state ρp (∈ Sp (A∗ ))”. If the quasi-product observable O1 × O2 of O1 and³
qp
O2 exists, the measurement is represented by MA (O1× O2 , S[ρp ] ) and not “MA (O1 , S[ρp ] )´
qp+ MA (O2 , S[ρp ] )” . If the quasi-product observable O1 × O2 does not exist, the measurement does not also exist. That is, the symbol “MA (O1 , S[ρp ] ) + MA (O2 , S[ρp ] )” isnonsense. Thus we can say that

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31(♯) only one measurement is permitted to be conducted even in the classical measurement
theory.
¡ qp¢
which is the well-known fact in quantum mechanics. The measurement MA O1× O2 , S[ρp ]is sometimes called a simultaneous measurement (or iterated measurement) of two observables O1 and O2 . That is, it is prohibited to take measurements twice in measurement
theory. For example, the following statement:¡¢¡¢• “Take two measurements MA O1 , S[ρp ] and MA O2 , S[ρp ] .”is prohibited.
[(III): Sample space]. Let ρm be a mixed state, i.e., ρm ∈ Sm (A∗ ). Applying Hopf
extension theorem (cf. [92]), we can get the measure space (X, F, ρm (F ( · )) ) such thatρm (F (Ξ)) = ρm (F (Ξ)) for all Ξ ∈ F where F is the smallest σ-ﬁeld that contains F. For
DE
simplicity, the ρm (F ( · )) is also denoted by ρm (F ( · )) or A∗ ρm , F ( · ) A . Axiom 1 makes³´
us call the measure space (X, F, ρp (F ( · )) ) or in short, (X, F, ρp (F ( · )) ) a sample
space concerning a measurement MA (O ≡ (X, F, F ), S[ρp ] ).
[(IV): Conditional probability]. Let O ≡ (X, F, F ) and O′ ≡ (Y, G, G) be observables
qp
b be a quasi-product observable of O and O′ , that is, Ob ≡ O × O′ =in A. Let O(X × Y, F × G, F × G). Assume that we know that the measured value (x, y) (∈ X × Y )b S[ρp ] ) belongs to Ξ × Y (∈ F × G). Then, it is clear
obtained by a measurement MA (O,qp

that the unknown measured value y (∈ Y ) is distributed under the conditional probabilityPΞ (·), where


qp
qp
p
×
×
G(Γ)⟩A  ρ (F (Ξ) G(Γ)) ∗ ⟨ρ , F (Ξ)
PΞ (Γ) = A=pρp (F (Ξ))⟨ρ , F (Ξ)⟩AA∗p(∀Γ ∈ G).

[(V): Commutativity and simultaneous measurability]. Let ρp be a pure state,i.e., ρp ∈ Sp (A∗ ). Let O ≡ (X, F, F ) and O′ ≡ (Y, G, G) be crisp observables in A. Now
we have the following problem:• What is the simultaneous measurability condition of O and O′ for the ﬁxed ρp ?
This is answered in [39] as follows:• ρp -commutativity, i.e., F (Ξ)G(Γ)ρp = G(Γ)F (Ξ)ρp

for all Ξ ∈ F, Γ ∈ G.

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However, in this book we are not concerned with such arguments.
[(VI): Schrödinger’s cat paradox].

Note that Schrödinger’s cat does not appear in

the world of MT. Let us explain it as follows: In 1935 (cf. [77]) Schrödinger publishedan essay describing the conceptual problems in quantum mechanics. A brief paragraphin this essay described the cat paradox.
• Suppose we put a cat in a cage with a radioactive atom, a Geiger counter, and apoison gas bottle; further suppose that the atom in the cage has a half-life of one
hour, a ﬁfty-ﬁfty chance of decaying within the hour. If the atom decays, the Geiger
counter will tick; the triggering of the counter will get the lid oﬀ the poison gas
bottle, which will kill the cat. If the atom does not decay, none of the above things
happen, and the cat will be alive. Now the question:(Q) We then ask: What is the state of the cat after the hour?
The answer according to quantum mechanics is that(A) the cat is in a state which can be thought of as half-alive and half-dead, that
“Fig.(a)” +“Fig.(b)”is, the state such as
2Fig.(a)Fig.(b)···poison gas

cattick !

cat

6

Of course, this answer (A) is curious. This is the so-called Schrödinger’s cat paradox.
This paradox is due to the fact that micro mechanics and macro mechanics are mixed in
the above situation. On the other hand, as seen in (2.38), micro mechanics (= quantum
measurement theory) and macro mechanics (= classical measurement theory) are always
separated in MT. Therefore, Schrödinger’s cat does not appear in the world of MT, though
this may be a surface solution of Schrödinger’s cat paradox.

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2.6

ExamplesAgain recall the (1.4), i.e.,“measurement theory (or in short, PMT)”=[measurements] + [the relation among systems]“Axiom 1 (2.37)”

in C ∗ -algebra A(2.39a)[Axiom 2 (3.26)](=(1.4a))or more precisely,= “Apply (2.39a) to every phenomenon by an analogy of quantum mechanics”(2.39b)(=(1.4b))Thus, in order to understand PMT, we need a little knowledge of quantum mechanics.
The following example is enough tested ,8 and thus, it is the most ﬁrm in PMT
Example 2.15. [(i): The spin observable concerning the z-axis, Stern and Gerlach’s
experiment]. Assume that we examine the beam (of silver particles) after passing throughthe magnetic ﬁeld. Then, as seen in the following ﬁgure, we see that all particles aredeﬂected either equally upwards or equally downwards in a 50:50 ratio.
“Stern and Gerlach’s experiment (1922)”

S

↑z

N

↓z

silver particle˛»√ –» √ –˛˛ 1/ 2 ED 1/ 2 ˛√√ ˛,ρp = ˛˛
1/ 2
1/ 2 ˛screenConsider the two dimensional Hilbert space V = C2 , And therefore, we get the noncommutative C ∗ -algebra A = B(V ), that is, the algebra composed of all 2 × 2 matrices.
8

A lot of tests of quantum mechanics have been conducted. Especially Aspect’s experiment [8] iswell authorized. (Cf. §2.9 Bell’s inequality)
Recall that “quantum system theory” ⊂ “PMT”. Thus,
quantum mechanics must be enough tested though the experimental test of PMT is generally meaningless.
(Cf. Remark 1.1(e).)

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Note that A = B(V ) = C(V ) = CI (V ) (cf. Example 2.3 and Remark 2.6 (i)) since thedimension of V is ﬁnite. Deﬁne Oz ≡ (Z, 2Z , F z ), the spin observable concerning thez-axis, such that, Z = {↑z , ↓z } and·¸
1 0F ({↑z }) =
,
0 0

·
¸
0 0
F ({↓z }) =
.
0 1

z

·
¸
0 0
F (∅) =,
0 0z

(2.40)·¸
1 0
F ({↑z , ↓z }) =
.
0 1
¡ z
¢
Z
z
2 ) O ≡ (Z = {↑z , ↓z }, 2 , F ), S[ρp ] , where
For example,
considerthemeasurementMB(C¯· ¸ ED· ¸¯
¯ α
α ¯¯
p
, |α|2 + |β|2 = 1. That is, considerρ = ¯¯
β
β ¯
z

³

z

¡
¢
the measurement MB(C2 ) Oz ≡ (Z = {↑z , ↓z }, 2Z , F z ), S[ρp ]zp´= “the measurement of the observable O for a particle with the state ρ ” .
Then, the probability that the measured value “↑z ” [resp. “↓z ”] is obtained by the mea¡¢
surement MB(C2 ) Oz , S[ρp ] is given by ρp (F z ({↑z })) = |α|2 [resp. ρp (F z ({↓z })) = |β|2 ].
¯· √ ¸ ED· √ ¸¯
¯ 1/ 21/√2 ¯¯p√, we see that ρp (F z ({↑z })) = 1/2 [resp. ρp (F z ({↓z }))Thus, if ρ = ¯¯
1/ 2
1/ 2 ¯= 1/2]. For the further argument, see §2.9 (Bell’s thought experiment).
[(ii): The other spin observables]. Also, we can deﬁne Ox ≡ (X, 2X , F x ), the spin observable concerning the x-axis, such that, X = {↑x , ↓x } and¸¸··
1/2 −1/2
1/2 1/2xx.
, F ({↓x }) =F ({↑x }) =−1/2 1/2
1/2 1/2(2.41)

And furthermore, we can deﬁne Oy ≡ (Y, 2Y , F y ), the spin observable concerning they-axis, such that, Y = {↑y , ↓y } and·¸
1/2 i/2yF ({↑y }) =,−i/2 1/2√
where i = −1.

·¸
1/2 −i/2
F ({↓y }) =,i/2 1/2y(2.42)¥
The following example (= “urn problem” ) is the most important in the classical PMT,
though it is somewhat artiﬁcial. That is, we believe that it is not too much to say that• the probability in Axiom 1 for classical systems is essentially the sameas the probability in the following urn problem.

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However, it should be noted that no serious test for the urn problem has been conducted.9
It is generally considered to be self-evident without serious experiments. Recall thattheoretical informatics does not require serious experiments (cf. §1.4).
Example 2.16. [The urn problem (i)]. There are three urns U1 , U2 and U3 . The urn U1[resp. U2 , U3 ] contains 8 white and 2 black balls [resp. 4 white and 6 black balls, 1 white
and 9 black balls]. That is,white balls
8
4
1

urn U1
urn U2
urn U3black balls
2
6
9(2.44)

Here, consider the following measurement M2c :M2c := “Pick out one ball from the urn U2 , and recognize the color of the ball”
In measurement theory, the “measurement” M2c is formulated as follows: Deﬁne the state
space Ω by Ω = {ω1 , ω2 , ω3 }. Here,ω1 = [8 : 2],

ω2 = [4 : 6],

ω3 = [1 : 9].

Thus, we see thatU1

· · · “the urn with the state ω1 ”

U2

· · · “the urn with the state ω2 ”

U3

· · · “the urn with the state ω3 ”

In this sense, we have the identiﬁcation;
U1 ≈ ω1 ,U2 ≈ ω2 ,U3 ≈ ω3 .

That is,9[Fuzzy statement and precise statement]. Such a test (i.e., the experimental test of an urn problem)
is usually considered to be no more than the good theme of a child’s homework. However, the question“Why is a serious test (concerning the urn problem) not required?” may be profound. The reason can beunderstood if we think that the urn problem is a model within theoretical informatics. Cf. §1.4. That is,any model, represented by a precise statement, must be tested in theoretical physics. On the other hand,a model in theoretical informatics is not required to be tested, that is, it suﬃces to be useful. Cf. (I14 ) in
§1.3. We can say that the urn problem is as true as the statement “A cat is stronger than a mouse”. Itshould be noted that the statement “A cat is stronger than a mouse” is “almost experimentally true” (cf.
(I9 )) in §1.2, though it is ambiguous, fuzzy, vague, etc.

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U1 ≈ ω1U2 ≈ ω2

U3 ≈ ω3

And further, deﬁne the observable O = ({w, b}, 2{w,b} , F ) in C(Ω) such that
F ({w})(ω1 ) = 0.8,

F ({b})(ω1 ) = 0.2,

F ({w})(ω2 ) = 0.4,

F ({b})(ω2 ) = 0.6,

F ({w})(ω3 ) = 0.1,

F ({b})(ω3 ) = 0.9,

(2.45)

where ‘w’ and ‘b’ mean white and black respectively. Then, we see thatM2c = MC(Ω) (O, S[δω2 ] ).

(2.46)

Of course, the probability that a measured value w [resp. b] is obtained is, by Axiom 1,
given byF ({w})(ω2 ) = 0.4[The urn problem (ii)][ resp. F ({b})(ω2 ) = 0.6]

(2.47)

Further, assume that the (white or black) balls in the urns U1 ,U2 and U3 are also made of “stone” or “metal”.

For example, assume that the urn U1[resp. U2 , U2 ] contains 4 stone and 6 metal balls [resp. 5 stone and 5 metal balls, 1 stone
and 9 metal balls]. That is,

urn U1
urn U2
urn U3stone balls
4
5
7

metal balls
6
5
3(2.48)

Here, consider the following measurement M2m :M2m := “Pick out one ball from the urn U2 , and recognize the materials of the ball”
The measurement M2m is formulated as follows: Deﬁne the state space Ω by Ω = {ω1 , ω2 , ω3 }.
Here,
ω1 = [4 : 6],

ω2 = [5 : 5],

ω3 = [7 : 3].

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37

Thus, we see thatU1

· · · “the urn with the state ω1 ”

U2

· · · “the urn with the state ω2 ”

U3

· · · “the urn with the state ω3 ”

In this sense, we have the identiﬁcation;
U1 ≈ ω1 ,U2 ≈ ω2 ,U3 ≈ ω3 .

And further, deﬁne the observable O′ = ({s, m}, 2{s,m} , G) in C(Ω) such that
G({s})(ω1 ) = 0.4,

G({m})(ω1 ) = 0.6,

G({s})(ω2 ) = 0.5,

G({m})(ω2 ) = 0.5,

G({s})(ω3 ) = 0.7,

G({m})(ω3 ) = 0.3.

(2.49)

Thus, we see:M2 = MC(Ω) (O′ , S[δω2 ] ).

(2.50)For example, the probability that a measured value s [resp. m] is obtained is, by Axiom
1, given byG({s})(ω2 ) = 0.5[The urn problem (iii)][ resp. G({m})(ω2 ) = 0.5].

(2.51)

However, it should noted that some information is not rep-resented in the tables (2.44) and (2.48). That is, the situation is, for example, statedprecisely as follows:(1) the urn U1 contains 10 balls such as

white balls
black balls

stone balls
40

metal balls
42(2.52)

metal balls
05(2.53)(2) the urn U2 contains 10 balls such as

white balls
black balls

stone balls
4
1

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CHAPTER 2. MEASUREMENTS (AXIOM 1)

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(3) the urn U3 contains 10 balls such asstone balls
1
6

white balls
black balls

metal balls
03(2.54)

Here, consider the following measurement M2cm :M2cm := “Pick out one ball from the urn U2 , and recognize the color andmaterials of the ball”.
The measurement M12 is formulated as follows: Put Ω = {ω1 , ω2 , ω2 }. Deﬁne the state
space Ω by Ω = {ω1 , ω2 , ω3 }. Here,
·
¸
4 4
ω1 =
,
0 2

·
¸
4 0
ω2 =
,
1 5

·
¸
1 0
ω3 =
.
6 3

Thus, we see thatU1

· · · “the urn with the state ω1 ”

U2

· · · “the urn with the state ω2 ”

U3

· · · “the urn with the state ω3 ”

In this sense, we have the identiﬁcation;
U1 ≈ ω1 ,U2 ≈ ω2 ,U3 ≈ ω3 .

That is,U1 ≈ ω1U2 ≈ ω1stone metalstone metal

U3 ≈ ω1stone metalqp

b = ({w, b} × {s, m}, 2{w,b}×{s,m} , H( ≡ F × G)) in
And further, deﬁne the observable O
C(Ω) such that
H({(w, s)})(ω1 ) = 0.4, H({(w, m)})(ω1 ) = 0.4, H({(b, s)})(ω1 ) = 0.0, H({(b, m)})(ω1 ) = 0.2,
H({(w, s)})(ω2 ) = 0.4, H({(w, m)})(ω2 ) = 0.0, H({(b, s)})(ω2 ) = 0.1, H({(b, m)})(ω2 ) = 0.5,
H({(w, s)})(ω3 ) = 0.1, H({(w, m)})(ω3 ) = 0.0, H({(b, s)})(ω3 ) = 0.6, H({(b, m)})(ω3 ) = 0.3,
(2.55)

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39

which is, of course, constructed by (2.52) + (2.53) + (2.54) . Then, we see thatb S[δω ] ).
M12 = MC(Ω) (O,2(2.56)

Of course, the probability that a measured value (w, s) [resp. (w, m), (b, s), (b, m)] is
obtained is, by Axiom 1, given by
F ({(w, s)})(ω2 ) = 0.4[ resp. F ({(w, m)})(ω2 ) = 0.0, F ({(b, s)})(ω2 ) = 0.1, F ({(b, m)})(ω2 ) = 0.5].

(2.57)¥

Example 2.17. [Gaussian observable10]. [(i): Gaussian observable in C(Ω)]. Put Ω =[a, b] (⊆ R, the real line), i.e., the closed interval And let σ be a ﬁxed positive real. Deﬁnethe normal observable (or Gaussian observable) OGσ ≡ (R, BR , Gσ ) in C(Ω) such that:Z(x−ω)2
1σ[G (Ξ)](ω) = √e− 2σ2 dx (∀Ξ ∈ BR , ∀ω ∈ Ω ≡ [a, b]),2πσ 2 Ξwhich will be often used in this book.
y
6
x2

1
y = √2πσ
e− 2σ2
2

−2σ

−σ

σ

2σ

-

x

68.3%
95.4%

Here,

√ 1
2πσ 2

Rσ

√

−σ

x2

1
e− 2σ2 dx = 0.683... and √2πσ
2

1
2πσ 2

Z 1.96σ−1.96σ2− x2

e 2σ dx ≈ 0.95,

R 2σ

√

−2σ

x2

e− 2σ2 dx = 0.954... Also, note that

1
2πσ 2

Z 1.65σ−∞x2

e− 2σ2 dx ≈ 0.95

(2.58)

10
Why is the Gaussian observable fundamental? We should not be too serious with the question.
That is because we do not necessarily need a complete reason in theoretical informatics (cf. Chapter 1),
though the diﬀerential geometrical reason must be indispensable for theoretical physics. In informatics,
what is important is “useful or not”. And we know that the Gaussian observable is quite useful. Also
recall that every equation (e.g., Boltzmann’s kinetic equation, Navier-Stokes equation, etc.) in theoreticalinformatics is somewhat empirical. As mentioned in (I9 ) in §1.2, we think that “useful” =⇒ “almost
experimentally true”.

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Borel ring (Rd , Bbd), where Bbd= {Ξ ∈ BRd : Ξ is a bounded Borel set in Rd }. And
Rd
Rd
deﬁne the d-dimensional Gaussian observable OΣ ≡ (Rd , Bbd, F Σ ) in C0 (Rd ) such that:RdΣ[F (Ξ)](⃗ω ) = √1d2π |Σ|1/2Z1
exp[ − (⃗x − ⃗ω )t Σ−1 (⃗x − ⃗ω )]d⃗x
2Ξ(∀Ξ ∈ Bbd
Rd ,∀⃗ω ∈ Rd ),(2.59)

where the Σ is a covariance (d×d)-matrix, i.e., a positive deﬁnite (d×d)-matrix. Of course,
the probability that a measured value obtained by the measurement MC0 (Rd ) (OΣ , S[δω⃗ 0 ] )belongs to Ξ (∈ Bbd) is given by [F Σ (Ξ)](⃗ω0 ).
Rd¥
Example 2.18. [Discrete Gaussian observable]. Put Ω ≡ [a, b] ( ⊆ R, the real line),
the closed interval. Let σ > 0. And let N be a suﬃciently large ﬁxed integer. Put
XN ≡ { Nk | k = 0, ±1, ±2, ..., ±N 2 }. And deﬁne the discrete Gaussian observable Oσ2 ,N≡ (XN , 2XN , Fσ,N ) in the commutative C ∗ -algebra C([a, b]) such that:[Fσ,N ({k/N })](ω)
 1 R∞(x−ω)2√]dx

1 exp[ −
2
2σ 2
N
−

 2πσ R k 2N1
2
+ 2N
N
√ 1
=
exp[ − (x−ω)
2 ]dx
k
1
2
2σ
−
2πσ

N
2N

1
 √ 1 R −N + 2N(x−ω)2
]dx
exp[
−
2
2σ 2
−∞
2πσ

(k = N 2 , ∀ω ∈ [a, b]),(∀k = 0, ±1, ±2, ..., ±(N 2 − 1),∀ω ∈ [a, b]),(k = −N 2 , ∀ω ∈ [a, b]).
(2.60)

And thus, for any Ξ ( ⊆ XN ), we deﬁne [Fσ,N (Ξ)](ω) =

P
k
∈Ξ
N

[Fσ,N ({k/N })] (ω). ThisOσ2 ,N , as well as the d-dimensional Gaussian observable OΣ (in Example 2.17), is themost important observable in classical measurements.

¥

Example 2.19. [Fuzzy numbers observable (= triangle observable = round error observable)].

Let ∆ be any positive number. Deﬁne the membership function (i.e., triangle³´
fuzzy number) Z∆ ∈ C0 (R), where R is the real line with the usual topology such
that:

ω
0≤ω≤∆
 1− ∆
ω+ 1 −∆ ≤ ω ≤ 0Z∆ (ω) = ∆0otherwise .

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z (ω)

1
6
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1

41

-

-1
©

0

1

ω

ª
k ∈ Z ≡ {0, ±1, ±2, ...} .

Deﬁne the C ∗ -observable OZ∆ ≡P∆∆(Z∆ , P0 (Z∆ ), ζ(·) ) in the commutative C ∗ -algebra C0 (R) such that ζΞ (ω) = x∈Ξ Z∆ (ωPut Z∆ ≡∆k:−x) (∀Ξ ∈ P0 (Z∆ ), ∀ω ∈ R). This C ∗ -observable is called a fuzzy numbers observable in C0 (R). Putting ∆ = 1, we frequently use the fuzzy numbers observable OZ ≡(Z, P0 (Z), ζ(·) ) in this book.
¥
Example 2.20.

[(i): Exact observable]. Let Z be the set of all integers, i.e., Z ={0, ±1, ±2, ...}. And put P0 (Z) = {A( ⊆ Z) | A is ﬁnite }. Consider a commutativeC ∗ -algebra C0 (Z). And deﬁne the exact observable OEXA ≡ (Z, P0 (Z), E(·) ) in C0 (Z) suchthat:
 1 n ∈ Ξ( ∈ P0 (Z))
EΞ (n) =

(2.61)0 n∈/ Ξ( ∈ P0 (Z))

which is called the exact observable (or, fundamental observable) in C0 (Z). Of course wewant to deﬁne the exact observable in C0 (R) (or, C([a, b])). However, it is impossible
in the C ∗ -algebraic formulation. For this, we must prepare the W ∗ -algebraic formulation(cf. Chapter 9).
[(ii): Approximate exact observable]. Though the exact observable in C([0, 1]) can not be
deﬁned, we have the approximate exact observable OA
EXA in C([0, 1]) as follows: Let N be
( ≡ 1)}. Deﬁne the approximate
a suﬃciently large integer. Put XN = { N1 , N2 , N3 , ..., NN
exact observable OA
EXA ≡ (XN , P(XN ), F ) in C([0, 1]) such that:

 1
1
2
[F ({ })](ω) =− N2 (ω − N1 ) + 12

N
0

 0
N
N2
[F ({ })](ω) =(ω − NN−1 ) + 212
N
1

(0 ≤ ω ≤ N1 − N12 )( N1 − N12 ≤ ω ≤ N1 + N12 )( N1 + N12 ≤ ω ≤ 1)
(0 ≤ ω ≤ NN−1 − N12 )( NN−1 − N12 ≤ ω ≤ NN−1 + N12 )N( NN−1 + N12 ≤ ω ≤ N− N12 )

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For n = 2, 3, ..., N − 1,

0


2


) + 12
 N2 (ω − n−1Nn1[F ({ })](ω) =

2
N


− N2 (ω − Nn ) + 120

− N12 )
(0 ≤ ω ≤ n−1
N( n−1− N12 ≤ ω ≤ n−1+ N12 )NN
( n−1
+ N12 ≤ ω ≤ Nn − N12 )N( Nn − N12 ≤ ω ≤ Nn + N12 )( Nn + N12 ≤ ω ≤ 1)

Note that the observable (i.e., fuzzy numbers observable) in Example 2.19 is also regarded
as “approximate exact observable”, if ∆ is suﬃciently small.
¥
Example 2.21. [Null observable]. Deﬁne the observable O(nl) ≡ ({0, 1}, 2{0,1} , F (nl) ) inA such that:
F (nl) (∅) ≡ 0, F (nl) ({0}) ≡ 0, F (nl) ({1}) ≡ 1A , F (nl) ({0, 1}) ≡ 1A

in A,

(2.62)

which may be called the null observable (or, existence observable). Then, we have themeasurement MA (O(nl) ≡ ({0, 1}, 2{0,1} , F (nl) ), S[ρp ] ). Note that:(♯) the probability that measured value (by MA (O(nl) , S[ρp ] )) is equal to 1 ( ∈ {0, 1}) is
given by 1. That is, the measured value is always equal to 1 ( ∈ {0, 1}).
Thus, we think that “to take the measurement MA (O(nl) , S[ρp ] )” is the same as “to assure
the existence of the system”.
¥

2.7

Operations of observables

Recall the identiﬁcation (2.36), that is, we have the following identiﬁcation:
Fbk←→ Ok = (R, BR , Fk ) in C(Ω)(real valued function on Ω)(k = 1, 2, ..., n).

(2.63)(crisp observable)

Note that Fb1 + Fb2 , Fb1 · Fb2 , etc. are meaningful in the ordinary sense since Fb1 and Fb2 arereal-valued functions. This makes us ask the following question.
• For each k = 1, 2, ..., n, consider an observable Ok ≡ (Xk , Fk , Fk ) in a C ∗ -algebra
A. Are O1 + O2 , O1 · O2 , etc. meaningful in general?

Or, how the operations ofobservables are deﬁned?

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43

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This will be answered in what follows.

For each k = 1, 2, ..., n, consider an observable Ok ≡ (Xk , Fk , Fk ) in a C ∗ -algebra A.

Put O = × k=1,2,...,n Ok . Let g : ×k=1 → Y be a measurable map, where Y has the subﬁeld
qpn

G of 2Y . Then we can deﬁne the observable (Y, G, G), which is symbolically representedby g(O1 , O2 , ..., On ), as follows:• the (Y, G, G) is the image observable of the quasi-product observable O ≡ ( ×k=1 Xk ,×n Fk , Fb) concerning g (if it exists). That is,n

k=1(Y, G, G) = g(O)(2.64)

i.e.,
G(Γ) = Fb(g −1 (Γ))(∀Γ ∈ G).

(2.65)

Example 2.22. [The addition of triangle observables]. Let OZ ≡ (Z, P0 (Z), ζ(·) ) be
the fuzzy numbers observable in C0 (R) (cf. Example 2.19). Now let us calculate OZ +OZ as follows: Note that the product observable OZ × OZ ≡ (Z2 , P0 (Z2 ), ζ(·) × ζ(·) ) isrepresented by(i) |m − n| ≥ 2[ζ{m} × ζ{n} ](ω) = 0(2.66)(ii) |m − n| = 1[ζ{m} × ζ{n} ](ω) =


 0


(x−m)(x−n)
20ω ≤ min{m, n}min{m, n} ≤ ω ≤ max{m, n}ω ≤ min{m, n}(iii) m = n


 0

(x − (m − 1))2
[ζ{m} × ζ{m} ](ω) =(x − (m + 1))2
0

ω ≤m−1m−1≤ω ≤mm≤ω ≤m+1
m+1≤ω

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Thus we see(ζ + ζ){n} (ω)
0ω ≤m−1






2

(ω − (m − 1))
m−1≤ω ≤m(ζ + ζ){2m} (ω) =

2

(ω
−
(m
+
1))
m≤ω ≤m+1
(when n = 2m)

m+1≤ω
=
0


ω≤m

 0


2

−(ω
−
(2m
+
1)/2)
+
1/2
m≤ω ≤m+1
(ζ
+
ζ)
(ω)
=

{2m+1}
(when n = 2m + 1)0
m+1≤ω(2.68)

n = 2m
m−1

m

m+1

-

ω

n = 2m + 1
-

m

m+1

ω

Therefore we get the OZ + OZ ≡ (Z, P0 (Z), (ζ + ζ)(·) ) in C0 (R), where(ζ + ζ)Ξ (ω) =X(ζ + ζ){n} (ω)(Ξ ∈ P0 (Z), ω ∈ Ω).

n∈Ξ

¥
Example 2.23. (χ2 -observable).

Consider the (1-dimensional) Gaussian observableσ
Oσ2 ≡ (R, Bbd
R , G ) in A ≡ C0 (R) such that:Z(x−µ)21σ
e− 2σ2 dx[G (Ξ)](µ) = √2πσ 2 Ξ(∀µ ∈ R ∀Ξ ∈ Bbd
R ),(where σ 2 is a variance). And further, for each ϕ (= 0, 1, 2, ...), deﬁne the product observable (Oσ2 )ϕ+1 such that
σ ϕ+1
(Oσ2 )ϕ+1 = (Rϕ+1 , Bbd)
Rϕ+1 , (G )( in A ≡ C0 (R)where(Gσ )ϕ+1 (Ξ1 × Ξ2 × · · · × Ξϕ+1 ) = Gσ (Ξ1 ) × Gσ (Ξ2 ) × · · · × Gσ (Ξϕ+1 ).

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Deﬁne the map g : Rϕ+1 → R such that
ϕ+1R

∋ (x1 , x2 , x3 , ..., xϕ+1 ) 7→

ϕ+1
X
(xk −
k=1

Pϕ+1

j=1 xj

ϕ+1
2
σ

)2

∈ R.

The image observable g((Oσ2 )ϕ+1 ) is called the χ2 -observable with ϕ, the degree of freedom.
¥

2.8

Frequency probabilities

The meaning of “probability” in Axiom 1 seems to be a matter of common knowledge
in quantum mechanics. However, we, in this section, study the relation between “theprobability in Axiom 1” and “frequency probability”.

¡¢
For each k = 1, 2, ..., n, consider a measurement MAk Ok ≡ (X, P(X), Fk ), S[ρpk ] in ab = Nn Ak , i.e.,C ∗ -algebra Ak , where we assume, for simplicity, that X is ﬁnite. Put A
k=1∗

the tensor product C -algebra of {Ak : k = 1, 2, ..., n}. Here, consider the tensor-productNNb (≡ Nn Ak ) such that:C ∗ -observable n Ok ≡ (X n , P(X n ), Fb ≡ n Fk ) in A
k=1

k=1

k=1

Fb(Ξ1 × Ξ2 × · · · × Ξn ) = F1 (Ξ1 ) ⊗ F2 (Ξ2 ) ⊗ · · · ⊗ Fn (Ξn ) (∀Ξk ∈ P(X), k = 1, 2, ..., n).
(2.69)Nn
N
Therefore, we get the measurement M⊗Ak ( nk=1 Ok , S[ Nnk=1 ρpk ] ) in
k=1 Ak , which isNnalso denoted by k=1 MAk (Ok , S[ρpk ] ) and called the repeated measurement (or, “parallelmmeasurement”) of MAk (Ok , S[ρpk ] )’s. Put M+1(X) = {ν : ν is a positive measure on X

such that ν(X) = 1 } and deﬁne the map g : X n → Mm
+1 (X) such that:[g(x1 , x2 , ..., xn )](Ξ) =♯[{k : xk ∈ Ξ}]n(∀Ξ ∈ P(X)),(2.70)

where ♯[B] = “the number of the elements of a set B”.
Then we have the following proposition.
Proposition 2.24. [The weak law of large numbers, cf [56]]. Suppose the above notab Ξ,ϵ ( ∈ P(X n ) ) bytions. For any ϵ > 0 and any Ξ ( ∈ P(X)), deﬁne D
n
¯
¯ o
n
1X p
¯
¯
n
b
x)](Ξ) −DΞ,ϵ = xb = (x1 , x2 , ..., xn ) ∈ X : ¯[g(bρk (Fk (Ξ))¯< ϵ .
n k=1

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Then we see that

³
´
1
p
n
b
b
1 − 2 ≤ ( ⊗k=1 ρk ) F (DΞ,ϵ ) ≤ 1, (∀Ξ ∈ P(X), ∀ϵ > 0, ∀n).
(2.72)
4ϵ nP
Proof. We easily see that [g(bx)](Ξ) = n1 nk=1 χΞ (πk (bx)) (∀bx = (x1 , x2 , ..., xn ) ∈ X n ),where πk : X n → X is deﬁned by πk (bx) ≡ πk (x1 , x2 , ..., xk , ..., xn ) = xk and χΞ : X → R
is the characteristic function of Ξ (i.e., χΞ (x) = 1 (x ∈ Ξ), = 0 (x ∈/ Ξ) ). Using theterms in Kolmogorov’s probability theory, we can say that χΞ (πk ( · )), k = 1, 2, ..., n, are
³
´
p
n
n
n
b
b
independent variables on a probability space X , P(X ), P (· ) ≡ (⊗k=1 ρk )(F (· )) . AlsoR
R
it is clear that X n χΞ (πk (bx))Pb(dbx) = X n [χΞ (πk (bx))]2 Pb(dbx) = ρpk (Fk (Ξ)) (k = 1, 2, ..., n).
Therefore, by Čebyšev inequality, we seePnp¯ Pn χ (π (b
¯ o´
³
´
³n
¯ k=1 Ξ k x)))nnk=1 ρk (Fk (Ξ)) ¯bbb
P X \ DΞ,ϵ = P xb∈X : ¯
−
¯≥ ϵ
n
n
Z
n ³
´
X
1
≤ 2 2
|
χΞ (πk (bx)) − ρpk (Fk (Ξ)) |2 Pb(dbx)
ϵ n X n k=1
n Z
1 X|χΞ (πk (bx)) − ρpk (Fk (Ξ))|2 Pb(dbx)= 2 2
ϵ n k=1 X n£¤
1
1≤ 2 max ρpk (Fk (Ξ))(1 − ρpk (Fk (Ξ))) ≤ 2 ,(2.73)
ϵ n 1≤k≤n
4ϵ nwhich implies (2.72). This completes the proof.
Now we can show the following theorem as an immediate consequence of Proposition
2.24. It clariﬁes the “probability” in Axiom 1 from the statistical point of view.
Theorem 2.25. [Frequency probability, cf. [42] ]. Put Ak = A, ρpk = ρp and Ok = O≡ (X, P(X), F ), k = 1, 2, ..., n, in Proposition 2.24. Consider the repeated measurement
NN
M⊗A ( nk=1 O, S[⊗nk=1 ρp ] ) in nk=1 A. Then, we see that
n
³ O
¯
¡
¢´
♯[{k : xk ∈ Ξ}] ¯¯1pnF ) {bx ∈ X n : ¯ρp (F (Ξ)) −1 − 2 ≤ ( ⊗k=1 ρ ) (< ϵ} ≤ 1,
4ϵ n
n
k=1

(∀Ξ ∈ P(X), ∀ϵ > 0, ∀n).
³ N
¡ ¢´b is the probability that a measured
Here note, by Axiom 1, that ( ⊗nk=1 ρp ) ( nk=1 F ) ΞNnb Therefore, if n is suﬃciently large, for avalue by M⊗A ( k=1 O, S[⊗nk=1 ρp ]) belongs to Ξ.
Nmeasured value x
b (= (x1 , x2 , ..., xn ) ∈ X n ) by M⊗A ( nk=1 O, S[⊗nk=1 ρp ] ), we can consider

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(in the sense of (2.72)) thatρp (F (Ξ)) ≈♯[{k : xk ∈ Ξ}].
n(2.74)¥The (2.74) says that• “probability in Axiom 1” = “frequency probability”.
Thus, there is a reason that the probability space (X, F, ρp (F ( · )) ) is called a sample¡¢
space obtained by a measurement MA O, S[ρp ] .
Remark 2.26. [“repeated measurement = iterated measurement” for S[δω0 ] ]. As seen inthis section, we think that)”“take a measurement Mω0 N times” ⇔ “take a measurement M⊗N( ⊗N
n=1 O, S[⊗N
n=1 δω0 ]n=1 C(Ω)

Thus, in classical measurements, we have the following identiﬁcation:) ” ⇔ “take a measurement MC(Ω) (ON , S[δω0 ] ) ”“take a measurement M⊗N( ⊗N
n=1 O, S[⊗N
n=1 δω0 ]n=1 C(Ω)

That is because it holds that
D

⊗N
n=1

E
N
N
⊗
δ
,
⊗
F
(Ξ
)
ω
n
n=1 0
n=1
M(Ω)
⊗N

n=1

D
E
N
δ
=
,
×
F
(Ξ
)
.
ω
n
0
n=1
M(Ω)
C(Ω)
C(Ω)

However, it should be noted that it does not always hold that “repeated measurement =iterated measurement” in statistical measurement theory (mentioned in Chapter 8) andquantum measurement theory.
¥Deﬁnition 2.27. [Semi-distance, moment method (inference for a pure state in repeatedmeasurement)].
[(i): Semi-distance].

Let Y be a set. If the map ∆ : Y × Y → R satisﬁes the following(a)∼(d):(a): ∆(x, y) ≥ 0 (∀x, y ∈ Y ),(b): “x = y” ⇒ ∆(x, y) = 0,(c): ∆(x, y) = ∆(y, x) (∀x, y ∈ Y ),(d): ∆(x, y) ≤ ∆(x.z) + ∆(z, y) (∀x, y, z ∈ Y ),then, the ∆ is called a semi-distance on Y . In addition, if “(b’): x = y ⇔ ∆(x, y) = 0” isassumed, then the ∆ is called a distance ( or metric ) on Y .

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CHAPTER 2. MEASUREMENTS (AXIOM 1)

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/¡¢[(ii): Moment method]. Assume the ρp0 (in MA O ≡ (X, F, F ), S[ρp0 ] ) is unknown. And
further, we get the sample space (X, F, ν0 ) from the measured value x
b (= (x1 , x2 , ...,
Nxn ) ∈ X n ) obtained by M⊗A ( nk=1 O, S[⊗nk=1 ρp0 ] ). That is, ν0 (Ξ) ≈ ♯[{k:xnk ∈Ξ}] . Note, by

11(2.74), that ρp (F (Ξ)) ≈ ν0 (Ξ) (∀Ξ ∈ F). Let ∆ be a semi-distance on Mm+1 (X). Then,

there is a very reason to infer the unknown ρp0 ( ∈ Sp (A∗ )) such that∆(ν0 , ρp0 (F ( · )) ) = p min∆(ν0 , ρp (F ( · )) ).
p∗
ρ ∈S (A )This method is called “generalized moment method” or “moment method”.

Cf. §9.4.

Note that the “semi-distance ∆ on Mm+1 (X)” is not always unique. In this sense, themoment method is somewhat artiﬁcial.
¥
Example 2.28. [The urn problem by the moment method]. There are two urns ω1 andω2 . The urn ω1 [resp. ω2 ] contains 8 white and 2 black balls [resp. 4 white and 6 blackballs]. Assume that they can not be distinguished in appearance. Choose one urn from thetwo. Assume that you do not know whether the chosen urn is ω1 or ω2 . Now you sample,
randomly, with replacement after each ball. In 7 samples, you get (w, b, b, w, b, w, b) insequence where “w” = “white”, “b” = “black”.
(Q) Which is the chosen urn, ω1 or ω2 ?
ω1

ω2

¡
¢
[Answer]. We regard Ω ≡ {ω1 , ω2 } as the state space. And consider the observable¡¢O ≡ (X ≡ {w, b}, 2{w,b} , F ) in C(Ω) where[F ({w})](ω1 ) = 0.8,

[F ({b})](ω1 ) = 0.2,

[F ({w})](ω2 ) = 0.4,

[F ({b})](ω2 ) = 0.6.

Note that we have the real sample space (X ≡ {w, b}, 2{w,b} , ν0 ) such that:ν0 (∅) = 0,ν0 ({w}) = 3/7,

ν0 ({b}) = 4/7,

ν0 ({w, b}) = 1.

11

The deﬁnition of the semi-distance ∆ may be too strong for the generalized moment method. However, in this book we focus on the above deﬁnition.

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49

Also, note that the measurement
MC(Ω) (O, S[δω1 ] )[resp. MC(Ω) (O, S[δω2 ] )]has the sample space(X ≡ {w, b}, 2{w,b} , [F ( · )](ω1 ))[resp. (X ≡ {w, b}, 2{w,b} , [F ( · )](ω2 ))].

Thus, it suﬃces to compare∆(ν0 , [F ( · )](ω1 )) and ∆(ν0 , [F ( · )](ω2 )),where ∆ is a certain distance on Mm+1 ({w, b}). For example deﬁne the distance ∆ suchthat:∆(ν1 , ν2 ) = |ν1 ({w}) − ν2 ({w})| + |ν1 ({b}) − ν2 ({b})| (∀ν1 , ν2 ∈ Mm
+1 ({w, b})).
Then, we see∆(ν0 , [F ( · )](ω1 )) = |3/7 − 8/10| + |4/7 − 2/10| = 52/70
and∆(ν0 , [F ( · )](ω2 )) = |3/7 − 4/10| + |4/7 − 6/10| = 10/70.
Thus, we can, by the moment method, infer that the unknown urn is ω2 .
¥2.9

Appendix (Bell’s thought experiment)(Continued from Example 2.15. Also see the footnote below12 )2.9.1

EPR thought experiment

Although the original “EPR experiment (cf. [22])” was proposed in the framework ofclassical mechanics (cf. Chapter 12), the following argument is the quantum form of the“EPR experiment”.13
12

All appendixes in this book can be skipped.
The argument in §2.9.1 is essentially the same as EPR-experiment (i.e., EPR-paradox,cf. [22]), which
will be again discussed in §12.7.
13

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Now consider the quantum system composed of two particles with the singlet state ρsNN(concerning z-axis) formulated in B(C2 C2 ), where C2 C2 is the tensor Hilbert space¡¢Nof C2 and C2 . The singlet state ρs is represented by ρs = |ψs ⟩⟨ψs | ∈ Sp (B(C2 C2 )∗ ) ,
where
· ¸
· ¸
¢
1 ¡
1
0
2
2
2
ψs = √ ⃗e1 ⊗ ⃗e2 − ⃗e2 ⊗ ⃗e1 ( ∈ C ⊗ C ). ⃗e1 =∈ C , ⃗e2 =∈ C2 . (2.75)
0
1
2
¡
2
And consider the measurement MB(C2 )⊗B(C2 ) Oz ⊗ Oz ≡ (Z 2 = {↑z , ↓z }2 , 2Z , F z ⊗ F z ),¢S[ρs ] , where
·
¸
·
¸
1 0
0 0
z
z
F ({↑z }) =, F ({↓z }) =
,
0 0
0 1
·
¸
0 0
F (∅) =
,
0 0
z

·
¸
1 0
F ({↑z , ↓z }) =
.
0 1
z

¡
¢
2
Taking the measurement MB(C2 )⊗B(C2 ) Oz ⊗ Oz ≡ (Z 2 = {↑z , ↓z }2 , 2Z , F z ⊗ F z ), S[ρs ] ,we see that(a) the probability that a measured value (↑z , ↑z ) is obtained is equal to³´=ρs F z ({↑z }) ⊗ F z ({↑z })
D 1 ¡
¢
¢E
1 ¡
=C2 ⊗C2 √ ⃗e1 ⊗ ⃗e2 − ⃗e2 ⊗ ⃗e1 , [F z ({↑z }) ⊗ F z ({↑z })] √ ⃗e1 ⊗ ⃗e2 − ⃗e2 ⊗ ⃗e1 C2 ⊗C22
2=0(b) the probability that a measured value (↑z , ↓z ) is obtained is equal to³´=ρs F z ({↑z }) ⊗ F z ({↓z })
D 1 ¡
¢
¢E
1 ¡
=C2 ⊗C2 √ ⃗e1 ⊗ ⃗e2 − ⃗e2 ⊗ ⃗e1 , [F z ({↑z }) ⊗ F z ({↓z })] √ ⃗e1 ⊗ ⃗e2 − ⃗e2 ⊗ ⃗e1 C2 ⊗C22
2=1/2(c) the probability that a measured value (↓z , ↑z ) is obtained is equal to³´=ρs F z ({↓z }) ⊗ F z ({↑z })
D 1 ¡
¢
¢E
1 ¡
=C2 ⊗C2 √ ⃗e1 ⊗ ⃗e2 − ⃗e2 ⊗ ⃗e1 , [F z ({↓z }) ⊗ F z ({↑z })] √ ⃗e1 ⊗ ⃗e2 − ⃗e2 ⊗ ⃗e1 C2 ⊗C2
2
2
=1/2

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(d) the probability that a measured value (↓z , ↓z ) is obtained is equal to³´=ρs F z ({↓z }) ⊗ F z ({↓z })
D 1 ¡
¢
¢E
1 ¡
=C2 ⊗C2 √ ⃗e1 ⊗ ⃗e2 − ⃗e2 ⊗ ⃗e1 , [F z ({↓z }) ⊗ F z ({↓z })] √ ⃗e1 ⊗ ⃗e2 − ⃗e2 ⊗ ⃗e1 C2 ⊗C22
2=0.
Here, it should be noted that we can assume that the x1 and the x2 (in (x1 , x2 ) ∈ {(↑z , ↑z ), (↑z , ↓z ), (↓z , ↑z ), (↓z , ↓z )}) are respectively obtained in Tokyo and in New York(or, in the earth and in the polar star).
(probability 12 )(probability 12 )

(b)

(c)
↑z

↓z

↓z

↑z

Tokyo

New YorkorTokyo

New York

This fact is, ﬁguratively speaking, explained as follows:• Immediately after the particle in Tokyo is measured and the measured value ↑z[resp. ↓z ] is observed, the particle in Tokyo informs the particle in New York “Yourmeasured value has to be ↓z [resp. ↑z ]”.
Therefore, the above fact implies that quantum mechanics says that there is something
faster than light. This is essentially the same as the de Broglie paradox (cf. [20]. Also see§9.3.3). That is,• if we admit quantum mechanics, we must also admit the fact that there issomething faster than light. (cf. [18, 78]).

(2.76)

Of course we admit PMT, and therefore, we believe that there is something faster thanlight.

2.9.2

Bell’s thought experiment

In this section, we review Bell’s thought experiment in (quantum) measurement theory.
(Cf. [9, 18, 78].) All the idea is, of course, owed to J.S. Bell [9]. Thus, we do not intend toassert our originality in this section. The argument is divided into two steps (i.e., [Step:I] and [Step: II]). [Step: I] is essentially the same as the previous section (i.e., §2.9.1).

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CHAPTER 2. MEASUREMENTS (AXIOM 1)

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/[Step: I]. Let a = (α1 , α2 ) be any element in R2 such that ∥a∥R2 ≡ (|α1 |2 + |α2 |2 )1/2 = 1.
Put

·

√ ¸
0√
α1 − α2 −1
σa =∈ B(C2 ),α1 + α2 −1
0

· ¸
1
⃗e1 =
∈ C2 ,
0

· ¸
0
⃗e2 =
∈ C2 .
1

It is easy to see that the self-adjoint matrix σa : C2 → C2 has a unique spectral represen(1)(−1)

tation : σa = Fa − Fa
that

(1)

, where Fa(−1)

and Faare orthogonal projections on C2 such

·
√ ¸
√ ¸
1
1√
α1 − α2 −1
1 √−α1 + α2 −1
(−1)
, Fa
=
.
1
1
2 α1 + α2 −1
2 −α1 − α2 −1¡¢(1)Deﬁne the observable Oa ≡ X ≡ {1, −1}, P(X), Fa in B(C2 ) such that Fa ({1}) = Fa1
Fa(1) =·(−1)

and Fa ({−1}) = Fa

Now consider the quantum system composed of two particles with the singlet state ρsNN(concerning z-axis) formulated in B(C2 C2 ), where C2 C2 is the tensor Hilbert space¡¢Nof C2 and C2 . The singlet state ρs is represented by ρs = |ψs ⟩⟨ψs | ∈ Sp (B(C2 C2 )∗ ) ,
where
¢
1 ¡
ψs = √ ⃗e1 ⊗ ⃗e2 − ⃗e2 ⊗ ⃗e1 ( ∈ C2 ⊗ C2 ).
2
Put a = (α1 , α2 ), b = (β1 , β2 ) ∈ R2 where ∥a∥R2 = ∥b∥R2 = 1. And deﬁne the tensor
Nproduct observable Oab ( ≡ Oa ⊗ Ob ) = (X 2 , P(X 2 ), Fa Fb ) in B(C2 ⊗ C2 ) such thatOO(FaFb )({(x1 , x2 )}) = Fa ({x1 })Fb ({x2 })(∀(x1 , x2 ) ∈ X 2 ≡ {−1, 1}2 ).
Thus we get a measurement MB(C2 ⊗C2 ) (Oab , S[ρs ] ) in B(C2 ⊗ C2 ). Axiom 1 says thatthe probability that a measured value x ( = (x1 , x2 )) ∈ X 2 (≡ {1, −1}2 ) obtained by
the measurement MB(C2 ⊗C2 ) (Oab , S[ρs ] ) belongs to a set B ( ⊆ X 2 ) is given by νEPR (B),¡¢P
where νEPR (B) = x≡(x1 ,x2 )∈B ρs (Fa ⊗ Fb )({(x1 , x2 )}) . Therefore, we see, for example,
that(♯) if we know that x1 = 1, quantum mechanics says that the probability that x2 = 1[resp. x2 = −1] is given by
νEPR ({1} × {1})νEPR ({1} × {1, −1})

h
resp.

νEPR ({1} × {−1}) i
νEPR ({1} × {1, −1})and further, if we know that x1 = −1, the probability that x2 = 1 [resp. x2 = −1]
is given by
νEPR ({−1} × {1})νEPR ({−1} × {1, −1})hνEPR ({−1} × {−1}) i
resp.
.
νEPR ({−1} × {1, −1})

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/[Step: II]. Let a1 (= (α11 , α21 )), a2 (= (α12 , α22 )), b1 (= (β11 , β21 )) and b2 (= (β12 , β22 )) be elements
in R2 such that ∥a1 ∥R2 = ∥a2 ∥R2 = ∥b1 ∥R2 = ∥b2 ∥R2 = 1. Further, consider the parallelNmeasurement i,j=1,2 MB(C2 ⊗C2 ) (Oai bj , S[ρs ] ) in ⊗i,j=1,2 B(C2 ⊗ C2 ) ( ≡ B( ⊗i,j=1,2 (C2 ⊗C2 ))), that is,O

MB(C2 ⊗C2 ) (Oai bj , S[ρs ] )i,j=1,2= MB(⊗i,j=1,2 (C2 ⊗C2 ))¡¡O× X 2, P( × X 2),

i,j=1,2i,j=1,2¢¢(Fai ⊗ Fbj ) , S[⊗i,j=1,2 ρs ] .

i,j=1,2Here note that ⊗i,j=1,2 ρs = ρs ⊗ ρs ⊗ ρs ⊗ ρs = |ψs ⊗ ψs ⊗ ψs ⊗ ψs ⟩⟨ψs ⊗ ψs ⊗ ψs ⊗ ψs | and¢¡× X 2 ∋ (x111, x112), (x121, x122), (x211, x212), (x221, x222) = x ∈ X 8 ≡ {−1, 1}8.

i,j=1,2Axiom 1 (2.37) says that the probability that a measured value x ∈ X 8 (≡ {1, −1}8 )N
obtained by the parallel measurement i,j=1,2 MB(C2 ⊗C2 ) (Oai bj , S[ρs ] ) belongs to a set B
¢
¡
P
Q
ij
( ⊆ X 8 ) is given by νBTE (B), where νBTE (B) = x∈B i,j=1,2 ρs (Fai ⊗ Fbj )({(xij1 , x2 )}) .
That is, we have the sample space (X 8 , P(X 8 ), νBTE ), which is induced by the parallelNmeasurement i,j=1,2 MB(C2 ⊗C2 ) (Oai bj , S[ρs ] ).
Deﬁne the {−1, 1}-valued functions gkij on X 8 , (i, j, k = 1, 2), such that
ij
11
12
12
21
21
22
22
gkij ((x111 , x2 ), (x1 , x2 ), (x1 , x2 ), (x1 , x2 )) = xk(∀i, ∀j, ∀k ∈ {1, 2}).

(2.77)

Note that it holds that¡¢¡¢
νBTE (g111 )−1 ({1}) = νBTE (g112 )−1 ({1}) ,¡¢¡¢νBTE (g211 )−1 ({1}) = νBTE (g212 )−1 ({1}) ,¡¢¡¢νBTE (g121 )−1 ({1}) = νBTE (g122 )−1 ({1}) ,¡¢¡¢νBTE (g221 )−1 ({1}) = νBTE (g222 )−1 ({1}) .

Here note that (cf. (3.42) in §3.7 later)
g111 ̸= g112 ,g121 ̸= g122 ,g211 ̸= g221 ,g212 ̸= g222 .

(2.78)

Moreover, deﬁne the correlation functions P (g1ij , g2ij ) (i, j = 1, 2) byZ
ijij
P (g1 , g2 ) ≡g1ij (x) · g2ij (x)νBTE (dx),(2.79)

X8

which may be also denoted by P (ai , bj ). A simple calculation shows that P (ai , bj ) =−(α1i β1j + α2i β2j ). Thus, putting
a1 = (0, 1),

1 1
b1 = ( √ , √ ),2 2

a2 = (1, 0)

and

1
1
b2 = ( √ , − √ ),2
2

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we see that√
|P (a1 , b1 ) − P (a1 , b2 )| + |P (a2 , b1 ) + P (a2 , b2 )| = 2 2.

(2.80)

This is precisely Bell’s calculation concerning Bell’s thought experiment.
NK ¡ NThe (2.80) can be tested by the repeated measurement
k=1
i,j=1,2 MB(C2 ⊗C2 )¢¡ 11 11
¢
12
21
21
22
22
K
(Oai bj , S[ρs ] ) . Let xb = { (x1,k , x2,k ), (x12
1,k , x2,k ), (x1,k , x2,k ), (x1,k , x2,k ) }k=1 be a measured
value of the repeated measurement. Then, we see that
1 X ij ij
x x
K k=1 1,k 2,kK

P (ai , bj ) ≈√for suﬃciently large K. Thus, the experimental test: “2 2 or not?” is possible. In fact,
Aspect’s experiment [8] is generally believed to guarantee the (2.80). It is, of course,
important since quantum mechanics must be always tested.
(Continued in §3.7 (Appendix(Bell’s inequality)))

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Chapter 3
The relation among systems (Axiom
2)
As mentioned in Chapter 1, (pure) measurement theory (PMT) is formulated as follows:
PMT = measurement + the relation among systems[Axiom 1 (2.37)]

in C ∗ -algebra[Axiom 2 (3.26)]

.

(3.1)
(=(1.4))In Chapter 2 we studied “measurement (= Axiom 1)”. In this chapter we intend to explain “therelation among systems (= Axiom 2)”.

3.1

Newton Equation and Schrödinger equation

In this section, we review the Newton equation and Schrödinger equation.
[I: Newtonian Mechanics]
Put A = C0 (Rsq ×Rsp ) and A∗ = M(Rsq ×Rsp ), where Rsq ×Rsp ≡ {(q, p) = (q1 , q2 , · · · , qs ,p1 , p2 , · · · , ps ) | qj , pj ∈ R, j = 1, 2, · · · , s} and (Rsq × Rsp ) is the 2s-dimensional space (cf.
Example 2.2). It is well known that the Newton equation is mathematically equivalent
to the following Hamilton equation:∂Hdqj (t) =(q(t), p(t), t),
dt
∂pj

d
∂H
pj (t) = −(q(t), p(t), t),dt∂qj(q(0), p(0)) ∈ Rsq × Rsp .

j = 1, 2, · · · , s

(3.2)

(3.3)

where H : Rsq × Rsp × R → R is a Hamiltonian. Using the solution of Newton equation(i.e., Hamilton equation (3.2)), we deﬁne the continuous map ψt1 ,t2 : Rsq × Rsp → Rsq × Rsp ,55
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which is equivalent to (3.2).
Put Ω = Rsq × Rsp . Also, put Ωt = Ω (∀t ∈ R), and ω00 = (q(0), p(0)) ( ∈ Ω0 ). Thus,
the pair [ω00 , {ψt1 ,t2 : Ωt1 → Ωt2 }t1 ≤t2 ] can be considered to be equivalent to “(3.3)+(3.2)”.
Using the continuous map ψt1 ,t2 : Ωt1 → Ωt2 (∀t1 ≤ ∀t2 ), we deﬁne the continuous
linear operator Φt1 ,t2 : C0 (Ωt2 ) → C0 (Ωt1 ) such that:(∀ft2 ∈ C0 (Ωt2 ), ∀ωt1 ∈ Ωt1 ).

[Φt1 ,t2 (ft2 )](ωt1 ) = ft2 (ϕt1 ,t2 (ωt1 ))And therefore, we can consider the following identiﬁcations:“(3.3)+(3.2)” ⇔ [ω00 , {ψt1 ,t2 : Ωt1 → Ωt2 }t1 ≤t2 ] ⇔ [δω00 , {Φt1 ,t2 : C0 (Ωt2 ) → C0 (Ωt1 )}t1 ≤t2 ]where δω00 is the point measure at ω00 . The pair [δω00 , {Φt1 ,t2 : C0 (Ωt2 ) → C0 (Ωt1 )}t1 ≤t2 ] will
be called “general system” (cf. Deﬁnition 3.1), and will play an important role in ourtheory, that is, it is a special case of “the relation among systems” in (3.1).
[II:Quantum Mechanics in C(L2 (Rq , dq))]
We begin with the classical mechanics. For simplicity, consider the one dimensionalcase, i.e., Rq = {q | q ∈ R}. Thus q(t), −∞ < t < ∞, means the particle’s position at) means the particle’s momentum at time t. Let R2q,ptime t, and thus, p(t) ( ≡ m dq(t)dt( ≡ {(q, p) | q, p ∈ R} be a phase space. Deﬁne a Hamiltonian H : R2q,p → R such that:
H(q, p) =´
p2 ³2=kinetic energy= 12 m( dq(t)
)
dt
2m

+

³
´
V (q) =potential energy .

(3.5)

Thus we seeE(total energy)= H(q, p) =

p2
2m

+

(kinetic energy)

V (q)

.

(3.6)

(potential energy)

Put H = L2 (Rq , dq), that is, the Hilbert space composed of all complex valued L2 R∞
functions f on Rq , i.e., ∥f ∥L2 (Rq ,dq) ≡ [ −∞ |f (q)|2 dq]1/2 < ∞. And put A = C(H) =C(L2 (Rq , dq)), (i.e., the algebra composed of all compact operators on H, cf. Example
2.3). Applying the quantumization:
E 7→ i~∂,∂t

p 7→ −i~∂,∂qq 7→ q(where i =√−1, ~ = “Plank constant” /2π) (3.7)

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to the (3.6), we obtain the Schrödinger equation:∂∂~2 ∂ 2= H(q, −i~ ) = −+ V (q)∂t∂q
2m∂q 2(3.8)∂~2 ∂ 2ψ(q, t) = −ψ(q, t) + V (q)ψ(q, t).
∂t2m∂q 2(3.9)

i~
or preciselyi~

This solution is, formally, written by
ψ(q, t) = e− ~ H(q,−i~ ∂q )t ψ(q, 0).
i∂

Put U (t) = e− ~ H(q,−i~ ∂q )t , and ψ(·, t) = ψt . Then, we see,i∂ψt = U (t)ψ0
Thus, the time-evolution of the state |ψt ⟩⟨ψt |(∥ψ0 ∥H = 1).
³´≡ (Ψ0t )∗ (|ψ0 ⟩⟨ψ0 |) is represented by

³
´
³
´
p
|ψt ⟩⟨ψt | = (Ψ0t )∗ |ψ0 ⟩⟨ψ0 | = |U (t)ψ0 ⟩⟨U (t)ψ0 |∈ T r+1(H) .
Let Ψ0t : C(H) → C(H) be the pre-adjoint operator of (Ψ0t )∗ . Let O0 = (X, F, F0 ) bea C ∗ -observable in C(H). Then, the time-evolution of the observable Ot = (X, F, Ft ) isrepresented by(X, F, Ft ) = (X, F, U (t)F0 U (t)∗ ) = (X, F, Ψ0t F0 ).
Putting Φt1 ,t2 = Ψ0t2 −t1 , we get the pair [|ψ0 ⟩⟨ψ0 |, {Φt1 ,t2 : C(H) → C(H)}t1 ≤t2 ]. Also,
it should be note that the above Ft is the solution of the following Heisenberg kinetic
equation:i~dFt= Ft H − HFt
dtin C(H) ,(3.10)hwhich is equivalent to the Schrödinger equation (3.9). (Cf. [84].) The pair |ψ0 ⟩⟨ψ0 |,i{Φt1 ,t2 : C(L2 (Rq , dq)) → C(L2 (Rq , dq))}t1 ≤t2 will be called “general system” (cf. Deﬁnition 3.1), and will play an important role in our theory, that is, it is also a special case of“the relation among systems” in (3.1).

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3.2

The relation among systems (Deﬁnition)

By the hint of the arguments in the previous section, we shall devote ourselves to “therelation among systems (i.e., Axiom 2)” in PMT (3.1) (=(1.4)).
Let A1 and A2 be C ∗ -algebras. A continuous linear operator Ψ1,2 : A2 → A1 is calleda Markov operator, if it satisﬁes that(i) Ψ1,2 (F2 ) ≥ 0 for any positive element F2 in A2 ,(ii) Ψ1,2 (I2 ) = I1 , where Ik is the identity in Ak (k = 1, 2).
Here note that, for any observable (X, F, F2 ) in A2 , the (X, F, Ψ1,2 F2 ) is an observable in
A1 , which is denoted by Ψ12 O2 . For example, it is easy to see that[Ψ1,2 F2 ](Ξ ∪ Ξ′ ) = Ψ1,2 (F2 (Ξ ∪ Ξ′ )) = Ψ1,2 (F2 (Ξ) + F2 (Ξ′ ))=[Ψ1,2 (F2 )](Ξ) + [Ψ1,2 (F2 )](Ξ′ ) (for all Ξ, Ξ′ (∈ F) such that Ξ ∩ Ξ′ = ∅).

(3.11)

Also, a Markov operator Ψ1,2 : A2 → A1 is called a homomorphism (or precisely, C ∗ homomorphism), if it satisﬁes that(i) Ψ1,2 (F2 )Ψ1,2 (G2 ) = Ψ1,2 (F2 G2 ) for any F2 and G2 in A2 ,(ii) (Ψ1,2 (F2 ))∗ = Ψ1,2 (F2∗ ) for any F2 in A2 .
Let Ψ∗1,2 : A∗1 → A∗2 be the dual operator1 of a Markov operator Ψ1,2 : A2 → A1 , that is,
it holds that
A∗
1

D
E
D
E
ρ1 , Ψ1.2 F2 A1 = A∗ Ψ∗1.2 ρ1 , F2 A2
2(∀ρ1 ∈ A∗1 , ∀F2 ∈ A2 ).

(3.12)

Then the following mathematical results are well known (cf. [50, 76, 82]).
(a) Ψ∗1,2 (Sm (A∗1 )) ⊆ Sm (A∗2 ),(3.13)(b) Ψ∗1,2 (Sp (A∗1 )) ⊆ Sp (A∗2 ) if Ψ1,2 : A2 → A1 is homomorphic.
Suppose that A1 and A2 are commutative unital C ∗ -algebras, i.e., A1 = C(Ω1 ) and A2= C(Ω2 ). Then, under the identiﬁcation that Sp (A∗1 ) = Mp+1 (Ω1 ) = Ω1 and Sm (A∗2 ) =∗Mm+1 (Ω2 ) (cf. §2.1), the above (a) implies that the dual operator Ψ1,2 of a Markov operator

The symbol ∗ is used in the three following ways (i) ∼ (iii) in this book. (i) involution operator (e.g.,
F ), (ii) dual operator (e.g., Ψ∗ ), (iii) dual space (e.g., A∗ ).
1∗

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Ψ12 can be identiﬁed with a transition probability rule M (ω1 , B2 ), (ω1 ∈ Ω1 , B2 ∈ BΩ2 ),such that M (ω1 , B2 ) = (Ψ∗1,2 (δω1 ))(B2 ). Also, under the identiﬁcation that Mp+1 (Ω1 ) = Ω1
and Mp+1 (Ω2 ) = Ω2 , the above (b) implies that the dual operator Ψ∗1,2 of a homomorphismΨ1,2 can be identiﬁed with a continuous map ψ1,2 from Ω1 into Ω2 such that:(Ψ1,2 f2 )(ω1 ) = f2 (ψ1,2 (ω1 ))(∀ω1 ∈ Ω1 , ∀f2 ∈ C(Ω2 )).

Ψ1,2 f2Ω1(3.14)f2

ω1

ψ1,2 (ω1 )Ω2

Let (T, ≤) be a tree-like partial ordered set, i.e., a partial ordered set such that “t1 ≤ t3
and t2 ≤ t3 ” implies “t1 ≤ t2 or t2 ≤ t1 ”. Put T≤2 = {(t1 , t2 ) ∈ T 2 : t1 ≤ t2 }. An elementt0 ∈ T is called a root if t0 ≤ t (∀t ∈ T ) holds. Since we usually consider the subtree
Tt0 ( ⊆ T ) with the root t0 , we assume that the tree-like ordered set has a root. In
this chapter, assume, for simplicity, that T is ﬁnite (though it is sometimes inﬁnite inapplications).
Deﬁnition 3.1. [Markov relation among systems, General systems, Sequential observable].

The pair S[ρpt ] ≡ [S[ρpt ] , {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 ] is called a general system
00with an initial state ρpt0 if it satisﬁes the following conditions (i)∼(iii).
(i) With each t (∈ T ), a C ∗ -algebra At is associated.
(ii) Let t0 (∈ T ) be the root of T . And, assume that a system S has the state ρtp0 (∈Sp (A∗t0 )) at t0 , that is, the initial state is equal to ρpt0 .
(iii) For every (t1 , t2 ) ∈ T≤2 , a Markov operator Φt1 ,t2 : At2 → At1 is deﬁned such that
Φt1 ,t2 Φt2 ,t3 = Φt1 ,t3 holds for all (t1 , t2 ), (t2 , t3 ) ∈ T≤2 .
The family {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 is also called a “Markov relation among systems”.
Let an observable Ot ≡ (Xt , 2Xt , Ft ) in a C ∗ -algebra At be given for each t ∈ T . The pair[{Ot }t∈T , {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 ] is called a “sequential observable”, which is

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denoted by [OT ], i.e., [OT ] = [{Ot }t∈T , {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 ].

¥3.3

Examples (Several tree structures)

Before we propose Axiom 2 (3.26), we prepare some notations and examples. For
simplicity, assume that T is ﬁnite, or a ﬁnite subtree of a whole tree. Let T ( ={0, 1, ..., N }) be a tree with the root 0. Deﬁne the parent map π : T \ {0} → T suchthat π(t) = max{s ∈ T : s < t}. It is clear that the tree (T ≡ {0, 1, ..., N }, ≤ )can be identiﬁed with the pair (T ≡ {0, 1, ..., N }, π : T \ {0} → T ). Also, note that,for any t ∈ T \ {0}, there uniquely exists a natural number h(t) (called the height oft ) such that π h(t) (t) = 0. Here, π 2 (t) = π(π(t)), π 3 (t) = π(π 2 (t)), etc. Also, put
0{0, 1, ..., N }2≤ = {(m, n) | 0 ≤ m ≤ n ≤ N }. Thus, the general system S[ρp0 ] ≡ [S[ρp ,]0Φπ(t),t0{Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈{0,1,...,N }2 ] is sometimes represented by [S[ρAπ(t) (
p , At]→≤0t ∈ {0, 1, ..., N } \ {0})]. Let Ot ≡ (Xt , Ft , Ft ) be an observable in At (∀t ∈ T ). The “measurement” of {Ot : t ∈ T } for the S[ρpt ] is symbolically described by M({Ot }t∈T , S[ρpt ] ).
00Φπ(t),t

The Markov relation {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 is also denoted by {At → Aπ(t) }t∈T \{0}The following Examples 3.2, 3.3 and 3.4 will promote the understanding of Axiom 2
later.
Example 3.2. [Series structures2 ]. Suppose that a tree (T ≡ {0, 1, ..., N }, π) has a“series” structure, i.e., π(t) = t − 1 (∀t ∈ T \ {0}). Consider a general system S[ρp0 ] ≡Φπ(t),t00[S[ρAπ(t) ( t ∈ T \ {0})] with the initial system S[ρp , Atp , that is,
]
]
→
0

0

Φ0,1

Φ1,2

Φ2,3

A0 ←− A1 ←− A2 ←− · · · · · · · · ·ΦN −2,N −1←−

ΦN −1,NAN −1 ←− AN .

(3.15)For each t ∈ T , consider an observable Ot ≡ (Xt , Ft , Ft ) in a C ∗ -algebra At . Thus,eN ( ≡
we have a sequential observable [{Ot }t∈T , {Φt,π(t) : At → Aπ(t) }t∈T \{0} ]. Put O(XN , FN , FeN )) = ON ( ≡ (XN , FN , FN )). According to the Heisenberg picture (cf. §3.5),
e N in AN −1 . Thus,
the observable ON in AN can be identiﬁed with the observable ΦN −1,N O
qp

e N −1 ≡ ON −1× ΦN −1,N ON ≡ (XN −1 ×
we can consider the quasi-product observable O
2

Most problems in dynamical system theory are formulated as the general systems with series trees(i.e., T =“time”) Cf. Kalman ﬁlter in §8.4.

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XN , FN −1 × Fn , FeN −1 ) in AN −1 , that is,qp¡¡¢
FeN −1 (ΞN −1 × ΞN ) = FN −1× ΦN −1,N FN ) (ΞN −1 × ΞN ),(3.16)(though the existence and the uniqueness are not guaranteed in general). By a similar way,qp
e N −2 ≡ ON −2× ΦN −2,N −1 O
e N −1 ≡ (XN −2 ×
we can deﬁne the quasi-product observable O
XN −1 × XN , FN −2 × FN −1 × Fn , FeN −2 ) in AN −2 , that is,qp¡¢
FeN −2 (ΞN −2 × ΞN −1 × ΞN ) = FN −2× (ΦN −2,N −1 FeN −1 ) (ΞN −2 × ΞN −1 × ΞN ).

(3.17)

Iteratively we get as follows:
[A0 ]

Φ

←−−−−−

F0
?
?
y

qp

e1 )
(F0 ×
× ΦF
e0
=F

Φ

Φ

[A1 ]

←−−−−− · · · ←−−−−−[AN −2 ]

F1
?
?
y

···

FN −2
?
?
y

e2 )
(F1 ×
× ΦF
e1
=F

←−−−−− · · · ←−−−−−

ΦΦ

eN −1 )× ΦF
(FN −2 ×
eN −2
=F

Φ

[AN −1 ]←−−−−− [AN ]

FN −1
?
?
y

FN
?
?
y

qp

qp

qp

Φ

←−−−−−

Φ←−−−−−

Φ←−−−−−eN )× ΦF(FN −1 ×
eN −1
=F

Φ

(F

←−−−−− =FeN)N

e 0 ≡ O0× Φ0,1 O
e 1 ≡ (×N Xt , ×N Ft ,And ﬁnally, we get the quasi-product observable O
t=0
t=0
e
F0 ) in A0 , that is,
qp

¡ qp
¢
Fe0 (Ξ0 × Ξ1 × Ξ2 × · · · × ΞN ) = F0× (Φ0,1 Fe1 ) (Ξ0 × Ξ1 × Ξ2 × · · · × ΞN ).

(3.18)

e 0 is a realization of the sequential observable [{Ot }t∈T , {Φt,π(t) : At → Aπ(t) }t∈T \{0}Here O]. Then, we have the “measurement” M({Ot }t∈T , S[ρp0 ] ) such ase 0 ≡ ( × Xt , × Ft , Fe0 ), S 0 p ).
M({Ot }t∈T , S[ρp0 ] ) = MA0 (O
[ρ ]
t∈T

t∈T

0

(3.19)

Also, note that the above arguments can be executed under the hypothesis that quasie n , n = 0, 1., , , .N ) exist. In other words, the existence of the
product observables (i.e., O
e 0.
“measurement” M({O}t∈T , S[ρp0 ] ) is equivalent to that of the observable O¥
Example 3.3. [Parallel structures3 ]. Suppose that a tree (T ≡ {0, 1, ..., N }, π) has a0“parallel” structure, i.e., π(t) = 0 (∀t ∈ T \ {0}). Consider a general system S[ρp0 ] ≡ [S[ρ
p ,
]
0

0
At Φπ(t),tAπ(t) ( t ∈ T \ {0})] with the initial system S[ρp , that is,→0]
3

Most problems in statistics are formulated as the general systems with parallel trees. Cf. Figure(6.12) in regression analysis.

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A1

Φ0,1
+

Φ0,2 A2)

A0············kAN

Φ0,N(3.20)For each t ∈ T , consider an observable Ot ≡ (Xt , Ft , Ft ) in a C ∗ -algebra At . Thus, wehave a sequential observable [{Ot }t∈T , {Φt,π(t) : At → Aπ(t) }t∈T \{0} ]. Then, we get thee 0 ≡ (×N Xt , ×N Ft , Fe0 ) in A0 such that:quasi-product observable O
t=0

t=0

¢
¡ qp
Fe0 (Ξ0 × Ξ1 × Ξ2 × · · · × ΞN ) = × Φ0,t Ft ) (Ξ0 × Ξ1 × Ξ2 × · · · × ΞN ).

(3.21)t∈T

e 0 is a realization of the sequential observable [{Ot }t∈T , {Φt,π(t) : At → Aπ(t) }t∈T \{0}Here O]. Then, we have the “measurement” M({Ot }t∈T , S[ρp0 ] ) such ase 0 ≡ ( × Xt , × Ft , Fe0 ), S 0 p ).
M({Ot }t∈T , S[ρp0 ] ) = MA0 (O
[ρ ]
t∈T

t∈T

0

(3.22)

Also, note that the above arguments can be executed under the hypothesis that quasiproduct observables exist. In other words, the existence of the “measurement” M({O}t∈T ,e 0.
S[ρp ] ) is equivalent to that of the observable O
0¥
Example 3.4.

[A simple general system, Heisenberg picture]. Suppose that a tree

(T ≡ {0, 1, ..., 6, 7}, π) has an ordered structure such that π(1) = π(6) = π(7) = 0,³´
π(2) = π(5) = 1, π(3) = π(4) = 2. See the ﬁgure (3.23).
Consider a general system
Φπ(t),tS[ρp0 ] ≡ [S[ρp0 ] , {At → Aπ(t) }t∈T \{0} ] with the initial system S[ρp0 ] .
Φ2,3

Φ1,2
Φ0,1
)
A0 i

kΦ0,6

Φ0,7

A1
A6

+

A2 )
k

A3

Φ2,4 A4

k Φ1,5A5

A7

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(3.23)

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Also, for each t ∈ {0, 1, ..., 6, 7}, consider an observable Ot ≡ (Xt , 2Xt , Ft ) in a C ∗ -algebra
At . Thus, we have a sequential observable [{Ot }t∈T , {Φt,π(t) : At → Aπ(t) }t∈T \{0} ]. Now
we want to consider the following “measurement”,
Φπ(t),t(♯) for a system S[ρp0 ] , take a measurement of “a sequential observable [{Ot }t∈T , {At →Aπ(t) }t∈T \{0} ]”, i.e., take a measurement of an observable O0 at 0( ∈ T ), and next,
take a measurement of an observable O1 at 1( ∈ T ), · · · · · · , and ﬁnally take ameasurement of an observable O7 at 7( ∈ T ),which is symbolized by M({Ot }t∈T , S[ρp0 ] ). Note that the M({Ot }t∈T , S[ρp0 ] ) is merely asymbol since only one measurement is permitted (cf. §2.5 Remark(II)). In what follows
let us describe the above (♯) (= M({Ot }t∈T , S[ρp0 ] )) precisely. Put
e t = Ot
O

and thus Fet = Ft(t = 3, 4, 5, 6, 7).

e 2 in A2 such asFirst we construct the quasi-product observable O
qp

e 2 = (X2 × X3 × X4 , 2X2 ×X3 ×X4 , Fe2 )O

qp

where Fe2 = F2 × ( × t=3,4 Φ2,t Fet ),if it exists. Iteratively, we construct the following:
Φ0,1

A0
qp

←−−−
qp

qp

qp

Φ0,1

(F0 × Φ0,6 Fe6 × Φ0,7 Fe7 × Φ0,1 Fe1 )A2

F1 × Φ1,5 Fe5


y
←−−−

qp

Φ1,2

←−−−

qp

F0 × Φ0,6 Fe6 × Φ0,7 Fe7


y
Fe0

A1

qp

Fe1

qp

(F1 × Φ1,5 Fe5 × Φ1,2 Fe2 )(3.24)
Φ1,2

←−−−

qp

Fe2

.

qp

(F2 × Φ2,3 Fe3 × Φ2,4 Fe4 )Q
e 1 ≡ (Q5 Xt , 2 5t=1 Xt , Fe1 ) of O1 , Φ1,2 O
e2That is, we get the quasi-product observable O
t=1
Q7Q
e 5 , and ﬁnally, the quasi-product observable O
e 0 ≡ ( 7 Xt , 2 t=0 Xt , Fe0 ) of O0 ,and Φ1,5 O
t=0

e 7 , if it exists. Here, Oe 0 is called the realization (or, the Heisenberg
e 6 and Φ0,7 O
e 1 , Φ0,6 O
Φ0,1 OΦπ(t),tpicture representation) of a sequential observable [{Ot }t∈T , {At → Aπ(t) }t∈T \{0} ]. Then,
we have the measurementYQ
e 0 ≡ ( Xt , 2 t∈T Xt , Fe0 ), S[ρp ] ),MA0 (O
0t∈T

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which is called the realization (or, the Heisenberg picture representation) of the symbol
M({Ot }t∈T , S[ρpt ] ).
0¥
Remark 3.5. Let (T ≡ {0, 1, ..., N }, π : T \ {0} → T ) be any tree with the root 0.
Let τ be any element of T . Consider a series structure Teτ such that Teτ = {π k (τ ) | k =0, 1, 2, ..., h(τ )} ( ⊆ T ), where h(τ ) is the height of τ , i.e., π h(τ ) (τ ) = 0. Note that Example
3.4 (i.e, diagram (3.24)) means that any general system (with a tree structure T ) can be
regarded as a general system with a series structure Teτ .
¥3.4

The relation among systems (Axiom 2)

Examining Example 3.4, we see as follows: Let (T ≡ {0, 1, ..., N }, π : T \ {0} → T ) be
Φπ(t),ta tree with root 0 and let S[ρp0 ] ≡ [S[ρp0 ] , At → Aπ(t) (t ∈ T \ {0})] be a general systemwith the initial system S[ρp0 ] . And, let an observable Ot ≡ (Xt , Ft , Ft ) in a C ∗ -algebra At
be given for each t ∈ T . Thus, we have a sequential observable [{Ot }t∈T , {Φt,π(t) : At →Qe s ≡ (QAπ(t) }t∈T \{0} ]. For each s ( ∈ T ), deﬁne the observable O
Xt ,Ft , Fes ) in
t∈Ts

t∈Ts

As such that:
(
es =
O

(if s ∈ T \ π(T ))

Os
qp qp

e t ) (if s ∈ π(T ))Os× ( × t∈π−1 ({s}) Φπ(t),t O(3.25)e 0 (i.e., the Heisenberg picture representation of theif possible. Then, if an observable O
sequential observable [{Ot }t∈T , {Φt,π(t) : At → Aπ(t) }t∈T \{0} ]) in A0 exists (such as in
Example 3.4), we have the measurement
e0 ≡ (
MA0 (O

Y
t∈T

Xt ,

Y

Ft , Fe0 ), S[ρp0 ] ),t∈Twhich is called the Heisenberg picture representation of the symbol M({Ot }t∈T , S[ρpt ] ).
0

Summing up the essential part of the above argument, we can propose the following
axiom, which corresponds to “the rule of the relation among systems” in PMT (1.4). Cf.
[43, 44, 46].

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65

AXIOM 2. [The Markov relation among systems, the Heisenberg picture]
The relation among systems is represented by a Markov relation {Φt1 ,t2 :At2 → At1 }(t1 ,t2 )∈T≤2 . Let Ot ( ≡ (Xt , Ft , Ft )) be an observable in At for eacht ( ∈ T ). If the procedure (3.25) is possible, a sequential observable [OT ]≡ [{Ot }t∈T , {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 ] can be realized as the observablee 0 ≡ (Q Xt , Q Ft , Fe0 ) in A0 .
O
(3.26)t∈Tt∈T

It is quite important to note that Axiom 2 is stated in terms of A (and not in terms ofA∗ ).4 Also, we must add the following statement:• Let S[ρpt ] ≡ [S[ρpt ] , {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 ] be a general system with an00(∈ Sp (A∗t0 )). Then, a measurement represented by the symbole 0 ≡ (Q Xt , Q Ft , Fe0 ), S[ρp ] ), if
M({Ot }t∈T , S[ρpt ] ) can be realized by MA0 (O
t∈T
t∈T
0
0
e
O0 exists.
initial stateρpt0

which explains the relation between Axiom 1 and Axiom 2.
Now we get the PMT (1.4). We have the following classiﬁcation in PMT:
deterministic PMT = “measurement” + “the deterministic relation among systems”.
[Axiom 1 (2.37)]
[ each Φt1 ,t2 is homomorphic in Axiom 2 (3.26)]

 stochastic PMT = “measurement” + “the Markov relation among systems”.
[Axiom 1 (2.37)][Axiom 2 (3.26)](3.27)
Remark 3.6. (i). Roughly speaking, Axiom 2 asserts Φ0,1 O1 is more fundamental thanO1 in the following identiﬁcation
Φ0,1 O1 (in A0 ) ←→ O1 (in A1 )where O1 is an observable in A1 and Φ0,1 : A1 → A0 is a Markov operator.
(ii). Also, it should be noted that Axiom 2 says that the time evolution of a system
satisﬁes the Markov property. Thus, automata theory and circuit theory are characterized
as special cases of measurement theory (especially, Axiom 2).
(iii). Axiom 2 has a great descriptive power. Note that “hysteresis” and “multiple Markov
properties” can be described in the framework of Axiom 2.
¥
4This fact makes us apply Axiom 2 to “statistical measurement theory” (in Chapter 8) as well as“PMT” (in this chapter).

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3.5

Heisenberg picture and Schrödinger picture

Now let us mention something about the relation between Heisenberg picture andSchrödinger picture.
Suppose that a simplest tree (T ≡ {0, 1}, π) has a “series” structure, i.e., π(1) = 0.
0,1Consider a general system S[ρp0 ] ≡ [S[ρp0 ] , A1 Φ→
A0 ] with the initial system S[ρp0 ] , that is,

Φ0,1

A0 ←− A1

(3.28)Let O1 = (X1 , F1 , F1 ) be an observable in A1 . Now we consider(M) the measurement of the observable O1 = (X1 , F1 , F1 ) for the general system S[ρp0 ] ≡0,1[S[ρp0 ] , A1 Φ→
A0 ]Under the following identiﬁcation:
Φ0,1 O1 in A0←→ O1 in A1(3.29)

we think that(M ) = MA0 (Φ0,1 O1 , S[ρp0 ] ).

(3.30)

This viewpoint is standard, and it is called the Heisenberg picture representation of (M ).
Axiom 1 says that0
• the probability that the measured value of the measurement (M) (i.e.,MA0 (Φ0,1 O1 , S[ρ
p ))
]
0

belongs to Ξ1 ( ∈ F1 ) is given byDE ¢¡ρp0 (Φ0,1 F (Ξ1 )) ≡ A∗ ρp0 , Φ0,1 F (Ξ1 ) A0 .

(3.31)

0

On the other hand, under the following identiﬁcation:
ρp0 in S(A∗0 )←→ Φ∗0,1 ρp0 in S(A∗1 ) ,we also consider that³(M ) = MA1 (O1 , S[Φ∗0,1 ρp0 ] )(3.32)

though Φ∗0,1 ρp0 is not in Sp (A∗ ) but in Sm (A∗ ) if Φ0,1 is not homomorphic. Cf. Chapter´
8 (statistical measurement theory), This viewpoint is called the Schrödinger picture
representation of (M ). We of course think that

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67• the probability that the measured value of the measurement (M) (i.e.,MA1 (O1 , S[Φ∗0,1 ρp0 ] ))belongs to Ξ1 is given byDE ¢¡ρp0 (Φ0,1 F (Ξ1 )) ≡ A∗ Φ∗0,1 ρp0 , F (Ξ1 ) A1 .

(3.33)

1

It should be noted that (3.31) = (3.33) holds. Thus it is usually and roughly said that
• Heisenberg picture (i.e., observable moves) and Schrödinger picture (i.e., state moves)
are equivalent,
though the Heisenberg picture is fundamental (and the Schrödinger picture representation
should be regarded as a kind of prescription). For the further arguments, see §6.2.

3.6

Measurability theorem

The following theorem is the most fundamental in classical PMT.
Theorem 3.7. [The measurability theorem of a general system, cf. [43]]. Let (T ≡{0, 1, ..., N }, π : T \ {0} → T ) be a tree with root 0 and let S[ρp0 ] ≡ [S[ρp0 ] , AtΦπ(t),t→Aπ(t) (t ∈ T \{0})] be a general system with the initial system S[ρp0 ] . And, let an observableOt ≡ (Xt , Ft , Ft ) in a C ∗ -algebra At be given for each t ∈ T . For each s ( ∈ T ), deﬁneQe s ≡ (QFt , Fes ) in As such that:Xt ,the observable O
t∈Ts

t∈Ts

(
es =
O

(if s ∈ T \ π(T ))

Os
qp qp

e t ) (if s ∈ π(T ))Os× ( × t∈π−1 ({s}) Φπ(t),t O

e 0 (i.e., the Heisenberg picture representation of theif possible. Then, if an observable O
sequential observable [{Ot }t∈T , {Φt,π(t) : At → Aπ(t) }t∈T \{0} ]) in A0 exists, we have themeasurementYY
e 0 ≡ ( Xt ,MA0 (OFt , Fe0 ), S[ρp0 ] ),
t∈T

³ N

t∈T Ft is sometimes denoted byQ(3.34)

t∈T

´
F
,
cf.
Deﬁnition
2.10, which is called thet∈T t

Heisenberg picture representation of the symbol M({Ot }t∈T , S[ρpt ] ). If the system is classi0

cal, i.e., At ≡ C(Ωt ) (∀t ∈ T ), then the measurement always exists, while the uniquenessis not always guaranteed. Also, it should be noted that, for each s( ∈ T ), it holds thatQQ
Φπ(s),s Fes (Ξt ) = Feπ(s) ((Πt∈T \Ts Xt ) × (Ξt )) (∀Ξt ∈ Ft (∀t ∈ T )).
t∈Ts

π(s)

t∈Ts

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Proof. It suﬃces to prove it in classical measurements. However it is clear since, inclassical measurements, the product observable of any observables always exists. Therefore the construction mentioned in Example 3.4 is always possible in classical systems.

Example 3.8. [Random walk]. Suppose that a tree (T ≡ {0, 1, ..., N }, π) has a “series”
structure, i.e., π(t) = t − 1 (∀t ∈ T \ {0}). Consider a general system S[δ0 ] ≡ [S[δ0 ] ,Aπ(t) ( t ∈ T \ {0})] with the initial system S[δ0 ] , that is,At Φπ(t),t
→
Φ0,1

Φ1,2

Φ2,3

A0 ←− A1 ←− A2 ←− · · · · · · · · ·ΦN −2,N −1←−

ΦN −1,NAN −1 ←− AN .

(3.35)Let Z be the set of all integers, i.e., Z = {0, ±1, ±2, ...}. Consider a commutative C ∗ algebra C0 (Z). Here, put
At = C0 (Z) (∀t ∈ {0, 1, ..., N })and deﬁne a Markov operator Φt−1,t ( ≡ Φ) : At ( ≡ C0 (Z)) → At−1 ( ≡ C0 (Z)) such that:(Φf )(n) = (Φt−1,t f )(n) =f (n + 1) + f (n − 1)2(∀f ∈ At ( ≡ C0 (Z)), ∀n ∈ Z).

Also, for each t = 0, 1, 2, ..., N , consider the exact observable Ot ≡ (Xt , Rt , E) ≡ (Z, P0 (Z), E)
in At ( = C0 (Z)) such that, (cf. Example 2.20),
 1 n ∈ Ξ( ∈ P0 (Z))[E(Ξ)](n) =
0 n∈/ Ξ( ∈ P0 (Z)).

(3.36)

e 0 ≡ (× Xt , × Ft , Fe0 ) ≡ (ZN +1 , P0 (ZN +1 ),Thus, we get the product observable O
t=0
t=0Fe0 ) in A0 ( ≡ C0 (Z)), that is,N

N

Fe0 (Ξ0 × Ξ1 × Ξ2 × · · · × ΞN ) = E(Ξ0 ) × Φ(E(Ξ1 ) × Φ(· · · · · · Φ(E(ΞN −1 ) × ΦE(ΞN )) · · · )).
Then, we have the “measurement” M({Ot }t∈T , S[δ0 ] ) such ase 0 ≡ (ZN +1 , P0 (ZN +1 ), Fe0 ), S[δ ] ).
M({Ot }t∈T , S[δ0 ] ) = MC(Z) (O
0
³
´
N +1
N +1
e
where δ0 is the point measure at 0 ( ∈ Z). The sample space Z, P0 (Z), [F0 (·)](0)is usually called a random walk.
¥
For the further arguments, see §10.4 (Brown motion).

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3.7

Appendix (Bell’s inequality)(Continued from §2.9 (Bell’s Thought Experiment))53.7.1

Deterministic evolution or Stochastic evolution?

Recall the following classiﬁcation (3.27) in PMT:
deterministic PMT = “measurement” + “the deterministic relation among systems”.
[Axiom 1 (2.37)]
[ each Φt1 ,t2 is homomorphic in Axiom 2 (3.26)]

 stochastic PMT = “measurement” + “the Markov relation among systems”.
[Axiom 1 (2.37)][Axiom 2 (3.26)]

However, we know that in classical (or quantum) mechanics, the general system S[ρp ]≡ [S[ρp ] , At Ψπ(t),tAπ(t) ( t ∈ T \ {0})] is always deterministic, that is, Ψπ(t),t is always→
homomorphic. (cf. “Newtonian mechanics and quantum mechanics” in §3.1.)
Recall (2.76), i.e., the de Broglie paradox (cf. [20]. Also see §9.3.3). That is,³• if we admit quantum mechanics = “Axiom 1 + Axiom 2 (homomorphic time´
evolution)” , we must admit the fact that there is something fasterthan light. (cf. [18, 78]).

(3.37)(=(2.76))

Of course we admit quantum mechanics, and therefore, we believe that there is something
faster than light. However, most people may hope that quantum mechanics is not true
rather than admit the fact that there is something faster than light. That is,(♯) Using the Schrödinger picture representation, they may assert that the singlet state³
ρs is not ﬁxed, but the Markov time evolution i.e., “the Markov relation
among systems (Axiom 2)” and not “the homomorphic relation among systems´
Axiom 2)” :Φ∗ρs ;ρm0(3.38)

should be considered.
5

Although Bell’s inequality is generally said to be one of the most profound discoveries in 20-th century
science, I could not understand the arguments (in [9, 18, 78, 8]), particularly, I had the question: “In
what framework is Bell’s inequality discussed (in [9, 18, 78])?”. I wonder if these arguments are confusing
physics with mathematics. Thus, I add this section, in which all arguments are discussed in the frameworkof PMT (Axioms 1 and 2).

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The purpose of the following section (i.e., §3.7.2) is to show that we must admit that
there is something faster than light, even under the above assumption (♯). That is, if³
we assert that PMT = “Axiom 1 + Axiom 2 (Markov time evolution)”, i.e., quantum´
mechanics with Markov (and not homomorphic) time evolution is true, we must admit
the fact that there is something faster than light.

3.7.2

Generalized Bell’s inequality in mathematics

First we prepare some mathematical inequalities. Of course, what is most importantis how to interpret these theorems in physics. This will be discussed in the next section.
In order to avoid to confuse physical results and mathematical ones, in this §3.7.2, wedevote ourselves to mathematical arguments.
Theorem 3.9. [Bell’s inequality, cf. [9, 78]]. Let (Y, G, m) be a probability space. Let g11 ,g12 , g21 , g22 be {−1, 1}-valued measurable functions on Y . Deﬁne the correlation function
P ′ (g1i , g2j ) such that:′Z

P (g1i , g2j ) =g1i (y)g2j (y) m(dy).

(3.39)Y

Then, it holds that|P ′ (g11 , g21 ) − P ′ (g11 , g22 )| + |P ′ (g12 , g21 ) + P ′ (g12 , g22 )| ≤ 2.

(3.40)

Proof. For completeness, we add the proof in what follows.
|P ′ (g11 , g21 ) − P ′ (g11 , g22 )| + |P ′ (g12 , g21 ) + P ′ (g12 , g22 )|
Z
Z
1
1
2
≤
|g1 (y)| · |g2 (y) − g2 (y)|m(dy) +|g12 (y)| · |g21 (y) + g22 (y)|m(dy)
4
Y
ZX
1
2
1
2
≤
|g2 (y) − g2 (y)| + |g2 (y) + g2 (y)|m(dy) = 2.
X4

This completes the proof.
Corollary 3.10. [Bell’s inequality]. Let (Y, G, m) be a probability space. Let g111 , g112 ,g121 , g122 , g211 , g212 , g221 and g222 be {−1, 1}-valued measurable functions on Y . Deﬁne thecorrelation function P (g1ij , g2ij ) such thatZijij
P (g1 , g2 ) =g1ij (y)g2ij (y) m(dy).
Y

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71

Further, assume thatg111 = g112 ,g121 = g122 ,g211 = g221 ,g212 = g222

(a.e. m)(3.42)|P (g111 , g211 ) − P (g112 , g212 )| + |P (g121 , g221 ) + P (g122 , g222 )| ≤ 2.

(3.43)¡¢
i.e., m {y ∈ Y : g111 (y) = g112 (y)} = 1, etc. Then, it holds that

Proof. It immediately follows from Theorem 3.9.

Next we present the following theorem, which can be regarded as a generalization of
the above corollary (cf. Remark 3.12 later).
Theorem 3.11. [Generalized Bell’s inequality]. Let (Y, G, m) be a probability space.
Let g111 , g112 , g121 , g122 , g211 , g212 , g221 and g222 be {−1, 1}-valued measurable functions on Y .
Assume that these satisfym[(g1ij , g2ij )−1 (B1 × B2 )] =X

αℓ µi1,ℓ (B1 ) µj2,ℓ (B2 ) (∀B1 , B2 ⊆ {−1, 1}, ∀i, j = 1, 2)ℓ∈L(3.44)
for some probability measures µik,ℓ , (k, i = 1, 2, ℓ ∈ L), on {−1, 1} and some nonnegativeP
sequence {αℓ }ℓ∈L such that ℓ∈L αℓ = 1. Then, it holds that
|P (g111 , g211 ) − P (g112 , g212 )| + |P (g121 , g221 ) + P (g122 , g222 )| ≤ 2,
where the correlation functions P (g1ij , g2ij ) are deﬁned by (3.41).
Proof. A simple calculation shows that
P (g1ij , g2ij ) =

X
ℓ∈L

=

X

αℓ [

X

x1 x2 µi1,ℓ ({x1 })µj2,ℓ ({x2 })](x1 ,x2 )∈{−1,1}2

αℓ (4µi1,ℓ µj2,ℓ + 1 − 2µi1,ℓ − 2µj2,ℓ ),ℓ∈L

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where µik,ℓ = µik,ℓ ({1}). Thus, we see that
|P (g111 , g211 ) − P (g112 , g212 )| + |P (g121 , g221 ) + P (g122 , g222 )|XX=|
αℓ (4µ11,ℓ µ12,ℓ + 1 − 2µ11,ℓ − 2µ12,ℓ ) −αℓ (4µ11,ℓ µ22,ℓ + 1 − 2µ11,ℓ − 2µ22,ℓ )|
ℓ∈L

+|
=|

X

X

ℓ∈L

αℓ (4µ21,ℓ µ12,ℓ + 1 − 2µ21,ℓ − 2µ12,ℓ ) +ℓ∈LXαℓ (4µ21,ℓ µ22,ℓ + 1 − 2µ21,ℓ − 2µ22,ℓ )|ℓ∈Lαℓ (4µ11,ℓ µ12,ℓ − 2µ12,ℓ − 4µ11,ℓ µ22,ℓ + 2µ22,ℓ )|

ℓ∈L

+|

X

αℓ (4µ21,ℓ µ12,ℓ + 2 − 4µ21,ℓ − 2µ12,ℓ + 4µ21,ℓ µ22,ℓ − 2µ22,ℓ )| ≡ |A| + |B|,ℓ∈L

and consequently,½ P| Pℓ∈L αℓ [2 − 4(µ21,ℓ + µ12,ℓ + µ11,ℓ µ22,ℓ − µ11,ℓ µ12,ℓ − µ21,ℓ µ12,ℓ − µ21,ℓ µ22,ℓ )]| (if A · B ≥ 0)=
| ℓ∈L αℓ [2 − 4(µ21,ℓ + µ22,ℓ + µ11,ℓ µ12,ℓ − µ11,ℓ µ22,ℓ − µ21,ℓ µ22,ℓ − µ21,ℓ µ12,ℓ )]| (if A · B ≤ 0)
½ P
α |2 − 4(µ21,ℓ + µ12,ℓ + µ11,ℓ µ22,ℓ − µ11,ℓ µ12,ℓ − µ21,ℓ µ12,ℓ − µ21,ℓ µ22,ℓ )| (if A · B ≥ 0)
≤ Pℓ∈L ℓ
2
2
1
1
1
2
2
2
2
1
(if A · B ≤ 0).
ℓ∈L αℓ |2 − 4(µ1,ℓ + µ2,ℓ + µ1,ℓ µ2,ℓ − µ1,ℓ µ2,ℓ − µ1,ℓ µ2,ℓ − µ1,ℓ µ2,ℓ )|Hence, it suﬃces to prove that 0 ≤ C(x, y, z, w) ≤ 1 (∀(x, y, z, w) ∈ [0, 1]4 ), whereC(x, y, z, w) = y + z + xw − xz − yz − yw. This is shown as follows:[Case 1; w − z ≥ 0].
0 ≤ y(1 − w) + z(1 − y) + x(w − z) ≡ C ≤ C + (w − z)(1 − x)
= y(1 − w) + w − yz ≤ 1 − yz ≤ 1.
[Case 2; w − z ≤ 0].
0 ≤ y(1 − z) + w(1 − y) = y + z + (w − z) − yz − yw
≤ y + z + x(w − z) − yz − yw ≡ C ≤ y + z − yz − yw ≤ y(1 − z) + z ≤ 1.
This completes the proof.
Remark 3.12. It is interesting to see that Corollary 3.10 can be regarded as a particularcase of Theorem 3.11. This can be easily shown as follows: Let (Y, G, m) and gkij be as inCorollary 3.10. Thus, we assume that the condition (3.42) holds. Put L = {−1, 1}4 .
For each ℓ ( ≡ (ℓ11 , ℓ21 , ℓ12 , ℓ22 ) ∈ L), deﬁne the αℓ ( ∈ [0, 1]) such that α(ℓ11 ,ℓ21 ,ℓ12 ,ℓ22 ) =¢¡Pm (g111 , g122 , g211 , g222 )−1 ({(ℓ11 , ℓ21 , ℓ12 , ℓ22 )}) . Clearly it holds thatℓ∈L αℓ = 1. Deﬁnethe probability measures µb1 and µb−1 on {−1, 1} such that µb1 ({−1}) = 0, µb1 ({1}) = 1

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¡
¢ P
and µ
b−1 = 1 − µb1 . It is easy to see that m (g111 , g122 , g211 , g222 )−1 ({(x11 , x21 , x12 , x22 )}) = ℓ∈Lαℓ µbℓ11 ({x11 })bµℓ21 ({x21 })bµℓ12 ({x12 }) µbℓ22 ({x22 }) (∀(x11 , x21 , x12 , x22 ) ∈ {−1, 1}4 ). Thus, puttingµik,(ℓ1 ,ℓ2 ,ℓ1 ,ℓ2 ) = µbℓik , we can immediately see that the {αℓ }ℓ∈L and the {µik,ℓ : i, k = 1, 2, ℓ ∈
1

1

2

2

L} satisfy the condition (3.44).
¥

3.7.3

Generalized Bell’s inequality in Measurements

Put X = {−1, 1}. Consider a measurement MA ( O ≡ (X 8 , P(X 8 ), G), S[ρ0 ] ) formu3
lated in arbitrary C ∗ -algebra A. Putting νBI( · ) = ρ0 (G( · )), we have the sample space3(X 8 , P(X 8 ), νBI), which is induced by the measurement MA ( O, S[ρ0 ] ). Consider the{−1, 1}-valued functions gkij on X 8 , (i, j, k = 1, 2). And deﬁne the correlation functions
P (g1ij , g2ij ) (i, j = 1, 2) by (3.41). Assume the condition (3.44) in Theorem 3.11. Then, wesee, by Theorem 3.11, that the following inequality holds:|P (g111 , g211 ) − P (g112 , g212 )| + |P (g121 , g221 ) + P (g122 , g222 )| ≤ 2.

(3.46)

Therefore, it may be viable to compare the measurement MA ( O, S[ρ0 ] ) with the measureN
ment i,j=1,2 MB(C2 ⊗C2 ) (Oai bj , S[ρs ] ) in Bell’s thought experiment, though it is also surethat these are not connected with each other. For example, some may, by some reason,
consider that the singlet state ρs in Bell’s thought experiment (cf. the formula (2.75)) is
reduced to a certain state ρ0 ( ∈ Sp (B(C2 ⊗ C2 )∗ )) such asρs ; ρ0 = |⃗e ⊗ f⃗⟩⟨⃗e ⊗ f⃗|(3.47)

for some ⃗e⊗f⃗ ( ∈ C2 ⊗C2 ) such that ∥⃗e∥C2 = ∥f⃗∥C2 = 1. If so, instead of the measurement
NNi,j=1,2 MB(C2 ⊗C2 ) (Oai bj , S[ρs ] ), we must consider the measurement
i,j=1,2 MB(C2 ⊗C2 )(Oai bj , S[ρ0 ] ), which has the sample space (X 8 , P(X 8 ), ν) such that:
Y
¡
¢
ij
ij
11
12
12
21
21
22
22
i ⊗ Fbj )({(x , x )})ν({(x11
,
x
,
x
,
x
,
x
,
x
,
x
,
x
)})
=
ρ
(F
0
a
1
2
1
2
1
2
1
2
1
2
Y £­
®­
®¤
⃗
⃗e, Fai ({xije f⃗, Fbj ({xij1 })⃗2 })f .

=

i,j=1,2i,j=1,2

Or more generally (or, in the sense of “ensemble”), using the adjoint operator Φ∗ of a
Markov operator Φ : B(C2 ⊗ C2 ) → B(C2 ⊗ C2 ), we may consider the following Markov
evolution:Φ∗ρs ;ρm
0 =

2 X
2
X

αmn |⃗em ⊗ f⃗n ⟩⟨⃗em ⊗ f⃗n |,n=1 m=1

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CHAPTER 3.

THE RELATION AMONG SYSTEMS (AXIOM 2)

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where {⃗em }2m=1 and {f⃗m }2i=1 are respectively the complete orthonormal basis in C2 , andPP
0 ≤ αmn ≤ 1 such that 2n=1 2m=1 αmn = 1. Thus we have the (statistical) measurementN88i,j=1,2 MB(C2 ⊗C2 ) (ΦOai bj , S[ρs ] ). Thus, we may have the sample space (X , P(X ), ν)such that:
11
12
12
21
21
22
22
ν({(x11
1 , x2 , x1 , x2 , x1 , x2 , x1 , x2 )}) =

=

=

Y

∗

¡

¢

Y

¡
¢
ij
ρs (ΦFai ⊗ Fbj )({(xij1 , x2 )})i,j=1,2Yij(Φ ρs ) (Fai ⊗ Fbj )({(xijρm
1 , x2 )}) =0i,j=1,2i,j=1,2
2 X
2
Y £X

¡

¢
ij
(Fai ⊗ Fbj )({(xij1 , x2 )})

­
®­
®¤
⃗
αmn ⃗em , Fai ({xijem f⃗n , Fbj ({xij1 })⃗2 })fn .

i,j=1,2 m=1 n=1Note that the probability space (X 8 , P(X 8 ), ν) and the gkij deﬁned by (2.77) satisfy thecondition (3.44) in Theorem 3.11. That is because it suﬃces to put L = {−1, 1}2 and­®
µ11,(m,n) ( · ) = ⃗em , Fa1 ( · )⃗em ,®
­µ12,(m,n) ( · ) = f⃗n , Fb1 ( · )f⃗n ,­®
µ21,(m,n) ( · ) = ⃗em , Fa2 ( · )⃗em ,®
­µ22,(m,n) ( · ) = f⃗n , Fb2 ( · )f⃗n ,for each (m, n) ( ∈ L ≡ {−1, 1}2 ). Thus, Theorem 3.11 says that such Markov evolution
as the above (3.47) or (3.48) does not occur in Bell’s thought experiment. Therefore wecan conclude that
• if we admit PMT (= “Axiom 1 + Axiom 2 (Markov relation)” ), we must also admit
the fact that there is something faster than light.

(3.49)

Of course we admit PMT, and therefore, we believe that there is something faster thanlight.

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Chapter 4
Boltzmann’s equilibrium statisticalmechanics

As mentioned in Chapters 2 and 3, we see that (pure) measurement theory (= PMT) is formulated
as follows:
PMT = measurement + the relation among systems[Axiom 1 (2.37)]

in C ∗ -algebra[Axiom 2 (3.26)]

.

(4.1)
(=(1.4))The purpose of this chapter1 is to understand Boltzmann’s equilibrium statistical mechanics 2 (i.e.,“the principle of equal a priori probability” and “the ergodic hypothesis”) as one of applications of
PMT. We believe that our approach completely justiﬁes the the thermodynamical weight method(i.e., the Gibbs method, cf. [26])3.

4.1

Introduction

In spite that equilibrium statistical mechanics is generally believed to be based on
Newtonian mechanics, the term “probability” frequently appears in equilibrium statisticalmechanics. Therefore, if we want to understand equilibrium statistical mechanics in theframework of Newtonian mechanics, a certain rule concerning “probability” should be
added. That is, we hope to understand equilibrium statistical mechanics such as:
1
It may be recommended that this chapter is skipped if readers want to study statistics in the framework of PMT ﬁrstly (cf. Chapters 5 and 6).
2
In this chapter readers are not required to have much knowledge of statistical mechanics.
3
In this book, we think that statistical mechanics should be understood as one of applications ofmeasurement theory and not theoretical physics, (cf. Table (1.7)). Thus, it should be noted that noserious test has been conducted in statistical mechanics. What we know is nothing but the fact thatstatistical mechanics is quite useful (cf. Table (1.8)). Or, statistical mechanics is “almost empiricallytrue” to such a degree that statistical mechanics is assured to be useful in usual situations. Cf. the (I9 )in §1.2.

75
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CHAPTER 4. BOLTZMANN’S EQUILIBRIUM STATISTICAL MECHANICSFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/“equilibrium statistical mechanics” = “Newton equation” + “probabilistic rule”[Axiom 2 (3.26)][Axiom 1 (2.37)](4.2)
in PMT.
First we must answer the following question:(Q1 ) What is the “probabilistic rule” in (4.2)?
Recall Example 2.16 (the urn problem), which is the most fundamental in the classicalmeasurement. Thus in order to understand “probabilistic rule (=Axiom 1) in (4.2)”, it
suﬃces to note the following simplest example:(A1 ) “Consider a box containing 7 × 1023 white balls and 3 × 1023 black balls, and choose
a ball at random from the box. Then the probability that the ball is white is given as
0.7.”
Even without the knowledge of measurement theory (in Chapters 2 and 3), every reader
surely agrees that the probability appearing in urn (i.e., box) problems is most typical instatistics.
Next we must refer to “Newtonian mechanics” in (4.2). Namely we must solve thefollowing question.
(Q2 ) What kinds of conditions are imposed on the Newton equation in (4.2)?
In equilibrium statistical mechanics, about 1024 (≈ 6.02 × 1023 : “Avogadro constant”)
particles, of course, move hard in a box such as the following ﬁgure:
K



*

R
z 


U
U

i 9
o
9

U
-i

R

U
i

Y

M
U


W

9
R

K

W K


z
7

U

U

O

K



)

w

R
7

*
I 9

N
z K

j) 9

W




K

z
: K
W

y



s

* 9
qy
3


N
U



q

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(4.3)

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77

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However it seems to be natural to think as follows:(A12 ) All particles are even, or on a level.

(A22 ) The motions of particles are (almost) independent of each other. In other words,
the information about a subsystem composed of some particles is invalid for theinference of the state of another subsystem.
This is our answer to the question (Q2 ). In §4.2, the (A12 ) and (A22 ) will be represented interms of PMT. Also, the (A1 ) will be discussed in §4.3.
Summing up, we think that equilibrium statistical mechanics is formulated as follows:“equilibrium statistical mechanics” =“probabilistic rule”(the probability such as in (A1 ))|+“Newton equation”{z(the conditions (A12 ) and (A22 ))}(+ “staying time interpretation”)(4.4)
in PMT. Or, equivalently,• An equilibrium statistical system can be regarded as an urn containing about 1024
particles. Also, the motions of particles are dominated by the Newtonian equation
with the conditions (A12 ) and (A22 ). Also, the “staying time interpretation” implies
the common sense such as it is almost impossible to ﬁnd a rare event.
And moreover, two conventional principles (i.e., “the principle of equal a priori probability”
and “the ergodic hypothesis”) will be completely clariﬁed in our proposal (4.4).
The ﬁrst attempt to understand equilibrium statistical mechanics in the framework of
PMT was executed in [45]. The content in [45] will be slightly modiﬁed and improved inthis chapter.
Note, for completeness, that our purpose is to understand equilibrium statistical mechanics as one of applications of PMT and not to derive equilibrium statistical mechanics
from Newtonian mechanics (cf. [75]). That is, we are in theoretical informatics and notin theoretical physics.4

4

We have no experimental evidence that the ergodic approach to statistical mechanics is proper.
However, in theoretical informatics, it suﬃces to ﬁnd a reason that many people do not doubt.

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CHAPTER 4. BOLTZMANN’S EQUILIBRIUM STATISTICAL MECHANICS

4.2

Dynamical aspects of equilibrium statistical mechanics

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In this section we shall devote ourselves to the mathematical description of the answers(A12 ) and (A22 ) mentioned in Section 4.1. Readers should note that all arguments inthis section are within Newtonian mechanics. Namely, it should be noted that it isprohibited to use the term “probability” in this section. For example, Lemma 4.9 (“thelaw of large numbers” in §4.5 Appendix) is not only most important in Kolmogorov’s
probability theory but also in this section (i.e., the derivation of the ergodic hypothesis(= Theorem 4.6)). Therefore, readers will see that Lemma 4.9 is used independently of
the concept of “probability”. This is the reason that the term “normalized measure” (andnot “probability measure” ) is used in Lemma 4.9.
Now let us begin with the well-known ergodic theorem (cf. [57, 83]). In Newtonianmechanics, any state of a system composed of N ( ≈ 1024 ) particles is represented by a¡¢point (q, p) ≡ (q1n , q2n , q3n , p1n , p2n , p3n )N
in a phase (or state) space R6N (cf. then=1
formula (2.8)). Let H : R6N → R be a Hamiltonian, i.e., a positive continuous function
on R6N . Deﬁne V (E), E ≥ 0, by “the volume of the set {(q, p) ∈ R6N | H(q, p) ≤ E}”,and deﬁne the measure νE on the energy surface SE (≡ {(q, p) ∈ R6N | H(q, p) = E})such thatZνE (B) =B|∇H(q, p)|−1 dm6N −1(∀B ∈ BSE , the Borel ﬁeld of SE )5

(4.5)

(E)
holds. Let
where dm6N −1 is the usual surface measure on SE . Note that νE (SE ) = dVdE

{ψ Et }−∞<t<∞ be the ﬂow on the energy surface SE induced by the Newton equation with
the Hamiltonian H. Liouville’s theorem (cf. [11]) says that the measure νE is invariant
concerning the ﬂow {ψ Et }−∞<t<∞ . Deﬁning the normalized measure ν E such that ν E =νE, we have the normalized measure space (SE , BSE , ν E ).
νE (SE )In order that equilibrium statistical mechanics must hold, we ﬁrst assume that the
Hamiltonian H satisﬁes the following ergodic hypothesis (EH):(EH) The ﬂow {ψ Et }−∞<t<∞ on the SE is ergodic. That is, there uniquely exists annormalized invariant measure ν E on SE such that ν E (B) = ν E (ψt (B)) ( − ∞ < ∀t <R
Or usually, νE (B) = h3N1 N ! B |∇H(q, p)|−1 dm6N −1 , where h is the Plank constant. In this book, forsimplicity, the constant h3N1 N ! will be omitted.
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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/∞, ∀B ∈ BSE ).

The ergodic theorem (cf. [11, 57]) says that the normalized measure ν E represents thenormalized averaging staying time, i.e., it holds that♯[{k | ψϵk ω ∈ B, k = 1, 2, ..., K}]K→∞Kν E (B) = limor generally,Z1f (ω)ν E (dω) = lim
T →∞ TΩ(space average)(∀ω ∈ SE , ∀ϵ > 0).

Z T

f (ψt (ω0 ))dt
0(time average)(∀f ∈ C(Ω),∀ω0 ∈ Ω),

(4.6)

which is equivalent to the (EH). Thus the normalized measure space (SE , BSE , ν E ) is called
the normalized averaging staying time space (cf. Remark 4.1 later).
We assert that
(STI) [Staying time interpretation of statistical mechanics]. Let N ∈ BSE such that thenormalized averaging staying time ν E (N) is quite small (i.e., ν E (N) ≪ 1). Then it
is almost impossible (or precisely, quite rare) to see that the state (q(t), p(t)) belongs
to the N.
We think that this (STI) is a common sense rather than a principle. The concept of “time”(or precisely “non-relativistic time”) is within Newtonian mechanics, and therefore thestatement (STI) (or “staying time”) can be understood within Newtonian mechanics.
Remark 4.1. [The probabilistic interpretation of (SE , BSE , ν E )]. The probabilistic interpretation is as follows:(PI) [Probabilistic interpretation of statistical mechanics]. The normalized averaging
staying time space (SE , BSE , ν E ) is regarded as Kolmogorov’s probability space.
That is, the probabilistic interpretation, which is usually called “the principle of equala priori probability”, means that the probability that the state of the system belongs toΞ( ∈ BSE ) is given by ν E (Ξ). If the probabilistic interpretation (PI) is assumed, the (STI)
obviously holds. However, the concept of “normalized staying time” is clearly diﬀerent
from that of “probability”. Note that:• the former (i.e., “the staying time interpretation”) is within Newtonian mechanics,
but the latter (i.e., “the probabilistic interpretation”) is not so.

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Thus, in this chapter we choose a common sense (i.e., “the staying time interpretation”)
rather than a principle (i.e., “the probabilistic interpretation”).6 This is the reason thatthe (SE , BSE , ν E ) is not called the probability space in this chapter. Again note that allarguments in this section are within Newtonian mechanics. In this chapter the (STI) will
be used instead of the (PI).
¥We introduce the following notation:
Notation 4.2. [In the sense of (STI)]. Let P(q, p) be a proposition concerning a state(q, p) ( ∈ SE ) such that P(q(t), p(t)) is true for every t ∈ SE \ N ( ≡ {ω | ω ∈ SE , ω ∈/ N}).
Assume that the normalized averaging staying time ν E (N) is quite small (i.e., ν E (N)≪ 1). Then we write it as³

P(q(t), p(t)) is true(almost every t in the sense of (STI)),´
Or, P(q(t), p(t)) is almost always true .

(4.7)

Also, when the probabilistic interpretation (cf. Remark 4.1) is added to the (SE , BSE , ν E ),we may write it asP(q(t), p(t)) is true(almost every t in the sense of (PR)).7

(4.8)

¥
As seen in Remark 4.1, it holds that (4.8)⇒(4.7). Throughout this chapter we, ofcourse, focus on the (4.7) and not (4.8).
Let ϵ > 0, f1 , f2 , ..., fK ∈ C0 (R6 ). Deﬁne the 0-neighborhood U in M(R6 ) (in thesense of weak∗ topology of M(R6 )) such that:U ( = Ufϵ1 ,...fK ) = {ρ ∈ M(R6 )(= C0 (R6 )∗ ) : |M(R6 ) ⟨ρ, fk ⟩C (R6 ) | < ϵ, k = 1, 2, ..., K}.
0
(4.9)
6
What is the most important is to recognize that statistical mechanics belongs to the category oftheoretical informatics and not that of theoretical physics. (cf. Table (1.7)). Thus, the present situationis the same as the following situation. Two ready-made suits (A) and (B) are on sale. The (A) issomewhat big, and the (B) is somewhat small. Which do you choose, (A) or (B)? Cf. (I15 ) in §1.3.
We must choose one from “the staying time interpretation” and “the probabilistic interpretation”. In
theoretical informatics, it can not be decided by experimental test. What we can say is we believe that“the staying time interpretation” will win more popularity than “the probabilistic interpretation”.
7
Note that this notation is diﬀerent from that of Kolmogorov’s probability theory, in which we use
the phrase “almost every t in the sense of (PR)” when ν̄(N) = 0.

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81

Put DN = {1, 2, ..., N ( ≈ 1024 )}. For each k ( ∈ DN ≡ {1, 2, ..., N ( ≈ 1024 )}), deﬁne
the map Xk : SE ( ⊂ R6N ) → R6 such that
Xk ((q1n , q2n , q3n , p1n , p2n , p3n )Nn=1 ) = (q1k , q2k , q3k , p1k , p2k , p3k )(4.10)

6N
for all (q, p) = (q1n , q2n , q3n , p1n , p2n , p3n )N). For any subset D ( ⊆ DN ≡n=1 in SE ( ⊂ R(·)

6
6{1, 2, ..., N ( ≈ 1024 )}), deﬁne the map RD : SE ( ⊂ R6N ) → Mm
+1 (R ) ( ≡ {ρ ∈ M(R ) :ρ ≥ 0, ρ(R6 ) = 1}) such that
(q,p)

RD

=

1 X
δX (q,p)♯[D] k∈D k(∀(q, p) ∈ SE ( ⊂ R6N )),(4.11)

where ♯[D] is the number of the elements of D and δx is a point measure at x ( ∈ R6 ).
Let U be a 0-neighborhood in M(R6 ) such as deﬁned in (4.9). For any (p, q) ( ∈ SE ),
put
£
¤
(p,q)
(p′ ,q ′ )HU (p, q) = kB log νE ({(p′ , q ′ ) ∈ SE | RDN − RDN ∈ U }) ,(4.12)(kB is the Boltzmann constant, i.e., kB = 1.381×10−23 J/K), which is called the U -entropyof a state (p, q).
6×♯[D0 ]Let D0 ⊆ DN . Deﬁne ν E ◦ ((Xk )k∈D0 )−1 ( ∈ Mm)) by the image measure+1 (Rconcerning the map (Xk )k∈D0 : R6N → R6×♯[D0 ] , that is,ν E ◦ ((Xk )k∈D0 )−1 ( × Ak ) = ν E ({(p, q) ∈ SE | Xk (p, q) ∈ Ak (k ∈ D0 )})(4.13)k∈D0

for any open set Ak ( ⊆ R6 ) (k ∈ D0 ).
In what follows we shall represent the conditions (A12 ) and (A22 ) (mentioned in §4.1)
in terms of mathematics. Cf. [45].
Deﬁnition 4.3. [Thermodynamical condition, equilibrium state].8

Let DN be a set{1, 2, ..., N (≈ 1024 )}. And let H, E, νE , ν E , Xk : SE → R6 be as in the above. A
Hamiltonian H on R6N (N ≈ 1024 ) is said to be thermodynamical (concerning energy E)
if the following condition (T ) is satisﬁed:(T ) {Xk : SE → R6 }N
k=1 is an almost independent sequence with the identical distribution.
8

Although this condition may be superﬁcial and not fundamental, we believe, from the measurement
theoretical point of view, that equilibrium statistical mechanics should start from this condition. Again
note that our purpose is to understand equilibrium statistical mechanics as one of applications of PMT
and not to derive equilibrium statistical mechanics from Newtonian mechanics.

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6
In other words, there exists a normalized measure ρE on R6 (i.e., ρE ∈ Mm
+1 (R )) such

that:
£
¤
(T 1 ) identical distribution, cf. (A12 ) in §4.1 it holds thatρE ≈ ν E ◦ Xk−1(∀k = 1, 2, ...., N ( ≈ 1024 )),(4.14)£¤(T 2 ) independence, cf. (A22 ) in §4.1 it holds thatO

ρE (: product measure) ≈ ν E ◦ ((Xk )k∈DN )−1 ,(4.15)k∈DN

though the condition (T 2 ) is too strong to assume it literally, (see Remark 4.4).
(q,p)

Here, a state (q, p) (∈ SE ) is called an equilibrium state if RDN ≈ ρE . 9¥
Let T be a suﬃciently large number. Assume that the closed interval [0, T ] has
the measure: dt/T (thus, the total measure of [0, T ] is equal to 1). For each k ( ∈DN ≡ {1, 2, ..., N ( ≈ 1024 )}), deﬁne the map wk : [0, T ] → R6 such that wk (t) =(q1k (t), q2k (t), q3k (t), p1k (t), p2k (t), p3k (t)) for all t ( ∈ [0, T ]). Assume that(♯) {wk | k ∈ DN } is a set composed of almost independent functions with the identical
distribution.
This assumption (♯) is essentially the same as (T ) in Deﬁnition 4.3.
R6

6

w1
w2

-

almost independent
identically distributed

T
wN

9

In our formulation, we do not assume that the “equilibrium state” is deﬁned by ν E since ν E is notassumed to have the probabilistic interpretation (cf. Remark 4.1).

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83

Remark 4.4. As mentioned in Deﬁnition 4.3, the condition (T 2 ) is too strong. Thus,
it should be understood symbolically and not literally. Therefore, we actually assume
some hypotheses, which are weaker than the (T2 ). For example we assume the following
conditions (T 2 )′ and (T 2 )′′ :£¤(T 2 )′ independence it holds thatO

ρE ≈ ν E ◦ ((Xk )k∈D0 )−1 ,(4.16)k∈D0(∀D0 ⊂ {1, 2, ..., N ( ≈ 1024 )} such that 1 ≪ ♯[D0 ] ≪ N ).
This is needed for the derivation of the ergodic hypothesis (cf. Theorem 4.6 later). Also,
we assume that£¤(T 2 )′′ independence it holds that³
´O³´
ν E ◦ ((Xk )k∈D1 )−1ν E ◦ ((Xk )k∈D2 )−1 ≈ ν E ◦ ((Xk )k∈D1 ∪D2 )−1 ,(4.17)

for any D1 , D2 (⊂ D) such that D1 ∩ D2 = ∅ and 1 ≪ ♯[D1 ], ♯[D2 ] ≤ N .
That is because, in equilibrium statistical mechanics, we usually assume that the interaction between the subsystem composed of the particles D1 and that of the particles D2
can be neglected.
¥
Remark 4.5. (i) If N0 is arbitrarily large (and thus N = ∞) and if the approximation
symbol “≈” is interpreted by the equality “=”, then (4.4) and (4.16) imply that thesequence {Xk }∞k=1 on the normalized averaging staying time space (SE , BSE , ν E ) is anindependent sequence with the identical distribution ρE . Thus, Lemma 4.9 (i.e., the lawof large numbers) says thatlim
N0 (=♯[D])→∞

(q,p)

RD

= ρE

( in the sense of the weak∗ topology of M(R6 ))(4.18)

holds for almost every (q, p) in (SE , B(SE ), ν E ). Note that Kolmogorov’s probability theory [56] is mathematics, and therefore, it is valid even if the probabilistic interpretation(cf. Remark 4.1) is not added to the normalized averaging staying time measure space(SE , BSE , ν E ). For completeness, again note that the terms: “identical distribution” in(T 1 ) and “independence” in (T 2 ) are not related to the concept of “probability” (but thatof “staying time”).

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CHAPTER 4. BOLTZMANN’S EQUILIBRIUM STATISTICAL MECHANICSFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(ii) The reader may doubt if the concepts of “identical distribution” and “independence”
are meaningful without the probabilistic interpretation. However, the following example
shows that these concepts are not only meaningful on a measure space but also on a
topological space. Let f : Ω → R be a continuous function on a topological space Ω.
For each n(= 1, 2, ..., N ), deﬁne the function fn : ΩN (= product topological space) → R
such that ΩN ∋ (ω1 , ω2 , ..., ωn , ..., ωN ) 7→ f (ωn ) ∈ R. Then we may say that {fn }N
n=1
is “an independent sequence with the identical distribution”. In fact we often say “Themotions of two particles are independent” in Newtonian mechanics (and not in statisticalmechanics).
¥
By an analogy of the arguments (i.e., the derivation of (4.18)) in the above Remark
4.5(i), we can assert that (4.14) and (4.16) imply that, if 1 ≪ N0 ( ≈ ♯[D0 ]) ≪ N ( ≈ 1024 ),(q(t),p(t))

R D0≈ ν E ◦ Xk−1 ( ≈ ρE )( almost every time t in the sense of (STI) ) (4.19)

holds for any k ( = 1, 2, ..., N ( ≈ 1024 )). Here consider the decomposition {D(1) , D(2) , ...,
D(L) } of DN ( ≡ {1, 2, ..., N ( ≈ 1024 )}) such that ♯[D(l) ] ≈ N0 (l = 1, 2, ..., L). Then wesee, by (4.19), thatPL(q(t),p(t))
R DN=(q(t),p(t))]
l=1 [♯[D(l) ] × RD(l)PL≈

l=1 [♯[D(l) ] × ρE ]N
N
( almost every time t in the sense of (STI) )= ν E ◦ Xk−1 ( ≈ ρE )holds for any k ( = 1, 2, ..., N ( ≈ 1024 )).
Summing up, we have the following theorem.
Theorem 4.6. (Ergodic hypothesis). Assume the thermodynamical condition (i.e., (T1 )in Deﬁnition 4.3 and (T 2 )′ in Remark 4.4). Then it holds that(q(t),p(t))

RDN≈ ν E ◦ Xk−1 ( ≈ ρE ) (k = 1, 2, ..., N ( ≈ 1024 ))(4.20)( almost every time t in the sense of (STI) )Thus, the state of the system is almost always equal to the equilibrium state (cf. Deﬁnition
4.3). That is, we see:

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4.2. DYNAMICAL ASPECTS OF EQUILIBRIUM STATISTICAL MECHANICSFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(q(t ),p(t1 ))• R DN 1(q(t ),p(t2 ))≈ R DN 285( almost every time t1 and t2 in the sense of (STI)).
(4.21)¥

This says that“the distribution of N ( ≈ 1024 ) particles at almost every time t” (in the sense of (STI))=“normalized averaging staying time of the k-th particle (∀k = 1, 2, ..., N ≈ 1024 )”(4.22)We believe that this is just what should be represented by the “ergodic hypothesis” :10“population average of many particles” = “time average of one particle”,(4.23)

that is, we see that (4.20)=(4.22)=(4.23).
Remark 4.7. [Another formulation of equilibrium statistical mechanics]. For completeness, note that the condition (T 2 )′ in (4.16) is assumed in order that (4.21) holds. Thus
some may assert that it suﬃces to start from the SE (with the measure νE which induces(STI)) and the (4.21). This formulation may be called the formulation without the ergodic
hypothesis. Also, see the formula (4.29) later.
¥
Remark 4.8. (i). If the probabilistic interpretation (i.e., the principle of equal a priori
probability) is assumed, in (4.20) we can replace “almost every time t in the sense of(STI)” to “almost every time t in the sense of (PR)”. However, if the (STI) is accepted as
a common sense, we can do well without this replacement, that is, the replacement does
not bring us any merit. Thus we think that the probabilistic interpretation is not needed.
Cf. Remark 4.5(ii). We may still have a question:• Why is the thermodynamical condition (i.e., (T 1 ) and (T 2 )) always satisﬁed in theusual circumstance of equilibrium statistical mechanics?
Though we do not know the ﬁrm answer,11 we can easily show, by (4.20), that the thermodynamical condition ((T 1 ) and (T 2 )) explains the following law (i.e.,“the law of increasing
10

In this book, the term “ergodic hypothesis” has two meanings. One is used in the sense of the formula(4.6). And the other is used in the sense of the formula (4.23) (or, Theorem 4.6).
11
If we think that statistical mechanics belongs to informatics and not physics (cf. in this book weconsider so), the ﬁrm answer may not be needed. If the thermodynamical condition is useful, it is enough.

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entropy” ).12(IE) the U -entropy HU (q(t), p(t)), (cf. (4.12)), is increasing concerning t, that is
HU (q(t), p(t)) ↑ log[ν(SE )] ( if t ↑ ∞)in the sense of (STI)(4.24)

for a suitable small 0-neighborhood U in M(R6 ).
That is because HU (q(t), p(t)) ≈ log[ν(SE )] holds for almost every time t in the sense of(STI) if the neighborhood U is chosen suitably. (How to choose the U suitably is our
future problem.)Therefore we consider that the law of increasing entropy is hidden

behind the thermodynamical condition ((T 1 ) and (T 2 )).
¥4.3

Probabilistic aspects of equilibrium statistical mechanics

In this section we shall study the probabilistic aspects of equilibrium statistical mechanics. Note that the (4.20) implies that the equilibrium statistical mechanical system
at almost every time t (in the sense of the (STI)) can be regarded as:(U ) an urn including about 1024 particles such as the number of the particles whosestates belong to Ξ ( ∈ BR6 ) is given by ρE (Ξ) × 1024 .
Recall the (A1 ) in §4.1, that is, the probability appearing in classical systems (or particularly, in the probabilistic rule in (4.2)) is essentially the same as the probability appearing
in urn problems. Therefore, we see, by the above (U),(A′1 ) if we choose a particle at random from the urn (=“box in Figure (4.3)”) at time t,then the probability that the state of the particle belongs to Ξ ( ∈ BR6 ) is given byρE (Ξ).
12
If my memory serves me right, in some book A. Einstein says: There is a possibility that someone
will ﬁnd his relativity theory is not true, but there is no possibility that someone will ﬁnd that the lawof increasing entropy is not true. We can understand what he wants to say, if we think that statisticalmechanics should be understood as an application of measurement theory, on the other hand, his relativity
theory belongs to theoretical physics. That is, we think that the law of increasing entropy is as “true” asthe statement “A cat is stronger than a mouse”. (Cf. footnote[9] in Chapter 2.) It should be noted thatthe statement “A cat is stronger than a mouse” is ambiguous, fuzzy, vague, etc, though it is “almost
experimentally true” (cf. (I14 ) in §1.3).

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In what follows, we shall represent this (A′1 ) in terms of measurements. Deﬁne the observable O = (R6 , BR6 , F ) in C(SE ) such that, (cf. (4.11)),³ ♯[{k | X (q, p) ∈ Ξ}] ´k(q,p)[F (Ξ)](q, p) = [RDN ](Ξ) ≡♯[DN ](∀Ξ ∈ BR6 , ∀(q, p) ∈ SE ( ⊂ R6N )).
(4.25)

Thus, we have the measurement MC(SE ) (O ≡ (R6 , BR6 , F ), S[δψt (q0 ,p0 ) ] ). Then we see that
(B1′ ) the probability that the measured value obtained by the measurement MC(SE ) (O ≡(R6 , BR6 , F ), S[δψt (q0 ,p0 ) ] ) belongs to Ξ(∈ BR6 ) is given by ρE (Ξ). That is because
Theorem 4.6 says that
[F (Ξ)](ψt (q0 , p0 )) = ρE (Ξ) (almost every time t in the sense of (STI))(4.26)

which is just the measurement theoretical representation of the (A′1 ).
Also, we see that(A′′1 ) if we choose N ′ particles at random from the urn (=“box in Figure (4.3)”), thenstatistics say that the distribution of the states of these particles is almost surely
expected to be approximately equal to ρE , where 1 ≪ N ′ ≤ N ( ≈ 1024 ).
′′′Here, consider the product observable ON = (R6N , BR6N ′ , F N ) in C(SE ). For each k′( ∈ KN ′ ≡ {1, 2, ..., N ′ ), deﬁne the map Xk : R6N → R6 such that′

Xk ((x1n , x2n , x3n , x4n , x5n , x6n )Nn=1 ) = (x1k , x2k , x3k , x4k , x5k , x6k )
′

′

′

6N
6
for all x = (x1n , x2n , x3n , x4n , x5n , x6n )N). Deﬁne the map GN ′ : R6N → Mm
n=1 in R+1 (R )

( ≡ {ρ ∈ M(R6 ) : ρ ≥ 0, ρ(R6 ) = 1}) such thatN′

1 X
δX (x)GN ′ (x) = ′N n=1 n′(∀x ∈ R6N ).

(4.27)′′

6N
6 , GN ′ (F)). And
Then we have the image observable GN ′ (ON ) ≡ (Mm+1 (R ), BMm+1 (R )

we see, by Theorem 4.6, that′

(B1′′ ) the measured value obtained by the measurement MC(SE ) (GN ′ (ON ), S[δψt (q0 ,p0 ) ] ) is
approximately equal to ρE .
which is just the measurement theoretical representation of the (A′′1 ).

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4.4

Conclusions

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In this chapter we assert that equilibrium statistical mechanics is formulated as follows:13“equilibrium statistical mechanics” = “probabilistic rule” + “Newton equation”|((B1′′ )(= Axiom 1))((T 1 ) and (T 2 )) under (EH)){z}(+ STI)(4.28)(=(4.4))in the framework of PMT.
It may be generally believed that the principle of equal a priori probability and theergodic hypothesis are two basic principles of statistical mechanics. However, our formulation (4.28) says that the principle of equal a priori probability is not needed (cf.
Remark 4.5 and Remark 4.8(i)), and moreover, the ergodic hypothesis is a consequenceof the thermodynamical condition (i.e., (T 1 ) and (T 2 ) under the (EH)), cf. the formulas(4.20)∼(4.23).
However we may assert that the following formulation is also possible:“equilibrium statistical mechanics” = “probabilistic rule” + “Newton equation”|((B1′′ )(= Axiom 1))((T 1 ) and (T 2 )) under (EH)){z}(+ PI)(4.30)
which is, strictly speaking, related to SMT (cf. Chapter 8, Statistical measurement theory).
Thus we have the question:• Which should be chosen, (4.28) or (4.30)?14
13

Or simply (cf. Remark 4.7), we may consider that“equilibrium statistical mechanics” = “probabilistic rule” + “Newton equation”|((B1′′ )(= Axiom 1)){z(4.29)

νE (in (4.5)) and (4.21)}(+ STI)

We believe that the term “economical” is one of the most important key-words of theoretical informatics(cf. Table (1.8b)). In this sense, the (4.29) should be also admitted though we did not focus on the (4.29)
in this chapter.
14
This situation is the same as the following situation. Two ready-made suits “the staying timeinterpretation” and “the probabilistic interpretation” are on sale. The former is too weak, and so somewhat ambiguous. The latter may be too strong. However, we must choose one from “the staying timeinterpretation” and “the probabilistic interpretation”. In theoretical informatics, we believe that it can

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4.4. CONCLUSIONS89

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The reason that we choose (4.28) is as follows: Recall quantum mechanics, in which
it is often emphasized that the concept of “probability” is not related to “Schrödinger
equation” but “Born’s quantum measurements”. Comparing quantum mechanics (1.3) andthe above (4.28), we have the reason to emphasize that the concept of “probability” is
not related to the thermodynamical condition but “probabilistic rule in (4.28)”.

That

is because we want to believe in the spirit that the term of “probability” should be usedcommonly in both classical and quantum systems, or, that there is no probability without
measurements. After all, we say that
• Our proposal (4.28) and quantum mechanics (1.3) are compatible.
On the other hand, the part “Newton equation ((T 1 ) and (T 2 )) under (EH))” in (4.30) is
related to the concept of “probability” under the assumption “probabilistic interpretationof ν E ”. Thus, we think that
• The (4.30) and quantum mechanics (1.3) are not compatible.
Thus, we do not choose the (4.30). However, we may choose the following (4.18):“equilibrium statistical mechanics” = “probabilistic rule” + “Newton equation”|((B1′′ )(= Axiom 1)){z}((T 1 ) and (T 2 )) under (EH))

|

(+PI)

{z

}

(+STI)(4.31)
This (4.31)15 and quantum mechanics (1.3) are compatible. Thus, the following questionis meaningful in measurement theory.
• Which should be chosen, (4.28) or (4.31)?
This may be a matter of opinion (though it is not serious as statistical mechanics is
assumed to belong to theoretical informatics in this chapter). If we are required to saysomething, we guess that the (4.28) will win more popularity than the (4.31). In fact,
not be decided by an experimental test. Or at least, we are convinced that it is not worthwhile deciding
it by an experimental test. That is because we believe that nobody wants to challenge the following
problem:• Decide (4.28) or (4.30) (or (4.31)) by an experimental test!
Thus, “(4.28) or (4.30)” should be chosen from the philosophical point of view, if we are urged to choose
one. Cf. (I15 ) in §1.3.
15
The part “probabilistic rule” in (4.31) is characterized as “Proclaim 1” in Chapter 8.
|((B1′′ )(= Axiom 1))

{z

}

(PI)

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CHAPTER 4. BOLTZMANN’S EQUILIBRIUM STATISTICAL MECHANICSFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/• we prefer (4.28) to (4.31),

since we do not want use the (PI) if possible.16 This is our opinion, though, in theoreticalinformatics, we must admit the case that opinion is divided.
We hope that our proposal (4.28) (or, (4.29), (4.31)) will be accepted as the standard
formulation of equilibrium statistical mechanics.

4.5

Appendix (The law of large numbers)

As a preparation of our main assertion (i.e., the derivation of the ergodic hypothesis(4.20)), we add the following well-known Lemma 4.9.
Lemma 4.9. [The strong law of large numbers, cf. [56]]. Let (S, BS , ν) be a measurespace such that ν(S) < ∞. Let {Xn }∞n=1 be a sequence of bounded measurable (orgenerally, L1 ) maps Xn : S → R6 such that there exists a normalized measure ρ on R6(i.e., ρ(R6 ) = 1, ρ(Γ) ≥ 0 (∀Γ ∈ BR6 )) such that:• (identical distribution)ν({x ∈ S | Xn (x) ∈ Γ})= ρ(Γ) (∀n = 1, 2, ...,
ν(S)∀Γ ∈ BR6 ),• (independence) for any positive integer N , it holds thatNν({x ∈ S | Xn (x) ∈ Γn (∀n = 1, 2, ..., N )})= × ρ(Γn )
ν(S)
n=1

(∀Γn ∈ BR6 ).

Then, there exists a measurable set N( ∈ BS ) such that ν(N) = 0 and
1 X
δXn (x) = ρ
N →∞ Nn=1
N

lim

in the sense of weak∗ topology of M(R6 ),6
for all x ∈ S \ N (≡ {x | x ∈ S, x ∈/ N}). Here δw ( ∈ Mm
+1 (R )) is a point measure atw( ∈ R6 ), i.e., δw (Γ) = 1 (if w ∈ Γ ∈ BR6 ), = 0 (if w ∈/ Γ ∈ BR6 ).
¥In the formula (4.18), readers should see that Lemma 4.9 is used in the part “Newton
equation” (and not “probability rule”) in our proposal (4.28), that is, Lemma 4.9 (the lawof large numbers) is used independently of the concept of “probability”.
16

Also, recall “Occam’s razor”, that is, “Given two equally predictive theories, choose the simplest”.

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Chapter 5
Fisher’s statistics I (under Axiom 1)
As mentioned in Chapters 2 and 3, measurement theory is formulated as follows:
PMT = measurement + the relation among systems[Axiom 1 (2.37)]

in C ∗ -algebra[Axiom 2 (3.26)]

.

(5.1)
(=(1.4))In this chapter we intend to understand Fisher’s statistics in Axiom 1. The reader will see that
Fisher’s maximum likelihood method is a direct consequence of Axiom 1.1 And further, we discuss“inference interval” and “testing statistical hypothesis” in Axiom 1. By the results obtained inthis chapter (and in the next chapter), we conclude that Fisher’s statistics is theoretically true.
(Cf. “Declaration (1.11)” in §1.4.)2

5.1

Introduction

The ﬁrst attempt of the measurement theoretical approach to statistics was proposedin [44]. Although the argument in [44] is not deep, at least it convinces us of the good
possibility of the axiomatic formulation (i.e., the measurement theoretical formulation) of
statistics.
Most statisticians consider that statistics is closely related to “measurements”, or,
statistics is the study to analyze “measured data” for some purpose. Therefore, PMT
should be immediately examined in comparison with statistics. The purpose of this chapter is to execute it, in other words, to propose a measurement theoretical formulation of
statistics. We think that statistics is mainly related to the following aspect of measurement theory:
1

Readers are not required to have much knowledge of statistics.
We believe that the philosophy of statistics should be more discussed in statistics, (Cf. [61]). That is
because it is indispensable for the understanding of “statistics (= mathematics + something)”. It should
be noted that “to formulate statistics in the framework of MT” implies “to introduce the philosophy of
MT into statistics”.
2

91
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CHAPTER 5. FISHER’S STATISTICS I (UNDER AXIOM 1)

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(♯) how to derive some useful information from the measured data obtained by a measurement.
¡¢
Let MA O ≡ (X, F, F ), S[ρp ] be a measurement formulated in a C ∗ -algebra A. Recall¢¡
the (III) in §2.5 [Remarks], that is, the measurement MA O ≡ (X, F, F ), S[ρp ] always³DE ´
determines the sample space X, F, A∗ ρp , F ( · ) A .
Here note that the mathematicaln D
E ostructure of the sample space A∗ ρp , F (Ξ) A
is the same as that of thep (A∗ ),Ξ∈F
ρp ∈So³nconventional formulation of statistics i.e, P (Ξ, θ), where, for each θ in a paθ∈Θ,Ξ∈F´
rameter space Θ, P (·, θ) is a probability measure on a measurable space (X, F), cf. [86] .
Therefore, there is good hope that statistics can be described in terms of measurements.
Also, this is precisely our motivation in this chapter. Following the common knowledgeof quantum mechanics, we believe that any scientiﬁc statement including the term “probability” is not meaningful without the concept of “measurement”. (cf. §2.5. Remarks).
As mentioned in the above, the term “state” in measurement theory corresponds to theterm “parameter” in statistics. The reason that we use the term “state” is due to that
we want to stress that PMT is constructed modeled on mechanics.3

5.2

Fisher’s maximum likelihood method

The purpose of this section is to study and understand “Fisher’s maximum likelihood
method” completely under Axiom 1 (of measurement theory). The following Problem 5.1
is the most typical in all examples of “Fisher’s maximum likelihood method”.

5.2.1

Fisher’s maximum likelihood method

Problem 5.1. [The urn problem by Fisher’s maximum likelihood method].

There aretwo urns U1 and U2 . The urn U1 [resp. U2 ] contains 8 white and 2 black balls [resp. 4
white and 6 black balls].
3This means that we study statistics by an analogy of “mechanics”. Note the following correspondence:system S[ρp ] (in PMT)
[represented by pure state]⇐⇒

population (in the conventional statistics)[represented by parameter]

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93

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U1

U2

Here consider the following procedures (P1 ) and (P2 ).
(P1 ) One of the two (i.e., U1 or U2 ) is chosen and is settled behind a curtain. Note, forcompleteness, that you do not know whether it is U1 or U2 .
(P2 ) Pick up a ball out of the urn chosen by the procedure (P1 ). And you ﬁnd that theball is white.

You do not know which the urn behind the curtain is, U1 or U2 .
Assume that you pick up a white ball from the urn.
The urn is U1 or U2 ?

Which do you think?

U1

U2

-

[∗]



Now we have the following question:(Q) Which is the chosen urn (behind the curtain), U1 or U2 ?
This is quite easy. That is, everyone will immediately infer “the urn behind the curtain =U1 ”. However, it is just “Fisher’s maximum likelihood method”. Cf. Example 5.8.
¥
We begin with the following deﬁnition.
Notation 5.2. [MA (O, S[∗] )]. Consider a measurement MA (O ≡ (X, F, F ), S[ρp ] ) formulated in a C ∗ -algebra A. In most measurements, it is usual to think that the stateρp (∈ Sp (A∗ )) is unknown. That is because the measurement MA (O, S[ρp ] ) may be taken

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in order to know the state ρp . Thus, when we want to stress that we do not know thestate ρp , the measurement MA (O, S[ρp ] ) is often denoted by MA (O, S[∗] ).
¥By using this notation, we can state our present problem as follows:(I) Infer the unknown state [∗] (∈ Sp (A∗ )) from the measured data obtained by themeasurement MA (O ≡ (X, F, F ), S[∗] ).
In order to answer this problem, in [44] we introduced Fisher’s method (precisely,³
Fisher’s max mum likelihood method) as follows: Strictly speaking, Theorem 5.3 should
not be called “theorem” but “assertion”, since it is not a purely mathematical result but´
a consequence of Axiom 1.
Theorem 5.3. [Fisher’s maximum likelihood method in classical and quantum measurements, cf. [44]]. Consider a measurement MA (O ≡ (X, F, F ), S[∗] ) formulated in a
C ∗ -algebra A. When we know that the measured value obtained by the measurement
MA (O, S[∗] ) belongs to Ξ (∈ F), there is a reason to infer that the state [∗] of the system
S is equal to ρp0 (∈ Sp (A∗ )) such that:A∗p⟨ρp0 , F (Ξ)⟩A = p max∗ ⟨ρ , F (Ξ)⟩A .
p∗ Aρ ∈S (A )(5.2)

Here, note, for completeness, that the state [∗] (in MA (O, S[∗] )) is the state before the³´measurement MA (O, S[∗] ). Cf. Corollary 5.6 later. Although the ρp0 in (5.2) is not
generally determined uniquely, in this book we usually assume the uniqueness.

Proof (or, Explanation).
ρp1 ” or “[ ∗ ] = ρp2 ”.

Let ρp1 and ρp2 be elements in Sp (A∗ ). Assume that “[ ∗ ] =And assume that ρp1 (F (Ξ)) < ρp2 (F (Ξ)). Then, Axiom 1 saysthat the fact that the measured value obtained by the MA (O, S[ρp1 ] ) belongs to Ξ happens
more rarely than the fact that the measured value obtained by the MA (O, S[ρp2 ] ) belongsto Ξ happens. Thus, there is a reason to regard the unknown state [∗] as the state ρp2
and not as the state ρp1 . Also, examining this proof, we can easily see that the state [∗](in MA (O, S[∗] )) is the state before the measurement MA (O, S[∗] ). This completes theproof.

Remark 5.4. [Radon-Nikodým derivative]. Assume that there exists a measure ν on

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5.2. FISHER’S MAXIMUM LIKELIHOOD METHODFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(X, F) (cf. (III) in §2.5) and f (·, ρp ) ∈ L1 (Ω, ν) (∀ρp ∈ Sp (A∗ )) such that:Zpρ (F (Ξ)) =f (x, ρp )ν(dx) (∀Ξ ∈ F, ∀ρp ∈ Sp (A∗ )).

95

(5.3)

Ξ

Then, even if Ξ = {x0 } and ρp (F ({x0 }) = 0 (∀ρp ∈ Sp (A∗ ) ) in Theorem 5.3, we may
calculate as follows:ρp1 (F ({x0 }))ρp1 (F (Ξ))
f (x0 , ρp1 )
= lim
=
.
ρp2 (F ({x0 })) Ξ→{x0 } ρp2 (F (Ξ))
f (x0 , ρp2 )(5.4)

In this sense (or, in the sense of “Radon-Nikodým derivative”), we can compare ρp1 (F ({x0 }))with ρp2 (F ({x0 })), even when ρp1 (F ({x0 })) = ρp2 (F ({x0 })) = 0. When we know that themeasured value x0 ( ∈ X) is obtained by the measurement MA (O, S[∗] ), by the same reason
in Theorem 5.3, we can infer that the state [∗] of the system S is equal to ρp0 (∈ Sp (A∗ ))such that:f (x0 , ρp0 ) = p max
f (x0 , ρp ).
p∗
ρ ∈S (A )Here, the map E : X → Sp (A∗ ), ( i.e., X ∋ x0 7→ ρp0 ∈ Sp (A∗ )), is called “Fisher’s
estimator”.
¥
We begin with the following corollary, which is used in the proof of Corollary 5.6 and³´
our main assertion i.e., Regression Analysis II (in Chapter 6) .
Corollary 5.5. [The conditional probability representation of Fisher’s method, cf. [55]].
b be a quasi-product
Let O ≡ (X, F, F ) and O′ ≡ (Y, G, G) be observables in A. Let O
qp
qp
b ≡ O × O′ = (X × Y, F N G, F × G). Assume that we
observable of O and O′ , that is, O
b S[∗] )know that the measured value (x, y) (∈ X × Y ) obtained by a measurement MA (O,N
belongs to Ξ × Y (∈ F G). Then, there is a reason to infer that the unknown measured
value y (∈ Y ) is distributed under the conditional probability PΞ (·), where


qp
qp
p
p
× G(Γ)) ∗ ⟨ρ , F (Ξ) × G(Γ)⟩A
= ρ0 (F (Ξ)
PΞ (Γ) = A 0 p(∀Γ ∈ G),p⟨ρ0 , F (Ξ)⟩A
ρ0 (F (Ξ))
A∗

(5.5)

where ρp0 (∈ Sp (A∗ )) is deﬁned byA∗p⟨ρp0 , F (Ξ)⟩A = p max∗ ⟨ρ , F (Ξ)⟩A .
p∗ Aρ ∈S (A )(5.6)

Proof. Since we know that the measured value (x, y) (∈ X × Y ) obtained by ab S[∗] ) belongs to Ξ × Y (∈ F N G), we can infer, by Theorem 5.3
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qp

b S[∗] )) is(Fisher’s method) and the equality F (Ξ) = F (Ξ) × G(Y ), that the [ ∗ ] (in MA (O,equal to ρp0 (∈ Sp (A∗ )). Thus, the conditional probability that PΞ ( · ) under the condition
that we know that (x, y) ∈ Ξ × Y is given byqp

qp

ρp0 (F (Ξ) × G(Γ))
PΞ (Γ) ==.
qp
ρp0 (F (Ξ))ρp (F (Ξ) × G(Y ))ρp0 (F (Ξ) × G(Γ))

(5.7)

0

This completes the proof.
The following corollary is the most essential in classical measurements. That is because
what we want to infer is usually the state after the measurement (cf. Theorem 5.3).
Corollary 5.6. [Fisher’s maximum likelihood method in classical measurements, cf.
[55]]. Let O ≡ (X, F, F ) be an observable in a commutative C ∗ -algebra C(Ω). Assume
that we know that the measured value obtained by a measurement MC(Ω) (O, S[∗] ) belongsto Ξ (∈ F). Then, we can assert the following (i) and (ii):(i) there is a reason to infer that the state [∗] of the system S³
i.e., “the state before

the measurement MC(Ω) (O, S[∗] )” cf. Fisher’s method in classical and quantum´measurements) is equal to δω0 (∈ Mp+1 (Ω)), where[F (Ξ)](ω0 ) = max[F (Ξ)](ω),

(5.8)

ω∈Ω

and,
(ii) there is a reason to infer that the state after the measurement MC(Ω) (O, S[∗] ) is also
regarded as the same δω0 (∈ Mp+1 (Ω)).
Summing up the above (i) and (ii), we see that(iii) there is a reason to infer that[ ∗ ] = “the state after the measurement MC(Ω) (O, S[∗] )” = δω0 .
1[F (Ξ)](ω)

0

ω0

Ω

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(5.9)

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97

Proof. The (i) is the special case of Fisher’s maximum likelihood method (cf. Theorem³
5.3), i.e., A = C(Ω). Thus it suﬃces to prove (ii) as follows: This (ii) will be, under
the deﬁnition of “S-state” (cf. Deﬁnition 6.7), proved in Remark 6.12 as a special case´of Lemma 6.11 later. In this sense, the proof mentioned here is temporary.
Let O′ ≡b be the product observable of O and O′ ,(Y, G, G) be any observable in C(Ω). Let Ob ≡ O × O′ = (X × Y, F N G, F × G). Consider a measurement MC(Ω) (Obthat is, O
N≡ (X × Y, F G, F × G), S[∗] ). And assume(A) we know that the measured value (x, y) (∈ X × Y ) obtained by the measurementb ≡ (X × Y, F N G, F × G), S[∗] ) belongs to Ξ × Y .
MC(Ω) (O
Corollary 5.5 says that there is a reason to infer that the unknown measured value y (∈ Y )is distributed under the conditional probability PΞ (·), wherePΞ (Γ)[F (Ξ)](ω0 ) = [G(Γ)](ω0 ) (∀Γ ∈ G),(5.10)

where ω0 (∈ Ω) is deﬁned in (5.8). Also note that the above (A) can be represented by³´
the following two steps (A1 ) and (A2 ) i.e., (A) = (A1 ) + (A2 ) :(A1 ) we know that the measured value by a measurement MC(Ω) (O ≡ (X, F, F ), S[∗] )belongs to Ξ (∈ F).
and(A2 ) And successively, we take a measurement of the observable O′ ≡ (Y, G, G), and geta measured value y ( ∈ Y ).
³
The above is somewhat metaphorical since “two measurements” seem to appear (cf.
´
§2.5[Remarks (II)]).
Comparing (A) and “(A1 ) + (A2 )”, we see, by (5.10), that“the probability that y belongs to Γ ( ∈ G) in (A2 )” = [G(Γ)](ω0 ) (∀Γ ∈ G)

(5.11)

That is, we get the sample space (Y, G, [G( · )](ω0 )). Therefore, we say, from the arbitrariness of O′ ≡ (Y, G, G), that
(A3 ) the state after the (A1 ) (i.e., the state after the measurement MC(Ω) (O, S[∗] )) isequal to δω0 .

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This completes the proof. This corollary does not hold in quantum measurements, sinceNb ≡ O×× O′ = (X × Y, F G, F × G) does not always exist. Thatthe product observable O
is, the concept of “the state after a measurement” is not always meaningful in quantum´
theory.
The “Bayes operator (in the following Remark 5.7)” is hidden in the above proof. This
will be more clariﬁed in Remark 6.12 later.
Remark 5.7. [Bayes operator].

Let O ≡ (X, F, F ) be an observable in C(Ω). For each(0,0)

Ξ ( ∈ F), deﬁne the continuous linear operator BΞ

O,(0,0)(or, BΞO , BΞ

) : C(Ω) → C(Ω)

such that:
(0,0)

BΞ

³
´
O,(0,0)(g) = F (Ξ) · g(g) ≡ BΞO (g) ≡ BΞ(∀g ∈ C(Ω)),(5.12)

which is called the Bayes operator (or, the simplest Bayes operator). Note that it clearlyholds thatb ≡ (X × Y, F N G,(i) for any observable O1 ≡ (Y, G, G), there exists an observable O
Fb) in C(Ω) such that:(0,0)
Fb(Ξ × Γ) = BΞ (G(Γ))(Ξ ∈ F, Γ ∈ G).

b by the product observable O × O1 . Deﬁne the map
That is because it suﬃces to deﬁne O
(0,0)

RΞ

m
: Mm
+1 (Ω) → M+1 (Ω) (called “normalized Bayes dual operator”) such that:(0,0)
RΞ (ν) =(0,0) ∗

where [BΞ(0,0) ∗

[BΞ

] (ν)

(0,0)
∥[BΞ ]∗ (ν)∥M(Ω)(∀ν ∈ Mm
+1 (Ω),(0,0)] : M(Ω) → M(Ω) is the dual operator of [BΞ ], that is,R[F (Ξ)](ω)ν(dω)(0,0)(∀D0 ∈ BΩ ).
[RΞ (ν)](D0 ) = RD0[F (Ξ)](ω)ν(dω)Ω(5.13)

Thus, we can describe the well known Bayes theorem (cf. [86]) such as
(0,0)

Mm
+1 (Ω) ∋ ν (= pretest state) 7→ (posttest state =)RΞ4(ν) ∈ Mm
+1 (Ω)(5.14)

Note that this says that (i)⇒(ii) in Corollary 5.6. That is because a simple calculation(0,0)

shows that RΞ
4(δω0 ) = δω0 in the case of Corollary 5.6. In §6.2, the reader will again

The pretest state [resp. posttest state] may be usually called “priori state” [resp. “posterior state”].

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study the Bayes operator in more general situations.

¥
Example 5.8. [Continued from Problem 5.1 (Urn problem)5 ]. Recall Example 5.1. Thatis, consider the following procedures (P1 ) and (P2 ).
(P1 ) One of the two (i.e., U1 or U2 ) is chosen and is settled behind a curtain. Note, forcompleteness, that you do not know whether it is U1 or U2 .6(P2 ) Pick up a ball out of the urn chosen by the procedure (P1 ). And you ﬁnd that theball is white.

You do not know which the urn behind the curtain is, U1 or U2 .
Assume that you pick up a white ball from the urn.
The urn is U1 or U2 ?

Which do you think?

U1

U2

[∗]

-



Now we have the following question:(Q) Which is the chosen urn (behind the curtain), U1 or U2 ?
[Answer]. Put Ω = {ω1 , ω2 }. Here,½
ω1 · · · · · · the state that the urn U1 is behind the curtain
ω2 · · · · · · the state that the urn U2 is behind the curtain.

(5.15)In this sense, we frequently use the following identiﬁcation:
U1 ≈ ω1 ,U2 ≈ ω2 .

5(5.16)As mentioned in Example 2.16, we believe that “urn problem” is the most fundamental in all examples
of statistics.
6
Here we are not concerned with SMTPEP (i.e., the principle of equal probability, cf. §11.4)

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And deﬁne the observable O ≡ (X ≡ {w, b}, 2{w,b} , F ) in C(Ω) where[F ({w})](ω1 ) = 0.8,

[F ({b})](ω1 ) = 0.2,

[F ({w})](ω2 ) = 0.4,

[F ({b})](ω2 ) = 0.6.

Since we do not know whether the state is ω1 or ω2 , we have the measurement MC(Ω) (O, S[∗] ).
Thus, out situation is• a measured value “w” is obtained by the measurement MC(Ω) (O, S[∗] ).
Then, we conclude, by Fisher’s maximum likelihood method, that• the urn behind the curtain is U1 .
That is because[F ({w})](ω1 ) = 0.8 = max{[F ({w})](ω1 ), [F ({w})](ω2 )}.
¥Example 5.9. [Urn problem]. Let Uj , j = 1, 2, 3, be urns that contain suﬃciently manycolored balls as follows:

urn U1
urn U2
urn U3

blue balls
60%
40%
20%

green balls
20%
20%
20%

red balls
10%
30%
40%

yellow balls
10%
10%
20%
(5.17)

Put U = {U1 , U2 , U3 }. We consider the state space Ω¡≡ {ω1 , ω2 , ω3 }¢with thediscrete topology, which is identiﬁed with U, that is, U ∋ Uj ↔ ωj ∈ Ω ≈ Mp+1 (Ω).7

7

U1 ≈ ω1U2 ≈ ω2

U3 ≈ ω3B B B G R

B B G R R

B G R R Y

B B B G Y

B B G R Y

B G R R Y

Strictly speaking, we must consider the identiﬁcation as (5.15).

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Deﬁne the observable O ≡ (X = {b, g, r, y}, P(X), F(·) ) in C(Ω) by the usual way. Thatis,
F{b} (ω1 ) = 6/10

F{g} (ω1 ) = 2/10

F{r} (ω1 ) = 1/10

F{y} (ω1 ) = 1/10

F{b} (ω2 ) = 4/10

F{g} (ω2 ) = 2/10

F{r} (ω2 ) = 3/10

F{y} (ω2 ) = 1/10

F{b} (ω3 ) = 2/10

F{g} (ω3 ) = 2/10

F{r} (ω3 ) = 4/10

F{y} (ω3 ) = 2/10.

(5.18)

Then we have the measurement MC(Ω) (O, S[∗] ).
[I] Now we consider the measurement MC(Ω) (O, S[∗] ). And assume that we get the measured value ‘b’ by the measurement MC(Ω) (O, S[∗] ). Then Fisher’s maximum likelihood
method (i.e., Corollary 5.6) says that there is a reason to infer that[∗] = ω1
since
F{b} (ω1 ) = 0.6 = max F{b} (ω) = max{0.6, 0.4, 0.2}.
ω∈ΩThat is, the unknown urn [∗] is U1 .
[II] Also, consider the (iterated) measurement MC(Ω) ( ×k=1 O ≡ (X 2 , P(X 2 ), ×k=1 F ),2

2

S[∗] ) where ( ×k=1 F )Ξ × Ξ (ω) = FΞ1 (ω) · FΞ2 (ω). Also, assume that21

2

• the measured value (b, r) is obtained by the iterated measurement MC(Ω) ( ×k=1 O, S[∗] ).
2

Applying Fisher’s method (= Corollary 5.6), we get the conclusion as follows: Put
E(ω) = F{b} (ω)F{r} (ω).
Clearly it holds that E(ω1 ) = 6 · 1/102 = 0.06, E(ω2 ) = 4 · 3/102 = 0.12 and E(ω3 ) =2 · 4/102 = 0.08. Therefore, there is a very reason to think that [ ∗ ] = δω2 , that is, theunknown urn [∗] is U2 .
[III; Remark (moment method)]. Here, let us consider the above [II] by the moment
method (cf. Deﬁnition 2.27). Deﬁne the distance ∆ on Mm+1 (X) such that:∆(ν1 , ν2 ) =X|ν1 ({x}) − ν2 ({x})|x∈X≡{b,g,r,y}=|ν1 ({b}) − ν2 ({b})| + |ν1 ({g}) − ν2 ({g})| + |ν1 ({r}) − ν2 ({r})| + |ν1 ({y}) − ν2 ({y})|.

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Note that M(Ω) δω1 , F{b} C(Ω) = δω1 (F{b} ) = F{b} (ω1 ) = 6/10, and similarly (cf. (5.18)),δω1 (F{b} ) = 6/10

δω1 (F{g} ) = 2/10

δω1 (F{r} ) = 1/10δω1 (F{y} ) = 1/10

δω2 (F{b} ) = 4/10

δω2 (F{g} ) = 2/10δω2 (F{r} ) = 3/10

δω2 (F{y} ) = 1/10

δω3 (F{b} ) = 2/10

δω3 (F{g} ) = 2/10

δω3 (F{r} ) = 4/10

δω3 (F{y} ) = 2/10.

Since the measured value (b, r) is obtained, we have the sample space (X, 2X , ν) such that
ν({b}) = 1/2,

ν({g}) = 0,

ν({r}) = 1/2,

ν({y}) = 0.

Then, we see that∆(δω1 (F{·} ), ν) = |6/10 − 1/2| + |2/10 − 0| + |1/10 − 1/2| + |1/10 − 0| = 8/10∆(δω2 (F{·} ), ν) = |4/10 − 1/2| + |2/10 − 0| + |3/10 − 1/2| + |1/10 − 0| = 6/10∆(δω3 (F{·} ), ν) = |2/10 − 1/2| + |2/10 − 0| + |4/10 − 1/2| + |2/10 − 0| = 8/10.
Thus, the moment method says that the unknown urn [∗] is U2 .
¥Example 5.10. [At a gun shop, [44]]. Let G ≡ {G1 , ..., G50 } be a set of guns in a gun
shop. Assume that


 80%
70%
the percentage of “hits of a gun Gj ” =10%

Gun Gj0if
if
if

1 ≤ j ≤ 30,
31 ≤ j ≤ 40,
41 ≤ j ≤ 50.

(5.19)Mark

Assume the following situation (i)+(ii):(i) Some one picks up a certain gun Gj0 from G. He does not know the information
concerning the j0 .
(ii) He shoots the gun Gj0 three times. First and second he hits the mark, and third hemisses the mark.

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Our present problem is to formulate the measurement (i)+(ii).

The above example is solved in what follows. Let Ω be a state space, which is identiﬁed
with the set G. That is, we have the identiﬁcation: G ∋ Gj ↔ ωj ∈ Ω. Deﬁne theobservable O ≡ (X = {0, 1}, P(X), F(·) ) in C(Ω) such that: 0.8 if 1 ≤ j ≤ 30,
0.7 if 31 ≤ j ≤ 40,
F{1} (ωj ) =
0.1 if 41 ≤ j ≤ 50

(5.20)and F{0} (ωj ) = 1 − F{1} (ωj ). Of course we think that(♯) “hit the mark by a gun Gj0 ” ⇔ “get the measured value 1 by the measurement
MC(Ω) (O, S[δωj ] )”0

¡ 3Here, consider the (three times) iterated measurement MC(Ω) ×k=1 O = (X 3 , P(X 3 ),¢×3k=1 F ), S[δωj ] in C(Ω) such that:03( × F )Ξ × Ξ × Ξ (ω) = FΞ1 (ω)FΞ2 (ω)FΞ3 (ω)
k=1

1

2

3

(∀Ξ1 × Ξ2 × Ξ3 ∈ P(X 3 ), ∀ω ∈ Ω).
(5.21)

Clearly, the above statement (ii) implies that the measured value (1, 1, 0) is obtained by
¡ 3
¢ ¡
¢
MC(Ω) ×k=1 O, S[∗] . The observer does not know that [∗] = δωj0 .
By a simple
calculation, we see 0.1280.147
F{1} (ωj )F{1} (ωj )F{0} (ωj ) =

0.009

if
if
if

1 ≤ j ≤ 30,
31 ≤ j ≤ 40,
41 ≤ j ≤ 50.

(5.22)

Therefore, by Fisher’s method (= Corollary 5.6), there is a very reason to consider that31 ≤ j0 ≤ 40.
¥
Example 5.11. [(i): Gaussian observable]. Consider a commutative C ∗ -algebra C0 (R).
σAnd deﬁne the Gaussian observable Oσ2 ≡ (R, Bbd
R , F(·) ) in C0 (R) such that:Z11σ2∀µ ∈ R).
(∀Ξ ∈ Bbd
exp[ − 2 (x − µ)2 ]dx
FΞ (µ) = √
R,
2σ
2πσ Ξ
2

3

(5.23)
2σ ,3Further, consider the product observable × k=1 O (or in short, O3σ2 ) ≡ (R3 , Bbd
R3 , F(·) ) in

C0 (R) such that:

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2

2

2

2

,3
FΞσ1 ×Ξ(µ) = FΞσ1 (µ) · FΞσ2 (µ) · FΞσ3 (µ)
2 ×Ξ3
Z
1
(x1 − µ)2 + (x2 − µ)2 + (x3 − µ)2= √exp[ −]dx1 dx2 dx32σ 2( 2πσ)3 Ξ1 ×Ξ2 ×Ξ3(∀Ξ = k ∈ Bbd∀µ ∈ R).
(5.24)R , k = 1, 2, 3,Here consider the measurement MC0 (R) (O3σ2 , S[∗] ). And assume that• the measured value (x01 , x02 , x03 ) ( ∈ R3 ) is obtained by the MC0 (R) (O3σ2 , S[∗] ).
Then, Fisher’s method (=Corollary 5.6) and Remark 5.4 say that there is a reason tothink that the unknown state [ ∗ ] = µ0 , where1(x0 − µ0 )2 + (x02 − µ0 )2 + (x03 − µ0 )2
exp[ − 1]2σ 2( 2πσ)3h1(x0 − µ)2 + (x02 − µ)2 + (x03 − µ)2 i= max √
exp[ − 1] ,µ∈R ( 2πσ)32σ 2√(5.25)

which is equivalent to(x01 − µ0 )2 + (x02 − µ0 )2 + (x03 − µ0 )2= min[(x01 − µ)2 + (x02 − µ)2 + (x03 − µ)2 ]µ∈R(5.26)and moreover, equivalently,µ0 = (x01 + x02 + x03 )/3.

(5.27)[(ii): Gaussian observable]. Consider a commutative C ∗ -algebra C([0, 100]), where [0, 100]σ
≡ {µ ∈ R | 0 ≤ µ ≤ 100}. And deﬁne the Gaussian observable Oσ2 ≡ (R, Bbd
R , F(·) ) in2

C([0, 100]) such that:Z11σ2FΞ (µ) = √exp[ − 2 (x − µ)2 ]dx
2σ
2πσ Ξ

(∀Ξ ∈ Bbd
R,∀µ ∈ [0, 100]).

(5.28)

2σ ,3Further, consider the product observable O3σ2 ≡ (R3 , Bbd
R3 , F(·) ) in C0 ([0, 100]) such that:2,3
FΞσ1 ×Ξ(µ)
2 ×Ξ3Z(x1 − µ)2 + (x2 − µ)2 + (x3 − µ)21
exp[ −]dx1 dx2 dx3= √2σ 2( 2πσ)3 Ξ1 ×Ξ2 ×Ξ3(∀Ξ = k ∈ Bbd∀µ ∈ [0, 100]).
(5.29)R , k = 1, 2, 3,Here consider the measurement MC([0,100]) (O3σ2 , S[∗] ). And assume that

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/• the measured value (x01 , x02 , x03 ) ( ∈ R3 ) is obtained by the MC([0,100]) (O3σ2 , S[∗] )Then, Fisher’s method and Remark 5.4 say that there is a reason to think that theunknown state [ ∗ ] = µ0 , where [ ∗ ] = µ0 , where1(x0 − µ0 )2 + (x02 − µ0 )2 + (x03 − µ0 )2√
exp[ − 1]2σ 2( 2πσ)3h1(x0 − µ)2 + (x02 − µ)2 + (x03 − µ)2 i√= max
exp[ − 1]µ∈[0,100] ( 2πσ)32σ 2(5.30)

which is equivalent to(x01 − µ0 )2 + (x02 − µ0 )2 + (x03 − µ0 )2= min [(x01 − µ)2 + (x02 − µ)2 + (x03 − µ)2 ](5.31)µ∈[0,100]and moreover, equivalently,

0





(x01 + x02 + x03 )/3
µ0 =





100

if x01 + x02 + x03 < 0
if 0 ≤ x01 + x02 + x03 ≤ 100(5.32)

if x01 + x02 + x03 > 100.
¥

5.2.2

Monty Hall problem in PMT

Problem 5.12. [Monty Hall problem, cf.[33]].
The Monty Hall problem is as follows:(P) Suppose you are on a game show, and you are given the choice of three doors (i.e.,“number 1”, “number 2”, “number 3” ). Behind one door is a car, behind theothers, goats.
You pick a door, say number 1, and the host, who knows what’s behind the doors,
opens another door, say “number 3”, which has a goat. He says to you, “Do you
want to pick door number 2?” Is it to your advantage to switch your choice of doors?

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Door
Number 1

?

?

Door
Number 2

Door
Number 3[Answer]. Put Ω = {ω1 , ω2 , ω3 }, whereω1 · · · · · · the state that the car is behind the door number 1
ω2 · · · · · · the state that the car is behind the door number 2
ω3 · · · · · · the state that the car is behind the door number 3.
Deﬁne the observable O ≡ ({1, 2, 3}, 2{1,2,3} , F ) in C(Ω) such that
[F ({1})](ω1 ) = 0.0,

[F ({2})](ω1 ) = 0.5,

[F ({3})](ω1 ) = 0.5, 8[F ({1})](ω2 ) = 0.0,

[F ({2})](ω2 ) = 0.0,

[F ({3})](ω2 ) = 1.0,

[F ({1})](ω3 ) = 0.0,

[F ({2})](ω3 ) = 1.0,

[F ({3})](ω3 ) = 0.0.

Thus we have a measurement MC(Ω) (O, S[∗] ). Here, note that(1) :“measured value 1 is obtained” ⇐⇒ The host says “Door (number 1) has a goat”,(2) :“measured value 2 is obtained” ⇐⇒ The host says “Door (number 2) has a goat”,(3) :“measured value 3 is obtained” ⇐⇒ The host says “Door (number 3) has a goat”.
The host said “Door (number 3) has a goat”. This implies that you get the measured
value “3” by the measurement MC(Ω) (O, S[∗] ). Therefore, Fisher’s maximum likelihood
method says that you should pick door number 2. That is because we see that
[F ({3})](ω2 ) = 1.0 = max{0.5, 1.0, 0.0}
= max{[F ({3})](ω1 ), [F ({3})](ω2 ), [F ({3})](ω3 )},and thus, [∗] = δω2 . However, this is not all of the Monty Hall problem. See Remark 5.13,
Problem 8.8 and Problem 11.13 later.
¥
8

Strictly speaking, F ({1})(ω1 ) = 0.5 and F ({2})(ω1 ) = 0.5 should be assumed in the problem (P).

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107

Remark 5.13. [Monty Hall problem by the moment method (cf. Deﬁnition 2.27)].
Here, consider Problem 5.12 by the moment method. Since you get measured value 3,
you get the sample space ({1, 2, 3}, 2{1,2,3} , νs ) such that νs ({1}) = 0, νs ({2}) = 0 andνs ({3}) = 1. For example deﬁne the distance ∆ such that: for any ν1 , ν2 ∈ Mm
+1 ({1, 2, 3}),∆(ν1 , ν2 ) = |ν1 ({1}) − ν2 ({1})| + |ν1 ({2}) − ν2 ({2})| + |ν1 ({3}) − ν2 ({3})|.
Then, we see∆(νs , [F ( · )](ω1 )) = |0 − 0| + |0 − 0.5| + |1 − 0.5| = 1,∆(νs , [F ( · )](ω2 )) = |0 − 0| + |0 − 0| + |1 − 1| = 0
and∆(νs , [F ( · )](ω3 )) = |0 − 0| + |0 − 1| + |1 − 0| = 2.
Thus, we can, by the moment method, infer that ω2 is most possible, that is, the car isbehind the door number 2.
¥

5.3

Inference interval

Let O( ≡ (X, F, F )) be an observable formulated in a C ∗ -algebra A. Assume that
X has a metric dX . And assume that the state space Sp (A∗ ) has the metric dS , which
induces the weak∗ topology σ(A∗ , A). Let E : X → Sp (A∗ ) be a continuous map, whichis called “estimator”. Let γ be a real number such that 0 ≪ γ < 1, for example, γ = 0.95.
For any ρp ( ∈ Sp (A∗ )), deﬁne the positive number ηργp ( > 0) such that:ηργp = inf{η > 0 : A∗Dp

ρ , F (E−1E
(B(ρ ; η)) A ≥ γ}p(5.33)

where B(ρp ; η) = {ρp1 ( ∈ Sp (A∗ )) : dS (ρp1 , ρp ) ≤ η}. For any x ( ∈ X), put
Dxγ = {ρp (∈ Sp (A∗ )) : dS (E(x), ρp ) ≤ ηργp }.
The Dxγ is called the (γ)-inference interval of the measured value x.
Note that,

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(A) for any ρp0 ( ∈ Sp (A∗ )), the probability, that the measured value x obtained by the¡¢measurement MA O ≡ (X, F, F ), S[ρp0 ] satisﬁes the following condition (♭), is larger
than γ (e.g., γ = 0.95).
(♭)

E(x) ∈ B(ρp0 ; ηργp )or equivalently0d(E(x), ρp0 ) ≤ ηργp .
0Assume that¡¢(B) we get a measured value x0 by the measurement MA O ≡ (X, F, F ), S[ρp0 ] .
Then, we see the following equivalences:(♭)⇐⇒

dS (E(x0 ), ρp0 ) ≤ ηργp0⇐⇒

Dxγ0 ∋ ρp0 .

Sp (A∗ )X

E(x0 )

ρp

Dxγ0

x0

Summing the above argument, we have the following theorem.
Theorem 5.14. [Inference interval]. Let O ≡ (X, F, F ) be an observable in A. Let ρp0¢¡
be any ﬁxed state, i.e., ρp0 ∈ Sp (A∗ ), Consider a measurement MA O ≡ (X, F, F ), S[ρp0 ] .
Let E : X → Sp (A∗ ) be an estimator. Let γ be such as 0 ≪ γ < 1 (e.g., γ = 0.95). For
any x( ∈ X), deﬁne Dxγ as in (5.34). Then, we see,(♯) the probability that the measured value x0 ( ∈ X) obtained by the measurement¡¢
MA O ≡ (X, F, F ), S[ρp0 ] satisﬁes the condition that
Dxγ0 ∋ ρp0 ,(5.35)is larger than γ.
¥

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109

Example 5.15. [The urn problem]. Put Ω = [0, 1], i.e., the closed interval in R. We
assume that each ω ( ∈ Ω ≡ [0, 1]) represents an urn that contains a lot of red balls andwhite balls such that:the number of white balls in the urn ω≈ωthe total number of red and white balls in the urn ω(∀ω ∈ [0, 1] ≡ Ω). (5.36)

Deﬁne the observable O = (X ≡ {r, w}, 2{r,w} , F ) in C(Ω) such that whereF (∅)(ω) = 0,

F ({r})(ω) = ω,F ({w})(ω) = 1 − ω,F ({r, w})(ω) = 1(∀ω ∈ [0, 1] ≡ Ω).

(5.37)

Here, consider the following measurement Mω :Mω := “Pick out one ball from the urn ω, and recognize the color of the ball”(5.38)

That is, we considerMω = MC(Ω) (O, S[δω ] ).

(5.39)

Moreover we deﬁne the product observable ON ≡ (X N , P(X N ), F N ), such that:[F N (Ξ1 × Ξ2 × · · · × ΞN −1 × ΞN )](ω)=[F (Ξ1 )](ω) · [F (Ξ2 )](ω) · · · [F (ΞN −1 )](ω) · [F (ΞN )](ω)(∀ω ∈ Ω ≡ [0, 1],∀Ξ1 , Ξ2 , · · · , ΞN ⊆ X ≡ {r, w}).

(5.40)As mentioned in Deﬁnition 2.27, we think that“take a measurement Mω N times” ⇔ “take a measurement MC(Ω) (ON , S[δω ] )”(5.41)
Deﬁne the estimator E : X N ( ≡ {r, w}N ) → Ω( ≡ [0, 1])♯[{n ∈ {1, 2, · · · , N } | xn = r}]NN(∀x = (x1 , x2 , · · · , xN −1 , xN ) ∈ X ≡ {r, w}N ).

E(x1 , x2 , · · · , xN −1 , xN ) =(5.42)For each ω( ∈ [0, 1] ≡ Ω), deﬁne the positive number ηωγ such that:
ηωγ

¯
o
n
¯ N
= inf η > 0 ¯ [F ({(x1 , x2 , · · · , xN ) | ω − η ≤ E(x1 , x2 , · · · , xN ) ≤ ω + η})](ω) > 0.95=

inf

[F N ({(x1 ,x2 ,··· ,xN )|ω−η≤E(x1 ,x2 ,··· ,xN )≤ω+η})](ω)>0.95

η.

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PutDxγ = {ω( ∈ Ω) : |E(x) − ω| ≤ ηωγ }.

(5.44)For example, assume that N is suﬃciently large and γ = 0.95. Then we see, by (2.58),
thatrηω0.95 ≈ 1.96

ω(1 − ω)N

and
Dx0.95 = [E(x) − η− , E(x) + η+ ](5.45)0.95
η− = ηE(x)−η
,−(5.46)

where0.95
η+ = ηE(x)+η
.
+

Under the assumption that N is suﬃciently large, we can consider thatr
0.95
η− ≈ η+ ≈ ηE(x)≈ 1.96E(x)(1 − E(x)).
N

Then we can conclude that• for any urn ω( ∈ Ω ≡ [0, 1])), the probability, that the measured value x = (x1 , x2 , · · · ,¡¢xN ) obtained by the measurement MA ON , S[δω ] satisﬁes the following condition(♯), is larger than γ (e.g., γ = 0.95).
(♯)qq
E(x)(1−E(x))E(x) − 1.96 E(x)(1−E(x))
≤
ω
≤
E(x)
+
1.96
.
N
N

where E is deﬁned by (5.42).
¥

5.4

Testing statistical hypothesis

Now we study “testing statistical hypothesis”, that is, answer the following question.

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Problem 5.16. [Testing statistical hypothesis]. Consider a measurement MC(Ω) (O ≡(X, F, F ), S[∗] ) formulated in C(Ω). Let E : X → Ω be Fisher’s estimator. Assume thefollowing hypothesis:(H) the unknown state [∗] belongs to a closed set CH ( ⊆ Ω).
And further assume that we see the following fact:(F) a measured value x0 ( ∈ X) is obtained by measurement MC(Ω) (O ≡ (X, F, F ), S[∗] ).
Here, our present purpose is to propose an algorithm that decides whether the above hypothesis (H) can be denied by the fact (F). This algorithm is called “the testing statistical
hypothesis”.
In the above problem, it is usually expected that the hypothesis (H) is not true. In
this sense, the above (H) is called the null hypothesis.
Now we provide two answers (i.e., Answer 1 and Answer 2). Answer 1 (likelihood ratiotest) is, of course, well-known and authorized. Also, in order to solve the question: “Isthere another answer?”, we add Answer 2 after Answer 1.
Answer 1. [Likelihood ratio test]. Consider a measurement MC(Ω) (O ≡ (X, F, F ), S[∗] )formulated in C(Ω). Let E : X → Ω be Fisher’s estimator, i.e., it is deﬁned byE(x) = lim ωn
Ξn →{x}(∀x ∈ X),

where ωn ( ∈ Ω) is chosen such that it satisﬁes[F (Ξn )](ωn )= 1.
maxω∈Ω [F (Ξn )](ω)³´
For the exact argument, see Remark 5.4 (Radon-Nikodým derivative).

Assumeboth (H) and (F) in Problem 5.16. Consider a real number α such that 0 < α ≪ 1(e.g. α = 0.05, which may be called a signiﬁcance level. Let ω be in Ω. Then, by
Axiom 1, we have a sample probability measure Pω on X (of the measurement MC(Ω) (O≡ (X, F, F ), S[δω ] )) such that:Pω (Ξ) = [F (Ξ)](ω) (∀Ξ ∈ F).

(5.47)

Here deﬁne the function ΛCH : X → [0, 1] such that:supω∈CH Pω (Ξ)Ξ→{x} supω∈Ω Pω (Ξ)ΛCH (x) = lim(∀x ∈ X).

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Also, for any ϵ (0 < ϵ ≤ 1), deﬁne [D]ϵCH ( ∈ F) such that:[D]ϵCH = {x ∈ X | ΛCH (x) < ϵ}.

(5.49)

1
ΛCH (x)ϵ0X[D]ϵCHThus we can deﬁne ϵ0.05
max such that:ϵϵ0.05
max = sup{ϵ | sup Pω0 ([D]CH ) ≤ 0.05}.
ω0 ∈CH(5.50)

Now we can conclude thatAnswer 1
ϵ0.05
, then the hypothesis (H) can be denied if x0 ∈ [D]Cmax
H

ϵ0.05
 if x ∈max
0 / [D]CH , then the hypothesis (H) can not be denied(5.51)

Next we shall propose “Answer 2”. Before this, we must prepare the following wellknown lemma.
Lemma 5.17. [Neyman-Pearson theorem, α-inﬂuential domain of ν1 for ν2 , cf. [59]]. Let(X, F) be a measurable space. Let ν1 and ν2 be probability measures on X. Deﬁne thedν1Radon-Nikodým derivative dν
: X → [0, ∞) such that:2dν1
ν1 (Ξ)(x) = lim
Ξ→x ν2 (Ξ)
dν2(x ∈ X).

(5.52)Put[D](ϵ,dν1
dν1) = {x ∈ X |(x) < ϵ},dν2dν2(0 ≤ ϵ ≤ ∞).

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Thus we can deﬁne ϵ̆0.05
max such that:³´dν1
α=0.05ϵ̆0.05≡ϵ̆= sup{ϵ | ν1 ([D](ϵ,)) ≤ 0.05}.
max
max
dν2

113

(5.54)

Now we have the[D](ϵ̆0.05
max ,

dν1
),
dν2

(5.55)

which is called “the 0.05-inﬂuential domain of ν1 for ν2 ”.
¥
Answer 2. [A test using Neyman-Pearson theorem]. Consider a measurement MC(Ω) (O≡ (X, F, F ), S[∗] ) formulated in C(Ω). Let E : X → Ω be Fisher’s estimator. Assume
both (H) and (F) in Problem 5.16. Consider a real number α such that 0 < α ≪ 1(e.g. α = 0.05 which may be also called a signiﬁcance level. Let ω be in CH . Consider
a measurement MC(Ω) (O ≡ (X, F, F ), S[δω ] ) . Let x be in X. Then, we have two sample
probability measures Pω and PE(x) on X such that:νω (Ξ) = Pω (Ξ) = [F (Ξ)](ω)(∀Ξ ∈ F)νE(x) = PE(x) (Ξ) = [F (Ξ)](E(x))(∀Ξ ∈ F).

and(5.56)

Thus, we have “the 0.05-inﬂuential domain of ν1 for ν2 ” such that:[D](ϵ̆0.05
max ,

dνω
).
dνE(x)(5.57)Putϵ̆0.05[D]Cmax= ∩ω∈CH [D](ϵ̆0.05
max ,
H ,x

dνω
).
dνE(x)(5.58)

Lastly, we put
ϵ̆0.05ϵ̆0.05[D]Cmax= {x ∈ X | x ∈ [D]Cmax
}.
H
H ,x

(5.59)

Now we can conclude thatAnswer 2ϵ̆0.05, then the hypothesis (H) can be denied if x0 ∈ [D]CmaxH
ϵ̆0.05
 if x ∈max
0 / [D]CH , then the hypothesis (H) can not be denied

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Remark 5.18. [Answers 1 and 2]. We believe that the above two answers 1 and 2
are proper though the meanings of “signiﬁcant level” is diﬀerent in each answer (cf. [II;CH = [0, ∞)] in Examples 5.16 and 5.17). We do not know whether there is anotherproper answer.
¥
Example 5.19. [Likelihood ratio test for the Gaussian observable]. Put Ω = R, A =σ
C0 (Ω), Oσ2 ≡ (R, Bbd
R , F(·) ) in C0 (Ω) such that:Z1(x − ω)2σ2FΞ (ω) = √exp[ −]du(∀Ξ ∈ Bbd
R,
2
2σ
2πσ Ξ
2

∀ω ∈ Ω = R).

(5.61)σσ
And thus. consider the product observable O2σ ≡ (R2 , Bbd
R2 , F(·) ) × F(·) ) in C0 (Ω). That
2

is,
2
(FΞσ1

×

2
FΞσ2 )(ω) =1√( 2πσ)2ZZ
Ξ1 ×Ξ2(∀Ξk ∈ Bbd
R (k = 1, 2),

exp[ −

2(x1 − ω)2 + (x2 − ω)2]dx1 dx2
2σ 2∀ω ∈ Ω = R).

(5.62)[Case(I): Two sided test, i.e., CH = {ω0 }]. Assume that CH = {ω0 }, ω0 ∈ Ω = R. Then,Λ{ω0 } (x1 , x2 ) =supω∈{ω0 } Pω (Ξ1 × Ξ2 )
Ξ1 ×Ξ2 →{(x1 ,x2 )} supω∈Ω Pω (Ξ1 × Ξ2 )lim

2 −ω0 )
exp[ − (x1 −ω0 ) 2σ+(x
]
2
2

=

2

2 −(x1 +x2 )/2)
exp[ − (x1 −(x1 +x2 )/2) 2σ+(x]22[(x1 + x2 )/2 − ω0 ]2[(x1 + x2 ) − 2ω0 ]
√
]
=
exp[
−
= exp[ −]4σ 22(σ/ 2)2(∀(x1 , x2 ) ∈ R2 ).
2

2(5.63)

Also, for any ϵ( > 0), deﬁne [D]ϵ{ω0 } ( ∈ F) such that:[D]ϵ{ω0 } = {(x1 , x2 ) ∈ R2 | Λ{ω0 } (x1 , x2 ) < ϵ}.

(5.64)

Thus we can deﬁne ϵ0.05
max such that:ϵϵ0.05
max = sup{ϵ | sup Pω ([D]{ω0 } ) ≤ 0.05}.
ω∈{ω0 }

Now we can conclude that
ϵ0.05

max
[D]{ω
0}

√
={(x1 , x2 ) ∈ R2 | (x1 + x2 )/2 ≤ ω0 − 1.96σ/ 2}[√{(x1 , x2 ) ∈ R2 | (x1 + x2 )/2 ≥ ω0 + 1.96σ/ 2}=“Slash part in the following ﬁgure”

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6x2√
2(ω0 + 1.96σ/ 2)√2(ω0 − 1.96σ/ 2)ω0√2(ω0 − 1.96σ/ 2)√2(ω0 + 1.96σ/ 2)

x1[Case(II): One sided test, i.e., CH = [ω0 , ∞)]. Assume that CH = [ω0 , ∞), ω0 ∈ Ω = R.
Then,supω∈[ω0 ,∞) Pω (Ξ1 × Ξ2 )
Ξ1 ×Ξ2 →{(x1 ,x2 )}supω∈Ω Pω (Ξ1 × Ξ2 )½
22 )−2ω0 ]2
exp[ − [(x1 +x4σ
] ( x1 +x< ω0 )
22=
1( otherwise )Λ[0,∞) (x1 , x2 ) =lim(5.66)

Also, for any ϵ( > 0), deﬁne [D]ϵ[ω0 ,∞) ( ∈ F) such that:[D]ϵ[0,∞) = {(x1 , x2 ) ∈ R2 | Λ[0,∞) (x1 , x2 ) ≤ ϵ}px1 + x2− ω0 < 4σ 2 log ϵ}.
= {(x1 , x2 ) ∈ R2 |2
0.05
Thus we can deﬁne ϵmax such that:
ϵ0.05
max = sup{ϵ |sup Pω0 ([D]ϵ[0,∞) ) ≤ 0.05}.

ω0 ∈[0,∞)

Therefore, we can conclude that
ϵ0.05

max
[D][0,∞)√={(x1 , x2 ) ∈ R2 | (x1 + x2 )/2 ≤ ω0 − 1.65σ/ 2}.
=“Slash part in the following ﬁgure”

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(cf. (2.58)).

(5.67)(5.68)

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6

x2

√
2(ω0 − 1.65σ/ 2)ω0√2(ω0 − 1.65σ/ 2)

x1¥
Example 5.20. [The test using Neyman-Pearson theorem for the Gaussian observable].
σ
2
2
bd
σ
σ
Put Ω = R, A = C0 (Ω), Oσ2 ≡ (R, Bbd
R , F(·) ) and Oσ 2 ≡ (R , BR2 , F(·) × F(·) ) in C0 (Ω)
222

are as in the above.
[Case(I): Two sided test, i.e., CH = {ω0 }]. Assume that CH = {ω0 }, ω0 ∈ Ω = R. Then,ν1ω0 (Ξ1 × Ξ2 ) = Pω0 (Ξ1 × Ξ2 ) = [F (Ξ1 × Ξ2 )](ω0 )(∀Ξ1 × Ξ2 ∈ Bbd
R2 )

and
E(x0 )

ν2= PE(x0 ) (Ξ1 × Ξ2 ) = [F (Ξ1 × Ξ2 )](E(x0 ))(∀Ξ1 × Ξ2 ∈ Bbd
R2 ). (5.69)

Thus, we have “the 0.05-inﬂuential domain of ν1 for ν2 ” such that:√½{(x1 , x2 ) | (x1 + x2 )/2 ≤ ω0 − 1.65σ/√2} (E(x0 ) < ω0 )0.05
[D](ϵ̆max , ϕν ω0 /ν E((x1 ,x2 )) ) =1
2{(x1 , x2 ) | (x1 + x2 )/2 ≥ ω0 + 1.65σ/ 2} (E(x0 ) > ω0 ).
Put
ϵ̆0.05

0.05
2
max
[D]{ω
= ∩ω0 ∈{ω0 } [D](ϵ̆0.05
max , ϕν ω0 /ν E((x1 ,x2 )) ) = [D](ϵ̆max , ϕν ω0 /ν E((x1 ,x2 )) ) (∀(x1 , x2 ) ∈ R ).
0 },(x1 ,x2 )
1

2

1

2

(5.70)

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117

Therefore, we can conclude thatϵ̆0.05ϵ̆0.05

max
max
[D]{ω
={(x1 , x2 ) ∈ R2 | (x1 , x2 ) ∈ [D]{ω}0}0 },(x1 ,x2 )√={(x1 , x2 ) | (x1 + x2 )/2 ≤ ω0 − 1.65σ/ 2}[√{(x1 , x2 ) | (x1 + x2 )/2 ≥ ω0 + 1.65σ/ 2}.

[Case(II): One sided test, i.e., CH = [ω0 , ∞)]. Assume that CH = [ω0 , ∞), ω0 ∈ Ω = R.
Then,
ν1ω0 (Ξ1 × Ξ2 ) = Pω0 (Ξ1 × Ξ2 ) = [F (Ξ1 × Ξ2 )](ω0 )(∀Ξ1 × Ξ2 ∈ Bbd
R2 )

and
E(x0 )

ν2(Ξ1 × Ξ2 ) = PE(x0 ) (Ξ1 × Ξ2 ) = [F (Ξ1 × Ξ2 )](E(x0 ))(∀Ξ1 × Ξ2 ∈ Bbd
R2 ).
(5.71)

Thus, we have “the 0.05-inﬂuential domain of ν1 for ν2 ” such that:√½{(x1 , x2 ) | (x1 + x2 )/2 ≤ ω0 − 1.65σ/√2} (E(x0 ) < ω0 )0.05
[D](ϵ̆max , ϕν ω0 /ν E((x1 ,x2 )) ) =2
1{(x1 , x2 ) | (x1 + x2 )/2 ≥ ω0 + 1.65σ/ 2} (E(x0 ) > ω0 ).
Put
ϵ̆0.05max[D][0,∞),(x= ∩ω0 ∈[0,∞) [D](ϵ̆0.05
max , ϕν ω0 /ν E(x) )1 ,x2 )
1

2(∀(x1 , x2 ) ∈ R2 ).

(5.72)

Therefore, we can conclude thatϵ̆0.05ϵ̆0.05max
max[D][0,∞)={(x1 , x2 ) ∈ R2 | (x1 , x2 ) ∈ [D][0,∞),(x}1 ,x2 )√={(x1 , x2 ) | (x1 + x2 )/2 ≤ ω0 − 1.65σ/ 2}.

¥

5.5

Measurement error model in PMT

Although we have several kinds of measurement error models in statistics (cf. Fuller[25], Cheng, etc. [16]), the following may be the simplest one (i.e., with normal distributions (= Gaussian distributions)): yen = θ0 + θ1 xn + en ,x
en = xn + un

2
2
)], 9
, σuu
(en , un ) ∼ NI[average(0, 0), variance(σee(n = 1, 2, ..., N ),Sample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

(5.73)

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which, of course, corresponds to the conventional statistics (i.e., the measurement equation in the dynamical system theory (1.2)). The ﬁrst equation is a classical regression
speciﬁcation, but the true explanatory variable xn is not observed directly. The observed
measure of xn , denoted by x
en , may be obtained by a certain measurement. Our present
concern is how to infer the unknown parameters θ0 and θ1 from the measured value{(exn , yen )}N
n=1 . Precisely speaking, the purpose of this section is to study this problem ingeneral situations (i.e., without the assumption of normal distributions).
Put A0 ≡ C(Ω0 ) and A1 ≡ C(Ω1 ). Let Θ be a compact space, which may be called
an index state space (or parameter space). Consider a parameterized continuous map ψ θ :Ω0 → Ω1 , θ ∈ Θ, which induces the parameterized homomorphism Ψθ : C(Ω1 ) → C(Ω0 )such that (cf. (3.14))(Ψθ f1 )(ω) = f1 (ψ θ (ω)) (∀f1 ∈ C(Ω1 ), ∀ω ∈ Ω0 ).
Consider observables O0 ≡ (X, F, F ) in C(Ω0 ) and O1 ≡ (Y, G, G) in C(Ω1 ). And recall
that Ψθ O1 can be identiﬁed with the observable in C(Ω0 ) (cf. Remark 3.6 (i)). Thus, wee θ = (X ×Y, F ×G, F × Ψθ G) in C(Ω0 ). Thus, we getcan consider the product observable Oe θ , S[δω ] ), (ω ∈ Ω0 ). Consider the N times repeated measurethe measurement MC(Ω0 ) (Oe θ , S[⊗N δ ] ). Here,e θ , S[δω ] ), which is represented by MC(ΩN ) ( NN O
ment of MC(Ω0 ) (On=1n=1 ωn0
NN e θ
N
p
N
N
N
N
N
⊗n=1 δωn = δ(ω1 ,ω2 ,...,ωN ) (∈ M+1 (Ω0 )) and n=1 O = (X × Y , F × GN , N
n=1 (F ×NNΨθ G)) in N
n=1 C(Ω0 ) ≡ C(Ω0 ), that is,N
O[((F × Ψθ G))(Ξ1 × · · · × ΞN × Γ1 × · · · × ΓN )](ω1 , ..., ωN )n=1=[F × Ψθ G(Ξ1 × Γ1 )](ω1 ) · [F × Ψθ G(Ξ2 × Γ2 )](ω2 ) · · · [F × Ψθ G(ΞN × ΓN )](ωN )(∀Ξn ∈ F, ∀Γn ∈ G, ∀(ω1 , ..., ωN ) ∈ ΩN
0 ).

(5.74)

Our present problem is as follows:(♯) Consider the measurement MC(ΩN0 ) (

NN

e θ̄
) where it is assumed that
n=1 O , S[⊗N
n=1 δω̄n ]ω̄1 , ω̄2 , ..., ω̄N and θ̄ (∈ Θ) are unknown. Assume that we know that the measured value (ex1 , ..., xeN , ye1 , ..., yeN ) (∈ X N × Y N ) obtained by the measurement
Q
NN e θ̄O , S[⊗N δ ] ) belongs to N (Ξn × Γn ). Then, infer the unknown
MC(ΩN ) (0

n=1

n=1 ω̄nn=1

ω̄1 , ω̄2 , ..., ω̄N and θ̄ (particularly, θ̄).
9

Independent random variables with normal distributions

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119

That is, for simplicity under the assumption that Ω0 = X, Ω1 = Y , we can illustrate thisproblem (♯) as follows:Ω1ω1 = ψ θ3 (ω0 )Question
From the measured data• (ex5 , ye5 )• (ex3 , ye3 )• (ex4 , ye4 )• (ex1 , ye1 )• (ex2 , ye2 )(ex1 , ..., xe5 , ye1 , ..., ye5 ) (∈ Ω50 × Ω51 ) ,infer the reasonable θ.
Answerω1 = ψ θ2 (ω0 )“Probably θ3 !”ω1 = ψ θ1 (ω0 )Ω0b ≡ (X N × Y N , FN × GN , H)b inThis problem is solved as follows: Deﬁne the observable O
b 1 × · · · × ΞN × Γ1 × · · · × ΓN )](ω1 , ..., ωN , θ) = (5.74). Note
C(ΩN × Θ) such that [H(Ξ0that we have the following identiﬁcation:b S[(⊗N δ )⊗δ ] ) = MC(ΩN ) (MC(ΩN0 ×Θ) (O,θ
n=1 ωn0NOe θ , S[⊗N δ ] ).
O
n=1 ωnn=1b S[(⊗N δ )⊗δ ] ) where it is assumed that we
Consider the measurement MC(ΩN0 ×Θ) (O,θ̄
n=1 ω̄ndo not know ω̄1 , ω̄2 , ..., ω̄N , θ̄. Then, we can, by Fisher’s maximum likelihood method (cf.
Corollary 5.6), infer the unknown state (⊗N
n=1 δω̄n ) ⊗ δθ̄ such that:b 1 × · · · × ΞN × Γ1 × · · · × ΓN )](ω̄1 , ..., ω̄N , θ̄)
[H(Ξ
=

max

(ω1 ,...,ωN ,θ)∈ΩN
0 ×Θ

b 1 × · · · × ΞN × Γ1 × · · · × ΓN )](ω1 , ..., ωN , θ).
[H(Ξ(5.75)

This is the answer to the above problem (♯). It should be noted that the problem (♯) isstated under the very general situations (i.e., Ω0 , Ω1 , X and Y are not necessarily thereal lines R).
In the following example, we apply our result (5.75) to the simple measurement error
model (5.73) with normal distributions.
Example 5.21. [The simple example of measurement error model (the case that θ0 , θ1 ,
ω1 , ..., ωN are unknown)]. Let L be a suﬃciently large number. Put Ω0 = [ − L, L],Ω1 = [ − L2 − L, L2 + L], Θ = [−L, L]2 , and deﬁne the map ψ (θ0 ,θ1 ) : Ω0 → Ω1 such that:ψ (θ1 ,θ2 ) (ω) = θ1 ω + θ0(∀ω ∈ Ω0 , ∀(θ0 , θ1 ) ∈ Θ.

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Also, put (X, F, F ) = (R, BR , Gσ1 ) in C(Ω0 ) and (Y, G, G) = (R, BR , Gσ2 ) in C(Ω1 )e (θ0 ,θ1 ) = (X × Y, F × G, H θ ),(cf. Example 2.17). Thus, we deﬁne the product observable O
where H θ ≡ F × Ψθ G, in C(Ω0 ) such that:ZZ
1(x − ω)2 (y − (θ1 ω + θ0 ))2θ
2√[H (Ξ × Γ)](ω) = ()
exp[ −
−
]dxdy
2σ12
2σ22
2πσ1 σ2Ξ×Γ(∀Ξ ∈ BR , ∀Γ ∈ BR , ∀ω ∈ Ω0 ).
b = (R2N , BR2N , H)b in C(ΩN × Θ) such that:
Thus, we have the observable O0b 1 × · · · × ΞN × Γ1 × · · · × ΓN )](ω1 , ..., ωN , θ0 , θ1 )
[H(Ξ
PN
PN
Z Z
2
2
n=1 (xn −ωn ) − n=1 (yn −(θ1 ωn +θ0 ))
−
1
2
2
2N
2σ1
2σ2
=( √
)
···
e
dx1 dy1 · · · dxN dyN .
2πσ1 σ2ΠN
n=1 (Ξn ×Γn )(5.76)
Assume the conditions in the problem (♯), and further add thatΞϵn = [exn − ϵ, x
en + ϵ],Γϵn = [eyn − ϵ, yen + ϵ] (for suﬃciently small positive ϵ).

Then, our main result (5.75) says thatb ϵ × · · · × Ξϵ × Γϵ × · · · × Γϵ )](ω1 , ..., ωN , θ)
[H(Ξ
1
N
1
N

max

(ω1 ,...,ωN ,θ0 ,θ1 )∈ΩN
0 ×Θ

N
N
£X
x
en ωn 2 X yenθ1 σ1 ωn θ0 2 ¤
⇐⇒
min
( −
) +
( −(
+ ))
(since ϵ is small)
σ1
σ1
σ2
σ2 σ1
σ2
(ω1 ,...,ωN ,θ0 ,θ1 )∈ΩN
0 ×Θ
n=1
n=1
¡
Here, note that the distance between a point ( xeσn1 , yσen2 ) and a line y = θ1σσ2 1 x + σθ02 is equal¢en −θ0 |n −θ1 x
to |ey√. Then, we see
22 2
σ2 +σ1 θ1

PN⇐⇒ min(θ0 ,θ1 )∈Θ( P⇐⇒

yn − θ1 x
en − θ0 )n=1 (e22 2
σ2 + σ1 θ12N(eyn − θ̄1 x
en − θ̄0 ) = 0
Pn=1N2
en σ1 + x
en σ22 − θ̄0 θ̄1 σ12 )(eyn − θ̄1 x
en − θ̄0 ) = 0
n=1 (θ̄1 y(5.77)(←(←∂(5.77) = 0),∂θ0∂(5.77) = 0).
∂θ1(5.78)
Thus, the unknown parameters θ̄1 and θ̄2 are inferred by the solution of this equation(5.78). Note that this is a direct consequence of our main result (5.75).
¥
Example 5.22. [The case that θ0 , θ1 , σ12 , σ22 , ω1 , ..., ωN are unknown]. Assume that θ0 ,
θ1 , σ12 , σ22 , ω1 , ..., ωN are unknown. The log-likelihood isL(θ0 , θ1 , σ12 , σ22 , ω1 , ..., ωN ) = log[(5.76)]
PN
PN
2
2
N log σ12
N log σ22n=1 (xn − ωn )
n=1 (yn − θ0 − θ1 ωn )
=−
−
−
−
.
2
2
2σ12
2σ22

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121

Taking partial derivatives with respect to θ0 , θ1 , σ12 , σ22 and ω1 , ..., ωN , and equating theresults to zero, gives the likelihood equations,
N
X

N
X

(yn − θ0 − θ1 ωn ) = 0,
n=1PN2
n=1 (xn ωn )= σ12 ,

N(xn ωn )2 (yn − θ0 − θ1 ωn )2
−
= 0,
2σ12
2σ22

(yn − θ0 − θ1 ωn )ωn = 0,
n=1PN2
n=1 (yn − θ0 − θ1 ωn )= σ22 ,
N(n = 1, 2, ..., N ).

Thus we can easily solve it as follows:
θ12 =

σ22
Syy
=
,
2
σ1
Sxx

θ0 = ȳ − θ1 x̄,Sxy, 2σ22 = Syy − Sxy θ1 ,
θ1yn − θ0yn − ȳn2ωn = xn += xn + x̄ +,θ1
θ1
2σ12 = Sxx −where
y1 + · · · + yN
x 1 + · · · + xN
,
ȳ =
,
N
N
(x1 − x̄)2 + · · · + (xN − x̄)2(y1 − ȳ)2 + · · · + (yN − ȳ)2Sxx =, Syy =,
NN
(x1 − x̄)(y1 − ȳ) + · · · + (xN − x̄)(yN − ȳ)
.
Sxy =
N
x̄ =

(Cf. Cheng, etc. [16]).
¥

5.6

Appendix (Iterative likelihood function method)

In this section we study the “Iterative likelihood function method (cf. [47])”, which
will be related to subjective Bayesian statistics (see §8.6 later).
Consider the “measurement” described in the following “step [1]” and “step [2]”,[1] First we take a measurement MC(Ω) (O1 ≡ (X, 2X , F ), S[∗] ), and we know that themeasured value is equal to x ( ∈ X).
[2] And successively, we take a measurement MC(Ω) (O2 ≡ (Y, 2Y , G), S[∗] ), and we knowthat the measured value is equal to y ( ∈ Y ).

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Note that “[1]+[2]” ie equal to the following [3]10 :[3] We take a measurement MC(Ω) (O1 × O2 ≡ ≡ (X × Y, F × G, H ≡ F × G) S[∗] ),and we know that the measured value obtained by MC(Ω) (O1 × O2 , S[∗] ) is equal to(x, y) ( ∈ X × Y ).
³

A non-negative (real-valued) continuous function F (Ξ) in an observable (X, F, F ) is´
called a likelihood function, or, a likelihood quantity. Then we can say:[♭] By Step [1], we get the likelihood function F ({x}). And further by step [2] (i.e.,by “[1]+[2]” (=[3])), we get the new likelihood function F ({x})G({y}) ( ≡ [F ×G]({x} × {y})).
Using the Bayes operator (cf. the formula (5.12)), this statement [♭] can be rewritten as
follows:
Step (1)

Step (2)

O1
B{x}

2
B{y}

I −−−−→ F ({x}) −−−−→ F ({x})G({y})

in C(Ω),

O(5.79)

where I( ∈ C(Ω)) is the identity element, i.e., the constant function such that I(ω) =1(∀ω ∈ Ω).

1

1[F ({x})G({y})](ω)

I

0

1

-

Ω

-

Step (1)

Step (2)

O1
B{x}

O2
B{y}

[F ({x})](ω)

0

Ω

0

Ω

It should be noted that:(F1 ) the constant likelihood function “I” (or “k × I” where k > 0) is the likelihood
function that represents the fact “we have no information about the system S[∗] ”.
Now we introduce the following notation. Cf. [47].
Notation 5.23. [S[∗] ((G))lq ]. The system S[∗] (formulated in C(Ω)) such that we know it
has the likelihood quantity G (G ∈ C(Ω), 0 ≤ G(ω) (∀ω ∈ Ω)) is denoted by S[∗] ((G))lq .
Recall §2.5 (Remarks(II)), that is, “Only one measurement is permitted to be conducted”.
“[1]+[2]” is a methodological explanation.
10

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123

¡
¢
¡
¢
Thus, the symbol MC(Ω) O, S[∗] ((kG))lq means “the measurement MC(Ω) O, S[∗] under
the condition that we know the likelihood quantity of the system S[∗] is equal to kG,
where G ∈ C(Ω), 0 ≤ G(ω) (∀ω ∈ Ω)”.
¥
Under this notation, the conventional Fisher’s maximum likelihood method (i.e., Corollary 5.6) says that:(F1′ ) Assume that we ﬁrst have no information about the system S[∗] . And we take a¡¢¡¢measurement MC(Ω) O, S[∗] , i.e., MC(Ω) O, S[∗] ((kI))lq . Then, from the fact that¡¢the measured value x ( ∈ X) is obtained by the MC(Ω) O, S[∗] ((kI))lq , we know³
that the likelihood quantity of the system S[∗] is equal to k[F ({x})](ω). Thus,
there is a reason to regard the unknown state [ ∗ ] as the state ω0 ( ∈ Ω) such that´
k[F ({x})](ω0 ) = maxω∈Ω k[F ({x})](ω).
However, it is usual to assume that we have a little bit of information before a measure¡¢ment. Thus, let us start from the measurement MC(Ω) O ≡ (X, 2X , F ), S[∗] ((G0 ))lq . Here
we have the following problem:(PG ) How to infer the new likelihood quantity of the system S[∗] from the fact that the¡¢measured value x ( ∈ X) is obtained by the MC(Ω) O, S[∗] ((G0 ))lq .
This is equivalent to the following problem:(P′G ) How to infer the likelihood quantity of the system S[∗] from the fact that the measured value (y0 , x) ( ∈ {y0 , y1 } × X) is obtained by the iterated measurement¡¢
MC(Ω) O0 × O, S[∗] ((kI))lq , where O0 = ({y0 , y1 }, 2{y0 ,y1 } , G) and G({y0 }) = G0 ,G({y1 }) = I − G0 .
Thus, from (F1′ ) and “(PG )↔(P′G )”, the problem (PG ) is solved as follows:(F2 ) (The answer of the (PG )): We know that the new likelihood quantity Gnew of theOOsystem S[∗] is equal to B{x}(G0 ). Here, Bayes operator B{x}: C(Ω) → C(Ω) isO
deﬁned by B{x}(G) = F ({x})G (∀G ∈ C(Ω)).

Thus we see:´`MC(Ω) O1 , S[∗] ((I))lq´`
MC(Ω) O2 , S[∗] ((F ({x})))lq

x is obtained

y is obtainedS[∗] ((I))lq −−−−−−−−−−−→ S[∗] ((F ({x})))lq −−−−−−−−−−−−−−→ S[∗] ((F ({x})G({y})))lq

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where O1 = (X, 2X , F ) and O2 = (Y, 2Y , G).
Summing up, we can symbolically describe it as follows:[F ] No information quantity ←→ kI( ∈ C(Ω)

 1
`
´
³
´
M
O, S[∗] ((G))lqO [F2 ] S[∗] ((G))lq −−C(Ω)−−−−−−−−−→ S[∗] ((B{x}G))lq = S[∗] ((F ({x})G))lq ,(5.80)

x is obtained

where O = (X, 2X , F ).
The following example will promote the understanding of “iterative likelihood function
method”.
Example 5.24. [The urn problem]. There are two urns ω1 and ω2 . The urn ω1 [resp.
ω2 ] contains 8 white and 2 black balls [resp. 4 white and 6 black balls]. Assume that they
can not be distinguished in appearance.
• Choose one urn from the two.
Now you sample, randomly, with replacement after each ball.
(i). First, you get “white ball”.
(Q1 ) Which is the chosen urn, ω1 or ω2 ?
(ii). Further, assume that you continuously get “black”.
(Q2 ) How about the case?

Which is the chosen urn, ω1 or ω2 ?

The illustration of MC(Ω) (O, S[∗] ) (or, MC(Ω) (O, S[∗] ((kI))lq ) )
ω1

ω2

-

[∗]



[Answers]. In what follows this problem is studied in the iterative likelihood function
method. Put Ω = {ω1 , ω2 }. O = ({w, b}, 2{w,b} , F ) where[F ({w})](ω1 ) = 0.8, [F ({b})](ω1 ) = 0.2, [F ({w})](ω2 ) = 0.4, [F ({b})](ω2 ) = 0.6.
(5.81)

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125

The situation of no information in Fisher’s method is represented by kI (k > 0). Thus,
it suﬃces to consider the measurement MC(Ω) (O, S[∗] ((kI))lq ). Since the measured value“w” was obtained, the new likelihood quantity Gnew is given as follows:³´
Gnew (ω1 ) = kI · [F ({w})](ω1 ) = 0.8k,³´
Gnew (ω2 ) = kI · [F ({w})](ω2 ) = 0.4k.

(5.82)

Thus, by Fisher’s maximum likelihood method, we see that
(A1 ) there is a reason to infer that [ ∗ ] = ω1 .
For the further case, it suﬃces to consider the measurement MC(Ω) (O, S[∗] ((Gnew ))lq ).
Thus we similarly calculate that³´G(ω1 ) = [Gnew ](ω1 ) · [F ({b})](ω1 ) = 0.16k,³´
Gnew2 (ω2 ) = [Gnew ](ω2 ) · [F ({b})](ω2 ) = 0.24k.
new2(5.83)

Thus we, by Fisher’s maximum likelihood method, see that
(A2 ) there is a reason to infer that [ ∗ ] = ω2 .
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Chapter 6
Fisher’s statistics II (related to
Axioms 1 and 2)

As mentioned in Chapters 2 and 3, measurement theory is formulated as follows:
PMT = measurement + the relation among systems[Axiom 1 (2.37)]

in C ∗ -algebra[Axiom 2 (3.26)]

.

(6.1)
(=(1.4))In this chapter we study the relation between Fisher’s statistics (mentioned in the previous chapter)and Axiom 2. Particularly we show that regression analysis can be completely understood within
the framework of Axioms 1 and 2. We expect that our result will make the readers notice that
regression analysis is more profound than they usually think. As mentioned in Chapter 1 (cf.
Declaration (1.11)), we assert that the results in Chapters 5 and 6 guarantee that “Fisher’s statistics
is theoretically true (in PMT)”. 1

6.1

Regression analysis I

6.1.1

Introduction

The purpose of this chapter is to study and understand “regression analysis” completely under Axiom 1 and 2 (of measurement theory). The following Example 6.1 is themost typical in all examples of “regression analysis”.
Example 6.1. [A typical example of regression analysis]. Let Ω ≡ {ω1 , ω2 , ..., ω100 } be a
set of all students of a certain high school. Deﬁne h : Ω → [0, 200] [resp. w : Ω → [0, 200]]
1

We believe that only “Fisher’s maximum likelihood method” and “regression analysis” are mostessential in statistics. Thus we believe that, in order to justify statistics, it suﬃces to show that the two(i.e., “Fisher’s maximum likelihood method” and “regression analysis”) are formulated in PMT.

127
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such that:h(ωn ) = “the height of a student ωn ”

hresp. w(ωn ) = “the weight of a student ωn ”

(n = 1, 2, ..., 100)i(n = 1, 2, ..., 100)

Ω
h(ω)
0

200

w(ω)

ω

0

³

100

Note that this is a special case of Fig. (3.20).

´

100

200

Assume that:(1) The principal of this high school knows the both functions h and w. That is, heknows the exact data of the height and weight concerning all students.
Also, assume that:(2) Some day, a certain student helped a drowned girl. But, he left without reporting
the name. Thus, all information that the principal knows is as follows:(i) he is a student of his high school.
(ii) his height [resp. weight] is about 170 cm [resp. about 80 kg].
Now we have the following question:• Under the above assumption (1) and (2), how does the principal infer who is he?
This is just what regression analysis says. For the solution, see Regression Analysis I (6.7)
later.
¥In order to explain our main assertion, let us begin with the following Example 6.2 (theconventional argument of regression analysis in Fisher’s maximum likelihood method),which is easy and well-known.
Example 6.2. [The conventional argument of regression analysis in Fisher’s method].
We have a rectangular water tank ﬁlled with water. Assume that the height of water at

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129

time t is given by the following function h(t):h(t) = α0 + β0 t,(6.2)

where α0 and β0 are unknown ﬁxed parameters such that α0 is the height of water ﬁlling
the tank at the beginning and β0 is the increasing height of water per unit time. The
measured height hm (t) of water at time t is assumed to be represented byhm (t) = α0 + β0 t + e(t),(6.3)

where e(t) represents a noise (or more precisely, a measurement error) with some suitableconditions. And assume that we obtained the measured data of the heights of water at
t = 1, 2, 3 as follows:hm (1) = 1.9,

hm (2) = 3.0,

hm (3) = 4.7.

(6.4)

6

h(t)
?

Under this setting, we consider the following problem:(i) Infer the true value h(2) of the water height at t = 2 from the measured data (6.4).
This problem (i) is usually solved as follows: From the theoretical point of view, we caninfer, by Fisher’s maximum likelihood method and regression analysis, that
(α0 , β0 ) = (0.4, 1.4).
³

(6.5)

´
For the derivation of (6.5) from (6.4), see Example 6.4 (6.16) later. And next, we caninfer thath(2) = 3.2,

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by the calculation: h(2) = 0.4 + 1.4 × 2 = 3.2. This is the answer to the problem (i).
¥The above argument in Example 6.2 is, of course, well known and adopted as the usual
regression analysis. Thus all statisticians may think that there is no serious problem inregression analysis. However it is not true. For example, we have the basic problem in
the argument of Example 6.2 as follows:(ii) What kinds of axioms are hidden behind the argument in Example 6.2? And moreover, justify the argument in Example 6.2 under the axioms.
It is important. If we have no answer to the question: “What kinds of rules are permitted
to be used in statistics?”, we can not prove (or, justify) that the argument in Example 6.2
is true (or not). That is because there is no justiﬁcation without an axiomatic formulation.
In this sense, we believe that the above question (ii) is the most important problem intheoretical statistics. Also, if some know the great success of the axiomatic formulationin physics (e.g., the three laws in Newtonian mechanics, or von Neumann’s formulation ofquantum mechanics, cf. [71], [84]), it is a matter of course that they want to understandstatistics axiomatically.
Trying to solve the problem (ii), some may consider as follows:(iii) Firstly, Fisher’s maximum likelihood method should be declared as an axiom (cf.
Corollary 5.6). Also, the derivation of the (6.6) from the (6.5) should be justiﬁed
under some axioms. That is, it must not be accepted as a common sense.
This opinion (iii) may not be far from our assertion proposed in this chapter. However,in order to describe the above (iii) precisely, we must make vast preparations.
Our standing point of this book is extremely theoretical (and not practical). However
we expect that many statisticians will be interested in our proposal. That is because webelieve that every statistician may want to know the justiﬁcation of both the (6.5) andthe (6.6) in Example 6.2.

6.1.2

Regression analysis I in measurements

By the results in the previous chapters (i.e., Theorem 3.7 and Corollary 5.6), we caneasily propose:

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131

REGRESSION ANALYSIS I [The conventional regression analysis in PMT].

(6.7)

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Let (T ≡ {0, 1, ..., N }, π : T \ {0} → T ) be a tree with root 0, and let S[∗] ≡ [S[∗] ; {Φπ(t),t :C(Ωt ) → C(Ωπ(t) )}t∈T \{0} ] be a general system with the initial system S[∗] . And, let ane0observable Ot ≡ (Xt , 2Xt , Ft ) in a C ∗ -algebra C(Ωt ) be given for each t ∈ T . Let O
be the Heisenberg picture representation of the sequential observable [{Ot }t∈T , {Φt,π(t) :C(Ωt ) → C(Ωπ(t) )}t∈T \{0} ] in C(Ω0 ). Then, we have a measurementYQ
e 0 ≡ ( Xt , 2 t∈T Xt , Fe0 ), S[∗] ).
MC(Ω0 ) (O(cf. Theorem 3.7).
t∈T

e 0 , S[∗] ) belongs to Q Ξt (∈Assume that the measured value by the measurement MC(Ω0 ) (O
t∈T
³
Q
Xt
t∈T
2
). Then, there is a reason to infer that the state [∗] of the system S i.e.,´e 0 , S[∗] ) , the state after the measurement
the state before the measurement MC(Ω0 ) (O³´e 0 , S[∗] ) and the δω0 (∈ Mp (Ω)) deﬁned by (6.9) are equal. That is, CorollaryMC(Ω0 ) (O+1
5.6 says that there is a reason to infer thate 0 , S[∗] )” = δω0 .
[ ∗ ] = “the state after the measurement MC(Ω0 ) (OHere the δω0 (∈ Mp+1 (Ω0 )) is deﬁned byY
Y[Fe0 ( Ξt )](ω0 ) = max[Fe0 ( Ξt )](ω).
t∈T

ω∈Ω0

(6.8)

(6.9)

t∈T

¥
Remark 6.3. [Regression analysis I]. The above regression analysis is quite applicable.
For example, note that the “Φπ(t),t : C(Ωt ) → C(Ωπ(t) )” is generally assumed to be
Markov operators (and not homomorphisms). In this sense, Regression analysis I may
not be “conventional”.
¥
Now we shall review Example 6.2 in the light of Regression Analysis I.
Example 6.4. [Continued from Example 6.2, the conventional argument of regression
analysis in Fisher’s method]. Put Ω0 = [0.0, 1.0] × [0.0, 2.0], and put Ω1 = Ω2 = Ω3 =[0.0, 10.0]. For each t (∈ {1, 2, 3}), deﬁne a continuous map ϕ0,t : Ω0 → Ωt such that:Ω0 ( ≡ [0.0, 1.0] × [0.0, 2.0]) ∋ ω ≡ (α, β) 7→ α + βt ∈ Ωt ( ≡ [0.0, 10.0]).
ϕ0,t(6.10)

Thus, for each t (∈ {1, 2, 3}), we have a homomorphism Φ0,t : C(Ωt ) → C(Ω0 ) such that:[Φ0,t ft ](ω) = ft (ϕ0,t (ω))(∀ω ∈ Ω0 , ∀ft ∈ C(Ωt )).

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It is usual to assume that regression analysis is applied to the system with a parallel³
structure such as in the ﬁgure (6.12). From the peculiarity of this problem, we can also´
assume that this system has a series structure. However, we are not concerned with it.
Φ0,1

C(Ω1 )+¾

C(Ω2 )

C(Ω0 )

Φ0,2

k

Φ0,3 C(Ω3 )(6.12)For each t ∈ {1, 2, 3}, consider the discrete Gaussian observable Oσ2 ,N ≡ (XN , 2XN , Fσ,N )in C(Ωt ), (cf.(2.60) in Example 2.18). That is,Ωt = [0.0, 10.0],

XN = {k| k = 0, ±1, ±2, ..., ±N 2 },N

and[Fσ,N ({k/N })](ω)
 1 R∞(x−ω)2√]dx

1 exp[ −
2
2σ 2
N
−

2πσ
2N

k
1
R
2
+ 2N
N
√ 1
=
exp[ − (x−ω)
]dx
k
1
2σ 2
− 2N
2πσ 2 N12
 √ 1 R −N + 2N
]dx
exp[ − (x−ω)
2σ 2
2πσ 2 −∞

(k = N 2 , ∀ω ∈ [a, b]),(∀k = 0, ±1, ±2, ..., ±(N 2 − 1),∀ω ∈ [a, b]),(k = −N 2 , ∀ω ∈ [a, b]).
(cf. (2.aa60) in Example 2.18)

e 0 ≡ (X 3 , 2XN3 , Fe0 ) in C(Ω0 ) such that:
Here, we deﬁne the observable O
N[Fe0 (Ξ1 × Ξ2 × Ξ3 )](ω) = [Φ0,1 Fσ2 ,N ](ω) · [Φ0,2 Fσ2 ,N ](ω) · [Φ0,3 Fσ2 ,N ](ω)=[Fσ2 ,N (Ξ1 )](ϕ0,1 (ω)) · [Fσ2 ,N (Ξ2 )](ϕ0,2 (ω)) · [Fσ2 ,N (Ξ3 )](ϕ0,3 (ω))(∀Ξ1 , Ξ2 , Ξ3 ∈ 2XN , ∀ω = (α, β) ∈ Ω0 = [0.0, 1.0] × [0.0, 2.0]).

(6.13)

e 0 , S[∗] ). The (6.4) says that the measured valueThen, we have the measurement MC(Ω0 ) ( Oe 0 , S[∗] ) is equal toobtained by the measurement MC(Ω ) (O0(1.9, 3.0, 4.7) (∈ XN3 ).

(6.14)

Here, Fisher’s method (Corollary 5.6) says that it suﬃces to solve the problem“Find (α0 , β0 ) such as max(α,β)∈Ω0 [Fe0 ({1.9} × {3.0} × {4.7}(α, β)”.

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Putting
Ξ1 = [1.9 −

1
1
1
1
1
1
, 1.9 +], Ξ2 = [3.0 −, 3.0 +], Ξ3 = [4.7 −, 4.7 +
],
2N
2N
2N
2N
2N
2N

we see, under the assumption that N is suﬃciently large, that
Z Z Z
(x −(α+β))2 +(x2 −(α+2β))2 +(x3 −(α+3β))2
1
[− 1
]
2σ 2
(6.15) ⇒ max √e
dx1 dx2 dx33(α,β)∈Ω0
2πσ 2 Ξ ×Ξ ×Ξ
1
2
3
³
´
⇒ max exp − [(1.9 − (α + β))2 + (3.0 − (α + 2β))2 + (4.7 − (α + 3β))2 ]/(2σ 2 )(α,β)∈Ω0⇒ min [(1.9 − (α + β))2 + (3.0 − (α + 2β))2 + (4.7 − (α + 3β))2 ](α,β)∈Ω0½⇒(by the least squares method)(1.9 − (α + β)) + (3.0 − (α + 2β)) + (4.7 − (α + 3β)) = 0(1.9 − (α + β)) + 2(3.0 − (α + 2β)) + 3(4.7 − (α + 3β)) = 0⇒ (α0 , β0 ) = (0.4, 1.4).

(6.16)

This is the conclusion of Regression Analysis I (6.7). Also, using the notations in Regression Analysis I, we remark that:Qe 0 ≡ (Q Xt , 2 t∈T Xt , Fe0 ), S[∗] ) is hidden behind the(R) the measurement MC(Ω0 ) (Ot∈T³´
inference (6.16) = (6.5) in Example 6.2 .

This fact will be important in §6.3.
¥
The above may be the standard argument of the conventional regression analysis inmeasurement theory. However, our problem (i) in Example 6.2 is not to infer the (α0 , β0 )but h(2). In this sense the above regression analysis I is not suﬃcient. As the answerof the problem (i) in Example 6.2, we usually consider that it suﬃces to calculate h(2)³´
≡ ϕ0,2 (0.4, 1.4) in the following:³h(2) = 0.4 + 1.4 × 2 = 3.2.

(6.17)

However, this is doubtful.

In fact, this (6.17) is not always true in general situations.
´(cf. Regression analysis II (6.51) later). We should not rely on “a common sense” but
Axioms 1 and 2. That is, we must solve the problem:³´• How can the above (6.17) = (6.6) in Example 6.2 be deduced from Axioms 1
and 2?
In order to do this, we will make some preparations in the next section.

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6.2

Bayes operator, Schrödinger picture, and S-states

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In order to improve Regression Analysis I (introduced in the previous section), in thissection we make some preparations (i.e., Bayes operator, Schrödinger picture, S-state,etc.). Our main assertion (Regression Analysis II) will be mentioned in §6.3. We begin
with the following deﬁnition, which is a general form of “Bayes operator” in Remark 5.7.
Deﬁnition 6.5.

[Bayes operator (or precisely, Bayes-Kalman operator)]. Let (T ≡
Φπ(t),t{0, 1, ..., N }, π : T \ {0} → T ) be a tree with root 0 and let S[∗] ≡ [S[∗] ; {C(Ωt ) →C(Ωπ(t) )}t∈T \{0} ] be a general system with the initial system S[∗] . And, let an observablee 0 ≡ (Q Xt , N Ft , Fe0 ) be
Ot ≡ (Xt , Ft , Ft ) in C(Ωt ) be given for each t ∈ T . Let O
t∈T

t∈T

e 0 is the Heisenberg picture
as in Theorem 3.7 in the case At = C(Ωt ) (∀t ∈ T ). That is, O
Φπ(t),trepresentation of the sequential observable [{Ot }t∈T ; {C(Ωt ) → C(Ωπ(t) )}t∈T \{0} ]. Let(0,τ )

τ be any element in T . If a positive bounded linear operator BΠt∈T Ξt : C(Ωτ ) → C(Ω0 )(0,τ )(0,τ )

satisﬁes the following condition (BO), we call {BΠt∈T Ξt | Ξt ∈ Ft (∀t ∈ T )} [resp. BΠt∈T Ξt ]a family of Bayes operators [resp. a Bayes operator]:b0 ≡(BO) for any observable O′τ ≡ (Yτ , Gτ , Gτ ) in C(Ωτ ), there exists an observable O´³Q
NN( t∈T Xt ) × Yτ , ( t∈T Ft ) Gτ ), Fb0 in C(Ω0 ) such thatb 0 is the Heisenberg picture representation of [{Ot }t∈T ; {C(Ωt )(i) OΦπ(t),t→C(Ωπ(t) )}′t∈T \{0} ], where Ot = Ot (if t ̸= τ ), = Oτ × Oτ (if t = τ ),Q
(0,τ )(ii) Fb0 (( t∈T Ξt ) × Γτ ) = BΠt∈T Ξt (Gτ (Γτ )) (∀Ξt ∈ Ft (∀t ∈ T ), ∀Γτ ∈ Gτ ),³´
QQ(0,τ )
e
b
Ξ
)
≡
B
(1
)
, (∀Ξt ∈ Ft (∀t ∈ T )), where 1τ
Ξ
)
×
Y
)
=
F
(
(iii) F0 (( t∈T tτ0Πt∈T Ξt τt∈T t
is the identity in C(Ωτ ).
(0,τ )

mAlso, deﬁne the map RΠt∈T Ξt : Mm
+1 (Ω0 ) → M+1 (Ωτ ) such that:(BΠt∈T Ξt )∗ (ν)(0,τ )(0,τ )
RΠt∈T Ξt (ν) =(0,τ )∥(BΠt∈T Ξt )∗ (ν)∥M(Ωτ )(∀ν ∈ Mm
+1 (Ω0 )),(6.18)

where (BΠt∈T Ξt )∗ : C(Ω0 )∗ → C(Ωτ )∗ is the adjoint operator of BΠt∈T Ξt : C(Ωτ ) → C(Ω0 ).
(0,τ )(0,τ )(0,τ )The map RΠt∈T Ξt is called a “normalized dual Bayes operator”. Bayes operator is also
called “Bayes-Kalman operator”.
¥

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135

We see(0,τ )

BΠt∈T Ξt (gτ ) ≤ Φ0,τ gτ

(∀gτ ∈ C(Ωτ ) such that gτ ≥ 0),(6.19)

because it holds, for any observable O′τ ≡ (Yτ , Gτ , Gτ ) in C(Ωτ ),(0,τ )
BΠt∈T Ξt (Gτ (Γτ )) = Fb0 ((YΞt ) × Γτ ) ≤ Fb0 ((t∈TYXt ) × Γτ )

t∈T

³
´
(0,τ )= Φ0,τ Gτ (Γτ ) = BΠt∈T Xt (Gτ (Γτ ))(∀Γτ ∈ Fτ ).

(6.20)

The following theorem is essential to Regression Analysis II later.
e0 ≡Theorem 6.6. [The existence theorem of the Bayes operator (cf. [46, 55])]. Let O
Q
Q( t∈T Xt , 2 t∈T Xt , Fe0 ) be as in Theorem 3.7 in the case At = C(Ωt ) (∀t ∈ T ). And,for any s (∈ T ), put Ts ≡ {t ∈ T | s ≤ t}. Assume that, for each s (∈ T ), there existsQ
e s ≡ (Qes ) in C(Ωs ) such that Φπ(s),s Fes (Qt∈Ts Xt , F
an observable O
X,
2tt∈Ts Ξt ) =
t∈T
s
³ Q
´
Q
Feπ(s) ( t∈Tπ(s) \Ts Xt ) × ( t∈Ts Ξt )(∀Ξt ∈ 2Xt (∀t ∈ T )), (cf. Theorem 3.7). Let τ be(0,τ )

any element in T . Then, there exists a family of Bayes operators {BΠt∈T Ξt | Ξt ∈ 2Xt (∀t ∈T )}.

Proof. See [46]. The proof in [46] is essentially true, but it is not complete. That is
because the deﬁnition of “Bayes operator” (i.e., Deﬁnition 6.5) was not mentioned in [46].
Thus, we add the complete proof in what follows. It will be proved by induction. Let O′τ= (Yτ , 2Yτ , Gτ ) be any observable in C(Ωτ ).
b (τ,τ )[Step 1] First, deﬁne the positive bounded linear operator B
Πt∈Tτ Ξt : C(Ωτ ) → C(Ωτ ) suchthat:b (τ,τ ) (gτ ) = Feτ (Πt∈Tτ Ξt ) × gτBΠt∈Tτ Ξt(∀gτ ∈ C(Ωτ )),(6.21)b τ ≡ ((Πt∈Tτ Xt ) × Yτ , 2Xτ ×Yτ , Fbτ ) in C(Ωτ ) such that:and deﬁne the observable Ob (τ,τ ) (Gτ (Γτ ))Fbτ (Πt∈Tτ Ξt × Γτ ) = B
Πt∈Tτ Ξt(∀Γτ ∈ 2Yτ ),(6.22)

which is clearly the Heisenberg picture representation of the sequential observable [{Ot }t∈Tτ ,Φπ(t),t{C(Ωt ) → C(Ωπ(t) )}t∈Tτ \{τ } ], where Ot = Ot (if t ̸= τ ), = Oτ × O′τ (if t = τ ). Thus,³eb (τ,τ )the operator B
:
C(Ω
)
→
C(Ω
)
is
the
Bayes
operator
induced
from
the
O
=
τ
τ
τ
Πt∈Tτ Ξt´(Πt∈Tτ Xt , 2Πt∈Tτ Xt , Feτ ) , which is uniquely determined.

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/b[Step 2] Let s be any element in T \ {0} such that s ≤ τ . Here, assume that B
Πt∈Ts Ξt´:³
e s = (Πt∈Ts Xt , 2Πt∈T′ s Xt , Fes ) .
C(Ωτ ) → C(Ωs ) is the Bayes operator induced from the O
Q( t∈Ts Xt )×Yτ b
b s ≡ ((QThat is, there exists an observable O, Fs ) in C(Ωs )t∈Ts Xt ) × Yτ , 2(s,τ )

such thatb s is the Heisenberg picture representation (cf. Theorem 3.7) of the sequential(i) Ob t }t∈Ts , {C(Ωt ) Φπ(t),tobservable [{O→ C(Ωπ(t) )}t∈Ts \{s} ], where Ot = Ot (if t ̸= τ ),= Oτ × O′τ (if t = τ ),b (s,τ )(ii) Fbs ((Πt∈Ts Ξt ) × Γτ ) = B
Πt∈Ts Ξt (Gτ (Γτ ))Q(iii) Fbs ((Πt∈Ts Ξt ) × Yτ ) = Fes ( t∈Ts Ξt )(Ξt ∈ 2Xt (∀t ∈ Ts ), ∀Γτ ∈ 2Yτ ),(Ξt ∈ 2Xt (∀t ∈ Ts )).

Let (xt )t∈Tπ(s) be any element in Πt∈Tπ(s) Xt . Note that {(xt )t∈Tπ(s) } = Πt∈Tπ(s) {xt }. Deﬁneb (π(s),τ )the positive bounded linear operator B: C(Ωτ ) → C(Ωπ(s) ) byΠt∈Tπ(s) {xt }b (s,τ )eπ(s) (Q[Ft∈Tπ(s) {xt })](ωπ(s) ) × [Φπ(s),s BΠt∈Ts {xt } (gτ )](ωπ(s) )(π(s),τ
)b[B
Q
Πt∈Tπ(s) {xt } (gτ )](ωπ(s) ) =[Feπ(s) ((Πt∈Tπ(s) \Ts Xt ) × t∈Ts {xt })](ωπ(s) )(∀gτ ∈ C(Ωτ ),∀ωπ(s) (∈ Ωπ(s) )).

(6.23)

Here, the above is assumed to be equal to 0 if the denominator of (6.23) is equal to´³Q
0 i.e., [Feπ(s) ((Πt∈Tπ(s) \Ts Xt ) × t∈Ts {xt })](ωπ(s) ) = 0 . And thus, we can deﬁne theb (π(s),τ ) : C(Ωτ ) → C(Ωπ(s) ) by
positive bounded linear operator BΠt∈Tπ(s) ΞtXb (π(s),τ ) =
B
Πt∈T
Ξt
π(s)

b (π(s),τ )B{(xt )t∈T(xt )t∈Tπ(s) ∈Πt∈Tπ(s) Ξt

π(s)

}.

( t∈T
Xt )×Yτb π(s) ≡ ((Qπ(s), Fbπ(s) ) in C(Ωπ(s) ) such
Deﬁne the observable Ot∈Tπ(s) Xt ) × Yτ , 2Qthat:b (π(s),τ ) (Gτ (Γτ )) (Ξt ∈ 2Xt (∀t ∈ Tπ(s) ), ∀Γτ ∈ 2Yτ ),Fbπ(s) ((Πt∈Tπ(s) Ξt ) × Γτ ) = B
Πt∈T
Ξt
π(s)

Φπ(t),twhich is clearly the Heisenberg picture representation of [{Ot }t∈Tπ(s) , {C(Ωt ) → C(Ωπ(t) )}t∈Tπ(s) \{π(s)} ], where Ot = Ot (if t ̸= τ ), = Oτ × O′τ (if t = τ ). Also, it holds that
Fbπ(s) ((Πt∈Tπ(s) Ξt ) × Yτ ) = Feπ(s) (

Y

Ξt )

(Ξt ∈ 2Xt (∀t ∈ Tπ(s) )).

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137

That is because we seeXb (π(s),τ ) (1τ ) =Fbπ(s) ((Πt∈Tπ(s) Ξt ) × Yτ ) = B
Πt∈T
Ξt
π(s)

b (π(s),τ )B
Πt∈T{xt } (1τ )(xt )t∈Tπ(s) ∈Πt∈Tπ(s) Ξtπ(s)Qb (s,τ )Feπ(s) ( t∈Tπ(s) {xt }) × Φπ(s),s B
Πt∈Ts {xt } (1τ )=Q
Feπ(s) ((Πt∈Tπ(s) \Ts Xt ) × t∈Ts {xt })(xt )t∈Tπ(s) ∈Πt∈Tπ(s) Ξt
Q
Q
Feπ(s) ( t∈Tπ(s) {xt }) × Feπ(s) ((Πt∈Tπ(s) \Ts Xt ) × t∈Ts {xt })X=eπ(s) ((Πt∈T \Ts Xt ) × Q
Ft∈Ts {xt })π(s)(xt )t∈Tπ(s) ∈Πt∈Tπ(s) ΞtX
YY=Feπ(s) ({xt }) = Feπ(s) (Ξt ).
(6.24)X(xt )t∈Tπ(s) ∈Πt∈Tπ(s) Ξtt∈Tπ(s)t∈Tπ(s)b
Therefore, we see that BΠt∈Tπ(s) Ξt : C(Ωτ ) → C(Ωπ(s) ) is the Bayes operator induced from³´e π(s) = (Πt∈T Xt , 2Πt∈Tπ(s) Xt , Feπ(s) ) . Thus, we can, by induction, ﬁnish the proof
the O
π(s)
(0,τ )b (0,τ ) .
since it suﬃces to put B=B(π(s),τ )Πt∈T ΞtΠt∈T0 ΞtQ
e 0 ≡ (Q Xt , 2 t∈T Xt , Fe0 ), O′ ≡ (Yτ , 2Yτ , Gτ ), {B (0,τ ) | Ξt ∈ 2Xt (∀t ∈ T )},Let OτΠt∈T Ξtt∈T
Q
Q(0,τ )( t∈T Xt )×Yτ bb
O0 ≡ (( t∈T Xt ) × Yτ , 2, F0 ) and {RΠt∈T Ξt | Ξt ∈ 2Xt (∀t ∈ T )} be as in

Deﬁnition 6.5. Assume thatQ
e 0 , S[δω ] )(C1 ) we know that the measured value (xt )t∈T (∈ ( t∈T Xt )) obtained by MC(Ω0 ) (O0Q
belongs to t∈T Ξt .
Note that this (C1 ) is the same as the following (C2 ).
(C2 ) we know that the measured value ((xt )t∈T , y) (∈ (b 0 , S[δω ] ) belongs to (Q Ξt ) × Yτ .
(O

Qt∈T Xt )×Yτ ) obtained by MC(Ω0 )t∈T0Thus we see that³

(C3 ) the probability distribution of unknown y¡¢´under the assumption (C2 ) =(C1 ) ,i.e., the probability that y (∈ Yτ ) belongs to Γτ , is represented byQ(0,τ )⟨δω0 , Fb0 (( t∈T Ξt ) × Γτ )⟩C(Ω0 ) ³ C(Ω0 )∗ ⟨δω0 , BΠt∈T Ξt (Gτ (Γτ ))⟩C(Ω0 ) ´C(Ω0 )∗. (6.25)≡(0,τ )b Q Ξt ) × Yτ )⟩(1τ )⟩∗ ⟨δω0 , F0 ((∗ ⟨δω , Bt∈T

C(Ω0 )

C(Ω0 )

C(Ω0 )

0Πt∈T Ξt

C(Ω0 )

A simple calculation shows:(BΠt∈T Ξt )∗ (δω0 )(0,τ )(6.25) = C(Ωτ )∗ ⟨(0,τ )∥(BΠt∈T Ξt )∗ (δω0 )∥M(Ω)(0,τ ), Gτ (Γτ )⟩C(Ωτ ) = C(Ωτ )∗ ⟨RΠt∈T Ξt (δω0 ), Gτ (Γτ )⟩C(Ωτ ) .

Therefore, we say that

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/³¢´(C4 ) the probability distribution of unknown y under (C2 ) =(C1 ) is represented by¡(0,τ )

C(Ωτ )∗⟨RΠt∈T Ξt (δω0 ), Gτ (Γτ )⟩C(Ωτ ) .

(6.26)

Let this (C4 ) be, as an abbreviation, denoted (or, called) by´³
e(C5 ) the S-state after the measurement MC(Ω0 ) (O0 , S[δω0 ] ) at τ (in T ) is equal to(0,τ )

RΠt∈T Ξt (δω0 ).
For completeness, again note that (C4 ) = (C5 ), i.e., (C5 ) is an abbreviation for (C4 ). Notethat the concept of “S-state” and that of “state” are completely diﬀerent. In measurement
theory, as seen in Axiom 1, the state always appears as the ρp in MA (O, S[ρp ] ). That is,
the state ρp is always ﬁxed and never moves. In this sense, the ρp may be called a “realstate”. On the other hand, the “S-state” is used in the abbreviation (C5 ) of (C4 ).
Summing up the above argument, we have the following deﬁnition.
Deﬁnition 6.7.

[S-state (= Schrödinger picture)]. Assume the above situation. If

the above statement (C4 ) holds, then we say “(C5 ) holds”, i.e., “the S-state (after theb 0 , S[δω ] )) at τ (∈ T ) is equal to R(0,τ ) (δω0 )”. The representation
measurement MC(Ω ) (O0Πt∈T Ξt0

using “S-state” is called the Schrödinger picture representation. The S-state is also calleda Schrödinger state or imaginary state.
¥
As seen in the above argument, we must note that the Bayes operator is always hidden
behind the Scrödinger picture representation.
¡¢
We sum up the above argument i.e., (C1 )⇒(C5 ) as the following lemma.
Q
e 0 ≡ (Q Xt , 2 t∈T Xt , Fe0 ), {B (0,τ ) | Ξt ∈ 2Xt (∀t ∈ T )}Lemma 6.8. [S-state]. Let O

Πt∈T Ξtt∈T(0,τ )
Xtand {RΠt∈T Ξt | Ξt ∈ 2 (∀t ∈ T )} be as in Deﬁnition 6.5. Assume that

• we know that the measured value (xt )t∈T (∈Q
belongs to t∈T Ξt .

Qet∈T Xt ) obtained by MC(Ω0 ) (O0 , S[δω0 ] )Then, we can say³(♯) the S-state´
e 0 , S[δω ] ) at τ (in T ) is equal toafter the measurement MC(Ω0 ) (O0(0,τ )

RΠt∈T Ξt (δω0 ).
¥The following lemma will be used as Theorem 6.13.

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Q
e 0 ≡ (Q Xt , 2 t∈T Xt , Fe0 ), {B (0,τ ) |[Inference and S-state]. Let O
Πt∈T Ξtt∈T

Lemma 6.9.

(0,τ )

Ξt ∈ 2Xt (∀t ∈ T )} and {RΠt∈T Ξt | Ξt ∈ 2Xt (∀t ∈ T )} be as in Deﬁnition 6.5. Assumethat( • ) we know that the measured value (xt )t∈T (∈Q
belongs to t∈T Ξt .

Qet∈T Xt ) obtained by MC(Ω0 ) (O0 , S[∗] )Then, there is a reason to infer that³´(0,τ )e(♯) the S-state after the measurement MC(Ω0 ) (O0 , S[∗] ) at τ (in T ) is equal to RΠt∈T Ξt (δω0 ).
Here the δω0 (∈ Mp+1 (Ω0 )) is deﬁned byY
Y[Fe0 ( Ξt )](ω0 ) = max[Fe0 ( Ξt )](ω).
ω∈Ω0

t∈T

(6.27)

t∈T

Proof. The proof is similar to that of Corollary 5.6. Let (Yτ , 2Yτ , Gτ ) be any observablein C(Ωτ ). Note that the above (•) is the same as the following:Q
b S[∗] )(•)’ we know the measured value ((xt )t∈T , y) (∈ ( t∈T Xt )×Yτ ) obtained by MC(Ω0 ) (O,Q
b 0 is as in Deﬁnition 6.5).
Ξt ) × Yτ (where O
belongs to (t∈TQ
Thus we can infer, by Theorem 5.3 (Fisher’s method) and the equality Fe0 ( t∈T Ξt ) =
³
´
³
Q
b
b
F0 (( t∈T Ξt )×Yτ ), that the unknown state [∗] in MC(Ω0 ) (O, S[∗] ) is equal to δω0 deﬁned´
by (6.27) . Thus the conditional probability PΠt∈T Ξt (·) under the condition that we know
Q
that ((xt )t∈T , y) ∈ ( t∈T Xt ) × Yτ is given byPΠt∈T Ξt (Γτ ) =Q(0,τ )⟨δω0 , BΠt∈T Ξt (Gτ (Γτ ))⟩C(Ω0 )⟨δω0 , Fb0 (( t∈T Ξt ) × Γτ )⟩C(Ω0 )C(Ω0 )∗=(0,τ )b Q Ξt ) × Yτ )⟩(1τ )⟩∗ ⟨δω0 , F0 ((∗ ⟨δω , B

C(Ω0 )∗
C(Ω0 )t∈T

C(Ω0 )(0,τ )= C(Ωτ )∗ ⟨RΠt∈T Ξt (δω0 ), Gτ (Γτ )⟩C(Ωτ )0

C(Ω0 )

Πt∈T Ξt

C(Ω0 )(∀Γτ ∈ 2Yτ ).

From the equivalence of (C4 ) and (C5 ), we can conclude the (♯).
Φ0,τNow we consider the simplest case that T ≡ {0, τ } and S[δω0 ] ≡ [S[δω0 ] ; C(Ωτ ) →(nl)

C(Ω0 )]. For each k = 0, τ , consider the null observable Ok≡ ({0, 1}, 2{0,1} , Fk ) inC(Ωk ) (cf. Example 2.21). Then, we have the measurement´³(nl)
2 {0,1}2(nl)eMC(Ω0 ) O0 ≡ ({0, 1} , 2, F0 × Φ0,τ Fτ ), S[δω0 ] .
Note that:

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e 0 , S[δω ] )) is equal to (1, 1) is(i) the probability that the measured value (by MC(Ω0 ) (O0given by 1. That is, the measured value is always (or surely) equal to (1, 1).
Thus,
b 0 , S[δω ] ) has always the form ((1, 1), y) (∈(ii) the measured value obtained by MC(Ω0 ) (O02
b{0, 1} × Yτ ). Here O0 is deﬁned by({0, 1}2 × Yτ , 2{0,1} ×Yτ , F02(nl)× Φ0,τ Fτ(nl) × Φ0,τ Gτ )(6.29)

for any any observable (Yτ , 2Yτ , Gτ ) in C(Ωτ ).
b 0 , S[δω ] ) and MC(Ω ) ((Yτ , 2Yτ , Φ0,τ Gτ ), S[δω ] ) are essentially the same.
Note that MC(Ω0 ) (O000e 0 , S[δ ] )” is essentially the same as “to take no meaThat is because “to take MC(Ω ) (Oω00surement” (cf. Example 2.21). Thus, the above (ii) implies that³¡¢´(iii) the probability distribution of unknown y under (ii) = (i) , i.e., the probability
that y ∈ Γτ , is represented byC(Ωτ )∗⟨Φ∗0,τ (δω0 ), Gτ (Γτ )⟩C(Ωτ )for any (Yτ , 2Yτ , Gτ ) in C(Ωτ ) and any Γτ (∈ 2Yτ ).
That is because it holds that(nl)

× Φ0,τ Fτ(nl)

× Φ0,τ Fτ

C(Ω0 )∗⟨δω0 , (F0C(Ω0 )∗⟨δω0 , (F0(nl)× Φ0,τ Gτ )({(1, 1)} × Γτ )⟩C(Ω0 )(nl)

× Φ0,τ Gτ )({(1, 1)} × Yτ )⟩C(Ω0 )=C(Ωτ )∗ ⟨Φ∗0,τ (δω0 ), Gτ (Γτ )⟩C(Ωτ ) .
Thus,we get the following (iv), which is short for (iii).
(iv) the S-state at τ (∈ T ≡ {0, τ }) is equal to Φ∗0,τ (δω0 ).
Thus we conclude that (i) ⇒ (iv). However, note that (i) always holds. Therefore, we
think that (iv) always holds.
From the above argument, we have the following lemma. This will be used in thestatement (6.33).
Lemma 6.10. [The Schrödinger picture representation]. Put T = {0, τ }. Let S[δω0 ] ≡Φ0,τ[S[δω0 ] ; {C(Ωτ ) → C(Ω0 )}] be a general system with an initial state S[δω0 ] . Then we see
that

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(♯) the S-state at τ (∈ T ≡ {0, τ } ) is Φ∗0,τ (δω0 ).

³Here it should be noted that the measurement MC(Ω0 ) ((Yτ , 2Yτ , Φ0,τ Gτ ), S[δω0 ] ) or, MC(Ω0 )´
b 0 , S[δω ] ), is hidden behind the assertion (♯).
(O
0
¥

³
Also, the following lemma is the formal representation of Corollary 5.6 (ii).
´mark 6.12.

Cf. Re-Lemma 6.11. [Inference and the Schrödinger picture representation]. Put T = {0, τ }.
Let S[∗] ≡ [S[∗] ; {Φ0,τ : C(Ωτ ) → C(Ω0 )}] be a general system with an initial state S[∗] . Let(nl)

O0 = (X0 , 2X0 , F0 ) be an observable in C(Ω0 ). And, let Oτ

= ({0, 1}, 2{0,1} , Fτ

(nl)

) be
e 0 (≡
the null observable in C(Ωτ ) (cf. Example 2.21). Consider a measurement MC(Ω0 ) (OO0 × Φ0,τ Oτ ), S[∗] ), which is essentially the same as MC(Ω0 ) (O0 , S[∗] ). Assume that(nl)e 0 ≡ O0 × Φ0,τ O(nl)• we know that the measured value obtained by MC(Ω0 ) (Oτ , S[∗] )belongs to Ξ0 × {1} (∈ 2X0 ×{0,1} ).
Then we see thate 0 , S[∗] ))(♯) there is a reason to infer that the S-state (after the measurement MC(Ω0 ) (Oat τ (∈ T ≡ {0, τ }) is Φ∗0,τ (δω0 ),where δω0 (∈ Mp+1 (Ω0 )) is deﬁned by[F0 (Ξ0 )](ω0 ) = max[F0 (Ξ0 )](ω).

(6.30)

ω∈Ω0(0,τ )(0,τ )

Proof. Let BΞ0 ×{1} : C(Ωτ ) → C(Ω0 ) and RΞ0m: Mm+1 (Ω0 ) → M+1 (Ωτ ) be as in

Deﬁnition 6.5. Here, note that, from the property of null observable, it holds that F0 (Ξ0 )×(nl)

Φ0,τ Fτ(0,τ )({1}) = F0 (Ξ0 ). Thus we see that BΞ0 ×{1} (gτ ) = F0 (Ξ0 ) × Φ0,τ gτ for any gτ (∈C(Ωτ )). By Lemma 6.9, it suﬃces to prove RΞ0 (δω0 ) = Φ∗0,τ (δω0 ). This is shown as(0,τ )

follows:(BΞ0 ×{1} )∗ (δω0 )(0,τ )(0,τ )⟨RΞ0 ×{1} (δω0 ), gτ ⟩C(Ωτ ) = C(Ωτ )∗ ⟨C(Ωτ )∗(0,τ )∥(BΞ0 ×{1} )∗ (δω0 )∥M(Ωτ ), gτ ⟩C(Ωτ )1[F0 (Ξ0 )](ω0 ) × [Φ0,τ gτ ](ω0 )(0,τ )⟨δω0 , BΞ0 ×{1} (gτ )⟩C(Ω0 ) =C(Ω0 )∗(0,τ )[F0 (Ξ0 )](ω0 )∥(BΞ0 ×{1} )∗ (δω0 )∥M(Ωτ )(∀gτ ∈ C(Ωτ )).
(6.31)=C(Ωτ )∗ ⟨Φ∗0,τ (δω0 ), gτ ⟩C(Ωτ )=Then, we see that RΞ0 ×{1} (δω0 ) = Φ∗0,τ (δω0 ). This completes the proof.
(0,τ )

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The following remark shows that Corollary 5.6 (ii) is a direct consequence of Lemma
6.11.
Remark 6.12. [Continued from Corollary 5.6 (Fisher’s maximum likelihood method inclassical measurements)]. As mentioned before, the proof of Corollary 5.6 is temporary.
Corollary 5.6 should be understood as a corollary of Lemma 6.11 as follows: In Lemma
6.11, put Ω0 = Ωτ = Ω+0 . And let Φ0,τ : C(Ω+0 ) → C(Ω0 ) be the identity map. Since“the S-state (after the measurement MC(Ω0 ) (O0 , S[∗] )) at τ (= +0)” = Φ0,τ (δω0 ) = δω0 , weeasily see that Corollary 5.6 is a consequence of Lemma 6.11. This should be regarded as
the formal proof of Corollary 5.6.
¥

6.3

Regression analysis II in measurements

Now let us explain the reason why we consider:¡¢¡¢(♯) it is worthwhile doubting the derivation of (6.6) = (6.17) from (6.5) = (6.16) ,i.e., the formula h(2) = 0.4 + 1.4 × 2 = 3.2.
Using the notations in Regression Analysis I (6.7), we recall the statement (R) of Example
6.4 as follows:e0 ≡ ((R) the measurement MC(Ω0 ) (O¡¢inference (6.5) =(6.16) .

Q

Qt∈T Xt , 2t∈T Xt, Fe0 ), S[∗] ) is hidden behind the

And we conclude, by Corollary 5.6 (or Remark 6.12), thate 0 , S[∗] )”[ ∗ ] = “the S-state after the measurement MC(Ω0 ) (O= δω0 .

(6.32)Q
Q
Here the δω0 (∈ Mp+1 (Ω0 )) is deﬁned by [Fe0 ( t∈T Ξt )](ω0 ) = maxω∈Ω0 [Fe0 ( t∈T Ξt )](ω).
On the other hand,³¡¢
i.e., the derivation of (6.6) = (6.17) from (6.5)• the map “δω0 7→¡¢´= (6.16)is due to the Schrödinger picture, behind which the measurementΦ∗0,τ (δω0 )”MC(Ω0 ) (Φ0,τ O′τ ≡ (Yτ , 2Yτ , Φ0,τ Gτ ), S[δω0 ] ) is hidden. Cf. Lemma 6.10.

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Thus, in order to conclude the assertion (6.6) = (6.17) , we need the above “twomeasurements”, that is,e 0 ≡ (Q“MC(Ω0 ) (Ot∈T Xt , 2Qt∈T Xt, Fe0 ), S[∗] ) ” and “ MC(Ω0 ) (Φ0,τ O′τ ≡ (Yτ , 2Yτ , Φ0,τ Gτ ), S[δω0 ] )”.
(6.34)

However, note that it is forbidden to conduct “two measurements” (cf. §2.5(II)). This¡¢is the reason that we think that it is worthwhile doubting (6.6) = (6.17) . In order toavoid this confusion, it suﬃces to consider the “simultaneous” measurement:YQb 0 ≡ (( Xt ) × Yτ , 2( t∈T Xt )×Yτ , Fb0 ), S[∗] ), (where O
b 0 is as in Deﬁnition 6.5),
MC(Ω0 ) (Ot∈T

(6.35)
instead of (6.34).
Then, we rewrite Lemma 6.9 as an main theorem as follows:Q
e 0 ≡ (Q Xt , 2 t∈T Xt ,Theorem 6.13. [= Lemma 6.9, Inference in Markov relation]. Let Ot∈TFe0 ) be as in Theorem 3.7 in the case At = C(Ωt ) (∀t ∈ T ). And consider a measureXt
e 0 , S[∗] ). Let τ be any element in T . Let {Rment MC(Ω0 ) (O(∀t ∈ T )}Πt∈T Ξt | Ξt ∈ 2(0,τ )

be as in Deﬁnition 6.5. Assume that we know that the measured value (obtained by
e 0 , S[∗] )) belongs to Q Ξt . Then, there is a reason to infer that
MC(Ω ) (O
t∈T

0

(♯)

e 0 , S[∗] )” = R“the S-state at τ (∈ T ) after MC(Ω0 ) (OΠt∈T Ξt (δω0 ).
(0,τ )(6.36)

Here δω0 (∈ Mp+1 (Ω)) is deﬁned byY
Y[Fe0 ( Ξt )](ω0 ) = max[Fe0 ( Ξt )](ω).
t∈T

ω∈Ω0

(6.37)t∈T¥
Lastly, we prove the following lemma, which justiﬁes the inference (6.6).
Q
e 0 ≡ (Q Xt , 2 t∈T Xt ,Lemma 6.14. [Some property of homomorphic relation]. Let Ot∈TFe0 ) be as in Theorem 3.7 in the case At = C(Ωt ) (∀t ∈ T ). Consider the family of Bayes(0,τ )

operators {BΠt∈T Ξt | Ξt ∈ 2Xt (t ∈ T )} such as in Deﬁnition 6.5. Let τ be any element in
T . Assume that Φπ(t),t : C(Ωt ) → C(Ωπ(t) ) (∀t ∈ T such that 0 < t ≤ τ ) is homomorphic.
Then, it holds that:
BΠt∈T Ξt (Gτ (Γτ )) = Fe0 ((0,τ )YΞt ) × Φ0,τ Gτ (Γτ ) (∀Ξt ∈ 2Xt (∀t ∈ T ), ∀Γτ ∈ 2Yτ ),t∈T

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for any observable (Yτ , 2Yτ , Gτ ) in C(Ωτ ). That is, we see that the Bayes operator BΠt∈T Ξt :C(Ωτ ) → C(Ω0 ) is determined uniquely under the homomorphic condition.
Proof. The proof is shown in the following three steps.
[Step 1]. Let ω0 be any element in Ω0 . And let gτ and hτ be in C(Ωτ ) such that:0 ≤ gτ ≤ 1, gτ (ϕ0,τ (ω0 )) = 0, 0 ≤ hτ ≤ 1, and hτ (ϕ0,τ (ω0 )) = 1.

(6.39)

where ϕ0,τ : Ω0 → Ωτ is deﬁned by (3.14). Then we see, by (6.19), that(0,τ )

0 ≤ [BΠt∈T Ξt (gτ )](ω) ≤ (Φ0,τ gτ )(ω) = gτ (ϕ0,τ (ω)) (∀ω ∈ Ω0 ).

(6.40)

Putting ω = ω0 in (6.40), we get, by (6.39), that(0,τ )[BΠt∈T Ξt (gτ )](ω0 ) = 0.

(6.41)

Also, from the linearity of Bayes operator and the condition (iii) of Deﬁnition 6.5, we get(0,τ )(0,τ )(0,τ )[BΠt∈T Ξt (1τ − hτ )](ω) = [BΠt∈T Ξt (1τ )](ω) − [BΠt∈T Ξt (hτ )](ω)
Y(0,τ )= [Fe0 ( Ξt )](ω) − [BΠt∈T Ξt (hτ )](ω) (∀ω ∈ Ω0 ).

(6.42)t∈T

Thus, putting ω = ω0 in (6.42), we get, by (6.39), that(0,τ )

0 ≤ [BΠt∈T Ξt (1τ − hτ )](ω0 )≤ [(Φ0,τ (1τ − hτ ))](ω0 ) = 1τ (ϕ0,τ (ω0 )) − hτ (ϕ0,τ (ω0 )) = 1 − 1 = 0.

(6.43)

Then, we obtainY(0,τ )[BΠt∈T Ξt (hτ )](ω0 ) = [Fe0 ( Ξt )](ω0 ).

(6.44)t∈T[Step 2]. Let ω0 be any ﬁxed element in Ω0 . Fix any f (∈ C(Ωτ )) such that 0 ≤ f ≤ 1.
Deﬁne gτ , hτ (∈ C(Ωτ )) such that:gτ (ωτ ) = max{0, f (ωτ ) − f (ϕ0,τ (ω0 ))}(∀ωτ ∈ Ωτ ),f (ωτ ), 1}(∀ωτ ∈ Ωτ ).
hτ (ωτ ) = min{f (ϕ0,τ (ω0 ))(6.45)The gτ and the hτ clearly satisfy (6.39). And moreover, we see, for any ωτ ∈ Ωτ , thatgτ (ωτ ) + f (ϕ0,τ (ω0 ))hτ (ωτ )= max{0, f (ωτ ) − f (ϕ0,τ (ω0 ))} + min{f (ωτ ), f (ϕ0,τ (ω0 ))}½(f (ωτ ) − f (ϕ0,τ (ω0 )) + f (ϕ0,τ (ω0 )), if f (ωτ ) ≥ f (ϕ0,τ (ω0 ))=0 + f (ωτ ),if f (ωτ ) ≤ f (ϕ0,τ (ω0 ))= f (ωτ ).

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/[Step 3]. Let ω0 be any element in Ω0 . Let Γτ be any element in 2Yτ . From the [step2], we see that there exist gbτ (∈ C(Ωτ )) and bhτ (∈ C(Ωτ )) such that Gτ (Γτ ) = gbτ +[Gτ (Γτ )](ϕ0,τ (ω0 ))bhτ , gbτ (ϕ0,τ (ω0 )) = 0, bhτ (ϕ0,τ (ω0 )) = 1. Then we seeh³´i(0,τ )(0,τ )[BΠt∈T Ξt (Gτ (Γτ ))](ω) = BΠt∈T Ξt gbτ + [Gτ (Γτ )](ϕ0,τ (ω0 ))bhτ (ω)=[BΠt∈T Ξt (bgτ )](ω) + [Gτ (Γτ )](ϕ0,τ (ω0 )) × [BΠt∈T Ξt (bhτ )](ω) (∀ω ∈ Ω0 ).
(0,τ )(0,τ )(6.47)

Putting ω = ω0 , we see, by (6.41) and (6.44), that [BΠt∈T Ξt (bgτ )](ω0 ) = 0 and [BΠt∈T Ξt (bhτ )](ω0 )Q= [Fe0 (Ξt )](ω0 ). And, we see, by (6.47), that(0,τ )(0,τ )t∈TY(0,τ )[BΠt∈T Ξt (Gτ (Γτ ))](ω0 ) = [Gτ (Γτ )](ϕ0,τ (ω0 )) × [Fe0 ( Ξt )](ω0 )t∈TY= [Φ0,τ Gτ (Γτ )](ω0 ) × [Fe0 ( Ξt )](ω0 ).
t∈TSince ω0 (∈ Ω0 ) is arbitrary, we obtain (6.38). This completes the proof.
Now we can propose our main assertion as follows:
REGRESSION ANALYSIS II [The new proposal of regression analysis, cf.[55]].
(6.48)
Φπ(t),tLet (T ≡ {0, 1, ..., N }, π : T \{0} → T ) be a tree with root 0, and let S[∗] ≡ [S[∗] ; {C(Ωt ) →C(Ωπ(t) )}t∈T \{0} ] be a general system with the initial system S[∗] . And, let an observableOt ≡ (Xt , 2Xt , Ft ) in a C ∗ -algebra C(Ωt ) be given for each t ∈ T . Then, we have ameasurement
e0 ≡ (MC(Ω0 ) (OYQ

Xt , 2 t∈T Xt , Fe0 ), S[∗] )(cf. Theorem 3.7).

(6.49)t∈T

e 0 , S[∗] ) belongs to Q Ξt (∈Assume that the measured value by the measurement MC(Ω) (Ot∈T
Q

2 t∈T Xt ). Also deﬁne δω0 (∈ Mp+1 (Ω0 )) such that:Y
Y[Fe0 ( Ξt )](ω0 ) = max[Fe0 ( Ξt )](ω).
ω∈Ω0

t∈T

(6.50)t∈T(0,τ )

Let τ be any element in T . Let {RΠt∈T Ξt | Ξt ∈ 2Xt (∀t ∈ T )} be as in Deﬁnition 6.5.
³´
(0,τ )The existence of {RΠt∈T Ξt | Ξt ∈ 2Xt (∀t ∈ T )} is assumed by Theorem 6.6. Then, we
see:

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(i). [The S-state at τ (∈ T )]. There is a reason to infer thate 0 , S[∗] )” = R(0,τ ) (δω0 ).
“The S-state at τ (∈ T ) after MC(Ω0 ) (OΠt∈T Ξt(♯)(6.51)

Also(ii). [The S-state at τ (∈ T ) for homomorphism Φ0,τ ]. Assume that Φ0,τ : C(Ωτ ) → C(Ω0 )³is homomorphic i.e., Φπ(t),t : C(Ωt ) → C(Ωπ(t) ) (∀t ∈ T such that 0 < t ≤ τ ) is´
homomorphic . Then there is a reason to infer thate 0 , S[∗] )” = Φ∗ (δω0 ).
“the S-state at τ (∈ T ) after MC(Ω0 ) (O0,τ(6.52)

Here note that Φ∗0,τ (δω0 ) = δϕ0,τ (ω0 ) where ϕ0,τ : Ω0 → Ωτ is deﬁned by (3.14).
Proof. (i). See Theorem 6.13 (= Lemma 6.9).
(ii). We see, by Lemma 6.14, that(0,τ )(BΠt∈T Ξt )∗ (δω0 )(0,τ )⟨RΠt∈T Ξt (δω0 ), gτ ⟩C(Ωτ ) = C(Ωτ )∗ ⟨ (0,τ ), gτ ⟩C(Ωτ )C(Ωτ )∗(BΠt∈T Ξt )∗ (δω0 )==

1(0,τ )C(Ω )∗(0,τ )∥(BΠt∈T Ξt )∗ (δω0 )∥M(Ωτ ) 0⟨δω0 , BΠt∈T Ξt (gτ )⟩C(Ω0 )

Y
1
e
⟨δ
,
F
(
Ξt ) × Φ0,τ gτ ⟩C(Ω0 )Q
ω00
C(Ω )∗[Fe0 ( t∈T Ξt )](ω0 ) 0t∈T=C(Ωτ )∗ ⟨Φ∗0,τ (δω0 ), gτ ⟩C(Ωτ )(by Lemma 6.14)(∀gτ ∈ C(Ωτ )).

Then, we see that RΠt∈T Ξt (δω0 ) = Φ∗0,τ (δω0 ).
(0,τ )

Remark 6.15. [(i) Continued from Example 6.2]. Note that our problem (i) in Example
6.2 was to infer the h(2) and not (α0 , β0 ). Regression analysis II (6.52) is applicable toour problem, that is, the above (6.52) says that there is a reason to calculate h(2) in thefollowing:h(2) = ϕ0,2 (0.4, 1.4) = 0.4 + 1.4 × 2 = 3.2.

(6.53)[(ii) Interesting logic]. It should be noted that, when τ = 0, the Regression Analysis II
is the same as the Regression Analysis I. Thus, we also conclude (6.5), i.e., (α0 , β0 ) =(0.4, 1.4). After all, the Regression Analysis II says that
(M1 ) as the result in the case that τ = 0, the conclusion (6.5) in Example 6.2 is reasonable,

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147

or(M2 ) as the result in the case that τ ̸= 0, the conclusion (6.6) in Example 6.2 is reasonable.
However, it should be noted that the Regression Analysis II does not guarantee that(M3 ) both (6.5) and (6.6) in Example 6.2 are (simultaneously) reasonable.
¡
That is because two measurements i.e., the measurement M1 behind (M1 ) and the mea¢
surement M2 behind (M2 ) are included in (M1 ) and (M2 ). If we want to conclude
this (M3 ), we must consider the simultaneous measurement of “measurement M1 ” and“measurement M2 ”, that is, we must generalize Deﬁnition 6.5 (Bayes operator) such as(0,(0,τ ))BΠt∈T Ξt : C(Ω0 ) × C(Ωτ ) → C(Ω0 ) satisfying similar conditions since only one measurement is permitted (cf. §2.5(II)). This is, of course, interesting, though it is not discussed
in this book.
¥

6.4

Conclusions

In this chapter we show that regression analysis can be completely understood in PMT
as follows (cf. [55]):
measurement theory½
Corollary 5.5 (conditional probability)
Axiom 1 ⇒Theorem 5.3⇒

Corollary 5.6 (classical Fisher’s method)(Fisher’s method)



=⇒

 Theorem 3.7 (measurability)





 Axiom 2 ⇒  Theorem 6.6 (the existence of Bayes operator)
Lemma 6.14 (some property of homomorphic relation).
And, using these results, we derive “regression analysis” as follows:(i) :(ii) :“Corollary 5.6” + “Theorem 3.7” ⇒ “Regression Analysis I ”,

Theorem 3.7 






“Corollary 5.5” + “Theorem 6.6” ⇒ “Theorem 6.13”⇒ “Regression Analysis II”.
(Markov inference) 





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We believe that Regression Analysis II is the best (i.e., precise, wide, deep etc.) in allconventional proposals of regression analysis (though it should be generalized as mentioned
in Remark 6.15.). It is surprising that both statistics and quantum mechanics can beunderstood in the same theory, i.e., measurement theory (6.1) (=(1.4)).
We believe that every statistician may want to know the justiﬁcation of (6.5) and(6.6) in Example 6.2. Thus we expect that many statisticians will be interested in our
axiomatic approach. That is because there is no justiﬁcation without axioms.
We think that the results in Chapters 5 and 6 guarantee that “Fisher’s statistics istheoretically true”, (cf. Declaration (1.11)).

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Chapter 7
Practical logic
It is certain that pure logic (cf. [89]) is merely a kind of rule in mathematics (or meta-mathematics).
However, if it is so, the logic is not guaranteed to be applicable to our world. For instance, (pure)logic does not assure the following famous statement:[♯] Since Socrates is a man and all men are mortal, it follows that Socrates is mortal.
That is, we think that the problem: “Is the [♯] true or not?” should be answered. Thus, the purposeof this chapter is to prove the statement [♯], or more generally, to propose “practical logic”, i.e.,“logic with an interpretation”, 1 which is formulated in the framework of the measurement theory:
PMT = measurement + the relation among systems[Axiom 1 (2.37)]

in C ∗ -algebra[Axiom 2 (3.26)]

.

(7.1)
(=(1.4))Firstly, the symbol “A ⇒ B” (i.e., “implication” ) is deﬁned in terms of measurements. And we
prove the standard syllogism for classical systems:“A ⇒ B, B ⇒ C” implies “A ⇒ C” 2

(7.2)

(This is not trivial, because the (7.2) does not necessarily hold in quantum systems.) We canassert, by “Declaration (1.11)” in §1.4, that this theorem (7.2) guarantees that the above (7.2) (or,the statement [♯]) is “theoretical true”. Several variants may be interesting. For example, under
the condition that “A ⇒ B, B ⇒ C”, we can assert a kind of conclusion such as “C ⇒ A”. For
completeness, “pure logic” and “practical logic” must not be confused. The former is a basic ruleon which mathematics is founded. On the other hand, the latter is a collection of theorems (whose
forms are similar to that of “pure logic” ) in MT. All results in this chapter are due to [41]. Also,
this chapter can be skipped if readers want to study statistics in the framework of SMT ﬁrstly (cf.
Chapters 8).

1

We have no conﬁdence for the naming “practical logic”. We may choose the other namings: “empirical
logic”, “applied logic”, “usual logic”. etc.
2
It is said that the syllogism is said to be, for the ﬁrst time, introduced by Aristotle (B.C.384-B.C.322)

149
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7.1

Measurement, inference, control andpractical logic

The PMT has various aspects.

For example, we believe that three concepts:“measurement”, “inference”, and “control” are diﬀerent aspects of the same thing.
¡¢
Let us explain it as follows: Let MA O ≡ (X, F, F ), S[ρp ] be a measurement formulated
in a C ∗ -algebra A. Note that Axiom 1 can be regarded as the answer to the following
problem:¡¢(M) What kind of measured value is obtained by a measurement MA O, S[ρp ] ?
¡¢¡As mentioned in Chapter 5, the measurement MA O, S[ρp ] is often denoted by MA O,¢
S[∗] , if we want to stress that we do not know the state ρp . Using this notation, we canrespectively characterize “inference (I)” and “control (C)” as follows:¡¢(I) Assume that we get a measured value x( ∈ X) by a measurement MA O, S[∗] .
Then, infer the state [ ∗ ],and¡(C) Assume that we want to get a measured value x( ∈ X) by a measurement MA O,¢
S[∗] . Then, settle the state [ ∗ ].
Of course, Fisher’s maximum likelihood method is one of the answers of the above problems (I) and (C).
Also, we think that(L) “Practical logic” is characterized as “a qualitative theory concerning conditional
probability (cf. §2.5 (IV)) in PMT”.
Thus “practical logic” is also one of the aspects of Axiom 1. Also, since “(practical)
logic” is a qualitative aspect of “inference”, we can say that “(practical) logic” [resp.
“inference”] is used in rough [resp. precise] arguments. For completeness, “pure logic” and“practical logic” must not be confused. The former is a basic rule on which mathematics isbuilt. And thus it is not related to our world. On the other hand, the latter is a collectionof theorems (whose forms are similar to that of “pure logic” ) in PMT. Since practical logic

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is regarded as a theorem in PMT, it automatically possesses the measurement theoreticalinterpretation. That is, we think that“practical logic” = “theorems (whose forms are similar to (pure) logic) in MT”.
Recall, throughout this book, that the measured value set (or, label set) X is assumed
to be ﬁnite if we write (X, 2X , F ) (or, (X, P(X), F ) and not (X, F, F ). In this chapter wealways assume that X is ﬁnite.

7.2

Quasi-product observables with dependence

We begin with the following deﬁnition.
Deﬁnition 7.1. [Marginal observable, quasi-product observable, consistency. (cf. Deﬁnition 2.10.)]. Let A be a C ∗ -algebras. Let K = {1, 2, ..., |K|}.

(i). Consider an observable O ≡ ( ×k∈K Xk , 2×k∈K Xk , F ) (with a label set ×k∈K Xk )in A. Let D be D ⊆ K. An observable OD ≡ ( ×k∈D Xk , 2×k∈D Xk , FD ) in A is calleda D-marginal observable of O if it satisﬁes that³
FD ( × Ξk ) = F ( × Ξk ) ×(
k∈D

k∈D

´

× Xk ) ,k∈K\D¯
for all Ξk ∈ 2Xk , k ∈ D. Also this OD is denoted by O¯D . Here note that the marginal¯
gDobservable O¯ is equal to the image observable O[g ] where ×k∈K Xk ∋ (xk )k∈K 7−→
D(xk )k∈D ∈ ×k∈D Xk .

D(ii). For each k ∈ K, consider an observable Ok ≡ (Xk , 2Xk , Fk ) in A. If there exists an¯
observable OK ≡ ( ×k∈K Xk , 2×k∈K Xk , F ) in A such that OK ¯
= Ok for all k ∈ K,{k}then [Ok : k ∈ K] is called consistent. Also, this OK is called a quasi-product observable ofK
KFk ), or ×k∈K
Ok
[O : k ∈ K], and is sometimes denoted by ( ×k∈K Xk , 2×k∈K Xk , ×k∈K
´
³ k
qp
qp
or, ( ×k∈K Xk , 2×k∈K Xk , × k∈K Fk ), or × k∈K Ok .

OO¥Note that the consistency of observables [(Xk , 2Xk , Fk ) : k ∈ K] in A is not guaranteed
in general. If the commutativity condition:
Fk1 (Ξk1 )Fk2 (Ξk2 ) = Fk2 (Ξk2 )Fk1 (Ξk1 ) (∀Ξk1 ∈ 2Xk1 , ∀Ξk2 ∈ 2Xk2 , k1 ̸= k2 )Sample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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holds, then we can construct a quasi-product observable O ≡ ( ×k∈K Xk , 2×k∈K Xk , F ≡×Ok∈K Fk ) such that:

F (Ξ1 × Ξ2 × · · · × Ξ|K| ) = F1 (Ξ1 )F2 (Ξ2 ) · · · F|K| (Ξ|K| ).
It is, of course, the case that the uniqueness is not guaranteed even under the above
commutativity condition.
Remark 7.2. [Only one measurement is permitted (cf. §2.5. Remarks (II))]. If we want
the data concerning both O1 and O2 for the system S[ρp ] , we must take a simultaneous
measurement MA (O12 ≡ O1 × 12 O2 , S[ρp ] ). Therefore, if a quasi-product observableO

O12 does not exist (i.e., [O1 , O2 ] is not consistent), the concept of “the data concerning
O1 and O2 for the system S[ρp ] ” is nonsense, i.e., it has no reality. This is a prevalent
notion in quantum theory as in the case that the concept “the momentum and positionof a particle” or “the trajectory of a particle” is meaningless in quantum theory. (For the
recent results, see [37, 40].)

It should be emphasized that the importance of this spirit(i.e., “the consistency of [O1 , O2 ]” ⇔ “the reality of data”) is essential.
¥

As the classical PMT is rather easy, people tend to overlook important facts in classicalsystems. Since quantum theory is moderately diﬃcult, it is rather handy compared toclassical theory.
Let X = {x1 , x2 , ...., xJ }. Let O ≡ (X, 2X , F ) be an observable in a commutative C ∗ algebra A (hence by Gelfand theorem, we can assume that A = C(Ω)). We can consider
the following identiﬁcation:(X, 2X , F ) ←→h

[F ({xj })](ω) : j = 1, 2, ..., Ji(where F ({xj }) ≡ [F ({xj })] ∈ C(Ω)), and therefore denotehiXjRep[O] = Rep[(X, 2 , F )] = [F ({x })](ω) : j = 1, 2, ..., J .
It is clear that0 ≤ [F ({x })](ω) ≤ 1 andjJX[F ({xj })](ω) = 1(∀ω ∈ Ω),

j=1

that is, Rep[(X, 2X , F )] is considered to be the resolution of the identity (cf. §2.3).

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Consider two observables O1 ≡ (X1 , 2X1 , F1 ) and O2 ≡ (X2 , 2X2 , F2 ) in C(Ω) suchthat:
X1 = {x11 , x21 , ..., xJ1 1 }and

X2 = {x12 , x22 , ..., xJ2 2 }.

Let O12 ≡ (X1 × X2 , 2X1 × X2 , F ≡ F1 × 12 F2 ) be a quasi-product observable with theOmarginal observables O1 and O2 . (The existence of O12 is guaranteed by Theorem 2.11
since C(Ω) is commutative.) Put[F ({(x11 , x12 )})](ω) [F ({(x11 , x22 )})](ω)
 [F ({(x2 , x1 )})](ω) [F ({(x2 , x2 )})](ω)
1
2
1
2

Rep[O12 ] = 
..
..

.
.
J1
J1
1
[F ({(x1 , x2 )})](ω) [F ({(x1 , x22 )})](ω)

[F ({(x11 , xJ2 2 )})](ω)[F ({(x21 , xJ2 2 )})](ω) 

.
..

.
J1
J2
. . . [F ({(x1 , x2 )})](ω)

...
...
..
.

Let X = {x1 , x2 , ...., xJ }. Let O ≡ (X, 2X , F ) be an observable in a C ∗ -algebra A. PutST
X = Ξy Ξn (where Ξy Ξn = ∅). Deﬁne the map g : X → X(2) ≡ {y, n} such thatg(x) = y (if x ∈ Ξy ), = n ( if x ∈ Ξn ). Here we can deﬁne the two-valued observable(X(2) ≡ {y, n}, 2X(2) , F(2) ) in A as the image observable O[g] . This two-valued observable
is also called yes-no observable or 1 − 0 observable. The following lemma says about theconditions that a quasi-product observable of yes-no observables should satisfy.
Lemma 7.3. [The existence condition of yes-no quasi-product observable]. Consider yesno observables O1 ≡ (X1 , 2X1 , F1 ) and O2 ≡ (X2 , 2X2 , F2 ) in a commutative C ∗ -algebraC(Ω) such that:
X1 = {y1 , n1 }and

X2 = {y2 , n2 }.

Let O12 ≡ (X1 × X2 , 2X1 × X2 , F ≡ F1 × 12 F2 ) be a quasi-product observable with theOmarginal observables O1 and O2 .
Put·¸[F ({(y1 , y2 )})](ω) [F ({(y1 , n2 )})](ω)
Rep[O12 ] =[F ({(n1 , y2 )})](ω) [F ({(n1 , n2 )})](ω)·¸α(ω)[F1 ({y1 })](ω) − α(ω)
=
,
(7.3)
[F2 ({y2 })](ω) − α(ω) 1 + α(ω) − [F1 ({y1 })](ω) − [F2 ({y2 })](ω)³
where α ∈ C(Ω). Note that [F ({(y1 , y2 )})](ω) + [F ({(y1 , n2 )})](ω) = [F1 ({y1 })](ω) and´[F ({(y1 , y2 )})](ω) + [F ({(n1 , y2 )})](ω) = [F2 ({y2 })](ω) .
That is,Sample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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1 + α(ω) − [F1 ({y1 })](ω) − [F2 ({y2 })](ω)

Then, it holds that
max{0, [F1 ({y1 })](ω) + [F2 ({y2 })](ω) − 1} ≤ α(ω) ≤ min{[F1 ({y1 })](ω), [F2 ({y2 })](ω)}(∀ω ∈ Ω).

(7.4)

Conversely, for any α ( ∈ C(Ω)) that satisﬁes (7.4), the observable O12 deﬁned by (7.3)
is a quasi-product observable with the marginal observables O1 and O2 . Also, note that
[F ({(y1 , n2 )})](ω) = 0 ⇔ α(ω) = [F1 ({y1 })](ω) ⇒ [F1 ({y1 })](ω) ≤ [F2 ({y2 })](ω).
(7.5)
Proof. Though this lemma is easy, we add a brief proof for completeness. Since 0 ≤[F ({(x11 , x22 )})] (ω) ≤ 1, (∀x1 , x2 ∈ {y, n}), we see, by (7.3), that0 ≤ α(ω) ≤ 1,

0 ≤ [F1 ({y1 })](ω) − α(ω) ≤ 1,

0 ≤ [F2 ({y2 })](ω) − α(ω) ≤ 1,

0 ≤ 1 + α(ω) − [F1 ({y1 })](ω) − [F2 ({y2 })](ω) ≤ 1,

(7.6)

which clearly implies (7.4). Conversely. if α satisﬁes (7.4), then we easily see (7.6). Also,(7.5) is obvious. This completes the proof.

Next we provide several examples, which will promote a understanding of our theory.
Example 7.4. [Tomatoes’ example]. Let Ω = {ω1 , ω2 , ...., ωN } be a set of tomatoes,
which is regarded as a compact Hausdorﬀ space with the discrete topology. Consider
yes-no observables ORD ≡ (XRD , 2XRD , FRD ) and OSW ≡ (XSW , 2XSW , FSW ) in C(Ω) suchthat:
XRD = {yRD , nRD } and XSW = {ySW , nSW },where we consider that “yRD ” and “nRD ” respectively mean “RED” and “NOT RED”.
Similarly, “ySW ” and “nSW ” respectively mean “SWEET” and “NOT SWEET”.
For example, the ω1 is red and not sweet, the ω2 is red and sweet, etc. as follows.

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···

ω1

ω2

ω3

ωN

yRD

yRD

nRD

···

nRD

nSW

ySW

ySW

···

nSW

We see that(∗) the probability that xRD ( ∈ XRD ≡ {yRD , nRD }), the measured value by the measurement MC(Ω) (ORD , S[δωn ] ), belongs to ΞRD ( ⊆ XRD ≡ {yRD , nRD }) is given byδωn (FRD (ΞRD )) (= [FRD (ΞRD )](ωn ) ) .
£¤That is, the probability that the tomato ωn is observed as “RED” resp. “NOT RED”£¤
is given by [FRD ({yRD })] (ωn ) resp. [FRD ({nRD })] (ωn ) . (Continued to Example 7.5).
¥
Example 7.5. [Tomatoes’ example; continued from Example 7.4]. Consider the quasiproduct observable as follows:
O = (XRD × XSW , 2XRD × XSW , F ≡ FRD × FSW ),Othat is,·¸[F ({(yRD , ySW )})](ω) [F ({(yRD , nSW )})](ω)
Rep[O] =[F ({(nRD , ySW )})](ω) [F ({(nRD , nSW )})](ω)·¸α(ω)[FRD ({yRD })](ω) − α(ω)=[FSW ({ySW })](ω) − α(ω) 1 + α(ω) − [FRD ({yRD })](ω) − [FSW ({ySW })](ω)
where α(ω) satisﬁes (7.4). Hence by Axiom 1, when we observe that the tomato ωn is“RED”, we can see that the probability that the tomato ωn is “SWEET” is given by[F ({(yRD , ySW )})](ωn ).
[F ({(yRD , ySW )})](ωn ) + [F ({(yRD , nSW )})](ωn )³´For the conditional probability, see §2.5(IV). Here note that (7.7) implies ;“[F ({(yRD , nSW )})](ωn ) = 0”

if and only if“RED” ⇒ “SWEET” ,Sample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

(7.7)

(7.8)

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which is also clearly equivalent to “NOT SWEET” ⇒ “NOT RED”.

¥
Being motivated by the above (7.8), we introduce the following deﬁnition of “implication” as a general form which is applicable to classical and quantum systems.
Deﬁnition 7.6. [Implication]. Let O1 ≡ (X1 , 2X1 , F1 ) and O2 ≡ (X2 , 2X2 , F2 ) be observables (not necessarily two-valued observables ) in a C ∗ -algebra A. Let O12 = (X1 × X2 ,2X1 × X2 , F1 × 12 F2 ) be a quasi-product observable of O1 and O2 . Let ρp ∈ Sp (A∗ ).
O

Let Ξ1 ∈ P(X1 ) and Ξ2 ∈ P(X2 ). Then, the condition´ρ (F1 × F2 )(Ξ1 ×(X2 \ Ξ2 )) = 0
p

³

O12

(7.9)

is denoted by
OΞ1 1

=⇒
MA (O12 ,S[ρp ] )OΞ2 2 .

(7.10)¥

Remark 7.7.

[Contraposition]. Assume that we get a measured value (x1 , x2 ) ( ∈X1 × X2 ) by the measurement MA (O12 , S[ρp ] ). And assume the condition (7.10). If weknow that x1 ∈ Ξ1 , then we can assure that x2 ∈ Ξ2 . Also, (7.9) is of course also equal to
X \Ξ1

O1 1

⇐=

MA (O12 ,S[ρp ] )

X \Ξ2

O2 2

since O12 = O{1,2} = O21 (i.e., K = {1, 2} is not regarded as
X \Ξ1

an ordered set). That is, “O1 1

7.3

⇐=

MA (O12 ,S[ρp ] )

X \Ξ2

O2 2

” is the contraposition of (7.10).
¥

Consistency and syllogism

In this section we study the consistent condition for observables. We show several
theorems of practical syllogisms (i.e., theorems concerning “implication” in Deﬁnition
7.6).

7.3.1

Consistent condition

Though we are not concerned with quantum theory in this chapter, our investigations
for classical systems become clearer in comparison with quantum theory. Therefore, theSample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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157

following deﬁnitions (Deﬁnitions 7.8 and 7.9) are common in both classical and quantum
theory.
Let A be a C ∗ -algebra.

For each k ∈ K ≡S{1, 2, ..., |K|−1, |K|}, consider a label set Xk . Consider D (⊆ 2K ) such that D∈D D = K.
Deﬁnition 7.8.

[Covering family].

Then, G ≡ [ OD ≡ ( ×k∈D Xk , 2×k∈D Xk , FD ) : D ∈ D ] is called a covering family ofobservables in A, if it satisﬁes the following condition:¯¯
OD1 ¯D1 ∩D2 = OD2 ¯D1 ∩D2(∀D1 , ∀D2 ∈ D such that D1 ∩ D2 ̸= ∅).

¯¯Note that, if G is a covering family, it holds that OD1 ¯{k} = OD2 ¯{k} for any k ∈ K and any
T
D1 , D2 ∈ D such that k ∈ D1 D2 . Thus, a covering family of observables G determines
a unique {k}-marginal observable Ok ≡ (Xk , 2Xk , Fk ) for each k ∈ K.
¥
The following deﬁnition is a generalization of Deﬁnition 7.1 (i.e., the case that D ={{1}, {2}, ..., {|K|}}).
Deﬁnition 7.9. [Consistent condition]. Let A be a C ∗ -algebra. A covering family ofobservable G ≡ [ OD ≡ ( ×k∈D Xk , 2×k∈D Xk , FD ) : D ∈ D ( ⊆ 2K) ] in A is called
consistent, if there exists an observable OK ≡ ( ×k∈K Xk , 2×k∈K Xk , F ) in A such that:¯OK ¯D = OD(∀D ∈ D).

(7.11)

Also, the above relation (7.11) is denoted by[OD : D ∈ D] < OK .

(7.12)¥

Remark 7.10. [Consistent condition]. Under the condition (7.12), the data concerning
G ≡ [OD : D ∈ D] for the system S[ρp ] is obtained by the simultaneous measurement
MA (OK , S[ρp ] ). Thus, a covering family G has no reality, if it is not consistent. Recall
the arguments in Remark 7.2, which correspond to the above deﬁnition for the case thatD = {{1}, {2}}.
¥Lemma 7.11. [Consistent condition]. Let A be a C ∗ -algebra. Let G1 ≡ [ O1D1 : D1 ∈ D1
( ⊆ 2K ) ] be a covering family of observables in A. And let G2 ≡ [ O2D2 : D2 ∈ D2 ( ⊆ 2K)] be a consistent covering family of observables in A. Assume that for any D1 ∈ D1 there

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exists an D2 ( ∈ D2 ) such that:
D1 ⊆ D2¯
and O1D1 = O2D2 ¯D1 .

(7.13)

Then, G1 is consistent.
Proof. Since a covering family G2 is consistent, there exists an observable OK ≡¯
( ×k∈K Xk , 2×k∈K Xk , FK ) in A such that O2D2 = OK ¯D2 (∀D2 ∈ D2 ). Let D1 be any¯
element in D1 . Then, by choosing D2 ( ∈ D2 ) satisfying (7.13), we see that O1D1 = O2D2 ¯D1¯ ¯¯= (OK ¯ )¯ = OK ¯ . This completes the proof.
D2

D1

D1

Lemma 7.12. [Consistent condition and quasi-product observables]. Let A be a commutative C ∗ -algebra (i.e., A = C(Ω)). Let D12 and D23 be subsets of K. Put D123 ≡
S
T
T
S
S
D12 D23 ≡ (D12 \ D23 )(D12 ∩ D23 )(D23 \ D12 ) ≡ D1 D2 D3 . Consider thefollowing observables in C(Ω) :OD12 ≡ ( × Xk , P( × Xk ), FD12 ) and OD23 ≡ ( × Xk , P( × Xk ), FD23 )
k∈D12

k∈D12

k∈D23

k∈D23

¯
¯
such that OD12 ¯D2 = OD23 ¯D2 . Then, there exists an observable OD123 ≡ ( ×k∈D123 Xk ,¯¯
P( ×k∈D123 Xk ), FD123 ) such that OD123 ¯D12 = OD12 and OD123 ¯D23 = OD23 .
T
T
Proof. Assume that D12 D23 ̸= ∅. (If D12 D23 = ∅, this lemma is trivial. Put
Q
1
2
jm
Mm
Ym = ×k∈Dm Xk = {ym, ym, ..., ym, ..., ym}, m = 1, 2, 3. (So, Mm = k∈Dm |Xk |.)Thus, we can put, by Y1 × Y2 = ×k∈D12 Xk and Y2 × Y3 = ×k∈D23 Xk , that
OD12 = (Y1 × Y2 , P(Y1 × Y2 ), F12 ≡ FD12 )andOD23 = (Y2 × Y3 , P(Y2 × Y3 ), F23 ≡ FD23 ).

Deﬁne the observable OD123 ≡ ( ×m=1 Ym , P( ×m=1 Ym ), F123 ) in C(Ω) such that:3[F123 ({(y1j1 , y2j2 , y3j3 )})](ω)
(
j1 j2

3

j

j

[F12 ({(y1 ,y2 )})](ω)·[F23 ({(y22 ,y33 )})](ω)
j[F2 ({y22 })](ω)=
0if [F2 ({y2j2 })](ω) ̸= 0
if [F2 ({y2j2 })](ω) = 0

for 1 ≤ ∀j1 ≤ M1 , 1 ≤ ∀j2 ≤ M2 , 1 ≤ ∀j3 ≤ M3 . Therefore, it is clear that this lemma¯= OD23 is easily seen as follows:
holds. For example, OD123 ¯
D23

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/[F123 (Y1 ×{(y2j2 , y3j3 )})](ω) =X

159[F123 ({(y1j1 , y2j2 , y3j3 )})](ω)

jy11 ∈Y1=

X [F12 ({(y j1 , y j2 )})](ω) · [F23 ({(y j2 , y j3 )})](ω)
1jy11 ∈Y1

=

2

2

3

[F2 ({y2j2 })](ω)[F12 (Y1 ×{y2j2 })](ω) · [F23 ({(y2j2 , y3j3 )})](ω)[F2 ({y2j2 })](ω) · [F23 ({(y2j2 , y3j3 )})](ω)=[F2 ({y2j2 })](ω)[F2 ({y2j2 })](ω)= [F23 ({(y2j2 , y3j3 )})](ω)(∀ω ∈ Ω, 1 ≤ ∀j2 ≤ M2 , 1 ≤ ∀j3 ≤ M3 ).

This completes the proof.
³
The following theorem is a kind of generalization of Theorem 2.11 which essentially´corresponds to the result for D = {{1}, {2}, ..., {|K|}} in the following theorem . Herenote that a covering family [OD : D ∈ D] is equivalent to [OD′ : D′ ∈ {D′ : D′ ⊆ D for¯some D ∈ D}] where OD′ = OD ¯ ′ for any D′ such that D′ ⊆ D.
D

Theorem 7.13. [Consistent condition and quasi-product observables]. Let D = {{1, 2},{2, 3}, ..., {|K| − 1, |K|}} ( ⊆ 2K ). Let G = [OD = ( ×k∈D Xk , 2×k∈D Xk , FD ) : D ∈ D] be³
a covering family of observables in a commutative C ∗ -algebra C(Ω). Here we can put G= [ Ok,k+1 ≡ (Xk × Xk+1 , P(Xk × Xk+1 ), Fk,k+1 ≡ Fk × k,k+1 Fk+1 ) : k = 1, 2, ..., |K|−1´].
Then, G = [ Ok,k+1 : k = 1, 2, ..., |K| − 1 ] is consistent.
O

Proof. Put D12 = {1, 2} and D23 = {2, 3}. By Lemma 7.12, we get O123 ( = OD123 )such that G3 = [O123 , O34 , O45 , ..., O|K|−1,|K| ] is a covering family in C(Ω) where O12 =¯¯O123 ¯
and O23 = O123 ¯. Iteratively, we get G|K|−1 = [O123···|K|−1 , O|K|−1,|K| ] and
{1,2}

{2,3}

G|K| = [O123···|K|−1,|K| ] ≡ [OK ], which is clearly consistent. So, by Lemma 7.11, we seethat G|K|−1 < OK . Therefore, we iteratively get G < OK . This completes the proof.
Remark 7.14. [Quantum PMT]. This theorem is due to the commutativity of a C ∗ algebra C(Ω). In general (particularly in quantum systems, i.e., A = C(V )), there exists
no O123 such that [O12 , O23 ] < O123 (i.e., [O12 , O23 ] is not consistent in general). Thus,
we have no simultaneous measurement MA (O123 , S[ρp ] ). Therefore, in general, we can
not get information (i.e., data) concerning the covering family [O12 , O23 ] for the quantum
system S[ρp ] . That is, in general, the covering family [O12 , O23 ] has no reality in quantum
mechanics.
¥

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The following notation is the preparation for Theorems 7.19 and 7.23.

Notation 7.15. [Preparation for Theorems 7.19 and 7.23]. Let G = [O12 , O23 , ...O|K|−1,|K| ]≡ [ (Xk × Xk+1 , P(Xk × Xk+1 ), Fk,k+1 ≡ Fk × k,k+1 Fk+1 ) : k = 1, 2, ..., |K| − 1 ] be aOcovering family of observables in a commutative C ∗ -algebra C(Ω). (So, G is consistent asin Theorem 7.13). Suppose that Xk = {yk , nk } for each k ∈ K. As in Deﬁnition 7.8, puthi hiRep[Ok ] = Rep[(Xk , 2Xk , Fk )] = [Fk ({yk })](ω), [Fk ({nk })](ω)] ≡ p1k (ω), p0k (ω)
for all k = 1, 2, 3, ..., |K|. And put
Rep[Ok,k+1 ] = Rep[(Xk × Xk+1 , 2Xk × Xk+1 , Fk,k+1 )]·¸[Fk,k+1 ({yk } ×{yk+1 })](ω) [Fk,k+1 ({yk } ×{nk+1 })](ω)=[Fk,k+1 ({nk } ×{yk+1 })](ω) [Fk,k+1 ({nk } ×{nk+1 })](ω)· 11¸
pk,k+1 (ω) p10
k,k+1 (ω)≡ 01
pk,k+1 (ω) p00
k,k+1 (ω)¸·
p11
p1k (ω) − p11
k,k+1 (ω)
k,k+1 (ω)
≡ 1
11
1
1
pk+1 (ω) − p11
k,k+1 (ω) 1 + pk,k+1 (ω) − pk (ω) − pk+1 (ω)
for all k = 1, 2, ..., |K| − 1, where p11
k,k+1 (ω) satisﬁes (7.4).
P( ×k∈K Xk ), FK ) be any observable in C(Ω) such that:(7.14)Let OK ≡ ( ×k∈K Xk ,[O12 , O23 , ...O|K|−1,|K| ] < OK .

(7.15)(The existence of OK is guaranteed by Theorem 7.13.) Put#"|K|
h j ,j ,...,ji
1 2
p1,2,...,|K||K| (ω) : j1 , j2 , ..., j|K| = 1, 0 ≡ [FK ( × {xjkk })](ω) : j1 , j2 , ..., j|K| = 1, 0 ,k=1(7.16)
where xjkk = yk (if jk = 1 ) and xjkk = nk (if jk = 0 ). Deﬁne O1,|K| ≡ (X1 × X|K| ,¯P(X1 × X|K| ), F1,|K| ) such that O1,|K| = OK ¯
. Put
{1,|K|}

Rep[O1,|K| ] = Rep[(X1 × X|K| , 2X1 × X|K| , F1,|K| )]·¸[F1,|K| ({y1 } ×{y|K| })](ω) [F1,|K| ({y1 } ×{n|K| })](ω)=[F1,|K| ({n1 } ×{y|K| })](ω) [F1,|K| ({n1 } ×{n|K| })](ω)
¸
· 11
¸ ·
p1,|K| (ω) p10
p11
p11 (ω) − p11
1,|K| (ω)
1,|K| (ω)
1,|K| (ω)
.
≡ 01
≡ 1
1
11
1
p|K| (ω) − p11
p1,|K| (ω) p00
1,|K| (ω) 1 + p1,|K| (ω) − p1 (ω) − p|K| (ω)
1,|K| (ω)(7.17)

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161(Continued to Lemmas 7.16 and 7.17 and Theorem 7.19 for K = {1, 2, 3}, and to Theorem
7.23 for general case).
¥
Lemma 7.16. [Continued from Notation 7.15]. Under Notation 7.15 for K = {1, 2, 3},
j1 j2 j3
1 j2 j3
101
we see, (putting pj123= p123(ω) in (7.16), p111
123 = A and p123 = B),

p111
123 = A(ω),

11p011
123 = p23 − A(ω),

11p110
123 = p12 − A(ω),

1
11
11
p010
123 = p2 − p12 − p23 + A(ω),

p101
123 = B(ω),

1
11
p001
123 = p3 − p23 − B(ω),

1
11
p100
123 = p1 − p12 − B(ω),

11
1
1
1
11
p000
123 = 1 − p1 − p2 − p3 + p12 + p23 + B(ω),(7.18)

where
11
11
11
max{0, −p12 (ω) + p11
12 (ω) + p23 (ω)} ≤ A(ω) ≤ min{p12 (ω), p23 (ω)}(7.19)

and11
max{0, p11 (ω) + p12 (ω) + p13 (ω) − p11
12 (ω) − p23 (ω) − 1}
1
11
≤ B(ω) ≤ min{p11 (ω) − p11
12 (ω), p3 (ω) − p23 (ω)}.

(7.20)

Proof. From (7.16), (7.15) and (7.14) for K = {1, 2, 3}, we see
110
11
p111
123 + p123 = p12 ,

100
10
1
11
p101
123 + p123 = p12 = p1 − p12 ,

010
01
1
11
p011
123 + p123 = p12 = p2 − p12 ,

000
00
11
1
1
p001
123 + p123 = p12 = 1 + p12 − p1 − p2 ,

011
11
p111
123 + p123 = p23 ,

010
10
1
11
p110
123 + p123 = p23 = p2 − p23 ,

001
01
1
11
p101
123 + p123 = p23 = p3 − p23 ,

000
00
11
1
1
p100
123 + p123 = p23 = 1 − p23 − p2 − p3 .

1 j2 j3
After a small computation, we get (7.18). Since 0 ≤ pj123(ω) ≤ 1, we see, from (7.18),

that
0 ≤ A ≤ 1,11
p11
23 − 1 ≤ A ≤ p23 ,11
p11
12 − 1 ≤ A ≤ p12 ,

11
1
11
11
− p12 + p11
12 + p23 ≤ A ≤ 1 − p2 + p12 + p23 ,0 ≤ B ≤ 1,

111
p13 − p11
23 − 1 ≤ B ≤ p3 − p23 ,111
p11 − p11
12 − 1 ≤ B ≤ p1 − p12 ,

11
11
1
1
1
11
p11 + p12 + p13 − p11
12 − p23 − 1 ≤ B ≤ p1 + p2 + p3 − p12 − p23 .

This implies (7.19) and (7.20). This completes the proof.

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Lemma 7.17. [Continued from Notation 7.15]. Under Notation 7.15 for K = {1, 2, 3},
we see11max{0, −p12 (ω) + p11
12 (ω) + p23 (ω)}
11
+ max{0, p11 (ω) + p12 (ω) + p13 (ω) − p11
12 (ω) − p23 (ω) − 1}

≤ p11
13 (ω)(7.21)

11
1
11
1
11
≤ min{p1112 (ω), p23 (ω)} + min{p1 (ω) − p12 (ω), p3 (ω) − p23 (ω)}.

(7.22)

111
101
Proof. Since p11
13 (ω) = p123 (ω)+ p123 (ω) = A(ω) + B(ω) in Lemma 7.16, by (7.19) and(7.20) we can easily get (7.21) and (7.22). This completes the proof.
Remark 7.18. [Comparison]. Let us compare the result in Lemma 7.17 with the result(7.4) in Lemma 7.3 (i.e., the result without consistent condition). Note that (7.4) implies
1
1
C1 ≡ max{0, p11 (ω) + p13 (ω) − 1} ≤ p11
13 (ω) ≤ min{p1 (ω), p3 (ω)} ≡ C2 .

Here we can easily see that C1 ≤ (7.21) and (7.22) ≤ C2 from the following trivial
inequalities:
max{0, α1 + α2 } ≤ max{0, max{0, α1 } + max{0, α2 }} = max{0, α1 } + max{0, α2 }andmin{α1 , α2 } + min{α3 , α4 } = min{ α1 + α3 , α1 + α4 , α2 + α3 , α2 + α4 }≤ min{ α1 + α3 , α2 + α4 } .
Therefore, we see in Lemma 7.17 that the value p11
13 (ω) is restricted under the consistentcondition of [O12 , O23 ].
¥

7.3.2

Practical syllogism

Now we show several theorems of practical syllogisms (i.e., theorems concerning “implication” in Deﬁnition 7.6) as the consequences of our arguments.
Theorem 7.19. [Practical syllogism, [41]]. Assume Notation 7.15 for K = {1, 2, 3}.
That is, [O12 , O23 ] is a covering family of observables in a commutative C ∗ -algebra C(Ω).

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163

Let δω0 ∈ Mp+1 (Ω) for any ﬁxed ω0 ∈ Ω. Let O123 (= OK ) be any observable such that¯
[O12 , O23 ] < O123 and let O13 = O123 ¯. (The existence of O123 is guaranteed by{1,3}

Theorem 7.13.) Then we have the following statements [1] ∼ [3]:[1]. Assume that
{y }

O1 1

{y }

O2 2 ,=⇒
MC(Ω) (O12 ,S[δ

[ω0 ]

])

{y }

O2 2

{y }

=⇒
MC(Ω) (O23 ,S[δω ] )O3 3 .

(7.23)

0Then, we see that
¸
· 11
¸ ·
p13 (ω0 ) p10
p11 (ω0 )
013 (ω0 )= 1,
00
p3 (ω0 ) − p11 (ω0 ) 1 − p13 (ω0 )p01
13 (ω0 ) p13 (ω0 )(7.24)

hence, we see that
{y }

O1 1

{y }

=⇒
MC(Ω) (O13 ,S[δω ] )O3 3 .

(7.25)

0[2]. Assume that
{y }

O1 1

{y }

⇐=

MC(Ω) (O12 ,S[δω ] )O2 2 ,

{y }

O2 2

0

{y }

=⇒
MC(Ω) (O23 ,S[δω ] )O3 3 .

(7.26)

0Then, we see that
· 11
¸ ·
¸
p13 (ω0 ) p10α(ω0 )
p11 (ω0 ) − α(ω0 )
13 (ω0 )= 100
p01
p3 (ω0 ) − α(ω0 ) 1 + α(ω0 ) − p11 (ω0 ) − p13 (ω0 )
13 (ω0 ) p13 (ω0 )wheremax{p12 (ω0 ), p11 (ω0 ) + p13 (ω0 ) − 1} ≤ α(ω0 ) ≤ min{p11 (ω0 ), p13 (ω0 )}.

(7.27)

Also (7.26) is equivalent to
{y }

O2 2

{(y ,y )}=⇒MC(Ω) (O123 ,S[δω ] )O13 1 3 .

(7.28)

0[3]. Assume that
{y }

O1 1

{y }

=⇒
MC(Ω) (O12 ,S[δω ] )0

O2 2 ,

{y }

O2 2

{y }

⇐=

MC(Ω) (O23 ,S[δω ] )O3 3 .

0Then, we see that
¸
¸ ·
· 11
α(ω0 )
p11 (ω0 ) − α(ω0 )p13 (ω0 ) p10
13 (ω0 )= 1
00
p3 (ω0 ) − α(ω0 ) 1 + α(ω0 ) − p11 (ω0 ) − p13 (ω0 )p01
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(7.29)

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where
max{0, p11 (ω0 ) + p13 (ω0 ) − p12 (ω0 )} ≤ α(ω0 ) ≤ min{p11 (ω0 ), p13 (ω0 )}.

(7.30)

Also (7.29) is equivalent to{(y ,y ),(y1 ,n3 ),(n1 ,y3 )}O13 1 3{y }=⇒MC(Ω) (O123 ,S[δω ] )O2 2 .

(7.31)

0

10
11
1
1
11
Proof. [1]. By (7.23) and (7.5), we see that p10
12 = p23 = 0, so, p12 = p1 ≤ p2 = p23 ≤11
11
p13 . Therefore, we see that (7.21) = p11
12 + max{0, p3 − 1} = p1 . And (7.22) = p1 + 0 = p1 .
1
10This implies that p11
13 = p1 , i.e., (7.24). Also, (7.25) follows from p13 = 0.
10
11
1
1
11
1
1
[2]. By (7.26) and (7.5), we see that p01
12 = p23 = 0, so, p12 = p2 ≤ p1 and p23 = p2 ≤ p3 .
1
1
1
1 1
1
Therefore, we see that (7.21) = p11
23 + max{0, p1 − p2 + p3 − 1} = max{p2 , p1 + p3 − 1}. And(7.22) = min{p12 , p12 } + min{p11 − p12 , p13 − p12 } = min{p11 , p13 }. This implies (7.27). Also, we
10
010
011
110
see that (7.26) ⇔ p01
12 = p23 = 0 ⇔ p123 = p123 = p123 = 0 ⇔ (7.28).
01
11
1
1
11
1
1
[3]. By (7.29) and (7.5), we see that p10
12 = p23 = 0, so, p12 = p1 ≤ p2 and p23 = p3 ≤ p2 .

Therefore, we see that (7.21) = max{0, p11 −p12 +p13 } + max{0, p12 −1} = max{0, p11 −p12 +p13 }.
01
101
And (7.22) = min{p11 , p13 }. This implies (7.30). Also, (7.29) ⇔ p10
12 = p23 = 0 ⇔ p123 =001
p100
123 = p123 = 0 ⇔ (7.31). This completes the proof.

Remark 7.20. [Practical logic and pure logic]. The reader must not confuse the result(for example, (7.23)⇒(7.25)) in Theorem 7.19 with pure logic (i.e., mathematical logic).
Theorem 7.19 is a consequence of Axiom 1. Note that Theorem 7.19 is due to Theorem
7.13, i.e., the commutativity of C ∗ -algebra C(Ω). That means the results in Theorem 7.19
can not be expected in quantum systems. In comparison with quantum theory, Theorem
7.19 becomes clearer. For example, in general, the syllogism is meaningless in quantum
systems. This is easily shown as follows. Put V = C5 , and A = B(C5 ). And
 
 
 
 
 
0
0
0
0
1
0
0
0
1
0
 
 
 
 
 
 , ⃗e2 = 0 , ⃗e3 = 1 , ⃗e4 = 0 , ⃗e5 = 0 ,0⃗e1 = 
 
 
 
 
 
0
1
0
0
0
1
0
0
0
0
and put f⃗4 = √⃗e42 + √⃗e52 , f⃗5 = √⃗e42 − √⃗e52 . Deﬁne the three observables O1 ≡ (X1 ≡{a1 , b1 , c1 }, 2X1 , F1 ), O2 ≡ (X2 ≡ {a2 , b2 , c2 }, 2X2 , F2 ) and O3 ≡ (X3 ≡ {a3 , b3 , c3 }, 2X3 , F3 )such that
F1 ({a1 }) = |⃗e1 ⟩⟨⃗e1 |,F1 ({b1 }) = |⃗e2 ⟩⟨⃗e2 | + |⃗e3 ⟩⟨⃗e3 | + |⃗e4 ⟩⟨⃗e4 |,F1 ({c1 }) = |⃗e5 ⟩⟨⃗e5 |,Sample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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F2 ({a2 }) = |⃗e1 ⟩⟨⃗e1 | + |⃗e2 ⟩⟨⃗e2 |,F2 ({b2 }) = |⃗e3 ⟩⟨⃗e3 |,F2 ({c2 }) = |⃗e4 ⟩⟨⃗e4 | + |⃗e5 ⟩⟨⃗e5 |,F3 ({b3 }) = |f⃗4 ⟩⟨f⃗4 |,

F3 ({a3 }) = |⃗e1 ⟩⟨⃗e1 | + |⃗e2 ⟩⟨⃗e2 | + |⃗e3 ⟩⟨⃗e3 |,165

F3 ({c3 }) = |f⃗5 ⟩⟨f⃗5 |.

Note that O1 and O2 [resp. O2 and O3 ] commute. Let O12 = (X1 ×X2 , 2X1 × X2 , F1 × F2 )be the product observable of O1 and O2 . And let O23 = (X2 × X3 , 2X2 × X3 , F2 × F3 ) be

the product observable of O2 and O3 . Let ρp be any pure state ( i.e., ρp ∈ Sp (B(C5 )∗ )).
Then, we have
{a }

O1 1

{a }

=⇒
MA (O12 ,S[ρp ] )O2 2 ,

{a }

O2 2

=⇒
MA (O23 ,S[ρp ] ){a }O3 3 .

since we see³´ρp (F1 × F2 )({a1 } ×({b2 , c2 })) = 0,³´
ρp (F1 × F2 )({a2 } ×({b3 , c3 })) = 0.

However, it should be noted that we have no product observable of O1 , O2 and O3 . Thus,
the implication:
{a }

O1 1

=⇒
MA (O13 ,S[ρp ] )

{a }

O3 3

is nonsense since O13 can not be deﬁned.
¥
Example 7.21. [Continued from Example 7.4, [41]]. Let Ω, C(Ω), O1 ≡ OSW ≡ (XSW ,2XSW , FSW ) and O3 ≡ ORD ≡ (XRD , 2XRD , FRD ) be as in Example 7.4. Let O2 ≡ ORP ≡(XRP , 2XRP , FRP ) be an observable in C(Ω) such that:
XRP = {yRP , nRP },where “yRP ” and “nRP ” respectively mean “RIPE” and “NOT RIPE”. Puthi
Rep[O1 ] = [FSW ({ySW })](ω), [FSW ({nSW })](ω) ,hiRep[O2 ] = [FRP ({yRP })](ω), [FRP ({nRP })](ω) ,hiRep[O3 ] = [FRD ({yRD })](ω), [FRD ({nRD })](ω) .
For example,

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ω1

ω2

ω3

···

ωN

nSW

ySW

ySW

···

nSW

yRP

nRP

yRP

···

nRP

yRD

yRD

nRD

···

nRD

Consider the following quasi-product observables:
O12 = (XSW × XRP , 2XSW × XRP , F12 ≡ FSW × FRP )O12andO23 = (XRP × XRD , 2XRP × XRD , F23 ≡ FRP × FRD ).
O23

Let δωn ∈ Mp+1 (Ω) for any ﬁxed ωn ∈ Ω. Assume that
{y }

O1 1

{y }

=⇒
MC(Ω) (O12 ,S[δω ] ){y }O2 2 ,

O2 2

0

{y }

=⇒
MC(Ω) (O23 ,S[δω ] )O3 3 .

(7.32)

0Then, we see, by Theorem 7.19 [1], that¸·
[F13 ({ySW } ×{yRD })](ωn ) [F13 ({ySW } ×{nRD })](ωn )Rep[O13 ] =[F13 ({nSW } ×{yRD })](ωn ) [F13 ({nSW } ×{nRD })](ωn )·¸[FSW ({ySW })](ωn )0=.
[FRD ({yRD })](ωn ) − [FSW ({ySW })](ωn ) 1 − [FRD ({yRD })](ωn )(7.33)

So, when we observe that the tomato ωn is “RED”, we can infer, by the fuzzy inferenceMC(Ω) (O13 , S[δωn ] ) (equivalently, MC(Ω) (O31 , S[δωn ] )), the probability that the tomato ωn
is “SWEET” is given by[F13 ({ySW } ×{yRD })](ωn )[FSW ({ySW })](ωn )=.
[FRD ({yRD })](ωn )[F13 ({ySW } ×{yRD })](ωn ) + [F13 ({nSW } ×{yRD })](ωn )(7.34)

Also, when we observe that the tomato ωn is “SWEET”, we can infer, by the fuzzy
inference MC(Ω) (O13 , S[δωn ] ), the probability that the tomato ωn is “RED” is given by[FRD ({yRD })](ωn )[F13 ({ySW } ×{yRD })](ωn )== 1. (7.35)[FRD ({yRD })](ωn )[F13 ({ySW } ×{yRD })](ωn ) + [F13 ({ySW } ×{nRD })](ωn )Note that (7.32) implies (and is implied by)“SWEET” =⇒ “RIPE”

and“RIPE” =⇒ “RED” .

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167(Recall (7.8)). So, it is “reasonable” to reach the conclusion:“SWEET” =⇒ “RED” ,(7.37)³
which is implied by the above (7.35).

Here we are afraid that the most important fact

may be overlooked. For completeness, note that the conclusion “(7.36) ⇒ (7.37)” is a´consequence of Theorem 7.19 (and therefore, our axiom). However, the result (7.34) isdue to the peculiarity of fuzzy inferences. That is, in spite of the fact (7.36), we get theconclusion (7.34) that is somewhat like“RED” =⇒ “SWEET” .

(7.38)

Note that the conclusion (7.37) is not valuable in the market. What we want in themarket is the conclusion such as (7.38) (or (7.34)).
¥
Example 7.22. [Continued from Example 7.21, [41]]. Instead of (7.32), assume that
{y }

O1 1

⇐=

{y }

MC(Ω) (O12 ,S[δωn ] ){y }O2 2 ,

O2 2=⇒
MC(Ω) (O23 ,S[δωn ] ){y }O3 3 .

(7.39)

Assume the notation (7.33). When we observe that the tomato ωn is “RED”, we caninfer, by the fuzzy inference MC(Ω) (O13 , S[δωn ] ), the probability that the tomato ωn is“SWEET” is given by
Q=[F13 ({ySW } ×{yRD })](ωn )[F13 ({ySW } ×{yRD })](ωn ) + [F13 ({nSW } ×{yRD })](ωn )which is, by (7.27), estimated as follows:½¾[FRP ({yRP })](ωn ) [FSW ({ySW })] + [FRD ({yRD })] − 1
max,[FRD ({yRD })](ωn )[FRD ({yRD })](ωn )[FSW ({ySW })](ωn )≤ Q ≤ min{, 1}.
[FRD ({yRD })](ωn )(7.40)(7.41)

Note that (7.39) implies (and is implied by)“RIPE” =⇒ “SWEET”

and“RIPE” =⇒ “RED” .

(7.42)

And note that the conclusion (7.41) is somewhat like“RED” =⇒ “SWEET” .

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Therefore, this conclusion is peculiar to “fuzziness”.

¥
The following theorem is a generalization of the ﬁrst part of Theorem 7.19.
Theorem 7.23. [Standard syllogism, cf. [41]]. Assume Notation 7.15. Let δω0 ∈ Mp+1 (Ω).
Assume that
{y }

Ok k

{y

=⇒
MC(Ω) (Ok,k+1 ,S[δω ] )

k+1
Ok+1

}

(∀k = 1, 2, ...., |K| − 1),(7.44)

0

Let OK be any observable as in Notation 7.15, i.e., G = [O12 , O23 , O34 , ..., O|K|−1,|K| ] <¯OK . Put O1,|K| = OK ¯
. Then, we see that
{1,|K|}¸· 11¸ ·p1,|K| (ω0 ) p10
p11 (ω0 )01,|K| (ω0 )= 1,
Rep[O1,|K| ]at ω0 = 01
p|K| (ω0 ) − p11 (ω0 ) 1 − p1|K| (ω0 )p1,|K| (ω0 ) p00
1,|K| (ω0 )(7.45)

hence, we see that
{y }

O1 1

{y

=⇒
MC(Ω) (O1,|K| ,S[δω ] )}O|K||K| .

(7.46)

0

Proof. Let OK be any observable such that G = [O12 , O23 , O34 , ..., O|K|−1,|K| ] < OK .
¯¯¯¯Thus, we see that [OK ¯, O34 , ..., O|K|−1,|K| ] < OK ¯
. Note that (OK ¯
)¯
=
{1,3}

K\{2}

{1,3}

{m}

Om , m = 1, 3. Also note, by (7.24), that·¸
p11 (ω0 )0]= 1,{1,3} at ω0p3 (ω0 ) − p11 (ω0 ) 1 − p13 (ω0 )¯Rep[OK ¯
{y }

{y }

and therefore O1 1=⇒
O3 3 . Hence, by induction, we see that Rep[O1,|K| ]MC(Ω) (OK |{1,3} ,S[δω ] )0¯≡ Rep[OK ¯{1,|K|} ] = (7.45) at ω = ω0 . This completes the proof.

7.4

Conclusion

It is certain that (pure) logic is merely a kind of rule in mathematics. However, if it
is so, the logic is not guaranteed to be applicable to our world. For instance, (pure) logic
does not assure the truth of the following famous statement:[♯] Since Socrates is a man and all men are mortal, it follows that Socrates is mortal.

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169

That is, we think that the problem: “Is this [♯] (theoretical) true or not?” is unsolved.
Thus, the purpose of this chapter was to prove the [♯], or more generally, to propose“practical logic”, i.e., a collection of theorems (whose forms are similar to that of “purelogic” ) in PMT.
Firstly, the symbol “A ⇒ B” (i.e., “implication” ) is deﬁned in terms of measurements(cf. Deﬁnition 7.6). And we prove the standard syllogism for classical systems:“A ⇒ B, B ⇒ C” implies “A ⇒ C”,(7.47)

which is the same as the above (♯). (This (7.47) is not trivial since it does not necessarily
hold in quantum systems.) We can assert, by “Declaration (1.11)” in §1.4, that PMT
guarantees that the above statement [♯] is true.
Several variants may be interesting. For example, under the condition that “A ⇒ B,
B ⇒ C”, we can assert a kind of conclusion such as “C ⇒ A”. That is,“A ⇒ B, B ⇒ C” implies “C ⇒ A”

in some sense.

(7.48)For completeness, “pure logic” and “practical logic” must not be confused. The former isa basic rule on which mathematics is founded. On the other hand, the latter is a collectionof theorems (whose forms are similar to that of “pure logic” ) in PMT.

7.5
7.5.1

Appendix (Zadeh’s fuzzy sets theory)
What is Zadeh’s fuzzy sets theory?

As mentioned in Chapter 1 (i.e., the footnote below Problem 1.2), one of motivationsof our research is motivated by Zadeh’s fuzzy sets theory. In 1965, L.A. Zadeh proposeda certain system theory, in which a membership function f : Ω → [0, 1], which is asserted
to represent “fuzziness”, plays an important role. The membership function is considered
as a kind of generalization of a characteristic function. Here, the characteristic function
χD of D ( ⊆ Ω) is deﬁned by χD : Ω → {0, 1} such that:½
χD (ω) =1 (ω ∈ D)
0 (ω ∈/ D).

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Consider the identiﬁcation:“characteristic function χD ”←→

“set D”,

which gives us the question “What is the following [♯]?”“membership function f ”

1←→ [♯].

χD (ω)
f (ω)

0

Ω
[♯] ?

D

The [♯] is called a fuzzy set by Zadeh. Thus we think that Zadeh’s fuzzy sets theory has
two aspects [A1 ] and [A2 ] as follows: [A1 ] : membership functions (analytic aspect),Zadeh’s fuzzy sets theory
[A2 ] : fuzzy sets (logical aspect).

(7.49)

Zadeh’s fuzzy sets theory acquired a lot of believers. In fact, his paper [93] is one of
the most cited papers in all ﬁelds of 20th century science. However, his theory seems“fuzzy” rather than “diﬃcult”. Thus, it is natural that the following problem arises:[♯1 ] Is Zadeh’s fuzzy sets theory true or not?
When we examine the problem, we are immediately confronted with the following problem:[♯2 ] What is “true or not”?

Or, if we want to assert “Zadeh’s fuzzy sets theory is true[or not]”, what do we say?
And when we study the problems [♯1 ] and [♯2 ], we immediately notice the fact that wehave not yet the clear answer to even the question: “Is Fisher’s statistics true or not?”. 3
As mentioned in Chapter 1, our research starts from the above questions [♯1 ] and [♯2 ].
And we conclude “Declaration (1.11)” in §1.4 as follows:• MT is entitled to check all theories in theoretical informatics. In other words, wecan, by using MT, introduce the criterion:“true or not” into theoretical informatics.
That is, MT can be regarded as “the Construction of theoretical informatics”.
3

In Chapters 5 and 6, it is proved that Fisher’s statistics is theoretically true.

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Now, consider an observable (X, 2X , F ) in C(Ω). Note that, for any Ξ ( ⊆ X), F (Ξ) is amembership function on Ω. Since F (Ξ) ∈ C(Ω), the F (Ξ), of course, has various analytic
aspects. Also, in this chapter we see that the membership function F (Ξ) has various
logical aspects. Thus, someone may conclude that Zadeh’s fuzzy sets theory (i.e., theanalytic aspect [A1 ] and the logical aspect [A2 ] in (7.49)) is understood in the framework ofmeasurement theory, that is, Zadeh’s fuzzy sets theory is true (cf. “Declaration (1.11)” in§1.4). We may agree with this opinion. In fact, these kinds of aspects [A1 ] and [A2 ] can
not be found in the conventional formulation of system theory (cf. (1.2)) such as“dyn. syst. theor.”= dx(t) dt = f (x(t), u1 (t), t), x(0) = x0

· · · (state equation),( measurement equation).

y(t) = g(x(t), u2 (t), t)(7.50)(=(1.2))That is because the conventional formulation (7.50) does not possess the concept of“observable in the sense of Deﬁnition 2.7”.
The believers of Zadeh’s fuzzy sets theory say too much (cf. [64]). And thus, we haveno ﬁrm answer to the question: “What is the essence of Zadeh’s theory?”.

If we canassume that:³(♯) Zadeh wanted to assert that DST (7.50) and “logic” are closely connected or pre´
cisely, “logic” is one of the aspects of DST (7.50) though the two are, in appearance,
independent,
then we can understand his assertion. That is because in this section we study “logic” inmeasurement theory, which is a kind of generalization of the system theory (7.50). This is
our opinion for Zadeh’s theory. Of course, there may be another opinion, that is, someone
may assert that Zadeh said something much more than the (♯). If it is so, we may notunderstand his theory in the framework of measurement theory.
Recall the arguments in Chapter 1 (particularly, “Declaration (1.11)” in §1.4, tables(1.7) and (1.8)). Now, we have only two options, i.e.,(i) Zadeh’s fuzzy sets theory is characterized as the theory concerning membershipfunctions in measurement theory.
(ii) Zadeh’s fuzzy sets theory is not characterized in measurement theory. Thus anotherfundamental theory (cf. The third mathematical scientiﬁc theory in (1.7)) should
be proposed.

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Although there is a possibility that (ii) is reasonable, that is, Zadeh’s fuzzy sets theory may
be understood in another fundamental theory (cf. The third mathematical scientiﬁc theoryin (1.7)), we should note that the proposal of another fundamental theory is much more
remarkable than the justiﬁcation of Zadeh’s fuzzy sets theory. Thus we choose the (i) even
if the essential part of Zadeh’s assertion (e.g., the scientiﬁc part asserted in [64]) can not
be characterized in MT. Thus we conclude that Zadeh’s assertion can not be completely
understood in measurement theory, i.e.,• Zadeh’s assertion is not completely “theoretical true” (cf. Declaration 1.11), thoughpractical logic somewhat has the property like “fuzzy set”.
This is our present opinion.

7.5.2

Why is Zadeh’s paper cited frequently?

Although we believe that the above argument in §7.5.1 is proper, it does not explain
the reason why Zadeh’s paper is cited frequently. As mentioned before, Zadeh’s paper [93]is one of the most cited papers of all scientiﬁc papers. This is an established fact. This
fact may imply that there is something interesting behind Zadeh’s assertion. Thus, wethink that the question “Why is Zadeh’s paper cited frequently?” is more important than
the question “What is Zadeh’s fuzzy sets theory?”. Thus we shall consider the question:• Why does the term “fuzzy” look attractive?
We think that the reason is that Zadeh’s spirit is regarded as the antithesis of the myth:“Science must be exact, clear, strict, etc”.

This myth seems to be due to Newtonianmechanics (and moreover, theoretical physics), which has been located in the center of allscience. That is, we think that• many people want another science, which is fuzzy, rough, vague, etc.
If it is so, we should recall Table 1.8 (in Chapter 1), which asserts mathematical scienceis classiﬁed as follows: theoretical physics (‘TOE’)

theoretical informatics (measurement theory)..... exact mathematical science..... fuzzy mathematical science.
(7.51)

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If it is true, we can understand the reason why the term “fuzzy” was accepted widely.
Thus we do not deny the following opinion:(♯) “measurement theory” = “fuzzy theory”.

(Cf. [42].) Or, the attractive parts of

Zadeh’s assertions are mostly included in measurement theory.
That is because we believe that(♭) Measurement theory is the very theory that represents the anti-spirit against themyth: “Science must be exact, clear, strict, etc”.
In fact, the terms• fuzzy statement (cf. the footnote below Example 2.16), ready-made, useful or not,subjective, popularity, likes or dislikes, (in “Theoretical informatics of Table (1.8)”)
seem to belong to the category of “fuzziness”. On the other hand, the terms• precise statement (cf. the footnote below Example 2.16), made to order, empiricaltrue or not, objective, truth, (in “Theoretical informatics of Table (1.8)”)
obviously belong to the category of “exactness”.

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Chapter 8
Statistical measurements inC ∗-algebraic formulation
As mentioned in the beginning of Chapter 2, measurement theory (MT) can be classiﬁed into two
subjects, i.e., “(pure) measurement theory (PMT)” and “statistical measurement theory (SMT)”.
That is, PMT (=“(pure) measurement theory”) in Chapters 2 ∼ 7
MT (=“measurement theory”)

SMT (=“statistical measurement theory”) in Chapters 8 ∼(8.1)
PMT is essential, and it is formulated as follows:
PMT = measurement + the relation among systems[Axiom 1 (2.37)]

in C ∗ -algebra[Axiom 2 (3.26)]

.

(8.2)
(=(1.4))

Here it should be noted that the state ρp is always assumed to be pure, i.e., ρp ∈ Sp (A∗ ). In thischapter we study the statistical measurement for a statistical state, i.e., the measurement in thecase that the state is distributed. The distribution (i.e., a statistical state) is represented by amixed state ρm ( ∈ Sm (A∗ )). The Statistical MT (i.e., SMT) is formulated as follows:
SMT = statistical measurement + the relation among systems[Proclaim 1 (8.10)]

in C ∗ -algebra ,(8.3)[Axiom 2 (3.26)]

where Proclaim 1 is characterized as follows:“Proclaim 1” = “Axiom 1” +“statistical state”(8.4)(the probabilistic interpretation of mixed state)Thus, the (8.3) is also rewritten such as
SMT =

PMT
(Axioms 1 and 2)+“statistical state”

in C ∗ -algebra .

(8.5)(the probabilistic interpretation of mixed state)

Therefore it should be noted that there is no SMT without PMT. Also, we add “belief measurement
theory” in §8.6 and “principal components analysis” in §8.7.

175
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8.1

Statistical measurements (C ∗-algebraic formulation)

8.1.1

General theory of statistical measurements¡¢
Axiom 1 (proposed in §2.4) says that the measurement of an observable O ≡ (X, F, F )for the system with the state ρp ( ∈ Sp (A∗ )) induces the sample space (X, F, P ( · ) ≡ρp (F ( · ))). That is, Axiom 1 says symbolically that:“observable”(X,F,F ) in A

and“state”
ρp ∈Sp (A∗ )=⇒measurement“sample space”(X,F,P (·)≡ρp (F (·))).

Here it should be noted that the state must be always pure, i.e., ρp ∈ Sp (A∗ ) in Axiom
1. However we sometimes want to generalize the concept of “state”, i.e., to introduce“statistical state”, which is represented by a mixed state ρm (∈ Sm (A∗ )). That is, we
assert (in Proclaim 1 later) that[♯] “statistical state” = “mixed state” + “probabilistic interpretation”.
(mathematics)

Also, it should be noted that we have already studied “S-states” in Chapter 6, which
is one of the aspects of the statistical state. Although the statistical state has various
aspects, we begin with the following example, which will promote a better understandingof the concept of “statistical state”.
Example 8.1. [Coin-tossing and urn problem]. There are two urns U1 and U2 . The urn
U1 [resp. U2 ] contains 8 white and 2 black balls [resp. 4 white and 6 black balls]. Under
the following identiﬁcation (cf. (5.16) in Example 5.8):
U2 ≈ ω2 ,¢¡≡ {ω1 , ω2 } as the state space. And consider the observable O ≡ (X ≡¢{w, b}, 2{w,b} , F ) in C(Ω) where

we regard Ω¡

U1 ≈ ω1 ,[F ({w})](ω1 ) = 0.8,

[F ({b})](ω1 ) = 0.2,

[F ({w})](ω2 ) = 0.4,

[F ({b})](ω2 ) = 0.6.

U1 ≈ ω1U2 ≈ ω2

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Here consider the following procedures (P1 ) and (P2 ).

(P1 ) One of the two (i.e., ω1 or ω2 ) is chosen by an unfair tossed-coin (Cp,1−p ), i.e.,
Head (100p%) → ω1 , Tail (100(1 − p)%) → ω2

(0 ≤ p ≤ 1).

(8.7)

The chosen urn is denoted by [∗](∈ {ω1 , ω2 }). Here deﬁne the mixed state ν0 (∈Mm+1 (Ω)) such that ν0 = pδω1 + (1 − p)δω2 (i.e., ν0 ({ω1 }) = p, ν0 ({ω2 }) = 1 − p),
which is considered to be “the distribution of [∗]”. Thus we call the ν0 a statisticalstate.
(P2 ) Take one ball, at random, out of the urn chosen by the procedure (P1 ). That is, wetake the measurement MC(Ω) (O, S[∗] ).
Then we have the following question:(Q) Calculate the probability that a measured value “w” [resp. “b”] is obtained by theabove measurement MC(Ω) (O, S[∗] ).
[Answer]. The “measurement” deﬁned in the above (P1 ) and (P2 ) is denoted byMC(Ω) (O, S[∗] ([δω1 ; p] ⊕ [δω2 ; 1 − p])).

(8.8)

This may be called a “probabilistic measurement”, and the symbol [δω1 ; p] ⊕ [δω2 ; 1 − p]may be called a “probabilistic state”. Note that:(i) the probability that [ ∗ ] = δω1 [resp. [ ∗ ] = δω2 ] is given by p [resp. 1 − p].
(ii) If [ ∗ ] = δω1 [resp. if [ ∗ ] = δω2 ], the probability that the measured value obtainedby MC(Ω) (O, S[∗] ) is equal to x ( ∈ {w, b}) is, by Axiom 1, given by­®
δω1 , F ({x}) C(Ω) = 0.8 ( if x = w),h®
­resp. M(Ω) δω2 , F ({x}) C(Ω) = 0.4 ( if x = w),
M(Ω)= 0.2 ( if x = b),= 0.6 ( if x = b)i.

Thus, under the condition (P1 ), the probability that the measured value obtained by themeasurement MC(Ω) (O, S[∗] ) is equal to x ( ∈ {w, b}) is given by
Z
­
®
­
®
P ({x}) =M(Ω) δω , F ({x}) C(Ω) ν0 (dω) = M(Ω) ν0 , F ({x}) C(Ω)½Ω
0.8p + 0.4(1 − p) ( if x = w),=
0.2p + 0.6(1 − p)) ( if x = b).
This is the answer to the above question (Q). Summing up, we see:

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178CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(♯) There is a reason that the “measurement” MC(Ω) (O, S[∗] ([δω1 ; p]⊕[δω2 ; 1−p])) is oneof interpretations of the “statistical measurement” MC(Ω) (O, S[∗] (ν0 )), (cf. Proclaim
1 (8.10) later). Here the mixed state ν0 (∈ Mm+1 (Ω)) is called a “statistical state”,
which represents the distribution of [ ∗ ]. And, the probability that the measured
value x ( ∈ {w, b}) is obtained by the measurement MC(Ω) (O, S[∗] (ν0 )), is given by
D
E
³ Z
D
E
´
ν
,
F
({x})
≡
δ
,
F
({x})
ν
(dω)
.
0
ω
C(Ω)
C(Ω) 0
C(Ω)∗
C(Ω)∗
Ω

Thus we consider thatprobabilistic form←−−−−−−−−−−S[∗] ([δω1 ; p] ⊕ [δω2 ; 1 − p])) −−−−−−−−−−−→ S[∗] (ν0 )(8.9)

statistical form

That is, the statistical state ν0 is the mixed state with probabilistic interpretation, or, themixed state generated by coin-tossing.
Thus, we see
The typical example of MC(Ω) (O, S[∗] (ν0 ))Pick up a ball from the urn behind the curtain

ω1

ω2

p

-

[∗]

1-p



On the other hand, we recall that
The typical example of MC(Ω) (O, S[∗] )Pick up a ball from the urn behind the curtain

ω1

ω2

-

[∗]



¥
Now, we introduce “statistical measurement MA (O, S[∗] (ρm ) )”. The mixed stateρm (with the probabilistic interpretation) is called an statistical state. We propose theSample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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179

following “Proclaim 1”, which should be read by the hint of the statement (♯) in Example
8.1.

PROCLAIM 1. [The probabilistic interpretationof mixed states, cf.¢¡
[44]]. Consider a statistical measurement MA O ≡ (X, F, F ), S[∗] (ρm )formulated in a C ∗ -algebra A. Then, the probability that x ( ∈ X), themeasured value obtained by the statistical measurement MA (O, S[∗] (ρm ) ),belongs to a set Ξ ( ∈ F) is given by
³
D
E ´
m
m
ρ (F (Ξ)) ≡ A∗ ρ , F (Ξ) A .
The statistical measurement MA (O, S[∗] (ρm ) ) is sometimes denoted by
MA (O, S(ρm ) ).
(8.10)
That is, Proclaim 11 asserts that[♯] “statistical state” = “mixed state” + “probabilistic interpretation” 2(mathematics)(8.11)(such as coin-tossing)

Note that the above “Proclaim 1” should be understood as“Proclaim 1” = “Axiom 1” +“statistical state”(the probabilistic interpretation of mixed state)

Therefore, the Statistical MT (i.e., SMT) is formulated as follows:
SMT =statistical measurement + the relation among systems[Proclaim 1 (8.10)]=

PMT(Axioms 1 and 2)+[Axiom 2 (3.26)]“statistical state”

in C ∗ -algebra.

(the probabilistic interpretation of mixed state)

Therefore, we stress:• there is no SMT without PMT.

(8.12)

Also, for the relation between PMT and SMT, see Remark 8.3 [hybrid measurement
theory] later.
The following deﬁnition is the same as Deﬁnition 3.1. Here, it should be noted that“Markov relation among systems (i.e., {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 )” and “sequential
1

Proclaim 1 is somewhat methodological. Thus, in [44], “Proclaim 1” was called “Method 1”.
As seen later (i.e., §8.7), Bertrand’s paradox is due to the confusion between mixed states (mathematical concept) and statistical states (measurement theoretical concept). In order to avoid this confusion,
it may be recommended to remember that there is always “coin-tossing” behind “statistical state”.
2

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observable (i.e., [{O}t∈T , {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 ])” are common to both PMT and
SMT. This implies that Axiom 2 is common to PMT and SMT.
Deﬁnition 8.2. [General systems in statistical measurements, cf. Deﬁnition 3.1]. Them
2 ] is called a general system with anpair S[∗] (ρmt0 ) ≡ [S(ρt0 ), {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤

initial state S(ρmt0 ) if it satisﬁes the following conditions (i)∼(iii).
(i) With each t (∈ T ), a C ∗ -algebra At is associated.
(ii) Let t0 (∈ T ) be the root of T . And, assume that a system S has the state ρmt0 (∈Sm (A∗t0 )) at t0 , that is, the initial state is equal to ρpt0 .
(iii) For every (t1 , t2 ) ∈ T≤2 , Markov operator Φt1 ,t2 : At2 → At1 is deﬁned such that
Φt1 ,t2 Φt2 ,t3 = Φt1 ,t3 holds for all (t1 , t2 ), (t2 , t3 ) ∈ T≤2 .
The family {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 is also called a “Markov relation among systems”.
Let an observable Ot ≡ (Xt , Ft , Ft ) in a C ∗ -algebra At be given for each t ∈ T . The pair[{O}t∈T , {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 ] is called a “sequential observable”.
¥
Again note that Axiom 2 is common to PMT and SMT. Thus we see,

PMT
SMT

measurements
Axiom 1 (2.37)
Proclaim 1 (8.10)

relation among systems
Axiom 2 (3.26)
Axiom 2 (3.26)

In what follows, we introduce some examples, which promote a better understandingof Proclaim 1. That is, readers will see that statistical states are not only generated by“coin-tossing” but also by several causes, for example, “Schrödinger picture”, “Bayes
theorem”, etc.
Remark 8.3. [(i) Axiom 1 and Proclaim 1, hybrid measurement theory (= “HMT” )].
For example, consider a pure state class Sp (C(Ω1 )∗ )) ( ≡ Mp+1 (Ω1 )) in Axiom 1 and amixed state class Sm (C(Ω2 )∗ )) ( ≡ Mm
+1 (Ω2 )) in Proclaim 1. Then we sometimes consider
the tensor state class Sp (C(Ω1 )∗ )) ⊗ Sm (C(Ω2 )∗ )), which is deﬁned by
¯
o
n
¯
m
m
m
m
δω1 ⊗ ρ1 ∈ M+1 (Ω1 × Ω2 ) ¯ ω1 ∈ Ω1 , ρ2 ∈ M+1 (Ω2 ) .
This is called a “hybrid state class”.

In applications, we often devote ourselves to thehybrid measurement theory (= HMT).

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181[(ii) Axiom 1 and Proclaim 1, hybrid measurement theory]. For each µ( ∈ R), consider ammixed state ρmµ ( ∈ M+1 (R)) such that1ρm
µ (D) = √

2πσ 2

Z
exp[−
D

(ω − µ)2]dω
2σ 2(∀D ∈ BR , Borel ﬁeld),

where σ is a ﬁxed positive number. Let O ≡ (X, F, F ) be an observable in C0 (R). Then,
we have the (statistical) measurement MC0 (R) (O, S(ρmµ )). On the other hand, deﬁne theb = (X, F, Fb) in C0 (R) such that:observable O
1[Fb(Ξ)](µ) = √2πσ 2Z[F (Ξ)](ω) exp[−R(ω − µ)2]dω
2σ 2(∀µ ∈ R, Ξ ∈ F).

Also note thatZ
DE
1(ω − µ)2
m
√
ρ
,
F
(Ξ)
=
]dω
[F
(Ξ)](ω)exp[−µ
C0 (R)
C0 (R)∗
2σ 2
2πσD2 REb
= C0 (R)∗ δµ , F (Ξ) C0 (R) ,which urges us to consider the following identiﬁcation:b
MC0 (R) (O, S(ρmµ )) ←→ MC0 (R) (O, S[δµ ] ) .
(statistical measurement)(pure measurement)[(iii): Axiom 1 and Proclaim 1, hybrid measurement theory]. Let Λ1 and Λ2 be compact spaces (or compact index sets). For each λ1 ( ∈ Λ1 ), consider a (parameterized)mmixed state ρm
λ1 ( ∈ M+1 (Ω)). And further, for each λ2 ( ∈ Λ2 ), consider a parameter-

ized observable Oλ2 ≡ (X, F, Fλ2 ) in C(Ω). Then, we have the (statistical) measurementb = (X, F, Fb) in C(Λ1 × Λ2 ) such
MC(Ω) (Oλ2 , S(ρm )) in C(Ω). Deﬁne the observable O
λ1

that:
D
E
m
b
[F (Ξ)](λ1 , λ2 ) = C(Ω)∗ ρλ1 , Fλ2 (Ξ) C(Ω)(∀(λ1 , λ2 ) ∈ Λ1 × Λ2 , Ξ ∈ F).

That is, we see
C(Ω)∗

D
E
D
E
b
δ
,
F
(Ξ)
ρm
,
F
(Ξ)
=
,
λ
∗
(λ
,λ
)
2
λ1
1 2
C(Ω)
C(Λ1 ×Λ2 )C(Λ1 ×Λ2 )which urges us to consider the following identiﬁcation:b
MC(Ω) (Oλ2 , S(ρmλ1 )) ←→ MC(Λ1 ×Λ2 ) (O, S[δ(λ1 ,λ2 ) ] ) .
(statistical measurement)(pure measurement)

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Such an identiﬁcation is often used in measurement theory. In this sense, the classiﬁcation¥(8.1) should be considered to be ﬂexible.

Remark 8.4. [Natural mixed state3 and statistical state, Bertrand’s paradox]. For
example, consider the square [0, 1] × [0, 1] ( ⊂ R2 ). This square has a natural measure m(which is usually called the Lebesgue measure) such that m([a, b] × [c.d]) = |b − a| · |d − c|(0 ≤ a ≤ b ≤ 1 and 0 ≤ c ≤ d ≤ 1). Here, it should be noted that m is a mixed
state (i.e., m ∈ Mm
+1 ([0, 1] × [0, 1])), however, it is not a statistical state. That is, thenatural mixed state is not always a statistical state. We should recall that there is no
statistical state without the probabilistic interpretation (such as coin-tossing). This isjust what Bertrand’s paradox (cf. [35], also see §8.7 Appendix (Bertrand’d paradox))
teaches us. That is because Bertrand’s paradox says that, if “the natural mixed state” isunreasonably regarded as “statistical state”, we encounter a serious paradox (since anatural mixed state is not always unique). Also, recall Chapter 4 (Boltzmann’s statisticalmechanics), in which the normalized invariant measure is not regarded as “probability”4
but “normalized staying time”.

(Continued to §8.7 Appendix (Bertrand’s paradox))
¥

8.1.2

Examples of statistical measurements

In Example 8.1, we showed “MC(Ω) (O, S[∗] ([δω1 ; p]⊕[δω2 ; 1−p]))” as the typical example
of statistical measurement MC(Ω) (O, S[∗] (ν0 )). In this section, we study the other typicalexamples.
The following example (Schrödinger picture) was already studied more precisely in
Chapter 6.
Example 8.5. [(i): Schrödinger picture I]. Let Ψ0,1 : A1 → A0 be a Markov operator.
Let ρp0 ∈ Sp (A∗0 ). That is, we consider the following general system:[A0 ](pure) stateρp0

Ψ0,1←− [A1 ].

(8.14)e0 =Also, consider any observable O1 ≡ (X1 , F1 , F1 ) in a C ∗ -algebra A1 . And put O
3The “natural mixed state ρ” usually means the “invariant mixed state ρ” for some “natural“homomorphism Φ : A → A. That is, it holds that Φ∗ (ρ) = ρ.
4Such probability may be called “a priori probability”. Thus we consider that the concept of “a priori
probability” is nonsense.

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183(X1 , F1 , Ψ0,1 F1 ). Thus we have the measurement
e 0 , S[ρp ] ).
MA0 (O0e 0 , S[ρp ] ) generates the sample space (X1 , F1 , P )Axiom 1 says that the measurement MA0 (O0such that:DP (Ξ1 ) = A∗
0

D

= A∗
1

E

ρp0 , Ψ0,1 F1 (Ξ1 )

E A0
p
∗
Ψ0,1 ρ0 , F1 (Ξ1 )A1(8.15)(∀Ξ1 ∈ F).

(8.16)

e 0 , S[ρp ] ) can be considered to be equal to theThis implies that the measurement MA0 (O0p∗e 0 , S[ρp ] ) is the representation
statistical measurement MA1 (O1 , S(Ψ ρ )). That is, MA0 (O
0,1 0

0

due to the Heisenberg picture, and MA1 (O1 , S[∗] (Ψ∗0,1 ρp0 )) is the representation due to the
Schrödinger picture. Summing up, we have the identiﬁcation:[the representation by Heisenberg picture]

MA0 (Ψ0,1 O1 , S[ρp0 ] )identiﬁcation←→(meaningful in the sense of Axiom 1)[the representation by Schödinger picture]
MA1 (O1 , S(Ψ∗0,1 ρp0 ))(meaningful in the sense of Proclaim 1)(8.17)

in which the left-hand side is understood in Axiom 1 and the right-hand side is understoodin Proclaim 1. For completeness, we explain the meaning of the identiﬁcation (8.17) as
follows: The left-hand side of (8.17) means that³(•1 ) Taking a measurement MA0 (Ψ0,1 O1 , S[ρp0 ] ) N-times that is, taking a measurement
MA0 (Ψ0,1 O1 , S[ρp0 ] ), and taking a measurement MA0 (Ψ0,1 O1 , S[ρp0 ] ),..., and taking a´measurement MA0 (Ψ0,1 O1 , S[ρp0 ] ) , we obtain measured values x1 , x2 ,...,xN . And
thus we have the sample space (X, F, ρp0 (Ψ0,1 F ( · ))) (= (8.15)).
The right-hand side of (8.17) means that(•2 ) Taking a statistical measurement MA1 (O1 , S(Ψ∗0,1 ρp0 )) N-times³
that is, taking ameasurement MA1 (O1 , S[∗1 ] (Ψ∗0,1 ρp0 )), and taking a measurement MA1 (O1 , S[∗2 ] (Ψ∗0,1 ρp0 ))´,..., and taking a measurement MA1 (O1 , S[∗N ] (Ψ∗0,1 ρp0 )) , we obtain measured values
x′1 , x′2 ,...,x′N . And thus we have the sample space (X, F, (Ψ∗0,1 ρp0 )(F ( · ))) (= (8.16)).
Since (8.15) = (8.16), we identify (•1 ) with (•2 ).5[(ii): Schrödinger picture II]. Let Ψ1,2 : A2 → A1 be a Markov operator. Let ρm
1 (∈
Sm (A∗1 )) be a statistical state. That is, we consider the following general system:
5

Strictly speaking. we must say “we regard (•2 ) as (•1 )”. That is because Axiom 2 says that Heisenberg
picture representation is more fundamental than Schrödinger picture representation.

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184CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/
Ψ0,1

[A1 ]
statistical stateρm
1←− [A2 ].

(8.18)

Here, let O2 ≡ (X2 , F2 , F2 ) be an observable in a C ∗ -algebra A2 .

e1 =And put Om∗(X2 , F2 , Ψ1,2 F2 ). Since ρm
1 (∈ S (A1 )) is a statistical state (i.e., the probabilistic in-terpretation is added), we have the statistical measurement
e 1 ≡ (X2 , F2 , Ψ1,2 F2 ), S(ρm )),MA1 (O1(8.19)

which generates the sample space (X2 , F2 , P ) such that:
DE
P (Ξ2 ) = A∗ ρp1 , Ψ0,1 F2 (Ξ2 ).

(8.20)

A1

1

This is equal to
D
A∗
2

E

Ψ∗0,1 ρm
1 , F2 (Ξ2 )A2,(8.21)

e 1 , S(ρm )) can be considered to bewhich implies that the statistical measurement MA1 (O1∗me 1 , S(ρm )) is theequal to the statistical measurement MA2 (O2 , S(Ψ ρ )). That is, MA1 (O
1,2 1

1

representation due to Heisenberg picture, and MA2 (O2 , S(Ψ∗1,2 ρm1 )) is the representation
due to Schrödinger picture. Summing up, we have the identiﬁcation:6[the representation by Heisenberg picture]
MA1 (Ψ1,2 O2 , S(ρm1 ))(meaningful in the sense of Proclaim 1)

identiﬁcation←→[the representation by Schödinger picture]
MA2 (O2 , S(Ψ∗1,2 ρm
1 ))(meaningful in the sense of Proclaim 1)(8.22)

in which the both sides are understood in Proclaim 1.
¥
The statistical state also appears in Bayes theorem, which was already studied in
Chapter 6.
Example 8.6. [A statistical state in Bayes theorem]. (continued from Example 8.1)
Assume the situation (P1 ) ∼ (P2 ) in Example 8.1 (Coin-tossing). That is, consider thefollowing statistical measurement MC(Ω) (O, S[∗] (ν0 )):6

Recall Axiom 2,
MA2 (O2 , S(Ψ∗1,2 ρm
1 )).

which

says

that

MA1 (Ψ1,2 O2 , S(ρm
1 ))

is

more

fundamental

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185

The picture of MC(Ω) (O, S[∗] (ν0 ))Pick up a ball from the urn behind the curtain

ω1

ω2

p

1-p

[∗]

-



Next, consider the following procedure.
(P3 ) We ﬁnd that the ball sampled in (P2 ) is a white one. That is, by the statisticalmeasurement MC(Ω) (O, S(ν0 )) in (P2 ), we obtain the measured value w(∈ {w, b}).
(P4 ) After the above (P3 ), we further take a “measurement” of an observable O1 ≡(Y, G, G). And, we know that the measured value belongs to Γ (∈ G).
In what follows we study the above (P3 ) and (P4 ). The procedures (P1 ) ∼ (P4 ) can becharacterized as the statistical measurement MC(Ω) (O × O1 , S(ν0 )). The probability thatthe measured value (w, y)(∈ {w, b} × Γ) obtained by MC(Ω) (O × O1 , S(ν0 )) belongs to Γ
is given by­®
ν0 , F ({w}) × G(Γ) .
Then, under the condition that we know (P3 ), the probability that the measured value y( ∈ Y ) is obtained in (P4 ) is given by the conditional probability
­
®
D
E ´
ν0 , F ({w}) × G(Γ) C(Ω) ³
F ({w}) × ν0
M(Ω)
­
®
­
®
= M(Ω), G(Γ) C(Ω) .
ν0 , F ({w}) C(Ω)
ν0 , F ({w}) C(Ω)
M(Ω)
M(Ω)

(8.23)

Since O1 ( ≡ (Y, G, G)) is arbitrary observable in C(Ω), this implies the following statereduction:
pretest state “ν0 ”
before “white” is obtained in (P2 )−→³
posttest state “ν1 ”
after “white” is obtained in (P2 )F ({w}) × ν0 ´
® .
=­
ν0 , F ({w})(8.24)

That is because the probability that the measured value obtained by MC(Ω) (O1 , S(ν1 ))belongs to Γ is given by
D
E
ν
,
G(Γ)
1
M(Ω)
C(Ω)

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(8.25)

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and it must hold that (8.23)=(8.25). Here, note that this new mixed state ν1 (∈ Mm+1 (Ω))
satisﬁesν1 ({ω}) =ν0 ({ω}) × [F ({w})](ω)ν0 (ω1 ) × [F ({w})](ω1 ) + ν0 (ω2 ) × [F ({w})](ω2 )(∀ω ∈ Ω ≡ {ω1 , ω2 }).
(8.26)

Then, it holds that
0.8p
2p
=
,
0.8p + 0.4(1 − p)1+p0.4(1 − p)1−p
ν1 ({ω2 }) ==.
0.8p + 0.4(1 − p)1+p
ν1 ({ω1 }) =(8.27)Since[ • ] the ν1 is the statistical state after the (P3 ),the “measurement” in (P4 ) is represented by the statistical measurement MC(Ω) (O2 , S(ν1 )),that is,2p
The picture of S([δω1 ; 1+p] ⊕ [δω2 ; 1−p]) (≈ S(ν1 ))
1+p

ω1

ω2
2p
1+p
-

[ ∗]

1−p
1+p

¥
Example 8.7. [(i): A statistical state in the repeated measurement]. Let ρm ∈ Sm (A∗ ).
p∗
By the Krein-Milman theorem (cf. [92]), we can choose a sequence {ρpk }N
k=1 in S (A )such that:
1 X pρ ≈ ρm
N k=1 kN(in the sense of the weak∗ -topology of Sm (A∗ )).

(8.28)

for a suﬃciently large natural number N . Consider an observable O ≡ (X, F, F ) in
¡
¢
NN
N
N
p
A. And consider the measurement M⊗A ⊗N
ρ
]
k=1 O ≡ (X , F ,k=1 F ), S[⊗N
k=1 k
N
N
m−1
N −m
A,
where
(
⊗
F
)(X
×
Ξ
×
X
)=
formulated in the tensor C ∗ -algebra N
m
k=1
k=1
N
( ⊗m−1
k=1 I) ⊗ F (Ξm ) ⊗ ( ⊗k=m+1 I) (∀Ξm ∈ F, 1 ≤ ∀m ≤ N ). For completeness, note

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187

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the measurement M⊗A ⊗N
is meaningful in the sense of Axiom 1. Letρ]
k=1 O, S[⊗N
k=1 k
¡
¢
p
(x1 , x2 , ..., xN ) be a measured value obtained by the measurement M⊗A ⊗N.
ρ
]
k=1 O, S[⊗N
k=1 k
Thus, by Axiom 1, we can “almost surely” expect thatρm (F (Ξ)) ≈♯[{k : xk ∈ Ξ}]N(∀Ξ ∈ F)(8.29)

holds for a suﬃciently large N , where ♯[B] is the number of the elements of a set B. That¡¢is because the probability that a measured value obtained by MA O, S[ρpk ] belongs to Ξ¡
( ∈ F) is given by ρp (F (Ξ)). In the above sense (8.29), the mathematical symbol MA O,
¢
¡
¢
P
p
S(ρm ) (or, MA O, S( N1 N
ρ)
k=1 k ) can be considered as the statistical measurement,
which may be called a “repeated measurement”.
[(ii)]. Let Ω be a ﬁnite set, i.e., Ω ≡ {ω1 , ω2 , ..., ωM }. Let O ≡ (X, F, F ) be an observNM
M C(Ω) ( ⊗Mable in C(Ω). Consider the repeated measurement M⊗Nn=1
n=1 O, S[⊗N
n=1 δω

modM [n]])(which may be called a cyclic measurement), where modM [n] is the integer such thatn = M j̇ + modM [n] and 0 ≤ modM [n] ≤ M − 1. Let (x1 , x2 , ..., xN M ) be a mea³NM
M C(Ω) ( ⊗NMsured value obtained by the cyclic measurement M⊗Nn=1
O,
S)=[⊗n=1 δωmod [n] ]
n=1
M
´
M
⊗N
n=1 MC(Ω) (O, S[δωmod [n] ] ) . Thus, by Axiom 1, we can “almost surely” expect that
M

Dδ
C(Ω)∗

ω1 + δω2 + · · · + δωMME♯[{k : xk ∈ Ξ}], F (Ξ) C(Ω) ≈NM(∀Ξ ∈ F)(8.30)

holds for a suﬃciently large N . In this sense,δ+δ+···+δ

ω2
ωM
ω1))• we often use the repeated statistical measurement ⊗N
n=1 MC(Ω) (O, S(
M³
M
or more precisely, the repeated probabilistic measurement ⊗N
n=1 MC(Ω) (O, S[∗] (⊕m=1´
NM
M C(Ω) ( ⊗n=1 O, S[⊗N M δ[δωm ; 1/M ])), cf. (8.8) as a substitute for M⊗Nn=1
] ).
n=1 ω
modM [n]That is, in the following table (in the case that Ω = {ω1 , ω2 }), the measured data
(x1 , x2 , ..., x2N ) and the measured data (y1 , y2 , ..., y2N ) have the same statistical properties(e.g., average, variance, etc.).

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188CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/
measurement······

measured|valuemeasurement · · · · · ·δω1 + δω2)) · · · · · · y12δω + δω2
MC(Ω) (O, S( 1)) · · · · · · y22δω + δω2
MC(Ω) (O, S( 1)) · · · · · · y32δω + δω2
MC(Ω) (O, S( 1)) · · · · · · y42······ ···δω + δω2
MC(Ω) (O, S( 1)) · · · · · · y2N −12δω + δω2)) · · · · · · y2NMC(Ω) (O, S( 12MC(Ω) (O, S[δω1 ] ) · · · · · · x1

| MC(Ω) (O, S(

MC(Ω) (O, S[δω2 ] ) · · · · · · x2|

MC(Ω) (O, S[δω1 ] ) · · · · · · x3|

MC(Ω) (O, S[δω2 ] ) · · · · · · x4|·········|

MC(Ω) (O, S[δω1 ] ) · · · · · · x2N −1 |MC(Ω) (O, S[δω1 ] ) · · · · · · x2N|

measured
value

¥

8.1.3

Problems (statistical measurements)

Problem 8.8. [Monty Hall problem, cf.[33]]. The Monty Hall problem is as follows (cf.
Problem 5.12, Remark 5.13 and Problem 11.13) :(P) Suppose you are on a game show, and you are given the choice of three doors (i.e.,“number 1”, “number 2”, “number 3” ). Behind one door is a car, behind theothers, goats.
(C) You know that the probability that behind the k-th door (i.e., “number k” ) is³
a car is given by pk (k = 1, 2, 3). For example, consider the two cases that´
p1 = p2 = p3 = 1/3, and p1 = 3/7, p2 = 1/7, p3 = 3/7.
You pick a door, say number 1, and the host, who knows what’s behind the doors,
opens another door, say “number 3”, which has a goat. He says to you, “Do you
want to pick door number 2?” Is it to your advantage to switch your choice of doors?
?

Door
Number 1?

Door
Number 2?

Door
Number 3

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189[Answer]. Put Ω = {ω1 , ω2 , ω3 }, whereω1 · · · · · · the state that the car is behind the door number 1
ω2 · · · · · · the state that the car is behind the door number 2
ω3 · · · · · · the state that the car is behind the door number 3.
Deﬁne the observable O ≡ ({1, 2, 3}, 2{1,2,3} , F ) in C(Ω) such that
[F ({1})](ω1 ) = 0.0,

[F ({2})](ω1 ) = 0.5,

[F ({3})](ω1 ) = 0.5, 7[F ({1})](ω2 ) = 0.0,

[F ({2})](ω2 ) = 0.0,

[F ({3})](ω2 ) = 1.0,

[F ({1})](ω3 ) = 0.0,

[F ({2})](ω3 ) = 1.0,

[F ({3})](ω3 ) = 0.0.

(8.31)

ν0 ({ω3 }) = p3

(8.32)

Deﬁne the statistical state ν0 ( ∈ Mm
+1 (Ω)) such that:ν0 ({ω1 }) = p1 ,

ν0 ({ω2 }) = p2 ,where p1 + p2 + p3 = 1, 0 ≤ p1 , p2 , p3 ≤ 1. Thus we have a statistical measurement
MC(Ω) (O, S[∗] (ν0 )). Note that(1) : “measured value 1 is obtained” ⇐⇒ the host says “Door (number 1) has a goat”(probability ←→ 0)(2) : “measured value 2 is obtained” ⇐⇒ the host says “Door (number 2) has a goat”(probability ←→ 0.5p1 + 1.0p3 )(3) : “measured value 3 is obtained” ⇐⇒ the host says “Door (number 3) has a goat”(probability ←→ 0.5p1 + 1.0p2 )Here, assume that
• By the statistical measurement MC(Ω) (O, S[∗] (ν0 )), you obtain a measured value 3.
which corresponds to the fact that the host said “Door (number 3) has a goat”. Then,the posttest state νpost ( ∈ Mm
+1 (Ω)) is given by
F ({3}) × ν0®.
νpost = ­
ν0 , F ({3})(8.33)

That is,p1

νpost ({ω1 }) = p1 2 ,
+ p2
2

p2
,
+ p2
2

νpost ({ω2 }) = p1

νpost ({ω3 }) = 0.

(8.34)

Thus,
7

Strictly speaking, F ({1})(ω1 ) = 0.5 and F ({2})(ω1 ) = 0.5 should be assumed in the problem (P).

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190CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/• if p1 = p2 = p3 = 1/3, then it holds that νpost ({ω1 }) = 1/3, νpost ({ω2 }) = 2/3,
νpost ({ω3 }) = 0, and thus, you should pick Door (number 2).
• if p1 = 3/7, p2 = 1/7 and p3 = 3/7, then it holds that νpost ({ω1 }) = 3/5, νpost ({ω2 }) =2/5, νpost ({ω3 }) = 0, and thus, you should not pick Door (number 2).
Also, more generally, we can say that
 if νpost ({ω1 }) ≤ νpost ({ω2 })(i.e.,p1 ≤ 2p2 ), then, you should pick Door (number 2)if νpost ({ω1 }) ≥ νpost ({ω2 })(i.e.,p1 ≥ 2p2 ), then, you should not pick Door (number 2).
¥

Remark 8.9. [P. Erdös]. I learnt the Monty Hall problem in the book [33] (“The Man
Who Loved Only Numbers, The story of Paul Erdös and the search for mathematical
truth”). This problem is famous as the problem in which even P. Erdös made a mistake.
I think that this problem is too profound to understand without measurement theory. In
fact, everyone may confuse the above Problem (P) for p1 = p2 = p3 = 1/3 with Problem
5.12 (i.e., the above problem (P) without the condition (C) ). In fact, in [33] (page 234),
it is written as follows:(Q) You’re on a game show and you’re given the choice of three doors. Behind one door
is a car, and behind the other two are goats. You choose, say, door 1, and the host,
who knows where the car is, opens another door, behind which is a goat. He now
gives you the choice of sticking with door 1 or switching to the other door? Whatshould you do?
If you read this description of the Monty Hall problem (in [33]), you may think that thecorrect answer should be due to Fisher’s likelihood method, i.e, the answer presented in
Problem 5.12. However, Problem 5.12, Remark 5.13 and Problem 8.8 are not all of theMonty Hall problem. See Problem 11.13 later (which may be my ﬁnal answer to theMonty Hall problem).
¥
Problem 8.10. [The problem of three prisoners].
Consider the following problem:(P) Three men, A, B, and C were in jail. A knew that one of them was to be set free
and the other two were to be executed. But he did not know who was the one to be

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191³

spared.

He knew that the probability that A [resp. B, C] will be set free is equal´
to 1/3 [resp. 1/3, 1/3], or more generally, pfa [resp. pfb , pfc ]. To the jailer who did
know, A said, “Since two out of the three will be executed, it is certain that either
B or C will be, at least. You will give me no information about my own chances if
you give me the name of one man, B or C, who is going to be executed.” Accepting
this argument after some thinking, the jailer said, “C will be executed.” Thereupon
A felt happier because now either he or C would go free, so his chance had increased
from 1/3 to 1/2. This prisoner’s happiness may or may not be reasonable. Whatdo you think?

J“C will be executed”
-

- A

B

C

[Answer]. Put Ω = {ωa , ωb , ωc }, whereωa · · · · · · the state that A will be set freeωb · · · · · · the state that B will be set freeωc · · · · · · the state that B will be set free .
Deﬁne the observable O ≡ ({xA , xB , xC }, 2{xA ,xB ,xC } , F ) in C(Ω) such that
[F ({xA })](ωa ) = 0.0,

[F ({xB })](ωa ) = 0.5,

[F ({xC })](ωa ) = 0.5, 8[F ({xA })](ωb ) = 0.0,

[F ({xB })](ωb ) = 0.0,

[F ({xC })](ωb ) = 1.0,

[F ({xA })](ωc ) = 0.0,

[F ({xB })](ωc ) = 1.0,

[F ({xC })](ωc ) = 0.0.

(8.35)

Deﬁne the statistical state ν0 ( ∈ Mm
+1 (Ω)) such that:ν0 ({ωa }) = pfa ,ν0 ({ωb }) = pfb ,ν0 ({ωc }) = pfc(8.36)

where pfa + pfb + pfc = 1, 0 ≤ pfa , pfb , pfc ≤ 1, though it may suﬃce to assume that pfa =pfb = pfc = 1/3. Here, note that the following (i) and (ii) are equivalent:
8

Strictly speaking, [F ({xB })](ωa ) = 0.5 and [F ({xC })](ωa ) = 0.5 should be assumed in the problem(P)

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192CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(i) The jailer said to A “C will be executed”.

(ii) By the statistical measurement MC(Ω) (O, S[∗] (ν0 )), A obtains a measured value xC
Thus, the posttest state νpost ( ∈ Mm
+1 (Ω)) is given by
F ({xC }) × ν0®.
νpost = ­
ν0 , F ({xC })(8.37)

That is,νpost ({ωa }) =

pfa
2
pfa
+ pfb
2

,

νpost ({ωb }) =

pfb
pfa
+ pfb
2

,

νpost ({ωc }) = 0.

(8.38)

Thus,• if pfa = pfb = pfc = 1/3, it holds that νpost ({ωa }) = 1/3, νpost ({ωb }) = 2/3,
νpost ({ωc }) = 0, and thus, the prisoner’s happiness is not reasonable. That is because pfa = 1/3 = νpost ({ωa }).
• if pfa = 3/7, pfb = 1/7, pfc = 3/7, it holds that νpost ({ωa }) = 3/5, νpost ({ωb }) = 2/5,
νpost ({ωc }) = 0, and thus, the prisoner’s happiness is reasonable. That is because
pfa = 3/7 < 3/5 = νpost ({ωa }).
• if pfa = 1/4, pfb = 1/2, pfc = 1/4, it holds that νpost ({ωa }) = 1/5, νpost ({ωb }) = 4/5,
νpost ({ωc }) = 0, and thus, the prisoner’s unhappiness is reasonable. That is because
pfa = 1/3 > 1/5 = νpost ({ωa }).
Also, more generally, we can say thatf
 if pfa ≤ νpost ({ωa })(i.e.,pfa + 2pb ≥ 1), the prisoner’s happiness is reasonable

if pfa ≥ νpost ({ωa })(i.e.,pfa + 2pfb ≤ 1), the prisoner’s unhappiness is reasonable.
¥

Remark 8.11. [(i).The problem of three prisoners in PMT]. Recall that the Monty Hall
problem is also studied in PMT, that is, Problem 5.12 (Fisher’s method) and Remark
5.13 (The moment method). On the other hand, it should be noted that the problem ofthree prisoners can not be solved in PMT.
[(ii): The relation between the Monty Hall problem and the problem of three prisoners].
Since the Monty Hall problem and the problem of three prisoners are similar, we add
something concerning the relation between the two. Consider the (P) (in Problem 8.8)
and the (Q) mentioned below.

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8.2. GENERAL STATISTICAL SYSTEM (EXAMPLE)

193

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(Q) (Continued from the (P) in Problem 8.10). There is a woman, who was proposed
to by the three prisoners A, B and C. She listened to the conversation between A
and the jailer. Thus, assume that she has the same information as A has. Then, we
have the following problem:(♯) Whose proposal should she accept?
[Answer]. For simplicity, consider the case that pfa = pfb = pfc = 1/3. Then we see that
νpost ({ωa }) = 1/3,

νpost ({ωb }) = 2/3,

νpost ({ωc }) = 0.

(8.39)

Thus, she should choose the prisoner B. Here it should be noted that the problem (♯) isthe same as the Monty Hall problem. That is, the problem:“(P) in Problem 8.10” + “(Q) in the above”
includes both the Monty Hall problem and the problem of three prisoners.
¥

8.2

General statistical system (Example)

As mentioned in the previous section, the Statistical MT (i.e., SMT) is formulated asfollows:
PMT = measurement + the relation among systems[Axiom 1 (2.37)]

andin C ∗ -algebra,[Axiom 2 (3.26)]

in C ∗ -algebra ,SMT = statistical measurement + the relation among systems[Proclaim 1 (8.10)][Axiom 2 (3.26)]

where it should be noted that
“Proclaim 1” = “Axiom 1” +“statistical state”.

(the probabilistic interpretation of mixed state)Thus we see
SMT =statistical measurement + the relation among systems[Proclaim 1 (8.10)]=

PMT(Axioms 1 and 2)+[Axiom 2 (3.26)]“statistical state”(the probabilistic interpretation of mixed state)

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in C ∗ -algebra .

(8.40)

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That is, Axiom 2 is common to PMT and SMT. This will be explicitly seen in the following
example (= Example 8.12), which should be compared with Example 3.4. Also recalling
Remark 8.3 [hybrid measurement theory (= HMT)], we say that
HMT =+ the relation among systems in C ∗ -algebra .

hybrid measurement[Axiom 1 (2.37) and Proclaim 1 (8.10)][Axiom 2 (3.26)](8.41)
Here note that PMT and SMT are respectively regarded as one of the aspects of HMT.
Since Axiom 2 is common to PMT and SMT, it is a matter of course that Example
3.4 (in PMT) and Example 8.12 (in SMT) are almost similar.
Example 8.12. [(Continued from Example 3.4) A simple general statistical system,
Heisenberg picture]. Suppose that a tree (T ≡ {0, 1, ..., 6, 7}, π) has an ordered structure³
such that π(1) = π(6) = π(7) = 0, π(2) = π(5) = 1, π(3) = π(4) = 2. See the ﬁgure´
Φπ(t),tm(8.42).
Consider a general system S(ρm0 ) ≡ [S(ρ0 ), {At → Aπ(t) }t∈T \{0} ] with theinitial system S(ρm
0 ).
Φ2,3

Φ1,2
Φ0,1
)
A0 i

kΦ0,6

Φ0,7

A1
A6

+

A2 )
k

A3

Φ2,4 A4

k Φ1,5A5

A7(8.42)

Also, for each t ∈ {0, 1, ..., 6, 7}, consider an observable Ot ≡ (Xt , 2Xt , Ft ) in a C ∗ -algebra
At . Thus, we have a sequential observable [{Ot }t∈T , {Φt,π(t) : At → Aπ(t) }t∈T \{0} ]. Now
we want to consider the following “measurement”,(♯) for a statistical system S(ρm0 ), take a measurement of “a sequential observable
Φπ(t),t[{Ot }t∈T , {At → Aπ(t) }t∈T \{0} ]”, i.e., take a measurement of an observable O0 at
0( ∈ T ), and next, take a measurement of an observable O1 at 1( ∈ T ), · · · · · · , andﬁnally take a measurement of an observable O7 at 7( ∈ T ),mwhich is symbolized by M({Ot }t∈T , S(ρm0 )). Note that the M({Ot }t∈T , S(ρ0 )) is merely

a symbol since only one measurement is permitted (cf. §2.5 Remark(II)). In what follows

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let us describe the above (♯) (= M({Ot }t∈T , S(ρm0 ))) precisely. Put
e t = Ot
O

and thus Fet = Ft(t = 3, 4, 5, 6, 7).

e 2 in A2 such asFirst we construct the quasi-product observable O
qp

e 2 = (X2 × X3 × X4 , 2X2 ×X3 ×X4 , Fe2 )O

qp

where Fe2 = F2 × ( × t=3,4 Φ2,t Fet ),if it exists. Iteratively, we construct the following:
Φ0,1

A0
qp

←−−−

A1

qp

qp

qp

F1 × Φ1,5 Fe5


y
Φ0,1

qp

A2

qp

F0 × Φ0,6 Fe6 × Φ0,7 Fe7


y
Fe0

Φ1,2

←−−−

←−−−

(F0 × Φ0,6 Fe6 × Φ0,7 Fe7 × Φ0,1 Fe1 )

qp

Fe1

qp

Φ1,2

←−−−

(F1 × Φ1,5 Fe5 × Φ1,2 Fe2 )

qp

Fe2

.

qp

(F2 × Φ2,3 Fe3 × Φ2,4 Fe4 )Q
e2e 1 ≡ (Q5 Xt , 2 5t=1 Xt , Fe1 ) of O1 , Φ1,2 O
That is, we get the quasi-product observable O
t=1Q
e 5 , and ﬁnally, the quasi-product observable O
e 0 ≡ (Q7 Xt , 2 7t=0 Xt , Fe0 ) of O0 ,and Φ1,5 O
t=0e 1 , Φ0,6 O
e 6 and Φ0,7 O
e 7 , if it exists. Here, Oe 0 is called the realization (or, the HeisenΦ0,1 O
Φπ(t),tberg picture representation) of a sequential observable [{Ot }t∈T , {At → Aπ(t) }t∈T \{0} ].
Then, we have the measurement
e0 ≡ (
MA0 (OY

Q

Xt , 2 t∈T Xt , Fe0 ), S(ρm
0 )),

t∈T

which is called the realization (or, the Heisenberg picture representation) of the symbol
M({Ot }t∈T , S(ρm
0 )).
¥

8.3

Bayes theorem in statistical MT

Now let us review “Bayes operator” (Deﬁnition 6.5 in §6.2), which plays an importantrole in SMT as well as PMT. Or, we may say that Bayes operator is more natural in STM
than in PMT.
Let (T ≡ {0, 1, ..., N }, π : T \ {0} → T ) be a tree with root 0 and let S[∗] ≡[S[∗] , C(Ωt )Φπ(t),t→C(Ωπ(t) ) (t ∈ T \ {0})] be a general system with the initial system

S[∗] . And, let an observable Ot ≡ (Xt , Ft , Ft ) in a commutative C ∗ -algebra C(Ωt ) be

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e 0 ≡ (Q Xt , N Ft , Fe0 ) be as in Theorem 3.7 in thegiven for each t ∈ T . Let Ot∈Tt∈Te 0 is the Heisenberg picture representation of thecase At = C(Ωt ) (∀t ∈ T ). That is, O
Φπ(t),tsequential observable [{Ot }t∈T , {C(Ωt ) → C(Ωπ(t) )}t∈T \{0} ]. Let τ be any element in(0,τ )

T . If a positive bounded linear operator BΠt∈T Ξt : C(Ωτ ) → C(Ω0 ) satisﬁes the following(0,τ )(0,τ )

condition (BO), we call {BΠt∈T Ξt : Ξt ∈ Xt (∀t ∈ T )} [ resp. BΠt∈T Ξt ] a family of Bayes
operators [ resp. a Bayes operator ]:b0 ≡(BO) for any observable O′τ ≡ (Yτ , Gτ , Gτ ) in C(Ωτ ), there exists an observable O
QNN(( t∈T Xt ) × Y, ( t∈T Ft ) Gτ , Fb0 ) in C(Ω0 ) such thatΦπ(t),tb 0 is the Heisenberg picture representation (cf. Theorem 3.7) of [{Ot }t∈T ; C(Ωt ) →(i) OC(Ωπ(t) ) (t ∈ T \ {0})], where Ot = Ot (if t ̸= τ ), = Oτ × O′τ (if t = τ ),(ii) Fb0 ((Πt∈T Ξt ) × Γτ ) = BΠt∈T Ξt (Gτ (Γτ ))(0,τ )(Ξt ∈ Ft (∀t ∈ T ), ∀Γτ ∈ Gτ ),Q(0,τ )(iii) Fb0 ((Πt∈T Ξt ) × Yτ ) = Fe0 ( t∈T Ξt ) = BΠt∈T Ξt (1τ ), (Ξt ∈ Ft (∀t ∈ T )), where 1τ is theidentity in C(Ωτ ).
(0,τ )

mAlso, deﬁne RΠt∈T Ξt : Mm
+1 (Ω0 ) → M+1 (Ωτ ) such that:[BΠt∈T Ξt ]∗ (ν)(0,τ )(0,τ )
RΠt∈T Ξt (ν) =(0,τ )∥[BΠt∈T Ξt ]∗ (ν)∥M(Ω0 )(∀ν ∈ Mm
+1 (Ω0 )),which is called “a normalized dual Bayes operator”.
¥(0,τ )

It is quite important to see that the Bayes operator BΠt∈T Ξt : C(Ωτ ) → C(Ω0 )is described in terms of the Heisenberg picture. This implies that the Bayes opera(0,τ )tor BΠt∈T Ξt : C(Ωτ ) → C(Ω0 ) is common to PMT and SMT. That is, the dual form
(0,τ )

m
RΠt∈T Ξt : Mm
+1 (Ω0 ) → M+1 (Ωτ ) can be applicable to both PMT and SMT and PMTPEW(i.e., subjective Bayesian PMT) mentioned later (in §6.4).

The following theorem is an analogy of Theorem 6.13. This theorem (= Theorem 8.13,
Remark 8.14) is also called “Bayes’ method”.
Theorem 8.13. [Generalized Bayes theorem, Bayes’ method, cf. [46]]. Let (T ≡ {0, 1, ...,
N }, π : T \ {0} → T ) be a tree with the root 0 and let S(ν0 ) ≡ [S(ν0 ), C(Ωt )Φπ(t),t→C(Ωπ(t) ) (t ∈ T \ {0})] be a general system with the initial system S(ν0 ). And, let an

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observable Ot ≡ (Xt , Ft , Ft ) in a C ∗ -algebra C(Ωt ) be given for each t ∈ T . Then, we
have a statistical measurement
YO
e 0 ≡ ( Xt ,MC(Ω0 ) (OFt , Fe0 ), S(ν0 )).
t∈T(cf. Theorem 3.7).

t∈T

e 0 , S(ν0 )) belongs
Assume that the measured value by the statistical measurement MC(Ω) (OQN
to t∈T Ξt (∈ t∈T Ft ). Let τ be any element in T . Then, we see(a)e 0 , S(ν0 ))” = R“the (statistical) S-state at τ ( ∈ T ) after MC(Ω0 ) (OΠt∈T Ξt (ν0 ).
(8.43)(0,τ )

Φπ(t),tProof. Since the sequential observable [{Ot }t∈T , {C(Ωt ) → C(Ωπ(t) )}t∈T \{0} ] is common to PMT and SMT, Theorem 3.7 is applicable. Also, by the same argument in
Theorem 6.13, the (8.43) immediately follows.
Remark 8.14. [(i): Bayes operator in Remark 5.7, Bayes’ method]. Let O ≡ (X, F, F )(0,0)

be an observable in C(Ω). For each Ξ ( ∈ F), deﬁne the continuous linear operator BΞ
O,(0,0)(or, BΞO , BΞ

) : C(Ω) → C(Ω) such that:
(0,0)

BΞ

(g) = F (Ξ) · g(∀g ∈ C(Ω)),(0,0)

which is called the Bayes operator (or, simplest Bayes operator). Deﬁne the map RΞ:m
Mm+1 (Ω) → M+1 (Ω) (called “normalized Bayes dual operator”) such that:
(0,0)

(B1 )

RΞ

(ν) =

(0,0)
[BΞ ]∗ (ν)(0,0) ∗∥[BΞ ] (ν)∥M(Ω)

that is,(∀ν ∈ Mm
+1 (Ω)),

R(0,0)[RΞ (ν)](D0 ) =[F (Ξ)](ω)ν(dω)
RD0[F (Ξ)](ω)ν(dω)Ω(∀D0 ∈ BΩ ).

Thus, we can describe the well known Bayes theorem (cf. [86]) such as
(0,0)

Mm
+1 (Ω) ∋ ν (= pretest state) 7→ (posttest state =)RΞ(ν) ∈ Mm
+1 (Ω).

(8.44)As a particular case of the above, assume that ν = δω0 ( ∈ Mp+1 (Ω)). Then we see that
(0,0)

Mp+1 (Ω) ∋ δω0 (= pretest state) 7→ (posttest state =)RΞ(δω0 ) = δω0 ∈ Mp+1 (Ω).

That is, a pure state δω0 is invariant.
[(ii): The conventional Bayes theorem in mathematics]. The above theorem should becompared with the following conventional Bayes theorem (B2 ).

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198CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(B2 ) Let (S, BS , P ) be a probability space. Let {E1 , E2 , ..., En } be a (measurable) de³´
composition of S, i.e., Ek ∈ BS , ∪nk=1 Ek = S, Ei ∩ Ek = ∅(if i ̸= k) . Let E ∈ BS .
Then
PE (Ek ) =P (Ek )PEk (E),P (E1 )PE1 (E) + ... + P (En )PEn (E)k)k)
where PE (Ek ) = P (E∩E
, PEk (E) = P P(E∩E
.
P (E)
(Ek )

The (B2 ) is, of course, a mathematical theorem. Thus, when we use the (B2 ), we must
add a certain interpretation to the (B2 ). In measurement theory, this is automaticallydone as follows:
(B1 )

=

(B2 )

+

“measurement theoretical interpretation”.

[(iii): The collapse (reduction) of wave packet in quantum mechanics]. The reduction suchas (8.44) may happen even in quantum mechanics. In fact, it is called “the collapse (reduction) of wave packet in quantum mechanics”. Assume that a measured value obtainedby a measurement MC(V ) ((X, F, F ), S(ρ)) belongs to Ξ ( ∈ F). Then, we may see thefollowing reduction (i.e., the collapse of wave packet):m
T r+1(V ) ∋ ρ (= pretest state) 7→ (posttest state =)F (Ξ)ρF (Ξ)
m∈ T r+1(V ).
∥F (Ξ)ρF (Ξ)∥T r(V )p(V ), the above reduction happens
Note that, even in the case that ρ = |u⟩⟨u| ∈ T r+1(i.e., not invariant). However, I believe that the collapse of wave packet is due to a nonstandard argument in quantum mechanics, though the collapse may be indispensable forthe intuitive understanding of “quantum Zeno eﬀect (cf. [65])”, etc. That is, I havean opinion that from the pure theoretical point of view quantum mechanics says nothing
after a measurement. That is because, from the theoretical point of view, we always
devote ourselves to the Heisenberg picture representation and not the Schrödinger picture
representation. And further, it should be noted that the collapse of wave packet inquantum mechanics is not a direct consequence of MT (i.e., Axioms 1 and 2, Proclaim³
1) though the (8.44) (i.e., the classical reduction) is a consequence of Theorem 8.13´
in MT . Thus, in this book we are not concerned with the collapse of wave packet inquantum mechanics.
¥

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8.4

Kalman ﬁlter in noise

As a consequence of Theorem 8.13 (and Theorem 6.13), in this section we reconsider
Kalman ﬁlter [51], and formulate “Kalman ﬁlter” in SMT, which is proposed in [55].
Consider the conventional Kalman ﬁlter in the following system:θ 1 (n + 1) s(n + 1)-+
- z −1 Is(n)- C(n)

6

-+

x(n)
-

6

ψn,n+1 ¾θ 2 (n)(Figure (8.45))

where s(n): L-dimensional state vector at time n(= 0, 1, ..., N ), x(n): M -dimensional
measured data vector, (ω ∈ Ω). In the framework of dynamical system theory (2.1), s(n)and x(n) are described by the following equations: for each ω ∈ Ω where (Ω, BΩ , P ) is a
probability space, s(n + 1, ω) = ψn,n+1 (s(n, ω)) + θ1 (n + 1, ω) : stochastic diﬀerence state equation
 x(n, ω) = C(n)s(n, ω) + θ (n, ω)2(n = 0, 1, ..., N − 1).
: measurement equation(8.46)
Here, it is assumed that ψn,n+1 , C(n), θ 1 (n, ·) (and its initial distribution) and θ 2 (n, ·) are
known where ψn,n+1 : K × K-dimensional transition matrix, θ 1 (n, ·): L-dimensional input
vector which represents a white noise, C(n): L × K-dimensional measurement matrix,
θ 2 (n, ·): L-dimensional vector which represents a measurement error. Here, our problem
is as follows:(♯) Let τ be any integer such that 0 ≤ τ ≤ N . Let Ξk ∈ BR (k = 0, 1, 2, ..., N ). Theninfer the state vector s(τ, ω) at time τ from the fact that(x(0, ω), x(1, ω), x(2, ω), ..., x(N, ω)) ∈ Ξ0 × Ξ1 × Ξ2 × · · · × ΞN .
Also, note the original equation of the stochastic diﬀerence equation (8.46) is the following
equation:s̄(n + 1) = ψn,n+1 (s̄(n)) (n = 0, 1, ..., N − 1).

(8.47)The problem (♯) was ﬁrstly answered in the framework of dynamical system theory (8.46).
Now, we consider the (♯) in the framework of SMT (8.3).

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8.4.1

The measurement theoretical formulation of Figure (8.45)

Firstly, we formulate the (8.45) in SMT, (or HMT in Remark 8.3). Assume, forsimplicity, that T (≡ {0, 1, ..., N }) is a tree with a series structure (though this assumption
is not needed). For each t (∈ T ), consider compact Hausdorﬀ spaces St and Θt . Although
it is natural to assume that S0 = S1 = · · · = SN and Θ0 = Θ1 = · · · = ΘN , we can dowell without this assumption. Now, consider the following two Markov relations amongsystems: [{Ψt1 ,t2 : C(St2 ) → C(St1 )}(t1 ,t2 )∈T≤2 ] and [{Υt1 ,t2 : C(Θt2 ) → C(Θt1 )}(t1 ,t2 )∈T≤2 ]such asΨ0,1

ΨN −2,N −1

Ψ1,2

ΨN −1,N[C(S0 )] ←−−− [C(S1 )] ←−−− · · · ←−−−−−− [C(SN −1 )] ←−−−−− [C(SN )](8.48)

where the initial state δs0 (∈ Mp+1 (S0 )) is assumed to be unknown, and
Υ0,1

Υ1,2

ΥN −2,N −1

ΥN −1,N[C(Θ0 )] ←−−− [C(Θ1 )] ←−−− · · · ←−−−−−− [C(ΘN −1 )] ←−−−−− [C(ΘN )]¡¢with the known initial state ν0Θ (∈ Mm(8.49)+1 (Θ0 )) .
Here, it should be noted that the above (8.48) [resp. (8.49)] is the measurement theoreticalformulation of (8.47) [resp. the θ 1 in (8.45)]. Also, note that the (8.48) is equivalent to
Ψ∗0,1

Ψ∗1,2

Ψ∗N −2,N −1

Ψ∗N −1,Nm[Mm−−−−−→ [Mm−−−−→ [Mm
+1 (S0 )] −−−→ [M+1 (S1 )] −−−→ · · · −+1 (SN −1 )] −+1 (SN )]mwhere Ψ∗n,n+1 : Mm
+1 (Sn ) → M+1 (Sn+1 )] is the dual operator of Ψn,n+1 : C(Sn+1 ) → C(Sn ).

Since the (8.48) corresponds to the conventional (8.47), it is natural to assume that the(8.48) is deterministic, i.e., Ψn,n+1 is homomorphic. Thus, for each n = 0, 1, ..., N − 1,there exists a continuous map ψn,n+1 : Sn → Sn+1 , i.e.,
[S0 ]

ψ0,1

−−−→

[S1 ]

ψ1,2

ψN −2,N −1−−−→ · · · −−−−−−→[SN −1 ]ψN −1,N

−−−−→

[SN ]

where
fn+1 (ψn,n+1 (sn )) = (Ψn,n+1 (fn+1 ))(sn ) (∀fn+1 ∈ C(Sn+1 ), ∀sn ∈ Sn ).
Next, consider a continuous map λn : Sn × Θn → Sn , that is,
Sn × Θn ∋ (sn , θn ) 7→ λn (sn , θn ) ∈ Sn

(n = 0, 1, ..., N )(8.50)

which should be regarded as the corresponding thing of the left ⊕ in (8.45). The continm
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such that:(Λn (νnS ⊗ νnΘ ))(Bn ) = (νnS ⊗ νnΘ )(λ−1n (Bn ))(∀(νnS ⊗ νnΘ ) ∈ Mm
+1 (Sn × Θn ), ∀Bn ⊆ Sn : open).

(8.51)mmb∗
Further, deﬁne the continuous map Φn,n+1 : M+1 (Sn × Θn ) → M+1 (Sn+1 × Θn+1 ), such
that
S
Θ
S
Θ
b∗
Mm
+1 (Sn × Θn ) ∋ νn ⊗ νn 7→Φn,n+1 (νn ⊗ νn )≡[Λn+1 (Ψ∗n,n+1 νnS ⊗ Υ∗n,n+1 νnΘ )] ⊗ Υ∗n,n+1 νnΘ ∈ Mm
+1 (Sn+1 × Θn+1 )mwhere Υ∗n,n+1 : Mm
+1 (Θn ) → M+1 (Θn+1 ) is a dual operator of Υn,n+1 : C(Θn+1 ) → C(Θn ).

That is,¡¢SΘb ∗n,n+1 (νnS ⊗ νnΘ )νn+1⊗ νn+1≡Φ=[Λn+1 (Ψ∗n,n+1 νnS ⊗ Υ∗n,n+1 νnΘ )] ⊗ Υ∗n,n+1 νnΘ

(n = 0, 1, ..., N − 1)

(8.52)

which (or, the following (8.53)) corresponds to the state equation (8.46). Thus, we have−1b n,n+1 : C(Sn+1 × Θn+1 ) → C(Sn × Θn )}N
the Markov relation [{Φn=0 ]:b 0,1
Φ

b N −1,NΦb N −2,N −1
Φ

b 1,2
Φ

[C(S0 × Θ0 )] ←−−−−− [C(S1 × Θ1 )] ←−−−−− · · · ←−−−−−−−− [C(SN −1 × ΘN −1 )] ←−−−−−− [C(SN × ΘN )](8.53)∗
b n,n+1 is the pre-dual operator of Φb∗bb∗
where Φn,n+1 (i.e., (Φn,n+1 ) = Φn,n+1 ). That is, the(8.53) is equivalent tob∗Φ
N −1,N

b∗
Φ
1,2

b∗
Φ
0,1

m
m
m
[Mm
+1 (S0 × Θ0 )] −−−−→ [M+1 (S1 × Θ1 )] −−−−→ · · · [M+1 (SN −1 × ΘN −1 )] −−−−−→ [M+1 (SN × ΘN )](8.53)′

Next, we consider the measurement theoretical characterization of the measurement
equation (8.46). That is, consider the following Markov relation:
Υ′0,1

Υ′N −2,N −1

Υ′1,2

Υ′N −1,N[C(Θ′0 )] ←−−− [C(Θ′1 )] ←−−− · · · ←−−−−−− [C(Θ′N −1 )] ←−−−−− [C(Θ′N )]′′(with the initial state ν0Θ (∈ Mm+1 (Θ0 ))),

which corresponds to the θ2 in (8.46). Also, for each n (∈ T ), consider an observable
On = (Xn , 2Xn , Fn ) in C(Sn ×Θ′n ), which corresponds to the measurement equation (8.46).
Note that the observable On = (Xn , 2Xn , Fn ) in C(Sn × Θ′n ) can be also regarded as anobservable in C(Sn × Θn × Θ′n ). Thus, we see that the (8.46) corresponds to the following:
b
b 0,1
Φ

b
b 1,2
Φ

b
b N −1,NΦ[C(S0 × Θ0 × Θ′0 )] ←−−−− [C(S1 × Θ1 × Θ′1 )] ←−−−− · · · ←−−−−− [C(SN × ΘN × Θ′N )]X0(X0 , 2

, F0 )

X1(X1 , 2

, F1 )···XN(XN , 2

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, FN )(8.54)

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202CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/′bb n,n+1 ≡ Φ
b n,n+1 ⊗ Υ′with the initial state δs0 ⊗ ν0Θ ⊗ ν0Θ where Φn,n+1 ). Here, note that′pΘ′ν0Θ (∈ Mm(∈ Mm+1 (Θ0 )) and ν0+1 (Θ0 )) are known, but δs0 (∈ M+1 (S0 )) is unknown.

Therefore, we have the correspondence:(8.46) in DST ↔ (8.54) in SMT (or precisely, HMT, cf. Remark 8.3).
Thus, we can skip to the next section §8.4.2. However, in what follows we add theconcrete form of the family {On = (Xn , 2Xn , Fn )}N
n=0 (in (8.54)), which corresponds tothe measurement equation (8.46) in detail.
Let S′n and S′′n be compact spaces. Let C : Sn → S′′n be a continuous map, whichmm′′induces the continuous map ΛC
n : M+1 (Sn ) → M+1 (Sn ) such that:
S
′
S
C −1
′
(ΛC
n (νn ))(An ) = νn ((λn ) (An ))′′′(∀νnS ∈ Mm
+1 (Sn ), ∀An ⊆ Sn : open).

And consider a continuous map λ′n : S′′n × Θ′n → S′n , which induces the continuous map′′′m′Λ′n : Mm
+1 (Sn × Θn ) → M+1 (Sn ) such that:
′′

′

′′

′

(Λ′n (νnS ⊗ νnΘ ))(Bn′ ) = (νnS ⊗ νnΘ )((λ′n )−1 (Bn′ ))
′′

′

′′
′
(∀(νnS ⊗ νnΘ ) ∈ Mm
+1 (Sn × Θn ),∀Bn′ ⊆ S′n : open).

For each n (= 0, 1, ..., N ), consider an observable O′n = (Xn , 2Xn , Fn′ ) in C(S′n ), which
may be an (approximate) exact observable (cf. Example 2.20). Thus, for each n (∈ T ),we can deﬁne the observable On = (Xn , 2Xn , Fn ) (in (8.54)) in C(Sn × Θ′n ) such that:′

C(Sn ×Θ′n )∗

′

S
Θ
′
⟨νnS ⊗ νnΘ , Fn (Ξn )⟩C(Sn ×Θ′ ) = C(S′ )∗ ⟨Λ′n (ΛCn (νn ) ⊗ νn ), Fn (Ξn )⟩C(S′ )
n

n

n

′
′
(∀(νnS ⊗ νnΘ ) ∈ Mm
+1 (Sn × Θn )).

8.4.2

Kalman ﬁlter in Noiseb n = Θn × Θ′ and ν Θb = ν Θ ⊗ ν Θ′ . And, we rewrite the (8.54) as
For simplicity, put Θ000nfollows:
b
b
Φ

b
b
Φ

b
b
Φ

0,1
1,2
N −2,N −1b 0 )] ←−−b 1 )] ←−−[C(S0 × Θ−−− [C(S1 × Θ−−− · · · ←−−−−−−−−(X0 , 2X0 , F0 )(X1 , 2X1 , F1 )···b N −1 )][C(SN −1 × Θ(XN −1 , 2XN −1 , FN −1 )bb
ΦN −1,Nb N )]←−−−−−− [C(SN × Θ(XN , 2XN , FN )

¡
b
b
m
Θ
b
with the initial state δs0 ⊗ ν0Θ , where ν0Θ (∈ Mm+1 (Θ0 )) is known that is, ν0 (∈ M+1 (Θ0 ))¢′p′and ν0Θ (∈ Mm+1 (Θ0 )) are known , but δs0 (∈ M+1 (S0 )) is unknown.

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8.4. KALMAN FILTER IN NOISE203

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/bb t1 ,t2 : C(St2 × Θ
b t2 ) →Now, we get the sequential observable [OT ] ≡ [{Ot }t∈T ; {Φe 0 ≡ (Q Xt , 2Πt∈T Xt , Fe0 )b t1 )}(t ,t )∈T 2 ]. Then, we can construct the observable O
C(St1 × Θ
1 2

t∈T

≤

b 0 ), which is the realization of the sequential observable [OT ], such asin C(S0 × Θ
b
b 0,1
Φ

b
b N −2,N −1
Φ

b
b 1,2
Φ

b
b N −1,NΦb 0 )] ←−−−−− [C(S1 × Θb 1 )] ←−−−−− · · · ←−−−−−−−− [C(SN −1 × Θ
b N −1 )] ←−−−−−− [C(SN × Θb N )][C(S0 × Θ
F0

F1

?
?
y

?
?
y

qp

¡

qp

b
b 0,1
Φ

b
bF
e1 )
(F0 ×
×Φ
e0
=F

···

←−−−−−

b
bF
e2 )
(F1 ×
×Φ
e1
=F

FN

?
?
y

?
?
y

qp

b
b N −2,N −1
Φ

b 1,2
Φ

FN −1

b
bF
eN )
(FN −1 × Φ
eN −1
=F

←−−−−− · · · ←−−−−−−−−bb N −1,NΦ←−−−−−−(8.55)

(FN )
eN
=F

¢
e 0 ≡ (Q Xt , 2Πt∈T Xt , Fe0 ) is assured by Theorem 3.7. Thus, we
The existence of the Ot∈Tb

can represent the “measurement” M({Ot }t∈T , S(δs0 ⊗ ν0Θ )) such ase 0 , S(δs0 ⊗ ν Θ )).
M({Ot }t∈T , S(δs0 ⊗ ν0Θ )) = MC(S0 ×Θb 0 ) (O0bbHere, assume that(♯) we know that the measured value (xt )t∈T (∈
Q
b
Θ
e
M
b (O0 , S(δs0 ⊗ ν )), belongs toC(S0 ×Θ0 )Qt∈T Xt ), obtained by the measurementt∈T Ξt .

0

Fisher’s maximum likelihood method (cf. Theorem 5.3, Corollary 5.6) says that there isa reason to infer that the unknown s0 (∈ S0 ) is determined by
Y
Y
b e
b e
Θ
Θ
Ξ
)⟩
=
max
Ξt )⟩C(S ×Θb ) .
t∗ ⟨δs0 ⊗ ν0 , F0 (∗ ⟨δs ⊗ ν0 , F0 (bbb
C(S ×Θ )C(S ×Θ )C(S ×Θ )
0

0

0

t∈T

Q

(0,τ )¡

Let τ ∈ T , and let {BΠt∈T Ξt |

s∈S0

0

t∈T Ξt

0

0

0

0

t∈T

∈ 2Πt∈T Xt } be a family of Bayes operators.

¢
The existence is assured by Theorem 6.6. Then, we see, by Lemma 8.9, that the newbb τ )) is deﬁned by
S-state ν S×Θτ (∈ Mm (Sτ × Θ
τ,new

+1

b

b

S×Θτ
ντ,new= RΠt∈T Ξt (δs0 ⊗ ν0Θ )(0,τ )(0,τ )mbbwhere RΠt∈T Ξt : Mm
+1 (S0 × Θ0 ) → M+1 (Sτ × Θτ ) is a normalized dual Bayes operator, i.e.,)∗ (ν)t∈T Ξt(0,τ )∥(BΠ)∗ (ν)∥t∈T Ξt(0,τ )(0,τ )

RΠt∈T Ξt (ν) =(BΠb(∀ν ∈ Mm
+1 (S0 × Θ0 )). Thus there is a reason to think thatS
the new S-state (in Mm+1 (Sτ )) is equal to ντ,new such that:Sτ ×ΘτSτb τ ) (∀Dτ (⊆ Sτ ) : open set).
(Dτ × Θ(Dτ ) ≡ ντ,newντ,newb

Remark 8.15. [Stochastic diﬀerential equation] It is important to generalize the stochastic diﬀerence state equation in (8.46) to the stochastic diﬀerential equation (1.2a). In orderto do it in SMT, we must prepare the W ∗ -algebraic formulation of SMT (in Chapter 9).
Thus we do not touch this problem in this book.

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204CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/

8.5

Information and entropy

As one of applications (of Bayes theorem), we now study the “entropy” of the measurement. Here we have the following deﬁnition.
Deﬁnition 8.16. [Information quantity, the entropy of measurement (= fuzzy entropy),¡¢
cf. [42]]. Consider a statistical measurement MC(Ω) O ≡ (X, 2X , F ), S(ρ0 ) in a commutative C ∗ -algebra C(Ω), where the label set X is assumed to be at most countable,i.e., X = {x1 , x2 , ..., xn , ...}. Then, the H(M), the (fuzzy) entropy of MC(Ω) (O, S(ρ0 )), isdeﬁned by³´
H MC(Ω) (O, S(ρ0 ))
Z
∞ ³Z
X
R
=
[F ({xn })](ω)ρ0 (dω)
n=1

=

∞
X

Ω

Ω

´
[F ({xn })](ω)[F ({xn })](ω)
Rlog
ρ0 (dω)[F ({xn })](ω)ρ0 (dω)[F ({xn })](ω)ρ0 (dω)ΩΩ
(8.56)·I({xn })n=1

Z
where, P ({xn }) = [F ({xn })](ω)ρ0 (dω)Ω³´= the probability that a measured value xn is obtainedZ[F ({xn })](ω)[F ({xn })](ω)
log R
ρ0 (dω)[F ({xn })](ω)ρ0 (dω)Ω Ω [F ({xn })](ω)ρ0 (dω)Ω
Z1=
[F ({xn })](ω) log[F ({xn })](ω)ρ0 (dω) − log P ({xn })P ({xn }) ΩI({xn }) =

³

R

´

= the information quantity when a measured value xn is obtained(8.57)¡¢
MC(Ω) O, S(ρ0 ) is the normalized W ∗ -algebraic representation of a C ∗ -measurement³´¡¢
MC0 (Ω) O ≡ (X, P0 (X), F ), S(ρ0 ) , the entropy H MC0 (Ω) (O, S(ρ0 )) is also deﬁned³¡¢´
by H MC(Ω) O, S(ρ0 ) .
¥The deﬁnition is derived from the following consideration. Assume that we get themeasured value x ( ∈ X) by the statistical measurement MC(Ω) (O, S(ρ0 )). Note that its

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8.5. INFORMATION AND ENTROPY

205

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DER
probability P ({x}) is given by P ({x}) = C(Ω)∗ ρ0 , F ({x}) C(Ω) = Ω [F ({x})](ω) ρ0 (dω).
Also, we consider, by (8.44) (or, (5.13)), that the new statistical state ρx ( ∈ Mm
+1 (Ω)) is

given byR
[F ({x})](ω)ρ0 (dω)ρx (D) = RD
[F ({x})](ω)ρ0 (dω)Ω(∀D ∈ BΩ ),Rxxwhose information quantity I(x) is of course determined by I({x}) = Ω dρ(ω) log dρρ (dω),dρ0dρ0 0({x})](ω)
xwhere the Radon-Nikodým derivative dρ(ω) is deﬁned by R [F[F. Thus, the avdρ0({x})](ω)ρ0 (dω)Ω

erage information quantity, i.e., entropy, is given by³´

H MC(Ω) (O, S(ρ0 )) =∞X

P ({xn }) · I({xn }),n=1

which is equal to (8.56). Also it should be noted that the formula (8.56) can easily
calculated as follows:
∞ Z
∞
X
X
H(M) =[F ({xn })](ω) log[F ({xn })](ω)ρ0 (dω) −P ({xn }) log P ({xn }). (8.58)
n=1

Ω

n=1

Also, if O is crisp, we see that H(M) = −P∞

n=1 P ({xn }) log P ({xn }).

Example 8.17. [Urn problem (in Example 8.1)]. There are two urns ω1 and ω2 . The
urn ω1 [resp. ω2 ] contains 8N white and 2N black balls [resp. 4N white and 6N black¡¢balls], where N is a suﬃciently large number. We regard Ω ≡ {ω1 , ω2 } as the state¡¢
space. And consider the observable O ≡ (X ≡ {w, b}, 2{w,b} , F ) in C(Ω) where[F ({w})](ω1 ) = 0.8,

[F ({b})](ω1 ) = 0.2,

[F ({w})](ω2 ) = 0.4,

[F ({b})](ω2 ) = 0.6.

Here deﬁne the statistical state ν0 (∈ Mm+1 (Ω)) such that ν0 ({ω1 }) = p, ν0 ({ω2 }) = 1 − p.
And consider a statistical measurement MC(Ω) (O, S[∗] (ν0 )).
The illustration of MC(Ω) (O, S[∗] (ν0 ))Pick up a ball from the urn behind the curtain

ω1

ω2

p

-

[∗]

1-p



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206CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/
Put
P ({x}) : the probability that a measured value x ( ∈ {w, b}) is obtainedI({x}) : the information quantity that is acquired when we know thata measured value x ( ∈ {w, b}) is obtained

ν1x : the posttest state after a measured value x ( ∈ {w, b}) is obtainedThen,
P ({w}) = 0.8p + 0.4(1 − p),

P ({b}) = 0.2p + 0.6(1 − p),

0.8p log 0.8 + 0.4(1 − p) log 0.4− log(0.8p + 0.4(1 − p)),0.8p + 0.4(1 − p)
0.2p log 0.2 + 0.6(1 − p) log 0.6
I({b}) =− log(0.2p + 0.6(1 − p)),0.2p + 0.6(1 − p)I({w}) =0.8p,
0.8p + 0.4(1 − p)
0.2p
ν1b ({ω1 }) =,0.2p + 0.6(1 − p)

ν1w ({ω1 }) =0.4(1 − p),
0.8p + 0.4(1 − p)0.6(1 − p)
ν1b ({ω2 }) =.
0.2p + 0.6(1 − p)
ν1w ({ω2 }) =Then, we see, by (8.58), that
H(MC(Ω) (O, S[∗] (ν0 )))=[F ({w})](ω1 ) log[F ({w})](ω1 )p + [F ({w})](ω2 ) log[F ({w})](ω2 )(1 − p)
+ [F ({b})](ω1 ) log[F ({b})](ω1 )p + [F ({b})](ω2 ) log[F ({b})](ω2 )(1 − p)
− P ({w}) log P ({w}) − P ({b}) log P ({b})=0.8(log 0.8)p + 0.4(log 0.4)(1 − p) + 0.2(log 0.2)p + 0.6(log 0.6)(1 − p)
− (0.8p + 0.4(1 − p)) log(0.8p + 0.4(1 − p)) − (0.2p + 0.6(1 − p)) log(0.2p + 0.6(1 − p)).
Assume that p = 1/2. Then, we see that
H(MC(Ω) (O, S[∗] (ν0 ))) = 0.6 − 0.3 log2 3 = 0.123 · · · (bit).
¥
Example 8.18. [Fuzzy information (fast or not fast), cf. [42]]. Let Ω ≡ {ω1 , ω2 , ..., ω100 }be a set of pupils in some school. Let Ob ≡ (X = {yb , nb }, 2X , b(·) ) be the crisp C ∗ observable in the commutative C ∗ -algebra C(Ω) such that b{yb } (ωn ) = 0 (n is odd), = 1

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207(n is even), and b{nb } (ωn ) = 1 − b{yb } (ωn ). Also, let Of ≡ (Y = {yf , nf }, 2Y , f(·) ) be
the C ∗ -observable in C ∗ -algebra C(Ω) such that f{yf } (ωn ) = (n − 1)/99 (∀ωn ∈ Ω) andf{nf } (ωn ) = 1 − f{yf } (ωn ). Let ρ0 ∈ Mm
+1 (Ω), for example, assume that ρ0 = νu , i.e., theequal weight on Ω, namely, νu ({ωn }) = 1/100 (∀n). Thus we have two measurements
MC(Ω) (Ob , S(νu )) and MC(Ω) (Of , S(νu )).
1f{nf }

f{yf }

0Ω
100

Then, we see, by (8.58), that³´
H MC(Ω) (Ob , S(νu ))= −∥b{yb } ∥L1 (Ω,νu ) log ∥b{yb } ∥L1 (Ω,νu ) − ∥b{nb } ∥L1 (Ω,νu ) log ∥b{nb } ∥L1 (Ω,νu )1
1 1
1= − log − log = log2 2 = 1 (bit),2
2 2
2(8.59)and³´

H MC(Ω) (Of , S(νu ))ZZ=
f{yf } (ω) log f{yf } (ω)νu (dω) + f{nf } (ω) log f{nf } (ω)νu (dω)ΩΩ− ∥f{yf } ∥L1 (Ω,νu ) log ∥f{yf } ∥L1 (Ω,νu ) − ∥f{nf } ∥L1 (Ω,νu ) log ∥f{nf } ∥L1 (Ω,νu )Z 1
1≈2
λ log2 λdλ + 1 = −+ 1 = 0.278 · · · (bit).
2 loge 20(8.60)(8.61)For example, assume that the symbol “yb ” [resp. “nb ”] in X is interpreted by “boy”[resp. “girl”]. And “yf ” [resp. “nf ”] in Y is interpreted by “fast runner” [resp. “notfast runner”]. When we guess the pure state (∗) of the system S ( = S(∗) (νu )) in theabove situation, the (8.60) and (8.61) say that the crisp information “boy or girl” is more
eﬃcient than the fuzzy information “fast or not fast”.
¥
Remark 8.19.

[Fuzzy information theory]. “Shannon’s entropy” is usually deﬁned

as follows (cf. [79]). Let (Ω, B, P ) be a probability space. Let D = {D1 , D2 , ...} be

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the countable decomposition of Ω. Then, the entropy H(D) of D is deﬁned by H(D) =P− ∞n=1 P (Dn ) log P (Dn ). Note that Deﬁnition 8.16 is the natural extension of Shannon’s
entropy if we regard the observable O as a “fuzzy decomposition”(cf. the formula (2.30)).
¥

8.6

Belief measurement theory (=BMT)In this section we study “belief measurement theory (=BMT)”, which is considered
to be closely related to “subjective Bayesian statistics”.9
Firstly let us consider the following problem:(P ) For example, consider a measurement MC(Ω) (O ≡ (X, 2X , F ), S[∗] ) formulated inC(Ω), where Ω = {ω1 , ω2 }, and further, assume that we have no information about
the [∗]. How do we represent “having no information about the [∗]” mathematically?
Or, how do we infer the statistical state?
We prepare three answers to the problem (P ) in this book. That is, we consider threekinds of “having no information about the [∗]” (or, “having no belief whether [∗] = ω1 or[∗] = ω2 ) as follows:(A1 ) Iterative likelihood function method in PMT. See MC(Ω) (O, S[∗] ((kI))lq ) in §5.5.
(A2 ) The principle of equal probability (= “PEP”). As seen later (i.e., Theorem 11.12),
this is essentially equivalent to the hypothesis that the [ ∗ ] is chosen by a faircoin-tossing (e.g., p = 0.5 in (8.7)). That is, it suﬃces to consider the statisticalmeasurement MC(Ω) (O, S[∗] (νu )), where νu ({ω1 }) = νu ({ω2 }) = 1/2.
(A3 ) The principle of equal weight(=“PEW” =Bayes’ postulate). See §8.6.2 later. Thismethod will be called “belief measurement theory” (or, “BMT”).
9This is not sure since my understanding of the subjective Bayesian statistics (cf. [21]) is not suﬃcient.

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Thus we may have the following classiﬁcation (and correspondence):
PMT= measurement + the relation among systems←→ (A1 )[Axiom 1 (2.37)][Axiom 2 (3.26)]






PMT
+
“statistical state”←→ (A2 )MT SMT = (Axioms
1 and 2)(the probabilistic interpretation of mixed state)







PMT
+
“belief weight”←→ (A3 ) BMT = (Axioms
1 and 2)(8.62)(the principle of equal weight)

8.6.1

The general argument about BMT

In §8.1∼§8.5, we studied SMT (i.e., Proclaim 1 (= the probabilistic interpretation of“mixed state” ) + Axiom 2), in which “mixed state” has the probabilistic interpretation.
In this section, we propose another interpretation of “mixed state”, which may be called“belief interpretation”. That is, we want to assert:
“probabilistic interpretation”[Proclaim 1 (8.10)]ρm ∈ Sm (A∗ ) · · ·(mixed state)
 “belief interpretation”→ “SMT”

in §8.1∼8.5(8.63)→ “BMT”

in this §8.6

[the principle of equal weight (8.72)]

The purpose of this section is, of course, to propose “belief measurement theory” (or,
“BMT”).
We begin with a simplest example as follows. Consider the statistical measurement
MC(Ω) (O ≡ ({w, b}, 2{w,b} , F ), S[∗] (ν0 )). Here[F ({w})](ω1 ) = 0.8, [F ({b})](ω1 ) = 0.2,

[F ({w})](ω2 ) = 0.4,

[F ({b})](ω2 ) = 0.6,

and, ν0 ({ω1 }) = 1/4 and ν0 ({ω1 }) = 3/4. Recall that this measurement is symbolicallydescribed as follows.
Pick up a ball from the urn behind the curtain

U1 (≈ ω1 )U2 (≈ ω2 )

1/4

-

[∗]

3/4



Figure(8.64)

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By a hint of the Figure (8.64), we can introduce “BMT” as follows. Assume that there
are 100 people. And moreover assume that10½
25 people (in 100 people) believe that [∗] = U1
75 people (in 100 people) believe that [∗] = U2
That is, we have the following picture (instead of Figure (8.64)):
Pick up a ball from the urn behind the curtain

U1 (≈ ω1 )U2 (≈ ω2 )-25 people believe that [∗] = U1 .

[∗]

75 people believe that [∗] = U2 .
Figure(8.65)

This is just the “belief measurement”, which is denoted by MC(Ω) (O, S[∗] ((ν0 ))bw ). Also,
the ν0 is called a belief weight (or, approval rate, conviction degree ).11
We add the following remark:(R1 ) Note that the [∗] (in MC(Ω) (O, S[∗] ((ν0 ))bw ) ) is assumed to be unknown. Thus,³
DE´
the triplet X, 2X , M(Ω) ν, F (·) C(Ω) is a merely mathematical symbol and nota sample space. In other words, it is nonsense to consider the probability thatthe measured value obtained by MC(Ω) (O ≡ (X, 2X , F ), S[∗] ((ν))bw ) belongs to Ξ(∈
2X ). That is, Proclaim 1(8.10) does not hold for a belief measurement MC(Ω) (O ≡(X, 2X , F ), S[∗] ((ν))bw ), or equivalently, a belief measurement has no sample space.
This (R1 ) is clear. That is because the argument mentioned in Example 8.1 is invalid
for a belief measurement, since ν ( in MC(Ω) (O, S[∗] ((ν))bw ) ) is a belief weight and not a
statistical state.
However (i.e., in spite of the fact that Proclaim 1(8.10) is invalid), we have the following
theorem:
Theorem 8.20. (Bayes theorem for belief measurements). Assume that we know thata measured value obtained by a belief measurement MC(Ω) (O ≡ (X, F, F ), S[∗] ((ν))bw )10

Recall “parimutuel betting” , which is very applicable. For example, we may consider the“probability” that life exists on Mars.
11
Thus, outsiders may think that MC(Ω) (O, S[∗] ((ν0 ))bw ) and MC(Ω) (O, S[∗] ) are the same. That is
because the number of the believers is not related to the measurement itself.

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belongs to Ξ (∈ F). Then, we have the “Bayes theorem” such that
(0,0)

Mm
+1 (Ω) ∋ ν (= priori belief weight) 7→ (posterior belief weight =)RΞ(ν) ∈ Mm
+1 (Ω).

(8.66)

where
R
(0,0)
[RΞ (ν)](D0 ) =[F (Ξ)](ω)ν(dω)
RD0[F (Ξ)](ω)ν(dω)Ω(∀D0 ⊆ Ω; Borel set ).

(8.67)

Proof. It suﬃces to prove a simple case since the proof of the general case is similar.
For example, consider the following ﬁgure, which is essentially the same as Figure (8.65).

Pick up a ball from the urn behind the curtain

U1 (≈ ω1 )U2 (≈ ω2 )-[∗]25 % people believe that [∗] = U1 .
20 % people guess that a white ball will be picked.
5 % people guess that a black ball will be picked.



75 % people believe that [∗] = U2 .
30 % people guess that a white ball will be picked.
45 % people guess that a black ball will be picked.

Assume that a “white ball ” is picked in the above picture. Then, we see:U1 (≈ ω1 )U2 (≈ ω2 )-[∗]25 % people believe that [∗] = U1 .
20 % people guess that a white ball will be picked.
5 % people guess that a black ball will be picked.



75 % people believe that [∗] = U2 .
30 % people guess that a white ball will be picked.
45 % people guess that a black ball will be picked.

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U1 (≈ ω1 )U2 (≈ ω2 )

-

[∗]



40 % people believe that [∗] = U1 .

60 % people believe that [∗] = U2 .

Thus we see that Bayes theorem holds for belief measurements. That is because Theorem
8.20 (Bayes theorem for belief measurements) says:
(0,0)

Mm
+1 (Ω) ∋ ν0 (= priori belief weight) 7→ (posterior belief weight =)RΞ
where
R
(0,0)
[R{w} (ν0 )]({ω}) =

{ω}

R

Ω

[F ({w})](ω)ν0 (dω)[F ({w})](ω)ν0 (dω)=(ν0 ) ∈ Mm
+1 (Ω). (8.68)


8
× 14
40
10


8
1
 10 × 4 + 104 × 34 = 100(if ω = ω1 )
(if ω = ω2 )

4
× 34
10
8
1
4
×
+
× 34
10
4
10

60
= 100

(8.69)
Although this proof is easy, it should be noted that this is diﬀerent from the proof ofBayes theorem for a statistical measurement. That is because Proclaim 1 (8.20) can not
be used in the proof of Theorem 8.26.
Remark 8.21.

(Extensive interpretation in theoretical informatics). Seeing Figure(8.65), some may think that the belief weight ν (in MC(Ω) (O ≡ (X, F, F ), S[∗] ((ν))bw )represents the only “public opinion”. However, this is wrong. Recall the spirit of theoretical informatics (in the footnote below the statement (1.12) in Chapter 1), i.e.,“extensive interpretation”. Thus, we consider that the belief weight ν (in MC(Ω) (O ≡(X, F, F ), S[∗] ((ν))bw ) often represents “personal belief”.
¥

8.6.2

The principle of equal weight

As mentioned in the previous section (i.e., §8.6.1) we have the following notation:

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213

¡
¢
The symbol MC(Ω) O, S[∗] ((ν))bw , (ν ∈¡¢Mm
+1 (Ω)), is assumed to represent the measurement MC(Ω) O ≡ (X, F, F ), S[∗] un-

Notation 8.22.

¡¢[MC(Ω) O, S[∗] ((ν))bw ].

der the hypothesis that the belief weight of the system S[∗] is ν. And it is called a belief
measurement.
¥
Now let us explain “Bayes postulate” (= “the principle of equal weight”). Assume
that Ω is ﬁnite (i.e., Ω = {ω1 , ω2 , ..., ωN }). Then, there is a reason to think that the mixed
state νu ( ∈ Mm
+1 (Ω)) deﬁned byνu (D) =♯[D]N(∀D ⊆ Ω)

represents “the loosest belief” or “knowing nothing about S[∗] ”.

(8.70)(The νu is called the“equal weight”. Cf. Remark 8.23 later). If Ω is inﬁnite, we have no ﬁrm opinion.12 Thusin this section we always assume that Ω is ﬁnite.
We add the following remark.
Remark 8.23. [Mathematical properties of equal weight νu , [42]]. Let Ω ≡ {ω1 , ω2 , ..., ωN }m
be a ﬁnite set with the discrete topology. Let ρm
0 be arbitrary belief weight (i.e., ρ0mm∈ Mm+1 (Ω)). Then, deﬁne the entropy H(ρ0 ) of the ρ0 by

³
H

ρm
0

´

=−

N
X

m
ρm
0 ({ωn }) log ρ0 ({ωn }).

n=1

Here, it is well known that
n ³ ´
o
m
m
(i) sup H ρm
:
ρ
∈
M
(Ω)
= log N ,00+1(8.71)m(ii) “ρm
0 ({ωn }) = 1/N (∀n)” ⇐⇒ “H(ρ0 ) = log N ”.

(iii) Let Tav : C(Ω) → C be the average functional on C(Ω), i.e., a linear positive
functional such that:(a)Tav (1) = 1(b)

Tav (f ) = Tav (f ◦ ϕ)(∀f ∈ C(Ω), ∀ bijection ϕ : Ω → Ω)

where (f ◦ ϕ)(ω) = f (ϕ(ω)).
For example, we may consider as follows: Let Ω be not ﬁnite. Let SΩ be a subset of {Φ | Φ : C(Ω) →C(Ω) is a Markov operator }. Assume that the SΩ has the unique invariant state νu ( ∈ Mm
+1 (Ω)), thatis, Φ∗ νu = νu (∀Φ ∈ SΩ ). And further assume that νu (U ) > 0 (∀U ( ⊆ Ω, open ). Then, we may say thatthe νu represents “no belief weight (concerning SΩ )” or “completely shuﬄed weight”. Also, see [47].
12

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214CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(iv) Tav is uniquely determined such as Tav (f ) =

³

R

Ω

f (ω)νu (dω)≡

PN

n=1 f (ωn )

N

´

(∀f ∈

C(Ω)).
¥
Therefore, we can assert:

The principle of equal weight (= “PEW” = Bayes’ postulate).
[The belief interpretation of mixed states]. Consider a system S[∗] formulated in C(Ω) where the state space Ω (≡ {ω1 , ω2 , ..., ωN }) is a ﬁnite
set. The belief weight is represented by a mixed state ν (∈ Mm+1 (Ω)). In
PN
1m
particular, the equal weight νu (≡ N n=1 δωn ∈ M+1 (Ω)) represents “theloosest belief”.
(8.72)
Thus BMT is summarized as follows.
[BMT1 ] the equal weight νu ( ∈ Mm
+1 (Ω)) represents “the most loosest belief”.
[BMT2 ] After we get the measured value x by a belief measurement MC(Ω) (O ≡ (X, 2X , F ),mm
S[∗] ((ρm0 ))bw ), the new belief weight of the system S[∗] is changed to ρnew ( ∈ M+1 (Ω))
R

[F ({x})](ω)ρm (dω)

0RB(∀B ∈ BΩ , Borel ﬁeld).
such that ρmnew (B) =[F ({x})](ω)ρm (dω)
Ω

0

(0,0)

m
Deﬁne the map [R{x} ] : Mm
+1 (Ω) → M+1 (Ω) such that:

R(0,0)[R{x} ](ρm ) =[F ({x})](ω)ν(dω)
RD0[F ({x})](ω)ν(dω)Ω(∀D0 ⊆ Ω; Borel set ).

(8.73)

Then, we can symbolically describe it as follows:[BMT1 ] the loosest belief weight[BMT] =`←→

ν u ( ∈ Mm
+1 (Ω))´O, S[∗] ((ρ))bw
M
(0,0)

 [BMT2 ] S[∗] ((ρ))bw −−C(Ω)−−−−−−−−−→ S[∗] (([R{x} ](ρ)))bw ,(8.74)

x is obtained

which should be compared with the characterization (5.80) of “Iterative likelihood function
method”.
Example 8.24. [= Example 5.24 (the urn problem)]. There are two urns ω1 and ω2 .
The urn ω1 [resp. ω2 ] contains 8 white and 2 black balls [resp. 4 white and 6 black balls].

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U1 (≈ ω1 )U2 (≈ ω2 )[∗]-50% people believe that [∗] = U1 .



50% people believe that [∗] = U2 .
Figure(8.75)

Assume that they can not be distinguished in appearance.
• Choose one urn from the two.

(8.76)

Now you sample, randomly, with replacement after each ball.
(i). First, you get “white ball”.
(Q1 ) Do you believe which the chosen urn is, ω1 or ω2 ?
(ii). Further, assume that you continuously get “black”.
(Q2 ) How about the case?

Do you believe which the chosen urn is, ω1 or ω2 ?

And further,(Q3 ) Also, study the case that the urn is chosen by a fair coin-tossing in (8.76).
[Answers]. In what follows this problem is studied in BMT. Put Ω = {ω1 , ω2 }. O =({w, b}, 2{w,b} , F ) where [F ({w})](ω1 ) = 0.8, [F ({b})](ω1 ) = 0.2, [F ({w})](ω2 ) = 0.4,
[F ({b})](ω2 ) = 0.6. The PEW (8.72) says that the loosest belief is represented by νu(i.e., νu ({ω1 }) = νu ({ω2 }) = 1/2)]. Thus we have the belief measurement MC(Ω) (O,S[∗] ((νu ))bw ).
(A1 ). Thus, consider MC(Ω) (O, S[∗] ((νu ))bw ). Since the measured value “w” was obtained,
the new belief weight ρmnewR[F ({w})](ω)νu (dω) ¢¡0.8 × 21
2
{ω1 }
R
({ω
})
=
ρm
=
1
new
1
1 = ,
3
[F ({w})](ω)νu (dω)
0.8 × 2 + 0.4 × 2Ω

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1ρm
new ({ω2 }) = .
3

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216CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(A2 ). Next, consider the measurement MC(Ω) (O, S[∗] (( ρmnew ))bw ). Since the measured
value “b” was obtained, the new belief weight ρmnew2 is represented byR³´[F ({b})](ω)ρm
0.2 × 232
new (dω){ω1 }m
ρnew2 ({ω1 }) = R
=
2
1 = ,
m
5
[F ({b})](ω)ρnew (dω)
0.2 × 3 + 0.6 × 3
R Ω³´[F ({b})](ω)ρm
0.6 × 133
new (dω)
{ω2 }
R
ρm
({ω
})
=
=
=
.
2
new
2
1
m (dω)
5[F({b})](ω)ρ
0.2
×
+
0.6
×
new
Ω
3
3
m
(A3 ) Also, when the urn is chosen by a fair coin-tossing, the above ρm
new and ρnew2 acquiremthe probabilistic interpretation. That is, ρm
new and ρnew2 are regarded as statisticalstates.
[Remark]. In order to make a belief measurement MC(Ω) (O, S[∗] ((νu ))bw ) change a statistical measurement MC(Ω) (O, S[∗] ( νu )), we have two methods. One is the fair coin-tossing
method as mentioned in the above (A3 ) ( and (Q3 )). Another will be proposed as SMTPEP
in §11.4, i.e., “the principle of equal probability”. Also, note that Theorem 11.12 says thatthe two methods are equivalent.
¥

8.6.3

Is BMT necessary?

Now we have the following classiﬁcation:

PMT= measurement + the relation among systems
[Axiom 1 (2.37)][Axiom 2 (3.26)]






PMT
+
“statistical state”
MT SMT = (Axioms
1 and 2)(the probabilistic interpretation of mixed state)







PMT
+
“belief weight”
 BMT = (Axioms
1 and 2)(8.77)(the principle of equal weight)

However, we must consider and answer the following question:(Q) Is BMT necessary?
In fact, some may think that(A) BMT is not necessary. It suﬃces to substitute SMT for BMT carefully. In theoreticalinformatics, the “economical” should come before the “exact”.

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I may agree with them. However, it should be remarked that(R) It is clear that we can not use SMT carefully without the understanding of therelation between SMT and BMT (i.e., without the understanding of the contents in§8.1 ∼ 8.6.2). Especially, note that Proclaim 1 (8.10) is not valid in BMT.
If this (R) is admitted, I agree to the above opinion (A). Thus, I recommend readers to
use BMT at least until becoming accustomed to BMT. Also, it should be noted that there
is a great confusion in the conventional statistics.
Remark 8.25. [The term: “subjectivity”]. Since the term “subjectivity” is frequently
used in statistics, we must be careful for the usage of “subjectivity”. For example, consider
the following phrase:• the probability that tomorrow is ﬁne.

(8.78)

The above term:“probability” is usually called a “subjective probability”. However, the“probability” in (8.78) is the same as the “probability” in the following problem (whichis due to Newtonian mechanics, and thus, deterministic). In spite of the deterministic
system, we have the following question:“Calculate the probability that the ball surmounts the mountain M.”(8.79)

That is, the case (a) or (b)?

⃗v -

(a)

Ball
x

M

(b)

bumpy

where the initial condition x(position) and ⃗v (velocity) are values with errors, and also,
the diﬀerential equation is not completely known. However, it should be noted that thisproblem is usual in engineering. Thus, if this is subjective (or, if a dearth of information
implies “subjective”), we consider that almost every problem in engineering is subjective.13
13

Recall the argument in Chapter 1. That is, in theoretical physics we must be in the objective standing
point. On the other hand, in theoretical informatics (and its applications) we are, more or less, in thesubjective standing point. Recall the engineer’s spirit “Use everything available”. Thus we may ask theexcellent bookmaker about the problem (8.79). However, it should be noted that the bookmaker may
calculate the “subjective probability in the sense of BMT (or, parimutuel betting among general people)”.

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There is a reason to consider that the probaility in the problem (8.79) can be regarded
as the “subjective probability in the sense of parimutuel betting among a certain setof specialists”. However, it is so, every probability may be regarded as the subjective
probability. Thus, in this book, the term: “subjective probability” is used in the casethat it is regarded as the probability in the sense of parimutuel betting.
¥Remark 8.26. [Diﬀerential geometry and operator algebra, cf. Table (1.8a)(4)]. In
mathematics, diﬀerential geometry is ﬂexible, but the theory of operator algebras (i.e,C ∗ -algebra and W ∗ -algebra) somewhat lacks adaptability. Thus, in MT14, we can notprepare so many ready-made theories. For example, we have two ready-made theories(i.e., BMT and SMTPEP (cf. §11.4)). This fact (i.e., few ready-made theories can beproposed) is just what we want. That is because to choose one from too many readymade theories is essentially the same as to create a made-to-order theory. On the other
hand, in order to create a made-to-order theory in theoretical physics, the ﬂexibility ofdiﬀerential geometry is essential.
¥

8.7

Appendix (Bertrand’s paradox)

As mentioned in Remark 8.4, a natural mixed state is not always a ststistical state.
In fact we see, in §8.6, that the no informational weight νu (∈ Mm+1 (Ω), where Ω is ﬁnite)
defnied by (8.70) can not be unconditionally regarded as the statistical state.15 (As seen
later (in §11.4), the term “unconditionally” is important.) In this section, we study
Bertrand’s paradox, which promote our understanding of the relation between a natural
mixed state and a ststistical state.

8.7.1

Review (Bertrand’s paradox)

Here, let us review the usual argument about Bertrand’s paradox (cf. [35]). Consider
14

Although Fisher information is closely related to Riemann manifold (in diﬀerential geometry, cf [5],[24]), it is not the axiom of MT but a kind of method.
15
The νu is invariant concerning any bijection ϕ on Ω, i.e., ϕ(νu ) = νu . In this sense, it is natural.

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219

the following problem:(P1 ) Given a circle with the radius 1. Suppose a chord of the circle is chosen at random.
√
What is the probability that the chord is longer than 3 (i.e., the side of an inscribedequilateral triangle)?
The problem has apparently several solutions as follows:(Fig.1)(Fig.2)A[First Solution (Fig.1)]. The “random endpoints” method: Choose a point A on thecircumference and rotate the triangle so that the point is at one vertex. Choose anotherpoint on the circle and draw the chord joining it to the ﬁrst point. For points on thearc between the endpoints of the side opposite the ﬁrst point, the chord is longer than aside of the triangle. The length of the arc is one third of the circumference of the circle,
therefore the probability a random chord is longer than a side of the inscribed triangle is
one third.
[Second Solution (Fig.2)]. The “random radius” method: Choose a radius of the circle androtate the triangle so a side is perpendicular to the radius. Choose a point on the radiusand construct the chord whose midpoint is the chosen point. The chord is longer thana side of the triangle if the chosen point is nearer the center of the circle than the point
where the side of the triangle intersects the radius. Since the side of the triangle bisects
the radius, it is equally probable that the chosen point is nearer or farther. Therefore theprobability a random chord is longer than a side of the inscribed triangle is one half.

8.7.2

Bertrand’s paradox in measurement theory

We assert that

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220CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(♯) If Bertrand’s paradox is a paradox (i.e., if the argument in §8.7.1 is considered to
be strange), it is due to the confusion between statistical states and mixed states(cf. (8.11)).
In what follows, we shall explain it. Consider the following problem:(P2 ) Given a circle with the radius 1. Deﬁne the state space Ω by the set composed of
all chords of this circle. Then, ﬁnd a natural mixed state ρ ( ∈ Mm
+1 (Ω)).
The reader will ﬁnd that the (P2 ) is essentially the same as the problem (P1 ) in §8.7.1.
Thus, the above problem has also apparently several solutions as follows:(Fig.0)l

Represent a chord lby a natural coordinate!

(Fig.1′ )(Fig.2′ )l(α,β)βαl(r,θ)rθ[First Solution (Fig.1′ )]. See Fig.0 (Represent a chord by a natural coordinate!). In
Fig.1′ , we see that the chord l is represented by a point (α, β) in the rectangle R1 ≡{(α, β) | 0 ≤ α ≤ 2π, 0 ≤ β ≤ π/2(radian)}. That is, we have the following identiﬁcation:Ω ∋ l(α,β) ←→ (α, β) ∈ R1 .
m
Under the identiﬁcation, we get the natural mixed state ρ1 ( ∈ Mm
+1 (Ω) ≈ M+1 (R1 )) such

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221

Area[A] = Area[A] (∀A ∈ B ), where “Area” = “Lebesgue measure”.
that ρ1 (A) = Area
R1
π2
[R ]
1

Therefore, we see√ρ1 ({l(α,β) ∈ Ω | ”the length of l(α,β) ” ≥ 3})
Area[{(α, β) | 0 ≤ α ≤ 2π, 0 ≤ β ≤ π/6}]=Area[{(α, β) | 0 ≤ α ≤ 2π, 0 ≤ β ≤ π/2}]
2π × (π/6)
1== .
2π × (π/2)
3(8.80)[Second Solution (Fig.2′ )]. See Fig.0 (Represent a chord by a natural coordinates). In
Fig.2′ , we see that the chord l is represented by a point (r, θ) in the rectangle R2 ≡{(r, θ) | 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π}. That is, we have the following identiﬁcation:Ω ∋ l(r,θ) ←→ (r, θ) ∈ R2 .
m
Under the identiﬁcation, we get the natural mixed state ρ2 ( ∈ Mm
+1 (Ω) ≈ M+1 (R2 )) such
Area[A] = Area[A] (∀A ∈ B ). Therefore, we seethat ρ2 (A) = Area
R2
2π
[R2 ]
√
ρ2 ({l(α,β) ∈ Ω | ”the length of l(r,θ) ” ≥ 3})
Area[{(r, θ) | 0 ≤ r ≤ 1/2, 0 ≤ θ ≤ 2π}]1=
= .
(8.81)
Area[{(r, θ) | 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π}]2Since the above argument is related to “mixed state” and not “statistical state”, wehave no paradox in the above arguments. That is, if Bertrand’s paradox is a paradox(in §8.7.1), it is due to the confusion between mixed states (mathematical concept) andstatistical states (measurement theoretical concept).
Some may assert that:• it suﬃces to test (8.80) or (8.81) experimentally.
However, it is not true. For completeness, we add the following remark.
Remark 8.27. [Mixed state and statistical state]. In the above arguments, note that ρ1m( ∈ Mm+1 (Ω)) and ρ2 ( ∈ M+1 (Ω)) are mixed states and not statistical states. In order toregard a mixed state ρ1 ( ∈ Mm
+1 (Ω)) as a statistical state, we must add the probabilistic
interpretation to the mixed state ρ1 . This is, for example, done as follows:(R1 ) Prepare two urns A and B, which respectively contain 100 balls (i.e., “ball 1”, “ball2”, ...,“ball 100” ). Pick out one ball from the urn A. Assume that the ball is “ballSample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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222CHAPTER 8. STATISTICAL MEASUREMENTS IN C ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/m”.

Next, pick out one ball from the urn B. Assume that the ball is “ball n”.

Deﬁne (α, β) in the rectangle R1 such that:
α=

2πm
,
100

β=

πn
.
200

Then, if (α, β) is chosen according to the above rule (R1 ), the mixed state ρ1 ( ∈ Mm
+1 (Ω))
acquires the probabilistic interpretation. And thus, it can be regarded as a statistical state.
In fact, if we take an exact measurement, we see that the probability that the length of√the chord is longer than 3 is given by 1/3. Of course, by a similar way, we can add
the probabilistic interpretation to the ρ2 (in the second solution). That is, it suﬃces tochoose a chord as follows.
(R2 ) Prepare two urns A and B, which respectively contain 100 balls (i.e., “ball 1”, “ball2”, ...,“ball 100” ). Pick out one ball from the urn A. Assume that the ball is “ballm”.

Next, pick out one ball from the urn B. Assume that the ball is “ball n”.

Deﬁne (r, θ) in the rectangle R1 such that:
r=

m
,
100

θ=

2πn
.
200
¥

Summing up, we conclude as follows. Consider the following problem:(P1 )′ Given a circle with the radius 1. And choose a chord. Find the probability that the√chord chosen is longer than 3 (i.e., the side of an inscribed equilateral triangle).
Then, we see:(A1 ) If we know that the chord was chosen by the rule (R1 ) in Remark 8.27, we can√conclude that the probability that the chord chosen be longer than 3 is 1/3.
(A2 ) If we know that the chord was chosen by the rule (R2 ) in Remark 8.27, we can√conclude that the probability that the chord chosen be longer than 3 is 1/2.
(A3 ) If we know that the chord was chosen by the physical experiment (conducted in[49]), we may conclude that the probability that the chord chosen be longer than√
3 is about 1/2 (cf. [49]).
(A4 ) etc.

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223

We consider that something like a (physical) coin-tossing (such as Brownian motion,
radioactive atom, etc.) is hidden behind the physical experiment (in (A3 )). Thus, weagain stress that
• A “coin-tossing” is always hidden behind a statistical state. Or there is no statisticalstate without a “coin-tossing” (or, “dice-throwing”, “urn problem”).
Also, it should be noted that we are in theoretical informatics and not in theoreticalphysics.

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Chapter 9
Statistical measurements inW ∗-algebraic formulation∗

The Statistical MT (= SMT) has two kinds of formulations. One is SMTC (i.e., the C ∗ −algebraic
formulation of SMT), which was introduced in the previous chapter, that is,∗in C ∗ -algebra.

SMTC = statistical measurement + the relation among systems[Proclaim 1 (8.10)][Axiom 2 (3.26)](9.1)(=(8.2))∗In this chapter we introduce another formulation of SMT (i.e., SMTW ), that is,∗in W ∗ -algebra,

SMTW = statistical measurement+ the relation among systems∗[ProclaimW 1 (9.9)]∗[ProclaimW 2 (9.23)]

∗

(9.2)

∗

which is called the W ∗ −algebraic formulation of SMT. Of course, “SMTC ” and “SMTW ” areessentially the same. The diﬀerence between the two is that of the mathematical tools (i.e.,∗∗C ∗ -algebra and W ∗ -algebra). Thus, “SMTW ” should be understood by an analogy of “SMTC ”.
Although the C ∗ -algebraic formulation is most fundamental, the W ∗ -algebraic formulation is rather
handy from the mathematical point of view.

9.1

Statistical measurements (W ∗-algebraic formulation)

The Statistical MT (= SMT) has two kinds of formulations. One is the C ∗ −algebraic∗

formulation of SMT (= SMTC ), which was introduced in the previous chapter. In orderto develop “Statistical MT”, in this chapter we introduce the W ∗ -algebraic formulation of∗∗∗

Statistical MT (= SMTW ).1 Here, it should be noted that “SMTC ” and “SMTW ” are∗Of course, the (pure) measurement theory (= PMT) has also two kinds of formulations, i.e., PMTC∗∗
and PMTW . However, the commutative
PMTW has a demerit such that a pure state can not be∗
represented in the commutative PMTW in general. (cf. the statement (9.3)). Thus, we usually focus∗∗∗
on SMTW and not
PMTW . However, it should be noted that as far as quantum mechanics, PMTW isC∗
superior to PMT . Cf. §9.3.
1

225
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essentially the same. The diﬀerence between the two is that of the mathematical tools(i.e., C ∗ -algebra and W ∗ -algebra).
The C ∗ -algebraic formulation stated in the previous chapter is, of course, most fundamental. However, from the mathematical (or technical) point of view, the topology ofa C ∗ -algebra A is somewhat too strong. Note that any C ∗ -algebra A can be imbedded
into B(V ), the algebra composed of all bounded linear operators on a Hilbert space V(cf. Theorem 2.4 (the GNS-construction in [50, 76, 82])). Thus, using the imbedding:A ⊆ B(V ), we may start from the weak∗ -closure A (of A) in B(V ). This A is called a W ∗ algebra. This method (i.e., to formulate measurement theory in terms of W ∗ -algebras) iscalled the W ∗ -algebraic formulation. Though this method is somewhat methodological,³
it is rather handy from the mathematical point of view. For example, this will be seen´
in Theorem 10.1 in Chapter 10.
Let N be a W ∗ -algebra, that is,[♯1 ] N is a weak∗ closed subalgebra of a certain B(V ).
It is well known (see, for example, [76]) that this is equivalent to[♯2 ] N is a C ∗ -algebra with the pre-dual Banach space N∗ (i.e., N = (N∗ )∗ ).
Also, it is well known that the uniqueness of the pre-dual Banach space N∗ is assured.
However, we may sometimes call the pair (N, N∗ ) a W ∗ -algebra.
An element F in N is called self-adjoint if it holds that F = F ∗ . A self-adjoint element
F in N is called positive (and denoted by F ≥ 0) if there exists an element F0 in N suchthat F = F0∗ F0 where F0∗ is the adjoint element of F0 . Also, a positive element F is calleda projection if F = F 2 holds.
Now we can deﬁne the normal state-class Sn (N∗ ) such as
Sn (N∗ ) ≡ {ρn ∈ N∗ : ∥ρn ∥N∗ = 1 and ρn ≥ 0 (i.e., ρn (T ∗ T ) ≥ 0 for all T ∈ N)}.
The element ρn (in Sn (N∗ )) is called a normal state (or, density state). The linear
DE
DE
functional ρn (T ) is sometimes denoted by ρn , T , or precisely, N∗ ρn , T N . Also, notethat
• a W ∗ -algebra N has a lot of projections,that is, the set of all ﬁnite linear combinations of projections is dense in N in the weak∗
topology σ(N; N∗ ). Also, note that

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227

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/• N has always the identity IN .

Example 9.1. [(i): Commutative W ∗ -algebras ; L∞ (Ω, µ))]. Let (Ω, BΩ , µ) be a measure³´
space. For any 1 ≤ p ≤ ∞, deﬁne Lp (Ω, µ) ≡ Lp (Ω, BΩ , µ) = {f : f is a complex³
Rvalued measurable function such that ∥f ∥Lp ≡ [ Ω |f (ω)|p µ(dω)]1/p < ∞ }. Here, of´
course, ∥f ∥L∞ = ess.sup {|f (ω)| : ω ∈ Ω}. Then, the N ≡ L∞ (Ω, µ) is a commutativeW ∗ -algebra with the pre-dual Banach space N∗ = L1 (Ω, µ). We see, of course, thatR
Sn (N∗ ) = L1+1 (Ω, µ) ≡ {ρn ∈ L1 (Ω, µ) : ρn ≥ 0, Ω ρn (ω)µ(dω) = 1, i.e., ρn is a density
function on Ω }. Also, it is well known that any commutative W ∗ -algebra N is represented
by some L∞ (Ω, µ). It should be noted that
• a “pure state” can not be generally represented in terms of the commutativeW ∗ -algebra L∞ (Ω, µ),(9.3)

since we see2 that δω0 (i.e., a point measure at ω0 ( ∈ Ω)) does not necessarily belong to
L1 (Ω, µ). Summing up (and recalling Example 2.2), we see,concrete form
dual space
pre-dual spacepure state
mixed (normal) statecharacteristics3

commutative C ∗ -algebraC(Ω)M(Ω) ( = C(Ω)∗ )nothingδω0 ∈ Mp+1 (Ω) ≈ Ων ∈ Mm+1 (Ω)
topological approachcommutative W ∗ -algebra
L∞ (Ω; µ)
not important
L1 (Ω; µ) ( = L∞ (Ω; µ)∗ )(no representation in general)ρ ∈ L1+1 (Ω, µ)measure theoretical approach[(ii): The case that Ω is countable or ﬁnite]. Of course, the above table is in the casethat Ω is general. In the case that Ω ≡ {ω1 , ω2 , ..., ωn } is ﬁnite, we can easily see that“commutative C ∗ -algebra” = “commutative W ∗ -algebra”, that is, we see the following
identiﬁcations:C({ω1 , ω2 , ..., ωn }) ≈ Cn (cf . the formula (2.15)) ≈ L∞ ({ω1 , ω2 , ..., ωn }, µ)(9.4)

where µ is a measure such that µ({ωk }) > 0 (∀k = 1, 2, ..., n). Next consider the case thatΩ ≡ {ω1 , ω2 , ..., ωk , ...} is countable inﬁnite. The commutative W ∗ -algebra N is deﬁned by
In this sense the W ∗ -algebraic formulation is ﬁt to SMT rather than PMT. However note our spirit(8.12) : “There is no SMT without PMT”. Thus we think that PMT (i.e., the concept of “pure state” )is not only hidden in the C ∗ -algebraic formulation of SMT but also in the W ∗ -algebraic formulation ofSMT.
3The Ω in C(Ω) is a topological space. On the other hand, the Ω in L∞ (Ω; µ) is a measure space. Cf.
Remark 9.14 later.
2

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L∞ (Ω, µ), where µ({ωk }) > 0 (∀k = 1, 2, ...). In this case, a pure state ρωk (k = 1, 2, ...),
is deﬁned by ρωk (ω) = µ({ω1 k }) (if ω = ωk ), = 0 (if ω ̸= ωk ).
¥Example 9.2.

[Non-commutative W ∗ -algebras; B(V )]. When N = B(V ), we seemthat N∗ = T r(V ) (cf. Example 2.3) and Sn (N∗ ) = T r+1(V ) ≡ {ρn ∈ T r(V ) : ρn ≥DEP0, ∥ρn ∥T r(V ) = 1}. Also, note that T r(V ) ρn , T B(V ) = tr[ρn ·T ]V . Here, tr[A]V ≡ λ∈Λ ⟨eλ , Aeλ ⟩Vwhere {eλ |λ ∈ Λ} is a complete orthonormal basis in V . Also, it is well known that thevalue tr[A]V is independent of the choice of a complete orthonormal basis {eλ |λ ∈ Λ} inPm
V . Further, any ρn ( ∈ T r+1(V )) is represented by ρn = λ∈λ αλ |eλ ⟩⟨eλ | (in the trace
norm ∥ · ∥T r(V ) ) for some complete orthonormal basis {eλ |λ ∈ Λ} in V and some sequenceP{aλ }λ∈Λ of non-negative numbers such that λ∈Λ aλ = 1. Also it should be noted thatany |v⟩⟨v|, (∥v∥ = 1), is just a pure state4 . Summing up (and recalling Example 2.3), we
see,

concrete form
dual space
pre-dual spacepure state
mixed (normal) state

non-commutative C ∗ -algebraC(V )T r(V ) ( = C(V )∗ )nothingp(V )|v⟩⟨v| ∈ T r+1
mmixed state: ρm ∈ T r+1(V )non-commutative W ∗ -algebraB(V )not important
T r(V ) ( = B(V )∗ )p
(V )|v⟩⟨v| ∈ T r+1mnormal state: ρn ∈ T r+1
(V )
(9.5)
¥

The following deﬁnition is the W ∗ -algebraic form of Deﬁnition 2.7 (C ∗ -observables).
Deﬁnition 9.3. [W ∗ -observables]. Let N be a W ∗ -algebra. A W ∗ -observable ( or inshort, observable, fuzzy observable) O ≡ (X, F, F ) in N is deﬁned such that it satisﬁes
that(i) [ σ-ﬁeld ]. (X, F) is a measurable space, that is, F ( ⊆ 2X ) is a σ-ﬁeld on X, i.e., itsatisﬁes that∅ ∈ F,

Ξk ∈ F (k = 1, 2, ...) =⇒ ∪∞k=1 Ξk ∈ F,

Ξ ∈ F =⇒ Ξc ∈ F,

This fact (i.e., a pure state can be represented in terms of W ∗ -algebra B(V )) is remarkable. Thus,
The W ∗ -algebra B(V ) has a power to describe quantum PMT as well as quantum SMT. Cf. §9.4.
4

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229(ii) for every Ξ ∈ F, F (Ξ) is a positive element in N (i.e., 0 ≤ F (Ξ) ∈ N) such thatF (∅) = 0 and F (X) = IN , where 0 is the 0-element and IN is the identity element
in N, and,(iii) [ countably additivity ]. For any countable decomposition {Ξ1 , Ξ2 , ..., Ξj , ...} of Ξ,³´∞
i.e., Ξ, Ξj ∈ F, ∪j=1 Ξj = Ξ, Ξj ∩ Ξi = ∅( if j ̸= i) , it holds that
F (Ξ) =∞X

F (Ξj )

j=1

where the series is convergent in the sense of the weak∗ -topology σ(N; N∗ ) in N.
If F (Ξ) is a projection for every Ξ ( ∈ F), a W ∗ -observable (X, F, F ) in N is called a crisp
W ∗ -observable in N. Also, a crisp observable O ≡ (R, BR , F ) (or, (Rn , BRn , F )) in N iscalled a quantity (or, Rn -valued quantity) in W ∗ -algebra N.
¥
Now we show several W ∗ -observables (in Example 9.4 ∼ 9.7).
Example 9.4. [Crisp W ∗ -observables in L∞ (Ω, µ)]. (i). As a typical crisp W ∗ -observable
in L∞ (Ω, µ), the exact observable OEXA ≡ (Ω, BΩ , χ(·) ) is frequently used where χΞ is the³´
characteristic function of Ξ ( ∈ BΩ ) i.e., χΞ (ω) = 1(ω ∈ Ξ), = 0 (otherwise) . Thisobservable is ﬁnest in L∞ (Ω, µ), i.e., it includes all projections.
(ii). Consider the commutative W ∗ -algebra L∞ (Ω, µ). Let a : Ω → R be a measurable
function. Then, we can deﬁne the crisp W ∗ -observable Oa = (R, BR , F ) in L∞ (Ω, µ)such that [F (Ξ)](ω) = χa−1 (Ξ) (ω) (∀Ξ ∈ BR , ∀ω ∈ Ω). Note that we can identify thereal-valued measurable function a( · ) with the Oa . That is, we see
a:Ω→R(real valued measurable function on Ω)←→ (R, BR , F ) in L∞ (Ω, µ).

(crisp observable)

That is because it holds that [F (( − ∞, λ))](ω) = 0 (if λ < a(ω)), = 1 (if λ ≥ a(ω)), andRR
therefore, the a(ω) is determined by the equality a(ω) = R λδa(ω) (dλ) = R λ[F (dλ)](ω)(a.e. µ). A real-valued measurable function on Ω is called a (classical ) quantity in
L∞ (Ω, µ) (though it is not always a bounded function).
¥
Example 9.5. [Gaussian W ∗ -observable]. Deﬁne the W ∗ -observable O ≡ (R, BR , F(·) )in N ≡ L∞ (R, dω) such that:

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1FΞσ (ω) = √

2πσ 2

Z

(u−ω)2e− 2σ2 du(∀ω ∈ R, ∀Ξ ∈ BR ). (σ 2 : variance).

ΞThis is, of course, the W ∗ -algebraic form of the Gaussian C ∗ -observable O ≡ (R, Bbd
R , F(·) )(cf. Example 2.17). Note that the BR in O is a σ-ﬁeld, and the Bbd
R in O is a σ-ring.
¥
Remark 9.6. [The vagueness of a crisp observable]. Let ν be a probability measure
on an index set Θ. For each θ ( ∈ Θ), consider a crisp observable Oθ ≡ (X, F, Eθ ) inW ∗ -algebra N. Deﬁne the observable O ≡ (X, F, F ) in W ∗ -algebra N such that:Z
F (Ξ) =Eθ (Ξ)ν(dθ)(∀Ξ ∈ F)Θwhich is not crisp but fuzzy in general. Thus we think that(F) “fuzzy observable“⇐⇒ “To understand a dearth of information concerning a crisp
observable by a fuzzy observable”.
This is one of the aspects of “fuzzy observable”. When we want to stress this statisticalaspect, the “observable” is often called a “fuzzy observable” (or, “random observable”).
This will be again discussed in §11.4.
¥
Example 9.7. [(i): Crisp W ∗ -observables in quantum B(V )]. Here, consider the quantum version of the (ii) in Example 9.4. Let A be a self-adjoint operator (not necessarily
Rbounded) on a Hilbert space V . Recall the spectral representation: A = R λEA (dλ).
Here, the spectral measure OA ≡ (R, BR , EA ) is of course the crisp W ∗ -observable inB(V ). Conversely, any crisp W ∗ -observable (R, BR , F ) in B(V ) determines a uniqueR
self-adjoint operator AF on V such that AF = R λF (dλ). Therefore, we have the identiﬁcation:A←→ OA = (R, BR , F ) in B(V )(self-adjoint operator on V )³´
i.e., A = R λF (dλ)
R.

(crisp observable)

A self-adjoint operator A on a Hilbert space V is called a (unbounded ) quantity in B(V )(though A is not always a bounded linear operator).
[(ii): Position quantity, momentum quantity].

Put V ≡ L2 (R; dq)), and deﬁne the(unbounded) self-adjoint operator Q [resp. P ], which is called the position quantity [resp.
momentum quantity ], such that:(Qψ)(q) = q · ψ(q),h~dψ(q) i
resp. (P ψ)(q) = −i.
dq

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By the following spectral representations,Z
Q=
λEQ (dλ)
and

231

Z
P =

R

λEP (dλ),R

we see the following identiﬁcations:←→ OQ = (R, BR , EQ ) in B(V )Q(self-adjoint operator on V )(crisp observable)

and←→ OP = (R, BR , EP ) in B(V )P(self-adjoint operator on V ).

(crisp observable)

Here note that
[EQ (Ξ)ψ](q) = χΞ (q) · ψ(q)(∀ψ ∈ L2 (R), ∀Ξ ∈ BR , q ∈ R)

EP (Ξ)ψ = F∗ (χΞ · (Fψ))(∀ψ ∈ L2 (R), ∀Ξ ∈ BR , q ∈ R),

andwhere the Fourier transform F : L2 (R, dx) → L2 (R, dy) is deﬁned byr Z~
ψ(x)e−i~xy dx.
(Fψ)(y) =2π RNote that both the position observable and momentum observable, which are most important in quantum mechanics, can not be deﬁned in the C ∗ -algebraic formulation.
[(iii): Glauber-Sudarshan representation]. Consider ψ0 ( ∈ V ≡ L2 (R; dq)) such that∥ψ0 ∥L2 (R) = 1 and⟨ψ0 , P ψ0 ⟩V = 0,

⟨ψ0 , Qψ0 ⟩V = 0.

If we deﬁne ϕx,y (q) = eixy ψ0 (q − x), then an elementary computation shows that⟨P ϕx,y , ϕx,y ⟩L2 (R) = y,⟨Qϕx,y , ϕx,y ⟩L2 (R) = x.

(9.6)

Here we can deﬁne the W ∗ -observable (R2 , BR2 , G) in B(L2 (R; dq)) such that:
ZZ
G(Ξ) =|ϕx,y ⟩⟨ϕx,y |dxdy(∀Ξ ∈ BR2 ).
ΞThis observable is essential in semi-classical mechanics (cf. [34]).
¥

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The following theorem is the W ∗ -algebraic form of Theorem 2.13. Since W ∗ -algebra
N has a lot of projections, it is much more useful than Theorem 2.13.
Theorem 9.8. [The W ∗ -algebraic form of Theorem 2.13, cf. [42]]. Let N be a W ∗ algebra. Let O1 ≡ (X1 , F1 , F1 ) and O2 ≡ (X2 , F2 , F2 ) be W ∗ -observables in N such that³´
at least one of them is crisp. So, without loss of generality, we assume that O2 is crisp .
Then, the following statements are equivalent:(i) There exists a quasi-product observable O12 ≡ (X1 × X2 , F1 × F2 , F1 × F2 ) withqp

marginal observables O1 and O2 .
(ii) O1 and O2 commute, that is, F1 (Ξ1 )F2 (Ξ2 ) = F2 (Ξ2 )F1 (Ξ1 ) (∀Ξ1 ∈ F1 , ∀Ξ2 ∈ F2 ).
Furthermore, if the above statements (i) and (ii) hold, the uniqueness of O12 is guaranteed.
(So, we can write that O12 = (X1 × X2 , F1 × F2 , F1 × F2 ) = O1 × O2 .)Proof. The proof is essentially the same as that of Theorem 2.13.
The purpose of this chapter is to propose the W ∗ -algebraic formulation of SMT, thatis,∗

SMTW = statistical measurement+ the relation among systems∗∗[ProclaimW 2 (9.23)][ProclaimW 1 (9.9)]

in W ∗ -algebra .
(9.7)(=(9.2))In order to do it, we must recall the C ∗ -algebraic formulation of SMT, that is,∗

SMTC = statistical measurement + the relation among systems[Proclaim 1 (8.10)]

in C ∗ -algebra .

[Axiom 2 (3.26)](9.8)(=(9.1))∗∗

As mentioned before, we want to understand SMTW by an analogy of SMTC . Here, it
should be recalled that• [Proclaim 1 (8.10), (The probabilistic interpretation of mixed states)]. Consider a¡¢statistical measurement MA O ≡ (X, F, F ), S(ρm ) formulated in a C ∗ -algebra A.
Then, the probability that x ( ∈ X), the measured value obtained by the statisticalmeasurement MA (O, S(ρm ) ), belongs to a set Ξ ( ∈ F) is given byρm (F (Ξ))

³

D
E ´
≡ A∗ ρm , F (Ξ) A .

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By an analogy of Proclaim 1, we can propose Proclaim

233

1 as follows: Cf [44].

W∗

PROCLAIM 1. [Statistical measurements in the W ∗ -algebraic formulation].
Consider a statistical measurement MN (O ≡ (X, F, F ), S(ρn ))formulated in a W ∗ -algebra N. The probability that x ( ∈ X),
the measured value obtained by the statistical measurement MN (O ≡(X, F, F ), S(ρn )), belongs to Ξ ( ∈ F) is given by
³
D
E ´
n
n
ρ (F (Ξ)) ≡ N∗ ρ , F (Ξ) N .
(9.9)

This will be easily read by the above [Proclaim 1] and the following [TABLE (Statisticalmeasurement theory)].

Statistical measurement theory(9.10)········································································[C ∗ -algebraic formulation]←→[W ∗ -algebraic formulation]········································································∗
Proclaim 1 (8.10)←→
ProclaimW 1 (9.9)
Sm (A∗ ) ∋ ρm←→ρn ∈ Sn (N∗ )C ∗ -observable O ≡ (X, F, F )←→W ∗ -observable O ≡ (X, F, F )MA (O ≡ (X, F, F ), S(ρm ))←→MN (O ≡ (X, F, F ), S(ρn ))Remark 9.9. [The W ∗ -algebraic formulation of PMT]. Though the commutative PMTW∗∗

has a demerit such that a pure state can not be represented in the commutative PMTW ingeneral (cf. the statement (9.3)), a pure state can be represented in the non-commutative∗

PMTW (i.e., in B(V ), cf. Example 9.2). Thus, it is worthwhile mentioning the following∗

AxiomW 1 (9.11). If N = B(V ) or N = L∞ (Ω, µ) (where Ω is ﬁnite or countable inﬁnite),the concept “pure state” is valid (cf. (9.4) and (9.5)). Thus, in this case, we can propose∗“AxiomW 1 (9.11)” (i.e., the W ∗ -algebraic formulation of Axiom 1) as follows:

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234CHAPTER 9. STATISTICAL MEASUREMENTS IN W ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/
∗

W
AXIOM
1. [The W ∗¢-algebraic formulation of Axiom 1]. Consider a measurement¡
MN O ≡ (X, F, F ), S [ρp ] formulated in a W ∗ -algebra N, where ρp is a pure state. Assume¡¢
that the measured value x ( ∈ X) is obtained by the measurement MN O, S [ρp ] . Then,.
¡ptheD probability
that
the
x
(
∈
X)
belongs
to
a
set
Ξ
(
∈
F)
is
given
by
ρ
(F
(Ξ))
≡
E ¢
p
ρ
,
F
(Ξ)
.
(9.11)N∗N¥
In the following example, we see that the C ∗ -algebraic formulation and the W ∗ algebraic formulation are essentially the same.
Example 9.10. [(i): The review of Example 8.1] .

There are two urns ω1 and ω2 .

The urn ω1 [resp. ω2 ] contains 8 white and 2 black balls [resp. 4 white and 6 black¡¢balls]. We regard Ω ≡ {ω1 , ω2 } as the state space. And consider the observable¡¢O ≡ (X ≡ {w, b}, 2{w,b} , F ) in C(Ω) where[F ({w})](ω1 ) = 0.8,

[F ({b})](ω1 ) = 0.2,

[F ({w})](ω2 ) = 0.4,

[F ({b})](ω2 ) = 0.6.

ω1

ω2Here consider the following procedures (P1 ) and (P2 ).
(P1 ) One of the two (i.e., ω1 or ω2 ) is chosen by an unfair tossed-coin (Cp,1−p ), i.e.,
Head (100p%) → ω1 , Tail (100(1 − p)%) → ω2

(0 ≤ p ≤ 1).

The chosen urn is denoted by [∗](∈ {ω1 , ω2 }). Note, for completeness, that we do
not know whether [∗] is ω1 or ω2 . Here deﬁne the mixed state ν0 (∈ Mm+1 (Ω)) suchthat ν0 ({ω1 }) = p, ν0 ({ω2 }) = 1 − p, which is considered to be “the distribution of[∗]”.
³(P2 ) Take one ball, at random, out of the urn chosen by the procedure (P1 ).
´
we take the measurement MC(Ω) (O, S[∗] ).

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235[(ii): Continued from the above (i): C ∗ -algebraic formulation]. As seen in Example 8.1,
• “(P1 ) + (P2 )” is notated by MC(Ω) (O ≡ (X, 2X , F ), S(ν0 )).
Of course, we see• the probability that the measured value x ( ∈ {w, b}) is obtained by the measurement MC(Ω) (O, S[∗] (ν0 )), is given by
D
E
³ Z
D
E
´
ν
,
F
({x})
≡
δ
,
F
({x})
ν
(dω)
0
ω
0
∗
∗
C(Ω)
C(Ω)
C(Ω)
C(Ω)
Ω
½
0.8p + 0.4(1 − p) ( if x = w),=
0.2p + 0.6(1 − p)) ( if x = b).

(9.12)[(iii): Continued from the above (i): W ∗ -algebraic formulation]. Deﬁne the measure µ onΩ, for example, such thatµ({ω1 }) = µ({ω2 }) = 1.
¡
Thus we have the commutative W ∗ -algebra L∞ (Ω, µ). And consider the observable O ≡¢(X ≡ {w, b}, 2{w,b} , F ) in L∞ (Ω, µ) where[F ({w})](ω1 ) = 0.8,

[F ({b})](ω1 ) = 0.2,

[F ({w})](ω2 ) = 0.4,

[F ({b})](ω2 ) = 0.6.

Also deﬁne the normal state ρn ( ∈ L1+1 (Ω, µ)) such that5ρn (ω1 ) = p,

ρn (ω2 ) = 1 − p.

Then, we have the W ∗ -measurement ML∞ (Ω,µ) (O, S(ρn )) in L∞ (Ω, µ). Of course, we see,• the probability that the measured value x ( ∈ {w, b}) is obtained by the measurement ML∞ (Ω,µ) (O, S(ρn )), is given by
E
³ Z
´
n
ρ , F ({x}) L∞ (Ω,µ) ≡ [F ({x})](ω)ρ (ω)µ(dω)
L1 (Ω,µ)Ω½
0.8p + 0.4(1 − p) ( if x = w),=
0.2p + 0.6(1 − p)) ( if x = b).
Dn(9.13)

Note that µ is arbitrary (cf. the formula (9.4)). If µ({ω1 }) = 1/3 and µ({ω2 }) = 2, it suﬃces todeﬁne that ρn (ω1 ) = 3p and ρn (ω1 ) = (1 − p)/2.
5

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Thus we see that MC(Ω) (O, S[∗] (ν0 )) and ML∞ (Ω,µ) (O, S(ρn )) are essentially the same (cf.
(9.12) and (9.13)).
Also, we see:
The illustration of ML∞ (Ω,µ) (O, S(ρn ))Pick up a ball from the urn behind the curtain

ω1

ω2

p

-

[ ∗]

1-p



¥
∗

∗

The relation among systems (ProclaimW 2 in SMTW )9.2∗

We mentioned “statistical measurement” [ProclaimW 1 (9.9)] in the previous section. Thus in this section, we devote ourselves to the “relation among systems (i.e.,∗

ProclaimW 2)” in the W ∗ -algebraic formulation of SMT 6 . That is, we want to propose∗

SMTW = statistical measurement+ the relation among systems∗[ProclaimW 1 (9.9)]∗[ProclaimW 2 (9.23)]

in W ∗ -algebra N .
(9.14)(=(9.2))

Let N1 and N2 with weak∗ -topologies σ(N1 , (N1 )∗ ) and σ(N2 , (N2 )∗ ) respectively. A
continuous linear operator Ψ1,2 : N2 → N1 is called a Markov operator, if it satisﬁes that(i) Ψ1,2 (F2 ) ≥ 0 for any positive element F2 in N2 ,(ii) Ψ1,2 (I2 ) = I1 , where Ik is the identity in Nk (k = 1, 2).
Here note that, for any observable (X, F, F2 ) in N2 , the (X, F, Ψ1,2 F2 ) is an observablein N1 , which is denoted by Ψ12 O2 . For example, it is easy to see that, for any countabledecomposition {Ξj }∞j=1 of Ξ, (Ξj , Ξ ∈ F),
If N = B(V ) or N = L∞ (Ω, µ) (where Ω is ﬁnite), the concept “pure state” is valid (cf. (9.4) and∗(9.5)). Thus, in the case, we can propose “PMTW ” (i.e., the W ∗ -algebraic formulation of PMT) as
follows:
6∗

P M T W = statistical measurement+ the relation among systems∗[AxiomW 1 (9.11)]∗[ProclaimW 2 (9.23)]

in W ∗ -algebra N .

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Sample PDF File (Low Resolution Printing)∗∗9.2. THE RELATION AMONG SYSTEMS (PROCLAIMW 2 IN SMTW )For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/[Ψ1,2 F2 ](Ξ) = (w∗ )- lim Ψ1,2 (F2 ( ∪nj=1 Ξj )) = (w∗ )- lim Ψ1,2 (n→∞=(w∗ )- lim

n→∞

n
X

n→∞

n
X

237

F2 (Ξj ))j=1[Ψ1,2 (F2 )](Ξj ).

j=1

Also, a Markov operator Ψ1,2 : N2 → N1 is called a homomorphism (or precisely, W ∗ homomorphism), if it satisﬁes that(i) Ψ1,2 (F2 )Ψ1,2 (G2 ) = Ψ1,2 (F2 G2 ) for any F2 and G2 in N2 ,(ii) (Ψ1,2 (F2 ))∗ = Ψ1,2 (F2∗ ) for any F2 in N2 .
Then the following mathematical result is well known.
(a) (Ψ1,2 )∗ (Sn ((N1 )∗ )) ⊆ Sn ((N2 )∗ ).
Let (Ψ1,2 )∗ : (N1 )∗ → (N2 )∗ be the pre-dual operator7 of a Markov operator Ψ1,2 : N2 →N1 , that is, it holds that
D(N1 )∗EDE
ρn1 , Ψ1.2 F2 N1 = (N2 )∗ (Ψ1.2 )∗ ρn1 , F2 N2(∀ρn1 ∈ (N1 )∗ , ∀F2 ∈ N2 ).

(9.15)

Suppose that N1 and N2 are commutative W ∗ -algebras, i.e., N1 = L∞ (Ω1 , µ1 ) and N2 =L∞ (Ω2 , µ2 ). Then, under the identiﬁcation that Sn (N1∗ ) = L1+1 (Ω1 , µ1 ) and Sn ((N2 )∗ ) =L1+1 (Ω2 , µ2 ) (cf. Example 9.2), the above (a) implies that the pre-dual operator (Ψ1,2 )∗of a Markov operator Ψ12 can be identiﬁed with a transition probability rule M (ω1 , B2 ),(ω1 ∈ Ω1 , B2 ∈ BΩ2 ), such that:ZZn[(Φ12 )∗ (ρ1 )](ω2 )µ2 (dω2 ) =M (ω1 , B2 )ρn1 (ω1 )µ1 (dω1 ) (∀ρn1 ∈ L1+1 (Ω1 , µ1 ), ∀B2 ∈ BΩ2 ).
B2Ω1

Also, it is well known that, a Markov operator Ψ1,2 : L∞ (Ω2 , µ2 ) → L∞ (Ω1 , µ1 ) is homomorphic, if and only if there exists a measurable map ψ1,2 from Ω1 into Ω2 such that:(Ψ1,2 f2 )(ω1 ) = f2 (ψ1,2 (ω1 ))(almost all µ1 )(9.16)

for all f2 ∈ L∞ (Ω2 , µ2 ).
The symbol ∗ is used in the three following ways (1) ∼ (v) in this book. (i) involution operator (e.g.,
F ), (ii) dual operator (e.g., Ψ∗ ), (iii) dual space (e.g., A∗ ), (iv) pre-dual operator (e.g., Ψ∗ ), (v) pre-dual
space (e.g., N∗ ).
7∗

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Let (T, ≤) be a tree-like partial ordered set, i.e., a partial ordered set such that “t1 ≤ t3
and t2 ≤ t3 ” implies “t1 ≤ t2 or t2 ≤ t1 ”. Put T≤2 = {(t1 , t2 ) ∈ T 2 : t1 ≤ t2 }. An elementt0 ∈ T is called a root if t0 ≤ t (∀t ∈ T ) holds. Note that the sub-tree Tt0 ≡ {t ∈ T | t ≥ t0 }has the root t0 . Thus we always assume that the tree-like ordered set (T, ≤) has a root.
³
We assume that T is not always ﬁnite. In the next Chapter 10, T is always assumed to´
be inﬁnite.
Deﬁnition 9.11. [General systems]. The pair S(ρnt0 ) ≡ [S(ρnt0 ), {Φt1 ,t2 : Nt2 → Nt1 }(t1 ,t2 )∈T≤2 ]is called a general system with an initial state S(ρnt0 ) if it satisﬁes the following conditions(i)∼(iii).
(i) With each t (∈ T ), a W ∗ -algebra Nt is associated.
(ii) Let t0 (∈ T ) be the root of T . And, assume that a system S has the normal stateρnt0 (∈ Sn ((Nt0 )∗ )) at t0 , that is, the initial state is equal to ρnt0 .
(iii) For every (t1 , t2 ) ∈ T≤2 , Markov operator Φt1 ,t2 : Nt2 → Nt1 is deﬁned such that
Φt1 ,t2 Φt2 ,t3 = Φt1 ,t3 holds for all (t1 , t2 ), (t2 , t3 ) ∈ T≤2 .
The family {Φt1 ,t2 : Nt2 → Nt1 }(t1 ,t2 )∈T≤2 is also called a “Markov relation among systems”.
Let an observable Ot ≡ (Xt , 2Xt , Ft ) in a W ∗ -algebra Nt be given for each t ∈ T . The
pair [{Ot }t∈T , {Φt1 ,t2 : Nt2 → Nt1 }(t1 ,t2 )∈T≤2 ] is called a “sequential observable”, which isdenoted by [OT ], i.e., [OT ] ≡ [{Ot }t∈T , {Φt1 ,t2 : Nt2 → Nt1 }(t1 ,t2 )∈T≤2 ].
¥∗Before we explain ProclaimW 2, we prepare some notations. For simplicity, assume
that T is ﬁnite, or a ﬁnite subtree of a whole tree. Let T ( = {0, 1, ..., N }) be a treewith the root 0. Deﬁne the parent map π : T \ {0} → T such that π(t) = max{s ∈T : s < t}. It is clear that the tree (T ≡ {0, 1, ..., N }, ≤ ) can be identiﬁed with thepair (T ≡ {0, 1, ..., N }, π : T \ {0} → T ). Also, note that, for any t ∈ T \ {0}, there
uniquely exists a natural number h(t) (called the height of t ) such that π h(t) (t) = 0.
Here, π 2 (t) = π(π(t)), π 3 (t) = π(π 2 (t)), etc. Thus, the general system S(ρn0 ) ≡ [S(ρn0 ),{Ψt1 ,t2 : Nt2 → Nt1 }(t1 ,t2 )∈{0,1,...,N }2 ] is sometimes represented by [S(ρn0 ), Nt Ψπ(t),tNπ(t) (→≤t ∈ {0, 1, ..., N } \ {0})]. Also, we deﬁne the Φ0,τ : Nτ → N0 such that Φ0,τ = Ψ0,τ , that is,Φ0,τ = Ψ0,πh(τ )−1 (τ ) Ψπh(τ )−1 (τ )),πh(τ )−2 (τ ) · · · Ψπ2 (τ ),π(τ ) Ψπ(τ ),τ .

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(9.17)

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239

Let Ot ≡ (Xt , Ft , Ft ) be an observable in Nt (∀t ∈ T ). The “measurement” of {Ot : t ∈ T }for the S(ρnt0 ) is symbolically described by M({Ot }t∈T , S(ρnt0 )). The Markov relation
Φπ(t),t{Φt1 ,t2 : Nt2 → Nt1 }(t1 ,t2 )∈T≤2 is also denoted by {Nt → Nπ(t) }t∈T \{0}Example 9.12. [A simple general system. Compared to Examples 3.4 and 8.12]. Supposethat a tree (T ≡ {0, 1, ..., 6, 7}, π) has an ordered structure such that π(1) = π(6) = π(7) =³´
0, π(2) = π(5) = 1, π(3) = π(4) = 2. See the ﬁgure (9.18). Consider a general system
Φπ(t),tS(ρn0 ) ≡ [S(ρn0 ), {Nt → Nπ(t) }t∈T \{0} ] with the initial system S(ρn0 ).

Φ2,3

Φ1,2
Φ0,1
)
N0 i

N1

+

N3

Φ2,4 N4k Φ1,5

N5

N6
kΦ0,6
Φ0,7

N2 )
k

N7

(9.18)

Also, for each t ∈ {0, 1, ..., 6, 7}, consider an observable Ot ≡ (Xt , 2Xt , Ft ) in a W ∗ -algebra
Nt . Thus, we have a sequential observable [{Ot }t∈T , {Φt,π(t) : Nt → Nπ(t) }t∈T \{0} ]. Now
we want to consider the following “measurement”,(♯) for a system S((ρn0 )) where ρn0 ∈ Sn ((N0 )∗ ), take a measurement of “a sequentialΦπ(t),tobservable [OT ] ≡ [{Ot }t∈T , {Nt → Nπ(t) }t∈T \{0} ]”, i.e., take a measurement ofan observable O0 at 0( ∈ T ), and next, take a measurement of an observable O1 at
1( ∈ T ), · · · · · · , and ﬁnally take a measurement of an observable O7 at 7( ∈ T ),which is symbolized by M({Ot }t∈T , S(ρn0 )). Note that the M({Ot }t∈T , S(ρn0 )) is merely asymbol since only one measurement is permitted (cf. §2.5 Remark (II)). In what follows
let us describe the above (♯) (= M({Ot }t∈T , S(ρn0 ))) precisely. Put
e t = Ot
O

and thus Fet = Ft(t = 3, 4, 5, 6, 7).

e 2 in N2 such asFirst we construct the quasi-product observable O
e 2 = (X2 × X3 × X4 , 2X2 ×X3 ×X4 , Fe2 )
O

qp

qp

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(9.19)

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if it exists. Iteratively, we construct the following:
Φ0,1

N0
qp

←−−−
qp

qp

←−−−

qp

N2

F1 × Φ1,5 Fe5


y
Φ0,1

qp

Φ1,2

←−−−

qp

F0 × Φ0,6 Fe6 × Φ0,7 Fe7


y
Fe0

N1

(F0 × Φ0,6 Fe6 × Φ0,7 Fe7 × Φ0,1 Fe1 )

qp

Fe1

qp

(F1 × Φ1,5 Fe5 × Φ1,2 Fe2 )(9.20)
Φ1,2

←−−−

qp

Fe2

.

qp

(F2 × Φ2,3 Fe3 × Φ2,4 Fe4 )Q
e2e 1 ≡ (Q5 Xt , 2 5t=1 Xt , Fe1 ) of O1 , Φ1,2 O
That is, we get the quasi-product observable O
t=1Q
e 5 , and ﬁnally, the quasi-product observable O
e 0 ≡ (Q7 Xt , 2 7t=0 Xt , Fe0 ) ofand Φ1,5 O
t=0e 1 , Φ0,6 O
e 6 and Φ0,7 O
e 7 , if it exists. Here, Oe 0 is called the realization (or, theO0 , Φ0,1 O
Φπ(t),tHeisenberg picture representation) of a sequential observable [OT ] ≡ [{Ot }t∈T , {Nt →Nπ(t) }t∈T \{0} ]. Then, we have the measurementYQ
e 0 ≡ ( Xt , 2 t∈T Xt , Fe0 ), S(ρn )),MN0 (O0(9.21)t∈T

which is called the realization (or, the Heisenberg picture representation) of the symbol
M({Ot }t∈T , S(ρn0 )).
¥Examining Example 9.12, we have the following arguments. Let (T ≡ {0, 1, ..., N }, π :Φπ(t),tT \ {0} → T ) be a tree with root 0 and let S(ρn0 ) ≡ [S(ρn0 ), Nt → Nπ(t) (t ∈ T \ {0})] bea general system with the initial system S(ρn0 ). And, let an observable Ot ≡ (Xt , Ft , Ft )in a W ∗ -algebra Nt be given for each t ∈ T . For each s ( ∈ T ), deﬁne the observableQe s ≡ (QO
Xt ,Ft , Fes ) in Ns such that:
t∈Ts

t∈Ts

es =
O

(

(if s ∈ T \ π(T ))

Os
qp qp

(9.22)e t ) (if s ∈ π(T ))Os× ( × t∈π−1 ({s}) Φπ(t),t O

e 0 (i.e., the Heisenberg picture representation of theif possible. Then, if an observable O
sequential observable [OT ] ≡ [{Ot }t∈T , {Φt,π(t) : Nt → Nπ(t) }t∈T \{0} ]) in N0 exists (suchas in Example 9.12), we have the measurementYY
e 0 ≡ ( Xt ,MN0 (OFt , Fe0 ), S(ρn0 )),
t∈T

t∈T

which is called the Heisenberg picture representation of the symbol M({Ot }t∈T , S(ρn0 )).
Summing up the essential part of the above argument, we can propose the following∗

axiom, which corresponds to “the rule of the relation among systems” in SMTW .

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PROCLAIMW 2. [The Markov relation among systems, the Heisenberg
picture] The relation among systems is represented by a Markov relation{Φt1 ,t2 : Nt2 → Nt1 }(t1 ,t2 )∈T≤2 . Let Ot ( ≡ (Xt , Ft , F )) be an observable
in Nt for each t ( ∈ T ). If the procedure (9.22) is possible, a sequential
observable [OT ] ≡ [{Ot }t∈T , {Φt1 ,t2 : Nt2 → Nt1 }(t1 ,t2 )∈T≤2 ] can be realizede 0 ≡ (Q Xt , Q Ft , Fe0 ) in N0 .
as the observable O(9.23)t∈T

t∈T

Also, we must add the following statement:• Let S(ρnt0 ) ≡ [S(ρnt0 ), {Φt1 ,t2 : Nt2 → Nt1 }(t1 ,t2 )∈T≤2 ] be a general system with an initialstate ρnt0 (∈ Sn ((Nt0 )∗ )). And then, a measurement represented by the symbole 0 ≡ (Q Xt , Q Ft , Fe0 ), S(ρn )), if
M({Ot }t∈T , S(ρn )) can be realized by MN0 (O
t0

t∈T

0

t∈T

e 0 exists.
O∗∗

which explains the relation between ProclaimW 1 and ProclaimW 2.
W∗

Remark 9.13. [How to read Proclaim 2]. For completeness, we mention how to read∗

ProclaimW 2 as follows: Recall Axiom 2 (3.26), that is,• [Axiom 2. (The Markov relation among systems, the Heisenberg picture)] The relation among systems is represented by a Markov relation {Φt1 ,t2 : At2 → At1 }(t1 ,t2 )∈T≤2 .
Let Ot ( ≡ (Xt , Ft , F )) be an observable in At for each t ( ∈ T ). If the procedure (3.25) is possible, a sequential observable [OT ] ≡ [{Ot }t∈T , {Φt1 ,t2 : At2 →e 0 ≡ (Q Xt , Q Ft , Fe0 ) in A0 .
At1 }(t ,t )∈T 2 ] can be realized as the observable O
1 2

≤

t∈T

t∈T

W∗

Using this and the following correspondence, we can easily read the above Proclaim 2.

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242CHAPTER 9. STATISTICAL MEASUREMENTS IN W ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/

Statistical measurement theory(9.24)········································································∗C∗[SMT (C ∗ -algebraic formulation)]←→[SMTW (W ∗ -algebraic formulation)]········································································∗
Proclaim 1 (8.10)←→
ProclaimW 1 (9.11)
Sm (A∗ ) ∋ ρm←→ρn ∈ Sn (N∗ )C ∗ -observable O ≡ (X, F, F )←→W ∗ -observable O ≡ (X, F, F )MA (O ≡ (X, F, F ), S(ρm ))←→MN (O ≡ (X, F, F ), S(ρn ))Axiom 2 (3.26)←→

Proclaim←→³

general systemS(ρm )³=[S(ρm ),{Ψ

³

´

W∗

t1 ,t2 :At2 →At1 }(t ,t )∈T 2 ]1 2≤

sequential observable[OT ]´

M({Ot }t∈T , S(ρ ))t1 ,t2 :Nt2 →Nt1 }(t ,t )∈T 2 ]1 2≤

sequential observable[OT ]
1 2

≤

m

³

´
´

=[{Ot }t∈T ,{Ψt1 ,t2 :Nt2 →Nt1 }(t ,t )∈T 2 ]=[{Ot }t∈T ,{Ψt1 ,t2 :At2 →At1 }(t ,t )∈T 2 ]1 2

general systemS(ρn )=[S(ρn ),{Ψ←→

2 (9.23)←→≤

M({Ot }t∈T , S(ρ ))nRemark 9.14. [The C ∗ -algebraic and the W ∗ -algebraic formulations]. Now we have two
formulations of SMT, i.e., the C ∗ -algebraic formulation and the W ∗ -algebraic formulation.
Recall that any commutative C ∗ -algebra [resp. commutative W ∗ -algebra] is represented
by some C(Ω) [resp. L∞ (Ω, µ)]. Thus, we can say that the C ∗ -algebraic formulationand the W ∗ -algebraic formulation are respectively topological and measure theoretical.
Therefore, from the mathematical point of view, the W ∗ -algebraic formulation is handy
for us to deal with “limit” or “convergence”. For example, this will be seen in Theorem
10.1 (the W ∗ -algebraic generalization of Kolmogorov’s extension theorem).8¥
The following theorem is essentially the same as Theorem 3.7.
Theorem 9.15. [The measurability of a general system; Compared to Theorem 3.7].
Let (T ≡ {0, 1, ..., N }, π : T \ {0} → T ) be a tree with root 0 and let S(ρn0 ) ≡ [S(ρn0 ),Φπ(t),t

Nt → Nπ(t) (t ∈ T \ {0})] be a general system with the initial system S(ρn0 ). And, let
an observable Ot ≡ (Xt , Ft , Ft ) in a C ∗ -algebra Nt be given for each t ∈ T . For each s
8

If readers have some knowledge of Riemann integral (deﬁned in terms of topology) and Lebesgue
integral (deﬁned in terms of measure, cf. [29]), they can easily understand the mathematical handinessof “measure theoretical approach”.

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For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/Qe s ≡ (Qe( ∈ T ), deﬁne the observable Ot∈Ts Xt ,t∈Ts Ft , Fs ) in Ns such that:(Os(if s ∈ T \ π(T ))
es =
qp qp
O
e t ) (if s ∈ π(T ))Os× ( × t∈π−1 ({s}) Φπ(t),t O
e 0 (i.e., the Heisenberg picture representation of theif possible. Then, if an observable O
sequential observable [{Ot }t∈T , {Φt,π(t) : Nt → Nπ(t) }t∈T \{0} ]) in N0 exists, we have themeasurement
e0 ≡ (
MN0 (O

Y
t∈T

Xt ,

Y

Ft , Fe0 ), S(ρn0 )),(9.25)t∈T

which is called the Heisenberg picture representation of the symbol M({Ot }t∈T , Sρnt0 )).
If the system is classical, i.e., Nt ≡ L∞ (Ω, µ) (∀t ∈ T ), then the measurement always
exists, while the uniqueness is not always guaranteed. Also, it should be noted that,Q
Q
for each s( ∈ T ), it holds that Φπ(s),s Fes (Ξt ) = Feπ(s) ((Πt∈T \Ts Xt ) × (
Ξt ))
t∈Ts

π(s)

t∈Ts

(∀Ξt ∈ Ft (∀t ∈ T )).
Proof. The proof is the same as that of Theorem 3.7.
Remark 9.16. [Summing up]. In Chapters 2 ∼ 8, we studied the C ∗ -algebraic formulation such that∗
PMTC = measurement + the relation among systems (In Chap. 2∼7)
[Axiom 1 (2.37)][Axiom 2 (3.26)]
∗
MTC

C∗

 SMT = statistical measurement + the relation among systems (In Chap. 8)[Proclaim 1 (8.10)][Axiom 2 (3.26)]

In this chapter, we study the W ∗ -algebraic formulation as follows:∗
PMTW = measurement+ the relation among systems (in N)∗∗[AxiomW 1 (9.11)]
[ProclaimW 2 (9.23)]
W∗
MT

W∗

+ the relation among systems (in N)
 SMT = statistical measurement∗
W∗[Proclaim

1 (9.9)][ProclaimW 2 (9.23)](9.26)
Here we add the remarks as follows:∗(i) MTC is fundamental,∗∗(ii) MTW should be understood by an analogy of MTC . Cf. Table (9.24).
∗∗(iii) From the mathematical point of view, SMTW is more handy than SMTC . (Cf.
Remark 9.14).

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244CHAPTER 9. STATISTICAL MEASUREMENTS IN W ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(iv) When N = B(V ) or N = L∞ (Ω, µ) (where Ω is ﬁnite or countable inﬁnite), PMTW∗

is meaningful (cf. Example 9.1).
∗∗(v) Most results in MTC hold in MTW . However, we omit “Fisher’s maximum like∗lihood method” and “Generalized Bayes theorem”, etc. in MTW since the proofs
are the same.
¥

9.3

Quantum mechanics in B(L2(R))

9.3.1

Schrödinger equation and Heisenberg kinetic equation

Recall the C ∗ -algebraic formulation (in C(L2 (R))) of quantum mechanics (cf. §3.1).
However, as far as quantum mechanics, the W ∗ -algebraic formulation (in B(L2 (R))) ismore handy than the C ∗ -algebraic formulation (in C(L2 (R))). (Cf. [71].) Thus, in thissection, we explain the W ∗ -algebraic formulation of quantum mechanics (cf. §3.1). though
it is similar to the C ∗ -algebraic formulation of quantum mechanics,
We begin with the classical mechanics. For simplicity, consider the one dimensionalcase, i.e., Rq = {q | q ∈ R}. Thus q(t), −∞ < t < ∞, means the particle’s position at) means the particle’s momentum at time t. Let R2q,ptime t, and thus, p(t) ( ≡ m dq(t)dt( ≡ {(q, p) | q, p ∈ R} be a phase space. Deﬁne a Hamiltonian H : R2q,p → R such that:p2(=kinetic energy)H(q, p) =2m+

V (q)(=potential energy).

(9.27)

Thus we seeE(total energy)= H(q, p) =

p2
2m

+

(kinetic energy)

V (q)

.

(9.28)(potential energy)

Put H = L2 (Rq , dq), i.e., the Hilbert space composed of all L2 -functions on Rq . And put
N = B(L2 (Rq , dq)). Applying the quantumization:
E 7→ i~∂,∂t

p 7→ −i~∂,∂qq 7→ q(9.29)

to the (9.27), we obtain the Schrödinger equation:i~∂∂~2 ∂ 2= H(q, −i~ ) = −+ V (q)∂t∂q
2m∂q 2

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245

or, preciselyi~∂~2 ∂ 2ψ(q, t) = −ψ(q, t) + V (q)ψ(q, t).
∂t2m∂q 2(9.31)

This solution is formally written by
ψ(q, t) = e− ~ H(q,−i~ ∂q )t ψ(q, 0).
i∂(9.32)

Put U (t) = e− ~ H(q,−i~ ∂q )t , and ψ(·, t) = ψt . Then, we see,i∂ψt = U (t)ψ0 .

(9.33)

Thus, the time-evolution of the state |ψt ⟩⟨ψt | is represented by³´
0|ψt ⟩⟨ψt | = (Φt )∗ |ψ0 ⟩⟨ψ0 | = |U (t)ψ0 ⟩⟨U (t)ψ0 |Let O0 = (X, F, F0 ) be a W ∗ -observable in B(H). Then, the time-evolution of theobservable Ot = (X, F, Ft ) is represented by(X, F, Ft ) = (X, F, U (t)F0 U (t)∗ ) = (X, F, Φ0t F0 ).

(9.34)

Also, it should be note that it holds thatdFt
= Ft H − HFt ,dt(9.35)

which is the Heisenberg kinetic equation. Put Ψt1 ,t2 = Φ0t2 −t1 , And let ρ be any elementmin T r+1(H), i.e, a normal state. Then, we get the general statistical system [S(ρ), {Ψt1 ,t2 :p
B(H) → B(H)}t1 ≤t2 ]. Also, let ρu be any element in T r+1(H), i.e, ρu = |u⟩⟨u|, a purestate. Then, we get the general system [S [ρu ] , {Ψt1 ,t2 : B(H) → B(H)}t1 ≤t2 ].
Although the two formulations (i.e., the W ∗ -algebraic formulation (in B(L2 (R))) andthe C ∗ -algebraic formulation (in C(L2 (R))) are similar, it should be noted that the position
observable and the momentum observable can not be represented in the C ∗ -algebraic
formulation but the W ∗ -algebraic formulation (cf. Example 9.7).

9.3.2

A simplest example of Schrödinger equation

Consider a particle with the mass m in the box (i.e., the closed interval [0, 2]) in the
one dimensional space R. The motion of this particle (i.e., the wave function of theparticle) is represented by the following Schrödinger equationi~~2 ∂ 2∂ψ(q, t) = −ψ(q, t) + V0 (q)ψ(q, t).
∂t2m∂q 2

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246CHAPTER 9. STATISTICAL MEASUREMENTS IN W ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/
where½
V0 (q) =0 (0 ≤ q ≤ 2)
∞ ( otherwise )V0 (q)∞
ψ(q, t)

-

0

2

q

R

Put
ϕ(q, t) = T (t)X(q)(0 ≤ q ≤ 2).

And consider the following equation:i~∂~2 ∂ 2ϕ(q, t) = −ϕ(q, t).
∂t2m∂q 2

Then, we see
iT ′ (t)X ′′ (q)=−= K(= constant ).
T (t)2mX(q)
Then,
³
´
p
p
ϕ(q, t) = T (t)X(q) = C3 exp(iKt) C1 exp(i 2mK/~ q) + C2 exp( − i 2mK/~ q).
π ~Since X(0) = X(2) = 0 (perfectly elastic collision), putting K = n 8m
, we see2 2ϕ(q, t) = T (t)X(q) = C3 exp(

in2 π 2 ~t
) sin(nπq/2)8m(n = 1, 2, ...).

Assume the initial condition:
ψ(q, 0) = c1 sin(πq/2) + c2 sin(2πq/2) + c3 sin(3πq/2) + · · · .
where

R
R

|ψ(q, 0)|2 dq = 1. Then we seeψ(q, t)=c1 exp(

iπ 2 ~ti4π 2 ~ti9π 2 ~t) sin(πq/2) + c2 exp(
) sin(2πq/2) + c3 exp(
) sin(3πq/2) + · · · .
8m
8m
8m

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9.3.3

247

The de Broglie paradox

Consider the same situation in §9.3.2, i.e., a particle with the mass m in the box (i.e.,the closed interval [0, 2]) in the one dimensional space R.
V0 (q)∞
ψ(q, t)

-

0

R

2

Now let us partition the box [0, 2]] into [0, 1]] and [1, 2]. That is, we change V0 (q) toV1 (q), where

0



∞
V1 (q) =0
∞(0 ≤ q < 1)
(q = 1)(1 < q ≤ 2)
( otherwise )ψ2 (q, t)∞

V1 (q)

0 ψ1 (q, t) 1

-

2

£
¤
£
Next, we carry the box [0, 1] resp. the box [1, 2] to New York (or, the earth) resp.
¤
Tokyo (or, the polar star) .

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New YorkTokyoψ2 (q, t1 )

ψ1 (q, t1 )
0

1

1

2

£
¤
Note that the probability that we ﬁnd the particle in the box [0, 1] resp. the box [1, 2]£¤RR
is given by R |ψ1 (q, t1 )|2 dq resp. R |ψ2 (q, t1 )|2 dq . Here, we open the box [0, 1] at New
York. And assume that we ﬁnd the particle in the box [0, 1]. Then, quantum mechanics
says that at the moment the wave function ψ2 vanishes.
New York

Tokyo“Vanish”
0

1

1

2

£
¤
£
¤
Note that New York resp. Tokyo may be the earth resp. the polar star . Thus, theabove argument implies that there is something faster than light. This is called “the de
Broglie paradox” (cf. §2.9.1, [78]).

9.4

The method of moments

9.4.1

The moment method

In this book we mainly devoted ourselves to Fisher’s maximum likelihood method (cf.
Corollary 5.6) in (pure) measurements, and Bayes’ method (Cf. Theorem 6.6 and Theorem
8.13) in statistical measurements. In this section we study “the method of moments” (or,the moment method ) in measurements theory (particularly, repeated measurements, cf.

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249

Deﬁnition 2.27).
In what follows, we shall review “the method of moments” (cf. Deﬁnition 2.27).
¡¢
Let MA O ≡ (X, F, F ), S[ρp0 ] be a (pure) measurement, which may be constructed¡¢
as in (8.13) of Remark 8.3. Assume the ρp0 (in MA O, S[ρp0 ] ) is unknown. And further, we get the sample space (X, F, ν0 ) from the measured value x
b (= (x1 , x2 , ...,¡¢³
xT ) ∈ X T ) obtained by the repeated measurement ⊗Tt=1 MA O ≡ (X, F, F ), S[ρp0 ]
=
´
³
´
N
P
M⊗A ( Tt=1 O, S[⊗Tt=1 ρp0 ] ) . That is, ν0 = T1 Tt=1 δxt i.e., ν0 (Ξ) = ♯[{k:xTt ∈Ξ}] . Theorem
2.25 says that that ρp (F (Ξ)) ≈ ν0 (Ξ) (∀Ξ ∈ F) if T is suﬃciently large. Therefore,• [Generalized moment method]; there is a very reason to infer the unknown ρp0 ( ∈Sp (A∗ )) such that:∆(ν0 , ρp0 (F ( · )) ) = p min∆(ν0 , ρp (F ( · )) ),p∗
ρ ∈S (A )(9.36)

where ∆ is a certain semi-distance on Mm+1 (X).
This method is called “generalized moment method” or “moment method”.
Note that the “semi-distance ∆ on Mm+1 (X)” is not always unique. In this sense, themoment method is somewhat artiﬁcial. If X is a ﬁnite set, it is usual to deﬁne the distance∆ on Mm+1 (X) such that:∆(ν1 , ν2 ) =X|ν1 ({x}) − ν2 ({x})| (∀ν1 , ν2 ∈ Mm
+1 (X)).

(9.37)x∈X

More generally, assume that X is an inﬁnite set (and moreover, a metric space). Let
fl : X → R, l = 1, 2, ..., L, be a continuous function on X. Then, the semi-distance∆{fl }Ll=1 on Mm+1 (X) is deﬁned by
L ¯Z
¯
X
¯
¯
∆{fl }Ll=1 (ν1 , ν2 ) =fl (x)(ν1 (dx) − ν2 (dx))¯ (∀ν1 , ν2 ∈ Mm¯
+1 (X)).

(9.38)X

l=1

The above argument is quite general. We usually use the following moment method.
Remark 9.17. [The simple case of (9.36)]. The minimization problem (9.36) may besomewhat troublesome. Thus, we often want to solve the equation ∆(ν0 , ρp0 (F ( · )) ) = 0(i.e., the case of “minρp ∈Sp (A∗ ) ∆(ν0 , ρp (F ( · )) ) = 0” ). That is, our concern is to solve
the following equation:
Z
L ¯Z
¯
X
¯
¯
p
fl (x)ν0 (dx) −fl (x)ρ0 (F (dx))¯ = 0.
¯
l=1

X

X

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250CHAPTER 9. STATISTICAL MEASUREMENTS IN W ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/
Or, equivalently,
 R
R
f
(x)ν
(dx)
=
f (x)ρp0 (F (dx))

1
0
X
X 1





R
R


f2 (x)ν0 (dx) =f2 (x)ρp0 (F (dx))

X









 R

X

(9.39).....
f (x)ν0 (dx) =
X L

R
X

fL (x)ρp0 (F (dx)).

This is usually called the method of moments.
¥

9.4.2

Example 1 [Normal distribution (= Gaussian distribution)]Let ρµ,σ be the Gaussian state in the commutative W ∗ -algebra L∞ (R, dω) such that:
ρµ,σ (ω) = √

1
2πσ 2

exp[−

(ω − µ)2] (∀ω ∈ R),
2σ 2

where the average µ and the variance σ 2 are assumed to be unknown. Let OEXA ≡(R, BR , χ(·) ) be the exact observable in L∞ (R, dω) (cf. Example 9.4 (i)).

(ω−µ)21
ρ̄µ,σ (ω) = √2πσ
e− 2σ2
2

x̃2

σ

σ

x̃5 x̃3 µx̃1-x̃4 x̃6

ω( = x)Consider the statistical measurement ML∞ (R,dω) (OEXA , S(ρµ,σ )), which may be identiﬁed
with the (pure) measurement MC0 (R×R+ ) (OG ≡ (R, BR , G), S[δ(µ,σ) ] ) in C0 (R × R+ ) (cf.
Remark 8.3 (hybrid measurements)), where OG is deﬁned by i.e.,Z1(x − µ)2√[G(Ξ)](µ, σ) =exp[−
]dx (∀Ξ ∈ BR , ∀(µ, σ) ∈ R × R+ ).
2
2
2σ
2πσ Ξ

(9.40)

Assume that we take the measurement ML∞ (R,dω) (OEXA , S(ρµ,σ )) T times, that is, we takethe measurement ML∞ (RT ,⊗Tt=1 dω) (⊗Tt=1 OEXA , S( ⊗Tt=1 ρµ,σ )), which may be identiﬁed with

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the (pure) measurement ⊗Tt=1 MC0 ((R×R+ )) (OG ≡ (R, BR , G), S[δ(µ,σ) ] )251³

i.e., MC0 ((R×R+ )T )´(⊗Tt=1 OG ≡ (RT , BRT , ⊗Tt=1 G), S[⊗Tt=1 δ(µ,σ) ] ) in C0 ((R × R+ )T ) (cf. Remark 8.3) . Again
note that the average µ and variance σ 2 are assumed to be unknown. Here, we have thefollowing problem:(P) Under the assumption that the measured value (x̃1 , x̃2 , ..., x̃T ) ( ∈ RT ) is
obtained by the measurement ⊗Tt=1 MC0 ((R×R+ )) (OG ≡ (R, BR , G), S[δ(µ,σ) ] ),infer the unknown average µ and variance σ 2 .

(9.41)[(i): Answer (Moment method)]. The problem (P) says that we have the sample
space (R, BR , ν0 ) such that:
ν0 =

´
1X ³
δx̃t ∈ Mm
(R)
.
+1
T t∈T

(9.42)

Thus, it suﬃces to solve the following equation:∆{f1 ,f2 } (ν0 , [G( · )](µ0 , σ0 ) ) = 0,(9.43)

where fk : R → R is usually deﬁned by f1 (x) = x and f2 (x) = x2 . That is, seeing (9.39),
we have to solve (1).

R



R

(2).

xν0 (dx) =
R
2

R

R

x ν0 (dx) =R

x[G(dx)](µ0 , σ0 )(9.44)

R

2

R

x [G(dx)](µ0 , σ0 ).

The above (1) clearly implies thatx̃1 + x̃2 + · · · + x̃T ³
µ0 =
≡ AT
T

´
say,

Also, calculating (2)- (1)×(1), we get thatr(x̃1 − AT )2 + (x̃2 − AT )2 + · · · + (x̃T − AT )2
σ0 =
.
T

(9.45)(9.46)

This is the answer by the moment method.
[(ii): Answer (Fisher’s maximum likelihood method)].
Next, we present the answer by Fisher’s likelihood method. Note that the observableb in C0 ((R × R+ )T ) is represented by⊗T OG = (RT , BbdT , ⊗T G ≡ G)
t=1

R

t=1

b 1 × · · · × ΞT )](µ1 , σ1 , µ2 , σ2 , · · · , µT , σT ) = ΠT [G(Ξt )](µt , σt ).
[G(Ξ
t=1

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Assume the condition in the above (P ), and further add thatΞϵt = [x̃t − ϵ, x̃t + ϵ],(for suﬃciently small positive ϵ).

Since we take the (pure) measurement MC0 ((R×R+ )T ) (⊗Tt=1 OG ≡ (RT , BRT , ⊗Tt=1 G),
S[⊗Tt=1 δ(µ,σ) ] ) in C0 ((R × R+ )T ), we see“maximum problem” :max(µ,σ)∈R×R+

b ϵ × · · · × Ξϵ )](µ, σ, µ, σ, · · · , µ, σ)
[G(Ξ
1
T

T
h X
1
(x̃t − µ)2 i⇐⇒“maximum problem” :
max
exp −
(since ϵ is small)(µ,σ)∈R×R+ σ T
2σ 2
t=1(9.47)
∂T(i) µ = x̃1 +x̃2 +···+x̃( ← ∂µ(9.47) = 0)

T


⇐⇒
(9.48)(x̃1 −µ)2 +(x̃2 −µ)2 +···+(x̃T −µ)2∂2( ← ∂σ(9.47) = 0)T
 (ii) σ =(where µ is deﬁned by in the above (i)) .

Thus, Fisher’s maximum likelihood method says that there is a reason to infer thatrx̃1 + x̃2 + · · · + x̃T(x̃1 − AT )2 + (x̃2 − AT )2 + · · · + (x̃T − AT )2µ=≡ AT ,σ=.
TT(9.49)
This is the answer by Fisher’s likelihood method

9.4.3

Example 2 (measurement error model in SMT)

Put Ω0 = Ω1 = R, Θ = R2 and deﬁne the map ψ (θ0 ,θ1 ) : Ω0 ( ≡ R) → Ω1 ( ≡ R) suchthat:
ψ (θ0 ,θ1 ) (ω) = θ1 ω + θ0(∀ω ∈ Ω0 ( ≡ R), ∀(θ0 , θ1 ) ∈ Θ ≡ R2 ).

(9.50)

σ1bd
σ2Also, put (X, F, F ) = (R, Bbd
R , G ) in C0 (Ω0 ) and (Y, G, G) = (R, BR , G ) in C0 (Ω1 )(cf. Example 2.17 (Gaussian observable)), that is,Z(x − ω)21
σiexp[ −]dx(∀Ξ ∈ Bbd
[G (Ξ)](ω) = √R,2σi22πσi Ξ∀ω ∈ R,i = 1, 2).

e (θ0 ,θ1 ) = (X × Y, F × G, H (θ0 ,θ1 ) ≡ Gσ1 × Ψ(θ0 ,θ1 ) Gσ2 ) such
Deﬁne the product observable O(σ1 ,σ2 )(σ1 ,σ2 )that:(θ ,θ )

[H(σ10 ,σ12 ) (Ξ × Γ)](ω)
Z Z
1(x − ω)2 (y − (θ1 ω + θ0 ))2=−]dxdydωexp[−(2π)2/2 σ1 σ2 Ω0 Ξ×Γ
2σ12
2σ22

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R,253∀ω ∈ Ω0 ≡ R),

where θ0 , θ1 and σ2 are assumed to be unknown, but σ1 is known.
Let νµ,σ3 be the Gaussian state in Mm+1 (Ω0 ) such that:1
νµ,σ3 (D) = √
2πσ3

Z
exp[−
D

(ω − µ)2]dω
2σ32(∀D ∈ BΩ0 ),(9.52)

where the average µ and the variance (σ3 )2 are assumed to be unknown.
e (θ0 ,θ1 ) , S(ρµ,σ )). Deﬁne the observable ObHere we have the measurement MC (Ω ) (O00(σ1 ,σ2 )3b in C0 (Θ × ((R ) × R)9 such that:= (X × Y, F × G, H)+ 3

DEb × Γ)](θ0 , θ1 , σ1 , σ2 , σ3 , µ) = M(Ω ) νµ,σ3 , H (θ0 ,θ1 ) (Ξ × Γ)[H(Ξ0C0 (Ω0 )(σ1 ,σ2 )Z Z
22(x − ω)1(y − (θ1 ω + θ0 ))(ω − µ)2
=
exp[−
−
−
]dxdydω(2π)3/2 σ1 σ2 σ3 Ω0 Ξ×Γ
2σ12
2σ22
2σ32
(9.53)(∀Ξ, ∀Γ ∈ Bbd
R,∀ω ∈ Ω0 ≡ R).

Thus we have the identiﬁcation:e 0 1 , S(νµ,σ3 )) ←→ MC (Θ×((R+ )3 ×R) (O,b S[δMC0 (Ω0 ) (O).
0(θ0 ,θ1 ,σ1 ,σ2 ,σ3 ,µ) ](σ1 ,σ2 )(θ ,θ )

(θ0 ,θ1 ,σ1 ,σ2 ,σ3 ,µ)Thus, we have the sample space (R2 , Bbd) such that:
R2 , ν

b × Γ)](θ0 , θ1 , σ1 , σ2 , σ3 , µ) (∀Ξ, ∀Γ ∈ BR ).
ν (θ0 ,θ1 ,σ1 ,σ2 ,σ3 ,µ) (Ξ × Γ) = [H(Ξ(9.54)

Here, we have the following problem:b S[δ(P) Assume that we take the measurement MC0 (Θ×((R+ )3 ×R) (O,) T(θ0 ,θ1 ,σ1 ,σ2 ,σ3 ,µ) ]times, and get the measured value (x̃1 , ỹ1 , x̃2 , ỹ2 , ..., x̃T , ỹT ) ( ∈ R2T ). Here it isassumed that θ0 , θ1 , σ2 , σ3 and µ are unknown (but σ1 is known). Then, infer θ0 andθ1 (and moreover σ2 , σ3 and µ) from the measured value (x̃1 , ỹ1 , x̃2 , ỹ2 , ..., x̃T , ỹT )( ∈ R2T ) and the known σ1 .

(9.55)[(i): Answer (Moment method)].
If Θ × (R+ )3 × R is required to be compact, it suﬃces to consider [−L, L]2 × [(1/L), L]3 × [−L, L](for suﬃciently large L) instead of Θ × (R+ )3 × R.
9

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Under the notation in the problem (P), putxe1 + xe2 + · · · + xeTye1 + ye2 + · · · + yeT,
AỸT =,(9.56)TT222(ex1 − AX̃x1 − AX̃xT − AX̃X̃ X̃
T ) + (eT ) + · · · + (e
T)
VT =
,
(9.57)T222(ey1 − AX̃y1 − AX̃yT − AX̃Ỹ ỸT ) + (eT ) + · · · + (e
T)
VT =
,
(9.58)
T(ex1 − AX̃y1 − AX̃x2 − AX̃y2 − AX̃xT − AX̃yT − AX̃
X̃ Ỹ
T )(e
T ) + (eT )(eT ) + · · · + (e
T )(e
T)
VT =
.
T
(9.59)AX̃
T =Recall (9.54), and putZµX = µ,σuu = σ1 ,σee = σ2 ,σXX = σ3 ,by[H(R× dy)].

µY =(9.60)

R

Then we see that
Z
Ỹ
b
A =
y[H(R
× dy)]( ≡ µY ) = θ0 + θ1 µX ,TZbx[H(dx× R)] = µX , (9.61)

AX̃
T =

R

R

and
Z

Ỹ Ỹ
VT

=

X̃ Ỹ

VT

ZR

2
2
b
(y − µY )2 [H(R× dy)] = θ12 σXX+ σee,(9.62)

2b(x − µX )2 [H(dx× R)] = θ1 σXX,(9.63)

22b(x − µX )(y − µY )[H(dx× dy)] = σXX+ σuu,(9.64)

=
Z

R

X̃ X̃
VT =

R2

which is easily solved. Thus, the moment method says that there is a reason to infer thatX̃ X̃θ1 = (V TX̃ Ỹ− σ12 )−1 V T ,X̃ X̃θ0 = AỸT − (V TX̃ Ỹ

− σ12 )−1 AX̃
TVT .

(9.65)[(ii): Answer (Fisher’s maximum likelihood method)].
Next, we shall answer the problem (P) by Fisher’s likelihood method. Put, for suﬃciently small positive ϵ,Ξϵt = [x̃t − ϵ, x̃t + ϵ],Γϵt = [ỹt − ϵ, ỹt + ϵ] (t = 1, 2, · · · , T ).

(9.66)The probability that the measured value (x̃1 , ỹ1 , x̃2 , ỹ2 , ..., x̃T , ỹT ) ( ∈ R2T ) belongs toΠTt=1 (Ξϵt × Γϵt ) is given byihb ϵ × Γϵ )](θ0 , θ1 , σ1 , σ2 , σ3 , µ) .
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Since ϵ is suﬃciently small, we see, for some ﬁxed σ1 , thatmaxb ϵt × Γϵt )](θ0 , θ1 , σ1 , σ2 , σ3 , µ)
ΠTt=1 [H(Ξmax

ΠTt=1 [

(θ0 ,θ1 ,σ2 ,σ3 ,µ)∈Θ×(R+ )2 ×R⇐⇒(θ0 ,θ1 ,σ2 ,σ3 ,µ)∈Θ×(R+ )2 ×R

Z

1
3/2
(2π) σ1 σ2 σ3e[−(ỹ −(θ1 ω+θ0 ))2(x̃t −ω)2(ω−µ)2
− t
−
2
2
2 ]
2σ1
2σ2
2σ3

dω].

Ω0 (≡R)(9.68)
Thus, Fisher’s maximum likelihood method says that it suﬃces to ﬁnd the (θ0 , θ1 , σ2 , σ3 , µ)such that:ΠTt=1 [=

1(2π)3/2 σ1 σ2 σ3
max

Z
e

[−

(ỹ −(θ1 ω+θ0 ))2(x̃t −ω)2(ω−µ)2
− t
−
2
2
2 ]
2σ1
2σ2
2σ3

R

(θ0 ,θ1 ,σ2 ,σ3 ,µ)∈Θ×(R+ )2 ×R

ΠTt=1 [

13/2(2π) σ1 σ2 σ3

Z
e

[−

dω]

(ỹ −(θ1 ω+θ0 ))2(x̃t −ω)2(ω−µ)2
− t
−
2
2
2 ]
2σ1
2σ2
2σ3

dω]. (9.69)

R

However, it may be diﬃcult to solve it analytically. Thus, the numerical computation
may be recommended.
Remark 9.18. Comparing (9.65) and (9.69), readers may consider that the moment
method is simple and powerful. However, it should be noted that the moment method issomewhat artiﬁcial since the semi-distance is not unique. Summing up, we see,(pure) measurement
statistical measurement
repeated measurement(product sample space)

Inference

ExampleFisher’s likelihood method(Theorem 5.3, Corollary 5.6)
Bayes’ method(Theorem 8.13, Remark 8.14)
moment method (Deﬁnition 2.27)
See §9.4.1

Examples 5.8 and 5.9
Regression analysis (6.7), (6.48)
Example 8.6
Generalized Bayes theorem (Theorem 8.13)
Normal distribution (§9.4.2)
measurement error model (§9.4.3)(9.70)¥

9.5

Principal components analysis in MT

Our present purpose is to study “principal components analysis” in the framework ofMT.
Consider the following two cases [I] and [II]:[I: Homomorphic type]. Let Ω be a compact space. For each k ( = 1, 2, ..., K), consider acontinuous map fk : Ω → R. For example, we may consider that(♯) the Ω ( = {ω1 , ω2 , ..., ωN } represents the class of students in some high school. And
further, assume that

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256CHAPTER 9. STATISTICAL MEASUREMENTS IN W ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(a) fh (ωn ) · · · the student ωn ’s height(b) fw (ωn ) · · · the student ωn ’s weightΩfh (ω)
0

100

200

fw (ω)

ω

0

100

200

[II: Markov type]. Let Ω be a compact space. For each k ( = 1, 2, ..., K), consider a map³∗
Φ∗k : Ω → Mm
(R)
in
the
C
-algebraic
formulationor, Φ∗k : Ω → L1+1 (R; dm) in the+1´
W ∗ -algebraic formulation . For example, we may consider that(♯) the Ω ( ≡ {ω1 , ω2 , ..., ωN }) represents the set of students in some high school. And
further, assume that(a) Φ∗P (ωn ) · · · the student ωn ’s scholastic ability of physics (or, the distributionof the student ωn ’s marks (e.g., deviation values) in physics)(b) Φ∗C (ωn ) · · · the student ωn ’s scholastic ability of chemistry (or, the distributionof the student ωn ’s marks (e.g., deviation values) in chemistry)ΩΦ∗P (ω)050100Φ∗C (ω)

ω

0

50

100

Here consider the following problem:(P) What kind of relation among the height and weight in [I] (or, the scores of physics
and chemistry in [II]) of the students of the high school can we ﬁnd?
This problem (P) is usually studied by “principal components analysis”.
∗

Thus, inwhat follows, we shall study it in the framework of PMTW (though it can be also studied

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in PMTC since a cyclic measurement is also formulated in PMTC ). Clearly the homomorphic type [I] is the special case of the Markov type [II]. Thus, from here, we devote
ourselves to the Markov type [II].
Let Ω be a ﬁnite set, i.e., Ω = {ω1 , ω2 , ..., ωN }, which is assumed to have the counting
measure νc , that is, νc (A) = ♯[A] (∀A ⊆ Ω). For each k ( = 1, 2, ..., K), consider aMarkov operator Φk : L∞ (R, m) → L∞ (Ω, νc ), where m is the Lebesgue measure on R.
b ≡Let O ≡ (R, BR , EEXA ) be the exact observable in L∞ (R, m). Deﬁne the observable O(RK , BRK , ×k=1 Φk EEXA ) in L∞ (Ω, νc ) such that
K

K

K

k=1

k=1

[ × Φk EEXA ](Ξ1 × Ξ2 × · · · × ΞK ) = × Φk EEXA (Ξk ) (Ξ1 × Ξ2 × · · · × ΞK ∈ BRK )∞∞K
which is realization of the sequential observable [{O}K
k=1 , {Φk : L (R, m) → L (Ω, νc )}k=1 ].
b S(ρ̄ω
Thus we have the cyclic measurement ⊗N L ML∞ (Ω,νc ) (O,)), where ρ̄ωs ∈ L1 (Ω, νc ),j=1

modN [j]+1(s = 1, 2, ..., N ), is deﬁned by ρ̄ωs (ω) = 1 (if ω = ωs ), = 0 (if ω ̸= ωs ).
b S(ρ̄ω
Assume that, by the cyclic measurement ⊗N L ML∞ (Ω,νc ) (O,³))
or, the rej=1
modN [j]´bpeated measurement ⊗LN
j=1 ML∞ (Ω,νc ) (O, S(1/N )), cf. Example 8.7 (ii) , we get a measured value (x1 , x2 , ..., xN L ), where















1
2
K

xN = (xN , xN , ..., xN )


1
2
K

xN +1 = (xN +1 , xN +1 , ..., xN +1 ) 


..
.


x2N = (x12N , x22N , ..., xK

2N )



..


.



1
2
K

x3N = (x3N , x3N , ..., x3N )



..


.



1
2
K
xLN = (xN L , xN L , ..., xN L )x1 = (x11 , x21 , ..., xK
1 ),x2 = (x12 , x22 , ..., xK
2 ),
..
.

(9.71)

Here, note that it holds:♯[{j ∈ {1, 2, ..., N L} : xj ∈ Ξ1 × Ξ2 × · · · × ΞK }]
lim
L→∞
NL
Z
K
[×k=1 Φk EEXA (Ξk )](ω)νc (dω)(∀Ξ1 × Ξ2 × · · · × ΞK ∈ BRK )
=
N
Ω
Put

PN L K ´³ PN L x1 PN L x2
j=1 j
j=1 j
j=1 xj(µ1 , µ2 , ..., µK ) =,,··· ,NL
NL
NL

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258CHAPTER 9. STATISTICAL MEASUREMENTS IN W ∗ -ALGEBRAIC FORMULATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/
and put
PN L
Cpq =

j
j
j=1 (xp − µp )(xq − µq )(9.73)NL − 1³´For simplicity, here we are not concerned with the normalization, though it is reasonable.

Then, we have the correlation matrix C such that:


C11
 C21
h i

C = Cpq=  ..
 .
1≤p,q≤K

C12
C22
..
.

CK1 CK2

. . . C1K
. . . C2K ..  ,..
.
. . . . CKK

(9.74)

which is represented by
C = P ΛP ∗where Λ is a diagonal matrix such that:λ1 0
 0 λ2Λ =  ..
..
.
.
0 0

...
...
..
.


0
0

.. 
. 

(λ1 ≥ λ2 ≥ ... ≥ 0)

. . . λK

and P is the orthonormal matrix such that:e11 e12hi  e21 e22
P = ⃗e1 , ⃗e2 , ..., ⃗eK =  ..
..
 .
.
eK1 eK2

. . . e1K
. . . e2K ..  ,..
.
. . . . eKK

 
e1k
e1k  ⃗ek =  ..  ,
 . 
e1k

where
½
⟨⃗ek , ⃗ek′ ⟩RK =1 ( if k = k ′ ).
0 ( if k ̸= k ′ )Here, ⃗ek is called the k-th principal component. Also, The k-contribution ratio is deﬁned
by PKλk λ .
i=1iRemark 9.19. [(i): Several interpretations of principal components analysis]. Principal
components analysis (i.e., {(⃗ek , λk )}K
k=1 ) has several interpretations, which are important.
For example, the following ﬁgure is frequently stated in usual books of statistics.

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6

R
M

⃗e2
⃗e1

1

x4
λ2

x2
x3

x1

x6

λ1
x5

-R

However, we are not concerned with it, because what we want to say here is the following(ii).
[(ii): Markov type and homomorphic type]. Note that the data (9.71) is obtained by the√exact measurement. Thus the C pp is not the error. In the case of Markov type, thefollowing calculation is wrong. However, if Φk : L∞ (R, m) → L∞ (Ω, νc ) is homomorphic,b has the form such as (RK .BRK , ×K Φk Gσk ) in L∞ (Ω, νc ) where
and if the observable O
1
GσΞk (µ) = p
2πσk2

k=1

Z

e

(u−µ)2−2σ 2k(∀µ ∈ R ≡ R, ∀Ξ ∈ BR ). (σk2 : variance),duΞ(cf. Example 9.5 and Example 2.17), then the following calculation should be recommended: Put³ PL−1 x1

PL−1 K ´
x21+lNxx̄1 =,, · · · , l=0 1+lN ,L
L
L
PL−1 2
PL−1 K ´³ PL−1 x1
x
x
2+lN
l=0
x̄2 =
, l=0 2+lN , · · · , l=0 2+lN ,
L
L
L
..
.
PL−1 2
PL−1 K³ PL−1 x1
´
x
N +lN
l=0
l=0 xN +lNx̄N =,, · · · , l=0 N +lN .
L
L
L
l=0

Put

PL−1

1+lN

l=0

PN K ´³ PN x̄1 PN x̄2
j=1 j
j=1 j
j=1 x̄j(µ̄1 , µ̄2 , ..., µ̄K ) =,,··· ,,
N
N
N(9.75)and put
PN
C̄pq =jj
j=1 (x̄p − µ̄p )(x̄q − µ̄q )N.

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Then, we have the correlation matrix C̄ such that:C̄11 C̄12
 C̄21 C̄22h i
C̄ = C̄pq=  ..
..
 .
1≤p,q≤K.
C̄K1 C̄K2

. . . C̄1K
. . . C̄2K ..  .
...
. . . . C̄KK

Thus, by a similar way, we can get the k-th principal component and the k-contribution
ratio, etc.
Note that it holds: 
C11 C12 . . . C1KC̄11 + (σ1 )2C̄122 C21 C22 . . . C2K  C̄C̄21
22 + (σ2 ) (9.74) =  ..
..
..  = 
..
..
..
 .
.
.
.  
.
.
CK1 CK2 . . . CKK
C̄K1
C̄K2


(σ1 )2
0
...
0
2
 0

(σ
)
.
.
.
0
2


=(9.77) +  ..

..
..
.
.
 .

.
.
.
2
0
0
. . . (σK )

...
...
..
.

C̄1K
C̄2K
..
.







. . . C̄KK + (σK )2though the situations are diﬀerent.
¥

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Chapter 10
Newtonian mechanics inmeasurement Theory

In the previous chapter, we propose the W ∗ −algebraic formulation of SMT:∗

SMTW = statistical measurement+ the relation among systems∗[ProclaimW 1 (9.9)]∗[ProclaimW 2 (9.23)]

in W ∗ -algebra .

(10.1)As mentioned in Remark 1.1 (b), in this book, “Newtonian mechanics” in MT is called the “classical system theory (or dynamical system theory)”. In this sense, we will study “Newtonian∗mechanics” in SMTW . We ﬁrst introduce “the W ∗ -algebraic generalization of Kolmogorov’s extension theorem”. This theorem is essential to MT just like Kolmogorov’s extension theorem is
so in his probability theory. Using this theorem, we can deﬁne “particle’s trajectory” by “thesequence of measured values”. And further we prove:(i) the existence of “particle’s trajectory” in Newtonian mechanics,(ii) the existence of Brownian motion.
Thus, we can understand the diﬀerence between the concepts of “particle’s trajectory” and “state’s
evolution” in both classical and quantum mechanics. Throughout this chapter, readers will see
that, from the mathematical point of view, the W ∗ −algebraic formulation is more handy than theC ∗ −algebraic formulation.

10.1

Kolmogorov’s extension theorem in W ∗-algebra

In this section we study “Kolmogorov’s extension theorem” in the (W ∗ -algebraic) Statistical MT. It is generally said that Kolmogorov’s extension theorem is most fundamental
in Kolmogorov’s probability theory. That is because this theorem assures the existenceof a probability space (i.e., sample space). On the other hand, our theorem (= Theorem
10.1, i.e., the W ∗ -algebraic generalization of Kolmogorov’s extension theorem) assures
261
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the existence of a measurement (or, observable). Recall the our spirit see Remark (in´§2.3(I)) :(♯) there is no probability without measurements.
Thus, in measurement theory, the concept of “measurement” is more fundamental thanthat of “sample space”.

Therefore, this theorem (i.e., the W ∗ -algebraic generalizationof Kolmogorov’s extension theorem) is very important in MT. That is, this theorem (=Theorem 10.1) is essential to MT just like Kolmogorov’s extension theorem is so in his
probability theory. Using this theorem, we can deﬁne “particle’s trajectory” by “thesequence of measured values”. And further we prove:(i) the existence of “particle’s trajectory” in Newtonian mechanics,(ii) the existence of Brownian motion.
Thus, we can understand the diﬀerence between the concepts of “particle’s trajectory” and“state’s evolution” in both classical and quantum mechanics.
b be an index set. For each λ ∈ Λ,b consider a set Xλ . For any subsets Λ1 ⊆ Λ2 ( ⊆Let Λb πΛ1 ,Λ2 is the natural projection such that:Λ),πΛ1 ,Λ2 : × Xλ −→ × Xλ .
λ∈Λ2

λ∈Λ1

b consider a W ∗ -observable (Xλ , Fλ , Fλ ) in
Especially, put πΛ = πΛ,Λb . For each λ ∈ Λ,W ∗ -algebra N. Note that the quasi-product observable O ≡ (×λ∈Λb Xλ , ×λ∈Λb Fλ , FΛb ) ofb } is characterized as the observable such that:{ (Xλ , Fλ , Fλ ) | λ ∈ Λ−1FΛb (π{λ}(Ξλ )) = Fλ (Ξλ )b(∀Ξλ ∈ Fλ , ∀λ ∈ Λ),(10.2)

though the existence and the uniqueness of a quasi-product observable are not guaranteed
in general. The following theorem says something about the existence and uniqueness ofthe quasi-product observable.
Theorem 10.1. [W ∗ -algebraic generalization of Kolmogorov’s extension theorem, cf.
b consider a Borel measurable space (Xλ , Fλ ), where Xλ is a separable[43]]. For each λ ∈ Λ,b such as P0 (Λ)
b ≡ {Λ ⊆ Λ
b | Λ is ﬁnite }.
complete metric space. Deﬁne the set P0 (Λ)©
Assume that the family of the W ∗ -observables OΛ ≡ ( ×λ∈Λ Xλ , ×λ∈Λ Fλ , FΛ ) | Λ ∈ªb
P0 (Λ)
in a W ∗ -algebra N satisﬁes the following “W ∗ -algebraic consistency condition”:

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263

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/b such that Λ1 ⊆ Λ2 ,• for any Λ1 , Λ2 ∈ P0 (Λ)¡¢¡¢
FΛ2 πΛ−11 ,Λ2 (ΞΛ1 ) = FΛ1 ΞΛ1(∀ΞΛ1 ∈ × Fλ ).

(10.3)

λ∈Λ1eb ≡Then, there uniquely exists the W ∗ -observable OΛ¡¢×λ∈Λb Xλ, ×λ∈Λb Fλ, FeΛb in N such

that:
¡
¢
¡ ¢
b
FeΛb πΛ−1 (ΞΛ ) = FΛ ΞΛ (∀ΞΛ ∈ × Fλ , ∀Λ ∈ P0 (Λ)).

(10.4)

λ∈Λ

Proof. Let ρ be any normal state, i.e., ρ ∈ Sn (N∗ ). Then, the ρ(FΛ (·)) is a probability³
bmeasure on the product measurable space (×λ∈Λ Xλ , ×λ∈Λ Fλ ) for all Λ ∈ P0 (Λ). If´
N = L∞ (Ω, µ), the existence is assured. It is clear that the family { (×λ∈Λ Xλ , ×λ∈Λ Fλ ,b } satisﬁes the “usual consistency condition” in Kolmogorov’sρ(FΛ ( · ))) | Λ ∈ P0 (Λ)
probability theory. Therefore, by Kolmogorov’s extension theorem[56] , there uniquely¡¢
exists a probability measure PΛbρ on the product measurable space ×λ∈Λb Xλ , ×λ∈Λb Fλsuch that:¡¢bPΛbρ πΛ−1 (ΞΛ ) = ρ(FΛ (ΞΛ )) (∀ΞΛ ∈ × Fλ , ∀Λ ∈ P0 (Λ)).

(10.5)λ∈ΛDeﬁne the subﬁeld ×λ∈Λb Fλ of ×λ∈Λb Fλ such that:♯♯b
× Fλ = {πΛ−1(ΞΛ) | ΞΛ ∈ × Fλ, Λ ∈ P0(Λ)}.

bλ∈Λ

(10.6)

λ∈Λ

Then, we see, by (10.5), that there uniquely exists the countably additive function FΛb♯ :×♯λ∈Λb Fλ → N (in the sense of weak∗-topology σ(N, N∗)) such that:
♯
¡ ¢
¡
¢
PΛbρ Ξ♯Λb = ρ(FΛb♯ Ξ♯Λb ) (∀Ξ♯Λb ∈ × Fλ ).

(10.7)bλ∈ΛDeﬁne the map FeΛb : ×λ∈Λb Fλ → N such that:¡ ¢FeΛb ΞΛb =

inf

{Ξ♯,k
}∞
b)
k=1 ∈Q(ΞΛ
b
Λ

∞
X
k=1

¡
FΛb♯ Ξ♯,kb ),Λ

(10.8)

n
o
♯
♯,k ∞
♯,k
♯,k
∞
where Q(ΞΛb ) ≡ {ΞΛb }k=1 | ΞΛb ⊆ ∪k=1 ΞΛb , ΞΛb ∈ ×λ∈Λb Fλ (∀ΞΛb ∈ ×λ∈Λb Fλ ). It
clearly holds that

¡ ¢
¡ ¢
¢
¡
FeΛb Γ1Λb ∪ Γ2Λb ≤ FeΛb Γ1Λb + FeΛb Γ2Λb(∀Γ1Λb , Γ2Λb ∈ × Fλ ).

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Also, we see that, for any ΞΛb in ×λ∈Λb Fλ ,PΛbρ¡ΞΛb ) =
=

∞
X

inf

{Ξ♯,k
}∞
b)
k=1 ∈Q(ΞΛ
b
Λ

inf

{Ξ♯,k
}∞
b)
k=1 ∈Q(ΞΛ
b
Λ

≥ρ

³

inf

k=1
∞
X
k=1

}∞
{Ξ♯,k
b)
k=1 ∈Q(ΞΛb¡ ¢= ρ(FeΛb ΞΛb )Λ¡PΛbρ Ξ♯,kb )Λ

(by Caratheodory theorem, cf. [29])¡¢ρ(FΛb♯ Ξ♯,kb )
Λ(by (10.7))

∞
X
k=1

FΛb♯

¡ ♯,k ¢´ΞΛb(by the property of N)(by (10.8)).

¡¡ ¢Similarly we see that PΛbρ ΞcΛb ) ≥ ρ(FeΛb ΞcΛb ) where ΞcΛb = ( ×λ∈Λb Xλ ) \ ΞΛb . Thus we see,by (10.9), that

¡ ¢
¡ ¢
¡
¢
¡
¡
1 = PΛbρ ΞΛb ) + PΛbρ ΞcΛb ) ≥ ρ(FeΛb ΞΛb ) + ρ(FeΛb ΞcΛb ) ≥ ρ(FeΛb × Xλ ) = 1.
b
λ∈Λ

¡
¡ ¢
This implies that PΛbρ ΞΛb ) = ρ(FeΛb ΞΛb ). Thus we see that³³ ¡ ¢´
¡ −1
¢´
¡ −1
¢
ρ
e
b
ρ FΛb πΛ (ΞΛ ) = PΛb πΛ (ΞΛ ) = ρ FΛ ΞΛ(∀ΞΛ ∈ × Fλ , ∀Λ ∈ P0 (Λ)),λ∈Λ

which implies (10.4). This completes the proof.

10.2

The deﬁnition of “trajectories”∗

Now we shall propose the deﬁnition of the “trajectories” in SMTW . Let S(ρ0 ) ≡[S(ρ0 ), {Ψt1 ,t2 : N → N}(t1 ,t2 )∈R2 ] be a W ∗ -general system with an initial system S(ρ0 ).
≤+Let O ≡ (X, F, F ) be a crisp observable in N. For each time t ∈ R ≡ {t ∈ R | t ≥ 0},
consider a W ∗ -observable Ot ≡ (Xt , Ft , Ft ) in N such that:+• (Xt , Ft , Ft ) = (X, F, F ) for all t ∈ R .

(10.9)+Let us represent the “measurement M({Ot }t∈R+ , S(ρ0 ))” in what follows. Let Λ ∈ P0 (R )³´+
≡ {Λ0 ∈ 2R : Λ0 is ﬁnite } , that is, Λ = {t1 , t2 , ..., tn } where 0 ≤ t1 < t2 < · · · < tn .
Then, we can uniquely deﬁne the observable OΛ ≡ (X Λ , FΛ , FΛ ) at time 0 such that:
³
³
´
¡
¢´
FΛ (Ξt1 × Ξt2 × · · · × Ξtn ) = Ψ0,t1 F (Ξt1 ) · · · Ψtn−2 ,tn−1 F (Ξtn−1 ) Ψtn−1 ,tn F (Ξtn ) · · · ,(10.10)Sample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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though the existence of OΛ is not always guaranteed except for the classical cases. For´
the uniqueness, recall Theorem 9.8.
Assume that the observable OΛ exists for any++Λ ∈ P0 (R ). It is clear that the family { OΛ | Λ ∈ P0 (R ) } satisﬁes the consistency++e + ≡ (X R , FR , Fe + )condition (10.3). Thus, by Theorem 10.1 we have the observable O
RR

in N, which is called a trajectory observable (concerning O ≡ (X, F, F )). Therefore, we
e + , S(ρ0 )) of M({Ot } + , S(ρ0 ) ).
get the Heisenberg picture representation MN (ORt∈R

Now we can propose the following deﬁnition, which is our main assertion in thischapter.
Deﬁnition 10.2. [Trajectory (= particle’s trajectory)]. Assume the above notations.
e + , S(ρ0 )) is called a trajectory
The measured value obtained by the measurement MN (OR∗(concerning O ≡ (X, F, F ) ) of the W -general system S(ρ0 ) ≡ [S(ρ0 ), {Ψt1 ,t2 : N →N}(t1 ,t2 )∈R2 ].
≤¥The diﬀerence of “particle’s trajectory” and “state’s evolution” is clear in Deﬁnition
10.2. That is, “state’s evolution”
“particle’s trajectory”· · · (Ψ0,t )∗ ρ0 ,(0 ≤ t < ∞),e + , S(ρ0 )).
· · · the measured value of MN (OR(10.11)e + is not usually guaranteed, andNote that in quantum mechanics, the existence of O
R
thus, the concept of “particle’s trajectory” is meaningless in general (cf. [37, 40]).
Recall DST(1.2a), that is, dx(t) dt = g(x(t), u1 (t), t), x(0) = x0 · · · (state equation) ,“dyn. syst. theor.” =y(t) = f (x(t), u2 (t), t)( measurement equation).
(10.12)(=(1.2a))In order to compare (10.11) and (10.12), we add the following remark.
Remark 10.3. [(i): The case that u2 = 0 in (10.12)] (The generalization of Deﬁnition
10.2). The condition (10.9) can be easily generalized as follows:+• (Xt , Ft , Ft ) is crisp for all t ∈ R .

(10.13)

Under the condition, by a similar way of (10.10) we can easily deﬁne a trajectory (concerning+{(Xt , Ft , Ft ) | t ∈ R }) of the W ∗ -general system S(ρ0 ) ≡ [S(ρ0 ), {Ψt1 ,t2 : N →Sample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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CHAPTER 10. NEWTONIAN MECHANICS IN MEASUREMENT THEORYFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/+N}(t1 ,t2 )∈R2 ]. Here, consider classical cases, i.e., N = L∞ (Ω, µ). And, for each t ∈ R ,≤consider a measurable function ft : Ω → Rm , which can be identiﬁed with a crisp observable (Rm , BRm , Ft ), (cf. (ii) in Example 9.4). Thus, by Theorem 10.1 we have the++
e + ≡ ((Rm )R , (BRm )R , Fe + ) in N, which is called a trajectory observobservable O
RR+able (concerning { Ot ≡ (R , BRm , Ft ) | t ∈ R }. Thus we can also deﬁne a trajectorym+(concerning {ft | t ∈ R }) of the W ∗ -dynamical system S(ρ0 ) as the trajectory concerning+{ (Rm , BRm , Ft ) | t ∈ R }[(ii): The case that u2 ̸= 0 in (10.12)] (The generalization of Deﬁnition 10.2). The
condition (10.13) can be easily generalized as follows:+• (Xt , Ft , Ft ) is not always crisp for all t ∈ R .

(10.14)e + ≡ ( × + Xt , × + F t ,By a similar way as in the above (i), we have the observable O
R
t∈R
t∈R
∞
e
F + ) in N ( = L (Ω; µ)), which is called a trajectory observable (concerning { Ot ≡R+

(Xt , Ft , Ft ) | t ∈ R }).
¥10.3

Trajectories and Newtonian mechanics

In the previous section, we proposed Deﬁnition 10.2, in which the concept of “particle’s trajectory” is characterized as a measured value of the measurement. Thus, ourconcern in this section is to show that the “particle’s trajectory” is represented by the
Newton equation. If it is true, we can completely understand “Newtonian mechanics” inmeasurement theory.
First we review Liouville’s equation. Put N = L∞ (Rsq × Rsp , m2s ) and N∗ = L1 (Rsq × Rsp ,m2s ), where Rsq × Rsp ≡ { (q, p) = (q1 , q2 , · · · , qs , p1 , p2 , · · · , ps ) | qj , pj ∈ R, j =1, 2, · · · , s } and (Rsq × Rsp , B(Rsq × Rsp ), m2s ) is the 2s-dimensional Lebesgue measurespace. Liouville’s equation with an initial density function ρ0 is as follows:s∂ρt (q, p) X³ ∂H(q, p, t) ∂ρt (q, p) ∂H(q, p, t) ∂ρt (q, p) ´
=
−
,
∂t
∂qj
∂pj
∂pj
∂qj
j=1

ρ0 ∈ Sn (N∗ ) ≡ {ρ : ∥ρ∥L1 = 1, ρ ≥ 0},

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(10.15)(10.16)

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267

where H : Rsq × Rsp × R → R is a Hamiltonian. By using the solution of (10.15), we candeﬁne the operator [Ψt1 ,t2 ]∗ : L1 (Rsq × Rsp , m2s ) → L1 (Rsq × Rsp , m2s ) such that:³´[Ψt1 ,t2 ]∗ ρt1 (q, p) = ρt2 (q, p) ∀(q, p) ∈ Rsq × Rsp , ∀(t1 , t2 ) ∈ R2≤ .

(10.17)

That is, the “state’s evolution” is represented by the Schrödinger picture {[Ψt1 ,t2 ]∗ | (t1 , t2 ) ∈R2≤ }, which is induced by Liouville’s equation (10.15) for states. And furthermore, puttingΨt1 ,t2 = ([Ψt1 ,t2 ]∗ )∗ , we get the Heisenberg picture {Ψt1 ,t2 | (t1 , t2 ) ∈ R2≤ }, which is also induced by Liouville’s adjoint equation (i.e., Liouville’s equation for observables). Thus, we
get the W ∗ -dynamical system S(ρ0 ) ≡ [S(ρ0 ), {Ψt1 ,t2 : N → N}(t1 ,t2 )∈R2 ]. Also, it should≤

be noted that the dynamical system S(ρ0 ) is deterministic, i.e., each Ψt1 ,t2 : N → N is(bijective) homomorphic.
It is well known that Liouville’s equation is mathematically equivalent to the following
Newton equation:d∂H
qj (t) =(q(t), p(t), t),
dt
∂pj

d
∂H
pj (t) = −(q(t), p(t), t),dt∂qj

j = 1, 2, · · · , s(q(0), p(0)) ∈ Rsq × Rsp .

(10.18)(10.19)Using the solution of the Newton equation (10.18), we deﬁne the continuous map ψt1 ,t2 :Rsq × Rsp → Rsq × Rsp such that:ψt1 ,t2 (q(t1 ), p(t1 )) = (q(t2 ), p(t2 ))(∀(q(t1 ), p(t1 )) ∈ Rsq × Rsp ).

(10.20)Thus we can get the (bijective) homomorphism Ψt1 ,t2 : L∞ (Rsq × Rsp , m2s ) → L∞ (Rsq × Rsp ,m2s ) such that:(Ψt1 ,t2 F )(q, p) = F (ψt1 ,t2 (q, p)) (∀(q, p) ∈ Rsq × Rsp , ∀F ∈ L∞ (Rsq × Rsp ), ∀(t1 , t2 ) ∈ R2≤ ).
(10.21)
Of course, this Ψt1 ,t2 is the same as the Ψt1 ,t2 derived from Liouville’s equation. Since
Liouville’s equation and Newton equation are mathematically equivalent, there is a reason
to say that the time evolution is also represented by Newton equation. However, notethat the term “Newton equation” [resp. “Liouville’s equation”] is, in this book, deﬁned
to be the equation that represents “particle’s trajectory” [resp. “time evolution of states
or observables”].

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For simplicity, we put (Ω, B, dω) = (Rsq × Rsp , B(Rsq × Rsp ), m2s ). And, put (N, N∗ )= (L∞ (Ω), L1 (Ω)). Consider the deterministic W ∗ -dynamical system S(ρ0 ) ≡ [S(ρ0 ),{Ψt1 ,t2 : N → N}(t1 ,t2 )∈R2 ], which is induced by Liouville’s equation (10.15) and (10.16).
≤¡¢
Deﬁne the state space observable (or, exact observable) O ≡ (Ω, B, F ) in N ≡ L∞ (Ω)
such that:
F (Ξ) = χΞ∀Ξ ∈ B,(10.22)

which is, of course, crisp. Thus, by the same arguments appearing above Deﬁnition 10.2,++
e + ≡ (ΩR , BR , Fe + ) concerning the state space
we can get the trajectory observable ORR

e + , S(ρ0 ))observable O ≡ (Ω, B, F ). And therefore, we get the measurement ML∞ (Ω) (OR(cf. Remark 10.3). Assume thate + , S(ρ0 )).
• a measured value ωb (= (ωt )t∈R+ ∈ ΩR ) is obtained by ML∞ (Ω) (OR+Note that the measured value ωb is precisely the “particle’s trajectory” in Deﬁnition 10.2.
+

Now we shall investigate the properties of the measured value ωb (= (ωt )t∈R+ ∈ ΩR ),that is, we shall show that the trajectory ωb is represented by the Newton equation (10.18)+

and (10.19). Let D = {t0 , t1 , t2 , · · · , tn } be a ﬁnite subset of R , where t0 = 0 < t1 <¡+¢b = ×D + Ξt ∈ BR where Ξt = Ω (∀t ∈/ D). Then, we see thatt2 < · · · < tn . Put Ξt∈Rb = ×D + Ξt is given by• the probability that ωb ( = (ωt )t∈R+ ) belongs to the set Ξ
t∈R
³
³
³
¡
¡
¢´ ´
¢
b =ρ0 F (Ξ0 )Ψ0,t F (Ξt ) · · · Ψt ,tρ0 FeR+ (Ξ)
F
(Ξ
)
Ψ
F
(Ξ
)
···
tn−1
tn−1 ,tn
tn
1
1n−2 n−1³´¡¢=ρ0 Πnk=0 Ψ0,tk F (Ξtk )(because each Ψtk−1 ,tk is homomorphic)³´¢−1=ρ0 Πnk=0 F (ψ0,t(Ξtk )kZ ³´=
Πnk=0 χψ−1 (Ξt ) (ω) ρ0 (ω)dω.
Ω

0,tk(10.23)k

Let Ξ0 be any element in B such thatRΞ0

ρ0 (ω)dω ̸= 0. Thus, under the hypothesis that

we know that ω0 ∈ Ξ0 , i.e., ωb (= (ωt )t∈R+ ) ∈ Ξ0 × ΩR (where R+ = (0, ∞)), we can+calculate the following conditional probability:
´
R ³ n
¡
¢
D
−1
Π
χ
(ω)
ρ0 (ω)dω
k=1 ψ0,t (Ξtk )ρ0 FeR+ (×t∈R+ Ξt )
Ξ0
k
R
.
¢=
¡
ρ (ω)dωρ0 Fe + (Ξ0 × ΩR+ )Ξ0 0
R

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(10.24)

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269

Thus, we see that¡
¢ ½
D
ρ0 FeR+ ( ×t∈R+ Ξt )
1
lim
¡
¢=
+
0
e + (Ξ0 × ΩR )
Ξ0 →{ω0 } ρ F0−1
if ω0 ∈ ∩nk=1 ψ0,t(Ξtk )kotherwise.

R(10.25)³
Though the above argument is somewhat rough from the mathematical point of view,´
we can easily check it in mathematics.
This implies thatωt = ψ0,t (ω0 )+(∀t ∈ R ).

(10.26)+

That is, the measured value ωb (= (ωt )t∈R+ ∈ ΩR ) is the solution of the Newton equation.
Also, note that the (10.25) is independent of the choice of the initial normal state ρ0 .
Summing up, we see,• In Newtonian mechanics, the state’s evolution is represented by Liouville equation,and the existence of the trajectory (concerning the state space observable) is always guaranteed. That is, it can be represented by the Newton equation. Also, in
quantum mechanics, the state’s evolution is represented by Schrödinger equation.
However, the existence of the trajectory is not always guaranteed.
That is,Newtonian mechanics
quantum mechanics

state’s evolution
Liouville equation
Schrödinger equation

particle’s trajectory
Newton equation(meaningless)1(10.27)

10.4

Brownian motions

As emphasized throughout this chapter, the concepts of “state’s evolution” and “particle’s trajectory” are completely diﬀerent. This is, of course, a matter of common knowledge in quantum mechanics. And moreover, we can point out that the diﬀerence is clear
in diﬀusion processes for classical systems. Therefore, in this section we examine diﬀusion∗

processes in SMTW . The examination will promote a better understanding of our theory.
1

For the measurement theoretical model of Wilson chamber and its numerical analysis, see [37, 40].

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Put N = L∞ (Rq , m) and N∗ = L1 (Rq , m), where (Rq , B(Rq ), m) is the 1-dimensionalLebesgue measure space. The diﬀusion equation with an initial density function ρ0 at thetime t = 0 is as follows:∂ρt (q)∂ 2 ρt (q)
=
,
∂t
∂q 2

(10.28)ρ0 ∈ {ρ ∈ L1 (Rq , m) : ∥ρ∥L1 = 1, ρ ≥ 0}.

(10.29)By using the solution of (12.28), we can deﬁne the operator [Ψt1 ,t2 ]∗ : L1 (Rq , m) →L1 (Rq , m) such that:Z ∞³´[Ψt1 ,t2 ]∗ (ρt1 ) (q) = ρt2 (q) =ρt1 (y)Gt2 −t1 (q − y)m(dy),−∞where Gt (q) is the Gaussian function, that is, Gt (q) =(∀(t1 , t2 ) ∈ R2≤ ) (10.30)√ 1 exp
2πt

h

2
− q2t

i
. The “state’s

evolution” is, of course, represented by the Schrödinger picture {[Ψt1 ,t2 ]∗ | (t1 , t2 ) ∈ R2≤ }.
For simplicity, we put (Ω, B, dω) = (Rq , B(Rq ), m). And therefore, put (N, N∗ ) =∞(L (Ω), L1 (Ω)). Putting Ψt1 ,t2 = ([Ψt1 ,t2 ]∗ )∗ , we get the Heisenberg picture {Ψt1 ,t2 | (t1 , t2 ) ∈R2≤ }, and consequently, the W ∗ -dynamical system S(ρ0 ) ≡ [S(ρ0 ), {Ψt1 ,t2 : N → N}(t1 ,t2 )∈R2 ].
≤¡¢Consider the state space observable O ≡ (Ω, B, F ) in N ≡ L∞ (Ω) such as in Example 9.4.(i). Thus, by a similar way in the previous section, we get the measurement
ML∞ (Ω) (OR+ , S(ρ0 )). Assume that+

• a measured value ωb (= (ωt )t∈R+ ∈ ΩR ) is obtained by ML∞ (Ω) (OR+ , S(ρ0 )).
Note that the measured value ωb is precisely the “particle’s trajectory” in Deﬁnition 10.2.
Also, it may be usually called a “path”.
By a similar way in the previous section, we shall investigate the properties of the+measured value ωb (= (ωt )t∈R+ ∈ ΩR ). Let D = {t0 , t1 , t2 , · · · , tn } be a ﬁnite subset of¡+¢+b = ×D + Ξt ∈ BR where Ξt = Ω
R , where t0 = 0 < t1 < t2 < · · · < tn . Put Ξ
t∈R

(∀t ∈
/ D). Then, by Proclaim

W∗2, we seeb ≡ ×D + Ξt is given by• the probability that ωb ( = (ωt )t∈R+ ) belongs to the set Ξ
t∈R
³
³
³
´
¡
¢
¡
¢´
b = ρ0 F (Ξ0 )Ψ0,t1 F (Ξt1 ) · · · Ψtn−2 ,tn−1 F (Ξtn−1 ) Ψtn−1 ,tn F (Ξtn ) · · ·ρ0 FeR+ (Ξ)
Z
Z
Z Y
n
³Z ³
´
´
=
ρ0 (ω0 )···((Gtk −tk−1 (ωk − ωk−1 )dωn )dωn−1 ) · · · dω1 dω0 .
Ξ0

Ξ1

Ξtn−1

Ξtn k=1(10.31)

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Let Ξ0 be any element in B such thatRΞ0

ρ0 (ω)dω ̸= 0. Suppose that we know that

ω0 ∈ Ξ0 . i.e., ωb ( ≡ (ωt )t∈R+ ) ∈ Ξ0 × ΩR . Under the hypothesis, we can calculate the+following conditional probability:
³R
´
R
R Qn
¢
D
ρ
(ω
)
·
·
·
G
(ω
−
ω
)dω
·
·
·
dω
e
0
k
k−1
n
1 dω0
ρ0 FR+ (×t∈R+ Ξt )k=1 tk −tk−1
Ξ0 0
Ξt1
Ξtn
R
=
.
¡
¢
ρ
(ω
)dω
0
0
ρ0 Fe + (Ξ0 × ΩR+ )
0
Ξ
¡

R

0

(10.32)
And therefore, we see that
¡
¢
Z
Z Y
D
n
ρ0 FeR+ (×t∈R+ Ξt )lim···
Gtk −tk−1 (ωk − ωk−1 )dωn · · · dω1 . (10.33)¡¢=
e + (Ξ0 × ΩR+ )Ξ0 →{ω0 } ρ F
Ξt
Ξtn
0
R

1

k=1

Thus, under the hypothesis that we know that ωb ( ≡ (ωt )t∈R+ ) ∈ {ω0 } × ΩR , the mea+

sured value ωb ( ≡ (ωt )t∈R+ ) has the property like Brownian motion with the initial valueω0 . Also note that the (10.33) is independent of ρ0 .
6Rωb ( ≡ (ωt )t∈R+ )
ω0

t

-

Remark 10.4. [Complex system theory]. Here I shall mention my opinion for the relation
between Brownian motions and “complex system theory” (or, “chaotic system theory” )as follows:[(i): Chaotic system theory]. It is a matter of course that Brownian motion is used
to analyze stochastic phenomena (cf. [32]). It should be noted that Brownian motion
is, from the computational point of view, generated by “pseudo-random number”. And

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moreover, it should be noted that random number generator is regarded as a kind ofchaotic equation ( cf. [19]). In this sense, we consider, from the computational point ofview, that Brownian motion analysis is regarded as a kind of chaotic equation. However,chaotic theory (or complex system theory, cf [87]) should not be overestimated as “thethird physics (i.e., relativity theory, quantum mechanics, complex system theory)” 2 .
Chaotic theory is not such a theory. This is easily seen if chaotic theory is investigated³in the framework of MT in which “probability” (related to Axiom 1) is never born from“equations” (related to Axiom 2), cf. Chapter 4 (“staying time interpretation” and not´“probabilistic interpretation”) and Remark 8.4 (Bertrand’s paradox) .
[(ii): Information compression]. Newtonian mechanics may be regarded as a kind of“information compression”. In fact, if we want to know the motion of particles, it suﬃces
to solve the Newtonian kinetic diﬀerential equation. Also, it should be noted that thediﬀerential equation is, numerically, solved by iteration method (= “loop (in computer
programming)” ). Thus, there is a reason to think that an iteration (= “loop” ), which ismainly related to Axiom 2, is regarded as a kind of information compression method of“analytic function”, “pseudo-random number”, “self-similar ﬁgure (Julia and Mandelbrotset)”, etc.

In other words, any ﬁgure (or graph) treated in mathematical science is always

generated by iteration. Thus, we assert that MT is also a kind of information compression
method. That is, mathematical science always has the aspect such as “mathematicalmethod of information compression”.
[(iii): Butterﬂy eﬀect]. “Butterﬂy eﬀect” is mentioned as follows:(♯) The ﬂutter of a butterﬂy’s wings in China could, in fact, actually eﬀect weather
patterns in New York City, thousands of miles away.
It is impossible to test the above (♯). In this sense, we do not tell whether the (♯) is trueor not. However, recall the spirit of the mechanical world view (1.12), i.e., “at any rate,
study every problem in the framework of MT”.

Thus, if a certain diﬀerential equation

2
This overestimation is like the proverb “It’s always darkest just beneath the lighthouse”. I have an
opinion that Einstein’s relativity theory, quantum mechanics and dynamical system theory (=DST(1.2))
are the most inﬂuential mathematical scientiﬁc theories in the 20th century, though DST is too familiar
to us. The dropping of two atomic bombs (Einstein’s relativity theory) is obviously one of the mosttragic events in World War II. Also, Kalman ﬁlter (DST) and IC technology (quantum mechanics) leadthe Apollo plan to success. This feat promoted the end of Cold War. And further, I think that thisopinion is improved in this book (i.e., “quantum theory” + “DST” =⇒ “MT” ) and it is realized in Table(1.7), in which we may assert that “relativity theory (or, TOE)” ↔ “the ﬁrst physics”, and “MT” ↔“thesecond physics”.

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273

suggests the above fact (♯), we have to agree that there is a possibility that the above (♯)
is true.
¥

10.5

Conclusions

Summing up, we conclude (cf. [43]),

Newtonian mechanics
quantum mechanics
diﬀusion processstate’s evolution (≈Axiom 2)
Liouville equation
Schrödinger equation
diﬀusion equation

particle’s trajectory(sample space)
Newton equation(meaningless)stochastic diﬀerential equation3(10.34)Thus there is a reason to say that the state equation in DST(1.2) should be called “trajectory equation”, though DST(1.2) is sometimes called “state space method”. Therefore,
in this book we say that DST(1.2) is the “sample space method”, in which the theory of
diﬀerential equations and Kolmogorov’s probability theory play essential roles.4 Thus we
can symbolically say:(our proposal)“MT” ←−−−−−−−−− “DST” + “statistics”(10.35)(sample space method)

Here we have the following problem:• Can we propose another mathematical scientiﬁc theory for data analysis?

(cf. thethird theory in Table (1.7))
I think that it is impossible to propose “the third theory” in mathematical science but
computer science. Cf. Remark 1.5.
Also, recall we are not concerned with “Newtonian mechanics” in physics (which isrepresented in terms of diﬀerential geometry) but “Newtonian mechanics” in MT (whichis represented in terms of operator algebra). Thus, it should be noted that our viewpoint(proposed in this book) is, of course, one-sided.
3

Recall (1.2). It should be noted that the stochastic state equation (= stochastic diﬀerential equation)in (1.2) is not “state equation” but “trajectory equation (i.e., the equation that represents particle’s
trajectory)”.
4
I believe that “Kolmogorov’s probability space” is essentially the same as “the sample space in MT”.

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Chapter 11
Measurement error

Let Q ≡ (R, B, G) and O ≡ (R, B, F ) be respectively a crisp W ∗ -observable (i.e., quantity) and a
W ∗ -observable in a W ∗ -algebra N such that Q and O commute. Under the³ assumption that ´O isregarded as the approximation of Q, we deﬁne the measurement error ∆ MN (Q × O, S(ρ)) by´ hZ Z∆ MN (Q × O, S(ρ)) =
³

¡
¢i1/2
|λ1 − λ2 |2 ρ (G × F )(dλ1 dλ2 ).

(11.1)

R2

The purpose of this ´chapter is toThis is also called the distance between Q and O concerning ρ. ³
investigate the measurement error. Readers will see that the ∆ MN (Q × O, S(ρ)) is superior
to the “conventional deﬁnition” such as |“true value” −“measured value” |.

11.1

Approximate measurements for quantities

Let N be a W ∗ -algebra. Let Q ≡ (R, B, G) be a crisp W ∗ -observable (i.e., quantity)
in N. Let O ≡ (R, B, F ) be a W ∗ -observable in N such that Q and O commute. Let Q ×O ≡ (R2 , B2 , G × F ) be the product observable of Q and O. Consider the simultaneous∗

measurement MN (Q×O, S(ρ)). According to ProclaimW 1 (9.9), the probability that themeasured value (λ1 , λ2 ) ( ∈ R2 ) belong to Ξ1 ×Ξ2 ( ∈ B2 ) is given by ρ((G×F )(Ξ1 ×Ξ2 )).
Thus, the variance of |λ1 − λ2 | is given byZZ¡¢|λ1 − λ2 |2 ρ (G × F )(dλ1 dλ2 )(11.2)

R2

Here we have the following deﬁnition.
Deﬁnition 11.1. [Error (or precisely, Measurement error), cf. [44]].

Assume the above

notations. And assume the situation that we hope to approximate Q ( ≡ (R, B, G)) by275
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O (≡ (R, B, F )), that is, O is the approximation of Q. Then the measurement error,³´∆ MN (Q × O, S(ρ)) , is deﬁned by´ hZ Z∆ MN (Q × O, S(ρ)) =³¢i1/2|λ1 − λ2 | ρ (G × F )(dλ1 dλ2 ).
2¡(11.3)

R2

This is also called the distance between Q and O concerning ρ (or, the error of O for Q
concerning ρ).
¥
It should be noted that every measurement is exact. Thus the above deﬁnition is basedon the following assumption:(♯) We want to take a measurement MN (Q, S(ρ)). But it is impossible for some reason.
Thus, instead of the MN (Q, S(ρ)), we take a measurement MN (O, S(ρ)). In thissense, we regard MN (O, S(ρ)) as the approximation of MN (Q, S(ρ)).
The following examples will promote the understanding of Deﬁnition 11.1.
Example 11.2. [(i): Gaussian observables]. Consider the exact observable OEXA ≡(R, BR , χ(·) ) and Gaussian observable OG ≡ (R, BR , Gσ ) in N ≡ L∞ (R, dµ) such that:Z(x−µ)21σ(∀µ ∈ R ∀Ξ ∈ BR ),(11.4)[G (Ξ)](µ) = √e− 2σ2 dx
2πσ 2 Ξ(where σ 2 is a variance). Then we see, for each density function ρ ( ∈ L1+1 (R, dµ)),´ hZ Z
³
¡
¢i1/2
σ
2
∆ MN (OEXA × OG , S(ρ)) =|λ1 − λ2 | ρ (G × G )(dλ1 dλ2 )
R2
Z
Z
Z
h
´i1/2
³
(x−µ)2
1
−
2
2
2σ
√
=
e
dx × ρ(µ)dµ|λ1 − λ2 |χdλ1 (µ)
2πσ 2 dλ2R2
R
=σ,(11.5)
which is independent of ρ.
[(ii): Triangle observable, cf. Example 2.19]. Let ϵ be any positive number. Deﬁne themembership function (i.e., triangle function) Zϵ : R → R such that: 1 − ωϵ 0 ≤ ω ≤ ϵω
+ 1 −ϵ ≤ ω ≤ 0Zϵ (ω) =(11.6) ϵ0otherwise .
©
ªϵPut Zϵ ≡ ϵk : k ∈ Z ≡ {0, ±1, ±2, ...} . Deﬁne the W ∗ -observable OT ≡ (R, BR , T(·) )Pϵin the commutative W ∗ -algebra L∞ (R, dω) such that TΞ (ω) = x∈Ξ∩Zϵ Zϵ (ω −x) (∀Ξ ∈Sample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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BR , ∀ω ∈ R). This W ∗ -observable OT is called a triangle observable in L∞ (R, dω).
Consider the exact observable OEXA ≡ (R, BR , χ(·) ) and the triangle observable OT ≡(R, BR , T(·) ) in N ≡ L∞ (R, dω). Then we see, for each density function ρ ( ∈ L1+1 (R, dω)),³´hZ
i1/2 ϵ∆ MN (OEXA × OT , S(ρ)) = ϵ(ω − [ω]G )(1 − [ω]G + ω)ρ(ω)dω≤2Rϵwhere [ω]G is the integer such that [ω]G ≤ ω < [ω]G + 1.

¥

Example 11.3. [Self-adjoint operators]. Let A1 and A2 be commutative self-adjoint
operators on a Hilbert space H. For each i ( = 1, 2), consider the crisp observable Oi ≡R
(R, BR , EAi ) in B(H) which is the spectral measure of Ai , i.e., Ai = R λEAi (dλ). Then,
we see that³
´ hZ
∆ MB(H) (O1 × O2 , S(|u⟩⟨u|)) =D
Ei1/2|λ1 − λ2 |2 u, EA1 (dλ1 )EA2 (dλ2 )uR2= ∥(A1 − A2 )u∥2 .

(11.7)¥

11.2

The estimation under loss function in statistics

Let Q ≡ (R, B, G) and O ≡ (X, F, F ) be a quantity (i.e., a crisp observable on R)
and a W ∗ -observable in a W ∗ -algebra N respectively. Consider the measurable map
h : X → R, and the image observable O[h] ≡ (R, B, F (h−1 ( · )) ) in N. This measurable
map h : X → R is called a statistic. Also assume that Q and O[h] commute. Thus, thedistance between Q and Oh (concerning ρ ∈ Sn (N∗ )) is deﬁned by ∆(MN (Q×Oh , S(ρ)) )as in the above deﬁnition.
Now we have the following problem:
Problem 11.4. [The estimation under loss function in statistics]. Assume the above
notations. Then our present problem is as follows:(♯) how to choose a proper image observable O[h] (i.e., O (≡ (X, F, F )) and h : X → R)
as the approximation of a quantity Q (≡ (R, B, G)).

Our interest is concentrated on the problem (♯), which is regarded as a kind of “inference”.
Note that this (♯) is entirely diﬀerent from Fisher’s spirit in Chapter 5, that is, how toinfer the unknown state from the measured data obtained by a measurement.

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Of course, it is desirable that O and h in the above (♯) satisfy the following (A1 ) and(A2 ).
(A1 ) (unbias condition). There exists a dense set D ( ∈ Sn (N∗ )) such that:ZZ
λ N∗ ⟨ρ, G(dλ)⟩N =λ N∗ ⟨ρ, F (h−1 (dλ))⟩N(∀ρ ∈ D)R

R

(A2 ) ∆(MN (Q × O[h] , S(ρ)) ) is small (where Q and O[h] commute ).
In what follows, we shall study Problem 11.4 in Example 11.5 and Problem 11.6.
Example 11.5. [Heisenberg’s uncertainty relation, cf. [31], [36], Chapter 12]. Let A1
and A2 be a position quantity and a momentum quantity respectively (i.e. A1 and A2
are self-adjoint operators on a Hilbert space H satisfying that A1 A2 − A2 A1 = i~, ~ is“Plank constant” /(2π)). As mentioned before, we identify Ai with the spectral measureR
Ai ≡ (R, B, Gi ) in B(H), i.e., Ai = R λGi (dλ). Since A1 and A2 do not commute, theproduct observable does not exist. Therefore, consider an observable O ≡ (X, F, F ) in
B(H) and measurable maps hi : X → R, (i = 1, 2), and deﬁne the image observables
O[hi ] ≡ (R, B, F (h−1i ( · )) ≡ Fi ( · )) in B(H). And furthermore, assume the conditions:(i) There exists a set D ( ⊂ H) such that D (≡“closure on D)= {u ∈ H | ∥u∥ = 1}R
and it holds that ⟨u, Ai u⟩H = R λ⟨u, Fi (dλ))u⟩H (∀u ∈ D, i = 1, 2(ii) Qi and O[hi ] commute (i = 1, 2).
Then we get the following inequality:³´³´∆ MB(H) (Q1 × O[h1 ] , S(ρ)) · ∆ MB(H) (Q2 × O[h2 ] , S(ρ)) ≥ ~/2

for all ρ ∈ Tr+1 (H).
(11.8)

This is just Heisenberg’s uncertainty relation, of which non-mathematical representation
was proposed by W. Heisenberg in the famous thought experiment of γ-rays microscope(cf. [31]). This will be discussed in Chapter 12.
¥
The following problem is a main part of this section. The reader should ﬁnd “estimation under loss function in statistics” in the following problem.
Problem 11.6. [= Example 5.9 (Urn problem)]. Let Uj , j = 1, 2, 3, be urns that containsuﬃciently many colored balls as follows:

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urn U1
urn U2
urn U3

blue balls
60%
40%
20%

green balls
20%
20%
20%

red balls
10%
30%
40%

279

yellow balls
10%
10%
20%

Put U = {U1 , U2 , U3 }. By the same argument in Example 5.9, we consider the state space¡¢Ω ≡ {ω1 , ω2 , ω3 } with the discrete topology, which is identiﬁed with U, that is, U ∋ Uj
↔ ωj ∈ Ω ≈ Mp+1 (Ω).
U1 ≈ ω1U2 ≈ ω2

U3 ≈ ω3B B B G R

B B G R R

B G R R Y

B B B G Y

B B G R Y

B G R R Y

Let Q be a quantity in C(Ω), i.e., Q : Ω (≈ Mp+1 (Ω)) → R is a real valued continuousfunction on Ω. For example we may consider in what follows. Assume that the weight of a³
blue ball is given by 10 (gram), and green 20, red 30 and yellow 10. Thus, we can deﬁne´
the map W : X → R such that W (b) = 10, W (g) = 20, W (r) = 30 and W (y) = 10.
Therefore, we can deﬁne the quantity Q : Ω → [0, 50] such that the average weight Q(ω1 ) of
the balls in the urn U1 is given by 14 (= (10·60+20·20+30·10+10·10)/100), and similarly,
Q(ω2 ) = 18 and Q(ω3 ) = 20. Deﬁne the observable O ≡ (X = {b, g, r, y, }, 2X , F(·) ) inC(Ω) by the usual way. That is,
F{b} (ω1 ) = 6/10

F{g} (ω1 ) = 2/10

F{r} (ω1 ) = 1/10

F{y} (ω1 ) = 1/10

F{b} (ω2 ) = 4/10

F{g} (ω2 ) = 2/10

F{r} (ω2 ) = 3/10

F{y} (ω2 ) = 1/10

F{b} (ω3 ) = 2/10

F{g} (ω3 ) = 2/10

F{r} (ω3 ) = 4/10

F{y} (ω3 ) = 2/10.

Now consider the iterated measurement MC(Ω) ( ×2k=1 O ≡ (X 2 , 2X , ×2k=1 F ), S[∗] ) where2( ×2k=1 F )Ξ1 ×Ξ2 (ω) = FΞ1 (ω) · FΞ2 (ω). Also, assume that• the measured value (b, r) is obtained by the simultaneous measurement MC(Ω) (×2k=1
O, S[∗] ).
Now we have the following problem.

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CHAPTER 11. MEASUREMENT ERRORFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(♯) How do we infer Q(∗) from the measured value (b, r) obtained by the simultaneous
measurement MC(Ω) ( ×2k=1 O, S[∗] ) ?
¥In what follows, we provide four answers to the above problem.
Answer 1. [Fisher’s method, cf. [44]]. Recall “[II]” in Example 5.8, in which we infer,
by Fisher’s method, that the unknown urn is U2 . That is, applying Fisher’s method (cf.
Corollary 5.6), we get the conclusion as follows: Put E(ω) = F{b} (ω)F{r} (ω). Clearly it
holds that E(ω1 ) = 6·1/102 = 0.06, E(ω2 ) = 4·3/102 = 0.12 and E(ω3 ) = 4·2/102 = 0.08.
Therefore, there is a very reason to think that [ ∗ ] = δω2 , that is, the unknown urn is
U2 . Since we inferred that [ ∗ ] = δω2 ( ↔ ω2 ) in Example 5.8(II), we can immediately
conclude that (or more precisely, Regression analysis II (6.48))
Q( ∗ ) = Q(ω2 ) = 18.
Answer 2. [Moment method] Recall “Remark” in Example 5.8, in which we infer, by the
moment method, that the unknown urn is U2 . Thus, we conclude that Q(∗) = Q(U2 ) = 18.
Answer 3. [Bayes’ method in SMTPEP ]. Next study the above problem (♯) in SMTPEP method (cf. §8.6.2, and Theorem 11.12 later). Thus, we assume that the [ ∗ ] is chosen
by a fair rule (e.g., a fair coin-tossing, a fair dice-throwing, etc.). Consider a statisticalmmmmeasurement MC(Ω) ( ×2k=1 O, S[∗] (ρm0 ) ), where we assume that ρ0 = ρuni , i.e., ρuni =P3
12
j=1 δωj on Ω. When we get the measured value (b, r) by the measurement MC(Ω) (×k=1
3

O, S[∗] (ρm0 ) ), we infer, by Bayes’ method (for example, (B1 ) in Remark 8.14, or more
precisely, Theorem 8.13), that the new state ρm
new is1(0.06 · δω1 + 0.12 · δω2 + 0.08 · δω3 )0.06 + 0.12 + 0.08
1=(6 · δω1 + 12 · δω2 + 8 · δω3 ).
6 + 12 + 8

ρm
new =

Thus there is a very reason to consider that
Q( ∗ ) is approximated byRΩ

14·6+18·12+20·8= 17.69 · · · .
Q(ω)ρmnew (dω) =6+12+8

Also, the variance σ 2 is given byσ=h (14 − 17.69)2 · 6 + (18 − 17.69)2 · 12 + (20 − 17.69)2 · 8 i1/2
6 + 12 + 8= 2.19...

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Answer 4.

[The estimation under loss function in statistics, cf. [44]]. Let MC(Ω)( ×2k=1 O, S[∗] (ρm0 )) and Q : Ω → [0, 50] be as in Problem 11.6. Put O = (X = {b, g, r, y},m
2X , F(·) ) in C(Ω) ( ≡ C({ω1 , ω2 , ω3 })) and ρm
0 is any mixed state ∈ M+1 (Ω). Consider

a measure ν on Ω, for example, ν({ωj }) = 1 (j = 1, 2, 3). Deﬁne the W ∗ -observable O
in L∞ (Ω, ν) such that O = O, and deﬁne the normal state ρ ( ∈ L1+1 (Ω, ν)) such thatR2mρm
0 (B) = B ρ(ω)ν(dω) for all B ( ⊆ Ω). Then, we can identify MC(Ω) ( ×k=1 O, S[∗] (ρ0 ))with ML∞ (Ω,ν) ( ×2k=1 O, S(ρ)). Note that Q is equivalent to the crisp observable Q ≡(R, B, GQ ) in L∞ (Ω, ν) such that GQ
Ξ (ω) = χ{ω′ ∈Ω:Q(ω′ )∈Ξ} (ω) for all Ξ ∈ B and all ω ∈ Ω.
Deﬁne the map h : X 2 → R such that:h(x1 , x2 ) =´
1³W (x1 ) + W (x2 )2(∀(x1 , x2 ) ∈ X 2 ≡ {b, g, r, y}2 )(11.9)

where W (b) = 10, W (g) = 20, W (r) = 30 and W (y) = 10. Consider the image observable³´(×2 O)h ≡ (R, B, Fb = (×2 F )h−1 (·) ). Then, ∆ ML∞ (Ω,ν) (Q × (×2 O)h , S(ρ)) , the
k=1

k=1

k=1

distance between Q and (×2k=1 O)h concerning ρ, is calculated as³∆=

ML∞ (Ω,ν) (Q × (×2k=1 O)h , S(ρ))

3
hX

X

hZ Z

´

i1/2
|λ1 − λ2 |2 ρ((GQ × Fb)(dλ1 dλ2 ))

=
R2

i1/2
ρ(ωj )|Q(ωj ) − h(x1 , x2 )|2 F{x1 } (ωj )F{x2 } (ωj )

j=1 (x1 ,x2 )∈X 2

hi1/2= 22ρ(ω1 ) + 38ρ(ω2 ) + 38ρ(ω3 ).
Therefore, we see that (11.10) ≤√(11.10)38 ≈ 6.17 for all ρ ∈ L1+1 (Ω, ν). Now we can also

answer the problem (♯) in Problem 11.6. That is, we see,Q( ∗ ) = 21 (W (r) + W (b)) = (30 + 10)/2 = 20,though it of course includes the error 6.17.
¥
The map h : X n → R, (n = 2), in (11.9) may be chosen by the hint of “the law of large
numbers”. That is, if n is suﬃciently large, the map h : X n → R (deﬁned by h(x1 , ..., xn )³´
P
= n1 nk=1 W (xk )) has a proper property, i.e., limn→∞ ∆ ML∞ (Ω,ν) (Q × (×nk=1 O)h , S(ρ))= 0 for all ρ ∈ L1+1 (Ω, ν). However, there are several ideas for the choice of h.
Deﬁnition 11.7. [Admissible]. Let Q ≡ (R, B, G) and O ≡ (X, F, F ) be a quantity andW ∗ -observable in a W ∗ -algebra N respectively. For each i = 1, 2, consider a measurable

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map hi : X → R, and the image observable Ohi ≡ (R, B, F (h−1i ( · )) ) in N. Also assume
that Q and Ohi commute.
(i) When it holds that∆(MN (Q × Oh1 , S(ρ)) ) ≤ ∆(MN (Q × Oh2 , S(ρ)) ) ∀ρ ∈ Sn (N∗ ),(11.11)

we say that Oh1 is better than Oh2 as the approximation of Q.
(ii) Also, Oh2 is called admissible as the approximation of Q, if there exists no h1 thatsatisﬁes (11.11) and the following condition:∆(MN (Q × Oh1 , S(ρ0 )) ) < ∆(MN (Q × Oh2 , S(ρ0 )) )for some ρ0 ∈ Sn (N∗ ).
(11.12)¥

As a well known result concerning “admissibility”, we mention the following example.
Example 11.8. [Gaussian observable and admissibility]. Let O ≡ (R, BR , Gσ ) be theGaussian observable in N ≡ L∞ (R, dµ), that is,Z(u−µ)21σGΞ (µ) = √e− 2σ2 du
2πσ 2 Ξ(∀µ, ∈ R, ∀Ξ ∈ BR ).

(11.13)

Consider the quantity Q : R → R such that Q(µ) = µ (∀µ ∈ R), which is identiﬁed withQ
the observable Q ≡ (R, BR , F(·)) where FΞQ (µ) = χΞ (µ). Consider the product observable

×nk=1 O ≡ (Rn , BRn , ×nk=1 Gσ ) in L∞ (R, dµ). Deﬁne the map h : Rn → R such thathn∈ R. Then, it is well known (cf. [86]) that ( ×nk=1 O)h is
Rn ∋ (λ1 , ..., λn ) 7→ λ1 +···+λnadmissible as the approximation of Q.
¥

11.3

Random observable

Recall the probabilistic measurement MC(Ω) (O, S[∗] ([δω1 ; p] ⊕ [δω2 ; 1 − p])) in Example
8.1 (8.8). Here, the symbol [δω1 ; p] ⊕ [δω2 ; 1 − p] is called a “probabilistic state”.

The

concept of “probabilistic state” urges us to propose the “random observable” as follows:
For simplicity, in this section we devote ourselves to the classical case (i.e., C(Ω) andL∞ (Ω, µ)).
Let O1 ≡ (X, F, F1 ), O2 ≡ (X, F, F2 ), · · · , ON ≡ (X, F, FN ) be observables in C(Ω).
In a similar way in the procedures (P1 ) and (P2 ) of Example 8.1, deﬁne the “random

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observable” ⊕N
n=1 [On ; pn ], wherePN

n=1 pn = 1 (0 ≤ pn ≤ 1 (n = 1, 2, ..., N )).

That is, we

assume that:•

To take a measurement MC(Ω) (⊕N
n=1 [On ; pn ], S[δω ] ).
´“random measurement”.

³
This measurement is called a⇐⇒• To take one of {MC(Ω) (On , S[δω ] ) | n = 1, 2, ..., N } according to the probabilistic rule
(p1 , p2 , ..., pN ). That is, to take the measurement MC(Ω) (On , S[δω ] ) with probability
pn .
Here, it should be noted thatb
• the statistical property of MC(Ω) (⊕N
n=1 [On ; pn ], S[δω ] ) is equal to that of MC(Ω) (O,b ≡ (X.F, Fb) is deﬁned by Fb(Ξ) = PN pn Fn (Ξ). That is, for eachS[δω ] ), where O
n=1

Ξ( ∈ F) and ω ( ∈ Ω),“the probability that a measured value obtained by MC(Ω) (⊕N
n=1 [On ; pn ], S[δω ] )belongs to Ξ”=

NX

pn [F (Ξ)](ω)(11.14)n=1

b S[δω ] ) belongs= “the probability that a measured value obtained by MC(Ω) (O,to Ξ ”,
which is easily seen by a similar argument such as stated in Example 8.1.
Again note that(1) to take a random measurement MC(Ω) (⊕N
n=1 [On ; pn ], S[δω ] )(11.15)⇐⇒to take a measurement MC(Ω) (On , S[δω ] ) with probability pn (n = 1, 2, ..., N ).
(2) to take a probabilistic measurement MC(Ω) (O, S[∗] ( ⊕N
n=1 [δωn ; pn ]))(11.16)⇐⇒
to take a measurement MC(Ω) (O, S[δωn ] ) with probability pn (n = 1, 2, ..., N ).

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In the case that N = ∞. it suﬃces to prepare a probability space (Λ, F(Λ), ν). And,for each λ( ∈ Λ), consider an observable Oλ ( ≡ (X, F, Fλ )) in C(Ω). Then, the random³´
H
H
observable ⊕N
[O
;
p
]
is
generalized
as
O
ν(dλ)
≡
(X,
F,
F
ν(dλ))
.
n n
λ
n=1
Λ
Λ λ
The following example is typical (though the description is due to the W ∗ -algebraic
formulation).
Example 11.9. [Gaussian observable as a random observable]. For each λ( ∈ R( ≡ Λ)),consider an observable Oλ ( ≡ (R( ≡ X), BR , Eλ )) in L∞ (R( ≡ Ω), dω) such that
[Fλ (Ξ)](ω) = χΞ (ω − λ) (∀Ξ ∈ BR ( ⊆ 2X ), ∀ω ∈ R( ≡ Ω), ∀λ ∈ R( ≡ Λ)).
Deﬁne the probability space (R( ≡ Λ), BR , ν) such that:Z
λ2
1
ν(S) = √e− 2σ2 dλ(∀S ∈ BR ).
2πσ 2 SThus, we have the random observable:II³´
Oλ ν(dλ) ≡ (R( ≡ X), BR ( ≡ F), Fλ ν(dλ))Λ(11.17)(11.18)Λwhich the probabilistic form of the Gaussian observable (R( ≡ X), BR ( ≡ F), Gσ ) in
L∞ (R( ≡ Ω), dω) such that:
Z
Z
λ2
1
σ
[G (Ξ)](ω) = [Fλ (Ξ)](ω)ν(dλ) =χΞ (ω − λ) √e− 2σ2 dλ
2πσ 2
Λ
Λ
Z
2
(x−ω)
1
(∀ω ∈ R( ≡ Ω) ∀Ξ ∈ BR ( ⊆ 2X )),=√e− 2σ2 dx
2
2πσ Ξ(11.19)(Cf. Example 11.8.)¥
Although the following problem is easy, its measurement theoretical answer is quite
important.
Problem 11.10. [Which hand is the coin under?]. The following problems (P1 ) and (P2 )are essentially the same.
(P1 ) A coin is, intentionally or unintentionally, put under my right hand or my left hand.
Suppose that you do not know which hand the coin is under, and you choose oneof my hands which you guess that the coin is under. Is it reasonable to believe thatthe probability that the ball is under the hand you choose is equal to 1/2. How doyou think about it?

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my right hand

my left hand
Coin

?9

z

?
Table

(P2 ) There are three boxes (i.e., Box 1, Box 2 and Box 3) and a ball. A ball is, intentionally or unintentionally, put in one box (i.e., Box 1 or Box 2 or Box 3). Suppose thatyou do not know which box contains the ball, and you choose one of three boxes
which you guess the ball is in. In this case, it is often believed that the probability
that the ball is in Box 1 [resp. in Box 2; in Box 3] is 1/3 [resp. 1/3; 1/3]. How doyou think about it?

Ball

?

?

?

Box 1

Box 2

Box 3

•[The experimental answer to Problem (P1 )]. We can easily say “Yes”, that is,(A1 ) the probability that the ball is under the hand you choose is equal to 1/2.
In fact, it can be easily tested experimentally. For example, it suﬃces to ask to 1000
persons “Which hand is the coin under?”.

About 500 persons will say “Right hand”,

and the other persons will say “Left hand”.

In either case, about 500 persons’ guess ishit. Thus the above (A1 ) is true. Although this (P1 ) is the easiest problem throughout
this book, what I want to say is the measurement theoretical answer mentioned in what
follows.
•[The measurement theoretical answer to Problem (P2 )]. Since the two (P1 ) and (P2 ) areessentially the same, it suﬃces to answer Problem (P2 ) from the measurement theoreticalpoint of view. When the conclusion is said ﬁrst, we can say that:(A2 ) the probability that the ball is in your chosen box is equal to 1/3.

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In what follows we shall explain it. Put Ω = {ω1 , ω2 , ω3 }, where ω1 [resp. ω2 , ω3 ] means
the state that the ball is in Box 1 [resp. Box 2; Box 3]. First we consider the case ω1 ,that is, the ball is in Box 1.
[(i): The case ω1 , that is, the ball is in Box 1].

Deﬁne three observables Oe1 ( =({0, 1}, 2{0,1} , F1e )), Oe2 ( = ({0, 1}, 2{0,1} , F2e )), Oe2 ( = ({0, 1}, 2{0,1} , F3e )) such that:[F1e ({0})](ω1 ) = 0,[F1e ({0})](ω2 ) = 1,[F1e ({0})](ω3 ) = 1,[F1e ({1})](ω1 ) = 1,[F1e ({1})](ω2 ) = 0,[F1e ({1})](ω3 ) = 0,[F2e ({0})](ω1 ) = 1,[F2e ({0})](ω2 ) = 0,[F2e ({0})](ω3 ) = 1,[F2e ({1})](ω1 ) = 0,[F2e ({1})](ω2 ) = 1,[F2e ({1})](ω3 ) = 0,[F3e ({0})](ω1 ) = 1,[F3e ({0})](ω2 ) = 1,[F3e ({0})](ω3 ) = 0,[F3e ({1})](ω1 ) = 0,[F3e ({1})](ω2 ) = 0,[F3e ({1})](ω3 ) = 1.

(11.20)(11.21)(11.22)

Note that we identify the following (S11 ) and (S21 ):(S11 ) We take a measurement MC(Ω) (Oe1 , S[δω1 ] ). And we obtain a measured value 1. (Or,
we obtain a measured value 0.)(S21 ) We open Box 1. And we ﬁnd the ball. (Or, we do not ﬁnd the ball.)(11.23)(11.24)

Similarly, we see the following identiﬁcation:(S123 ) We take a measurement MC(Ω) (Oe2 , S[δω1 ] ) [resp. MC(Ω) (Oe3 , S[δω1 ] )]. And we obtaina measured value 1. (Or, we obtain a measured value 0.)(S223 ) We open Box 2. [resp. Box 3.]. And we ﬁnd the ball. (Or, we do not ﬁnd the ball.)Since “the state ω1 ” = “the case that the ball is in Box 1”, we can assume that• the measured value obtained by MC(Ω) (Oe1 , S[δω1 ] ) [resp. MC(Ω) (Oe2 , S[δω1 ] ); MC(Ω) (Oe3 ,S[δω1 ] )] is 1 [resp. 0; 0].
Since you have no information about the [∗], your choice is the same as the choice by afair coin-tossing. That is, we assume that“decision without having information” ⇐⇒ “decision by a fair coin-tossing”,

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which is the fundamental spirit of “the principle of equal probability” in the following
section. Thus, it is reasonable to consider thatthe probability that Box 1 is opened = the probability that Box 2 is opened=the probability that Box 3 is opened = 1/3.

(11.26)

Therefore, we see that(a) the probability that the measured value obtained by MC(Ω) ( ⊕3k=1 [Oek ; 1/3], S[δω1 ] )is 1 [resp. 0] is given by 1/3 [resp. 2/3].
[(ii): The case ω2 , that is, the ball is in Box 2]. Similarly we see that(b) the probability that the measured value obtained by the “measurement”
MC(Ω) ( ⊕3k=1 [Oek ; 1/3], S[δω2 ] ) is 1 [resp. 0] is given by 1/3. [resp. 2/3].
[(iii): The case ω3 , that is, the ball is in Box 3]. Similarly we see that(c) the probability that the measured value obtained by the “measurement”
MC(Ω) ( ⊕3k=1 [Oek ; 1/3], S[δω3 ] ) is 1 [resp. 0] is given by 1/3. [resp. 2/3].
[(iv): The case that we do not know which box contains the ball]. By the above (a), (b)and (c), we see that• the probability that the measured value obtained by the “measurement”
MC(Ω) ( ⊕3k=1 [Oek ; 1/3], S[∗] ) is 1 [resp. 0] is given by 1/3 [resp. 2/3].
Note that “measured value 1 is obtained” ⇔ “open the box that contains the ball”. Thus,
we can believe that the probability that the ball is in Box 1 [resp. in Box 2; in Box 3] is
1/3 [resp. 1/3 ; 1/3].
[Remark]. Recall BMT (in §8.6). Then, the system in Problem (P2 ) is clearly represented
by S[∗] ((νu ))bw , cf. §8.6.1. Here, νu ({ωk }) = 1/3 (k = 1, 2, 3). However, in the above
argument, we conclude that the “probability” that the ball is in Box 1 [resp. in Box 2; inBox 3] is 1/3 [resp. 1/3; 1/3]. Therefore, we have the following question:• Is the system represented by S[∗] (νu ) (as well as S[∗] ((νu ))bw )?
This will be discussed in the following section.

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11.4

The principle of equal probability

Consider a measurement MC(Ω) (O ≡ (X, F, F ), S[∗] ), where Ω is ﬁnite, i.e., Ω ≡{ω1 , ω2 , ..., ωN }. There may be several deﬁnitions of “Having no information about the[∗]”.

As mentioned in §8.6, in this book we introduce three deﬁnitions of “Having no

information about the [∗]” such as: (a). iterative likelihood function method in §5.6,(b). SMTPEP in SMT in this section and §11.4,
(c). BMT in §8.6.
We want to change S[∗] ((νu ))bw (belief weight) to S[∗] (νu ) (statistical state). This will bedone according to the spirit (11.25), that is,“decision without having information” ⇐⇒ “decision by a fair coin-tossing”,
which assures that the principle of equal probability holds. This is the purpose of thissection.
Let Ω be a ﬁnite set, i.e., Ω = {ω1 , ω2 , ..., ωN }. A map ϕ : Ω → Ω is said to be ergodic,
if it is a bijection and if it holds that Ω = {ϕn (ω) | n = 0, 1, ..., N − 1} for any ω ( ∈ Ω).
Also, a homomorphism Φ : C(Ω) → C(Ω) is said to be ergodic, if there exists an ergodic
bijection ϕ : Ω → Ω such that(Φf )(ω) = f (ϕ(ω)) (∀f ∈ C(Ω), ∀ω ∈ Ω).

(11.27)

Theorem 11.12. [The principle of equal probability (=“PEP”), SMTPEP method].
Consider a measurement MC(Ω) (O ≡ (X, F, F ), S[∗] ), where Ω is ﬁnite, i.e., Ω ≡ {ω1 , ω2 , ...,ωN }.
−1n
And consider the measurement MC(Ω) ( ⊕N
n=0 [Φ O; 1/N ], S[∗] ) (where Φ : C(Ω) → C(Ω)

is ergodic), which is called an unintentional random measurement.1 Then we seenN−1MC(Ω) ( ⊕N
n=0 [Φ O; 1/N ], S[∗] ) ⇐==⇒ MC(Ω) (O, S[∗] ( ⊕n=1 [δωn ; 1/N ]))identiﬁcation(11.28)andprobabilistic form←−−−−−−−−−−
MC(Ω) (O, S[∗] ( ⊕N
n=1 [δωn ; 1/N ])) −−−−−−−−−−−→ MC(Ω) (O, S[∗] (νu ))statistical form

1

(11.29)(=(8.9))Also, it is called a “completely random measurement”, “coin-tossing measurement”, “no information
measurement”.

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where νu = N1

PN

n=1 δωn . That is, we can assert that:−1nMC(Ω) ( ⊕N
n=0 [Φ O; 1/N ], S[∗] ) ⇐==⇒ MC(Ω) (O, S[∗] (νu )).
identiﬁcation(11.30)

Proof. Let ω ∈ Ω. Then we see that:N −1
to take an unintentional random measurement MC(Ω) ( ⊕n=0
[Φn O; 1/N ], S[δω ] )⇐⇒to take a measurement MC(Ω) (Φn O, S[δω ] )with probability 1/N , (n = 1, 2, ..., N )⇐⇒to take a measurement MC(Ω) (O, S[δϕn (ω) ] ) with probability 1/N(n = 0, 1, 2, ..., N − 1)⇐⇒(Note that Ω = {ϕn (ω) | n = 0, 1, ..., N − 1}.)

to take a measurement MC(Ω) (O, S[δωn ] ) with probability 1/N , (n = 1, 2, ..., N )⇐⇒−1
to take a probabilistic measurement MC(Ω) (O, S[∗] ( ⊕N
n=0 [δωn ; 1/N ]))⇐⇒to take a measurement MC(Ω) (O, S[∗] (νu )).
Thus we see that:−1nMC(Ω) ( ⊕N
n=0 [Φ O; 1/N ], S[∗] ) ⇐==⇒ MC(Ω) (O, S[∗] (νu )).
identiﬁcation(11.31)

Problem 11.13. [Monty Hall problem, cf.[33]].
The Monty Hall problem is as follows (cf. Problem 5.12, Remark 5.13, Problem 8.8) :(P) Suppose you are on a game show, and you are given the choice of three doors (i.e.,“number 1”, “number 2”, “number 3” ). Behind one door is a car, behind theothers, goats.
(C) The host knows the fact that the probability that the car was set behind thek-th door (i.e., “number k” ) is given by pk (k = 1, 2, 3), for example, p1 = 3/7,
p2 = 1/7, p3 = 3/7. But you do not know this fact.

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You pick a door (strictly speaking, you pick a door at random), say number 1, andthe host, who knows what’s behind the doors, opens another door, say “number 3”,
which has a goat. He says to you, “Do you want to pick door number 2?” Is it toyour advantage to switch your choice of doors?
?

?

Door
Number 1?

Door
Number 2

Door
Number 3[Answer]. Put Ω = {ω1 , ω2 , ω3 }, where ω1 [resp. ω2 , ω3 ] means the state that the car
is behind the door number 1 [resp. the door number 2, the door number 3]. Deﬁne theobservable O ≡ ({1, 2, 3}, 2{1,2,3} , F ) in C(Ω) such that
[F ({1})](ω1 ) = 0.0,

[F ({2})](ω1 ) = 0.5,

[F ({3})](ω1 ) = 0.5, 2[F ({1})](ω2 ) = 0.0,

[F ({2})](ω2 ) = 0.0,

[F ({3})](ω2 ) = 1.0,

[F ({1})](ω3 ) = 0.0,

[F ({2})](ω3 ) = 1.0,

[F ({3})](ω3 ) = 0.0,

(11.32)Thus, we have the unintentional random measurement MC(Ω) (⊕2n=0 [Φn O; 1/3], S[∗] ) (whereΦ : C(Ω) → C(Ω) is ergodic). Theorem 11.12 says that
MC(Ω) ( ⊕2n=0 [Φn O; 1/3], S[∗] ) ⇐⇒ MC(Ω) (O, S[∗] (νu ))(11.33)where νu ({ω1 }) = νu ({ω2 }) = νu ({ω3 }) = 1/3. Thus, it suﬃces to consider the statisticalmeasurement MC(Ω) (O, S(νu )). Here, note that
• By the statistical measurement MC(Ω) (O, S[∗] (νu )), you obtain a measured value 3,
which corresponds to the fact that the host said “Door (number 3) has a goat”. Then,the posttest state νpost ( ∈ Mm
+1 (Ω)) is given by
F ({3}) × νu®.
νpost = ­νu , F ({3})
2(11.34)

Strictly speaking, F ({1})(ω1 ) = 0.5 and F ({2})(ω1 ) = 0.5 should be assumed in the problem (P)Sample PDF File (Low Resolution Printing)Mathematical Foundations of Measurement Theory © 2006 Shiro Ishikawa, Keio University Press Inc.

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291

That is,νpost ({ω1 }) = 1/3,

νpost ({ω2 }) = 2/3,

νpost ({ω3 }) = 0,

(11.35)and thus, you should pick door number 2.
¥
Remark 11.14. [ Four answers to Monty Hall problem]. In this book four answers to theMonty Hall problem are presented in Problem 5.12, Remark 5.13, Problem 8.8, Problem
11.13. However, I believe that the Monty Hall problem in Problem 11.13 is the mostnatural.
¥Problem 11.15. [The problem of three prisoners, cf. Problem 8.10 and Remark 8.11].
Consider the following problem:(P) Three men, A, B, and C were in jail. A knew that one of them was to be set free
and the other two were to be executed. But he did not know who was the one tobe spared. To the jailer who did know, A said, “Since two out of the three will
bee executed, it is certain that either B or C will be, at least. You will give me
no information about my own chances if you give me the name of one man, B or
C, who is going to be executed.” Accepting this argument after some thinking, thejailer said, “C will be executed.” Thereupon A felt happier because now either heor C would go free, so his chance had increased from 1/3 to 1/2. This prisoner’s
happiness may or may not be reasonable. What do you think?

W

J“C will be executed”
-

- A

B

C

(Q) (Continued from the above (P)). There is a woman, who was proposed to by thethree prisoners A, B and C. She listened to the conversation between A and thejailer. Thus, assume that she has the same information as A has. Then, we havethe following problem:

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CHAPTER 11. MEASUREMENT ERRORFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/(♯) Whose proposal should she accept?

[Answer to (P)]. Let Ω (≡ {ωa , ωb , ωc }) and O ≡ (X ≡ {xA , xB , xC }, 2{xA ,xB ,xC } , F ) beas in Problem 8.10. Since A has no information, the unintentional random measurement MC(Ω) (⊕2k=0 [Φk O; 1/3], S[∗] (ν0 )) (where Φ : C(Ω) → C(Ω) is ergodic) is considered.
Theorem 11.12 asserts the following identiﬁcation:MC(Ω) (⊕2k=0 [Φk O; 1/3], S[∗] ) ⇐==⇒ MC(Ω) (O, S[∗] (ν0 ))identiﬁcation(11.36)

where ν0 ( ∈ Mm
+1 (Ω)) is deﬁned byν0 ({ωa }) = 1/3,

ν0 ({ωb }) = 1/3,

ν0 ({ωc }) = 1/3.

(11.37)

Thus, we can assume that the (P) in the above is the same as the (P) in Problem 8.10.
Therefore, we get that
νpost ({ωa }) =

ν0 ({ωa })2= 1/3,
ν0 ({ωa })+ ν0 ({ωb })2

ν0 ({ωb })νpost ({ωb }) = ν0 ({ωa })= 2/3,+ ν0 ({ωb })2

νpost ({ωc }) = 0.

(11.38)Therefore, we conclude that• the prisoner’s happiness is not reasonable. That is because ν0 ({ωa }) = 1/3 =νpost ({ωa }).
[Answer to (Q)]. In the above (11.38), we see that
νpost ({ωa }) = 1/3,

νpost ({ωb }) = 2/3,

νpost ({ωc }) = 0.

(11.39)νpost ({ωc }) = 0 < νpost ({ωa }) = 1/3 < νpost ({ωb }) = 2/3.

(11.40)Thus, we conclude that• she should choose the prisoner B. That is because¥

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Chapter 12
Heisenberg’s uncertainty relation
Quantum mechanics is surely one of the most successful theories in all science. In fact, most of
the Nobel prizes of physics and chemistry are due to quantum mechanics. Also, as recent topics(particularly, related to measurements), we see quantum computer [80], quantum cryptography[91], quantum teleportation [10], etc. Although these are quite interesting and promising, in thischapter, we devote ourselves to Heisenberg’s uncertainty relation, which is the most fundamental
in quantum mechanics.
Heisenberg’s uncertainty relation (cf. [31]).
(i) The particle position q and momentum p can be measured “simultaneously”, if the “errors”∆(q) and ∆(p) in determining the particle position and momentum are permitted to be
non-zero.
(ii) Moreover, for any ϵ > 0 , we can take the above “approximate simultaneous” measurementof the position q and momentum p such that ∆(q) < ϵ (or ∆(p) < ϵ ).
(iii) However, the following Heisenberg’s uncertainty relation holds:∆(q) · ∆(p) ≥~,2(12.1)

for all “approximate simultaneous” measurements of the particle position and momentum.
However, it should be noted that some ambiguous terms (i.e., “approximate simultaneous”, “error”)
are included in the above statement, Thus, we believe that it is not a scientiﬁc statement but a“catch phrase” that was used to promote the paradigm shift from classical mechanics to quantum
mechanics. Thus, in this last chapter1 we try to describe this uncertainty relation precisely interms of mathematics and further to derive it in the framework of the W ∗ -algebraic formulationof MT. For this, we ﬁrst give the mathematical deﬁnitions of “∆(q)” (or “∆(p)”) and “approximate
simultaneous measurement”, etc. in terms of MT.

1

Every result mentioned in this chapter was published in [36], which was the oldest result in our studyof “measurement theory”. That is, our research of “measurement theory” starts from the paper [36]. On
the other hand, the philosophical assertion mentioned in Chapter 1 is the latest result in our study. In
this sense, the progress of our research is symbolically summarized as“quantum” (physics) −→ “classical” (engineering) −→ “philosophical” (epistemology)(in Chapter 12)(in Chapters 2∼11)(in Chapter 1).

293
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12.1

Introduction

Although the uncertainty relation (discovered by Heisenberg in 1927) has a long history, the various discussions about its interpretations are even now continued. Mainly
there are two interpretations of uncertainty relations. One is the statistical interpretation. By repeating the exact (i.e. the “error” ∆(q) = 0) measurements of the position q ofparticles with same states, we can obtain its average value q̄ and its variance var(q). Also,
by repeating the exact (i.e. the “error” ∆(p) = 0) measurements of the momentum p ofthe same particles, we can similarly get its average value p̄ and its variance var(p). Fromthe simple mathematical deduction, we can obtain the following uncertainty relation:

1

1[var(q)] 2 · [var(p)] 2 ≥~,2(12.2)

where ~ =“Plank’s constant”/2π. This is the statistical aspect of the uncertainty relation.
The mathematical derivation of the uncertainty relation (12.2) was proposed by Kennard
in 1927 (or more generally, Robertson 1n 1929). Cf. [54, 73]. Thus, this inequality (12.2)
is called Robertson’s uncertainty relation.
On the other hand, Heisenberg’s uncertainty relation is rather individualistic. Most
physicists will agree that the content of Heisenberg’s uncertainty relation is roughly asstated in the following proposition (though it includes some ambiguous sentences as well
as some ambiguous words, i.e. “approximate simultaneous” and “error”).

Proposition 12.1. [Heisenberg’s uncertainty relation, cf. [31]].2(i) The particle position q and momentum p can be measured “approximately” and“simultaneously”, if the “errors” ∆(q) and ∆(p) in determining the particle position
and momentum are permitted to be non-zero.
(ii) Moreover, for any ϵ > 0 , we can take the “approximate simultaneous” measurementof the position q and momentum p such that ∆(q) < ϵ (or ∆(p) < ϵ ).
2

It may be usually considered that the (12.2) is the mathematical representation of the (12.3). However, it is not true. In fact, in [84], J. von Neumann pointed out the diﬀerence between Robertson’s
uncertainty relation (= (12.2)) and Heisenberg’s uncertainty relation (= (12.3)).

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295(iii) However, the following Heisenberg’s uncertainty relation holds:∆(q) · ∆(p) ≥~,2(12.3)

for all “approximate simultaneous” measurements of the particle position and momentum.
¥
It should be noted that the above “proposition (= Heisenberg’s assertion)” is ambiguous, that is, it is not a scientiﬁc statement but a “catch phrase” that was used topromote the paradigm shift from classical mechanics to quantum mechanics. In fact, theabove “proposition” is powerless to solve the paradox (i.e., the paradox between EPRexperiment and Heisenberg’s uncertainty relation), cf. §12.7.
Several authors have contributed to the problem to deduce Heisenberg’s uncertainty
relation. In [2] (Ali and Emach, 1974), [3] (Ali and Prugovec̆ki, 1977), these were done bymeans of the concept of (generalized) observable which has been developed by E.B. Davies[17] (cf. Deﬁnition 9.3 for B(V )). Hence, a certain part of this problem has been alreadysolved. In particular, the statements (i) and (ii) in the above Proposition 12.1 were deduced satisfactorily. However, concerning the statement (iii), there still seems to be somequestions. The mathematical formulation and derivation of the Heisenberg’s uncertainty
relation (iii) (in the above Proposition 12.1) was proposed by M. Ozawa [67], S. Ishikawa[36] independently. We believe that this is the ﬁnal version of Heisenberg’s uncertainty
relation concerning measurement errors. Thus, in this chapter we shall introduce thisformulation and derivation of the above Proposition 12.1.
Remark 12.2.

[(i): A classical understanding of Heisenberg’s uncertainty relation].

Let us explain the classical understanding of Heisenberg’s uncertainty relation (which isessentially equal to the thought experiment of γ-rays microscope (cf. [31])). In orderto know the position q(t0 ) and momentum p(t0 ) of a particle A at time t0 , it suﬃces tomeasure the position q(t0 ) of a particle A at time t0 (i.e., light L1 is irradiated at theparticle at time t0 ), and continuously (i.e., after δ seconds), measure the position q(t0 + δ)
at time t0 + δ. That is because (q(t0 ), p(t0 )(≡ mdq(t0 ))) is approximately calculated bydt
0 ))(q(t0 ), m(q(t0 +δ)−q(t).
δ

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q1 (t0 + δ)1

q(t0 )-q(t0 + δ)--Light L1[a]. However, if we want to know the exact position q(t0 ) (i.e., if we want ∆q ≈ 0), the“Plank constant” ×“lightspeed”
wavelength λ of the light L1 must be short (i.e., the energy (=)λ
of the light L1 must be large), and therefore, the particle A is strongly perturbed. Thus,
the position of the particle A at time t0 + δ will be changed to q1 (t0 + δ). Thus we observe
0 )), which is awaythat the momentum of the particle A at time t0 is equal to m(q1 (t0 +δ)−q(tδ
0 ))from p(t0 )(≡ mdq(t0 ) ≈ m(q(t0 +δ)−q(t) (i.e., ∆p is large).
dtδ[b]. Also, if we want to know the exact momentum p(t0 ) (i.e., if we want ∆p ≈ 0),
the wavelength λ of the light L1 must be long, and therefore, the particle A is weakly
perturbed. Although the position of the particle A at time t0 + δ will be changed toq1 (t0 + δ), it is almost the same as q(t0 + δ). Thus we observe that the momentum of
0 )), which is near p(t0 )(≡ mdq(t0 ) ≈
the particle A at time t0 is equal to m(q1 (t0 +δ)−q(tδdtm(q(t0 +δ)−q(t0 ))) (i.e., ∆p is small) if δ is large. However it should be noted that ∆q is largeδsince the wavelength λ of the light L1 is long.
[c]. Therefore, ∆p ≈ 0 and ∆q ≈ 0 are not compatible, that is, the inequality “∆p · ∆q >constant” always holds. Although this explanation is, of course, rough, there is something
thought-provoking in the above argument.
[(ii): EPR-experiment [22]]. Let A and B be particles with the same masses m. Consider
the situation described in the following ﬁgure:


-

A

B

where “the velocity of A” = −“the velocity of B”. The position qA of the particle A can bemeasured, and moreover, the velocity of vB of the particle B can be measured. Thus, we

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297

can conclude that the position and momentum of the particle A are respectively equal toqA and −mvB . Is this contradictory to Heisenberg’s uncertainty relation? This questionis signiﬁcant though their (i.e. Einstein, Podolosky and Rosen ) interest is concentrated
on “the reality of physics”.
¥

12.2

Example due to Arthurs-Kelley

Here, we mainly consider the following identiﬁcation:
L2 (R, dx) ∋ u←→(∥u∥L2 (R,dx) =1, u≈eiθ u)

identiﬁcationp|u⟩⟨u| ∈ T r+1(L2 (R, dx)).

We ﬁrst introduce Robertson’s uncertainty relation, which generally seems to be understood (or, misunderstood) as the mathematical representation of Heisenberg’s uncertainty
relation. By repeating the exact (i.e. the “error” ∆(q) = 0) measurements of the position
q of particles with same states, we can obtain its average value q̄ and its variance var(q).
Also, by repeating the exact (i.e. the “error” ∆(p) = 0) measurements of the momentum
p of the same particles, we can similarly get its average value p̄ and its variance var(p).
A simple calculation shows:
Z ¯
Z
¯2
¯2 ´
h ~d
i
³ Z ¯
¯
¯
¯
¯
u(x)
q̄ =
x¯u(x)¯ dx and p̄ =u(x) dx=p¯ũ(p)¯ dp(12.4)
idx
R
R
R
q R
³
´
~
−i~xp
where ũ is the Fourier transform of u, that is, ũ(p) =
u(x)e
dx
. And
2π R
further, we see,
Z

Z
¯
¯2
¯
¯2
¯
¯
¯
var(q) =|x − q̄| ¯u(x)¯ dx =|x|2 ¯u(x)¯ dx − q̄ 2 ,
ZR
ZR
¯
¯2
~d
¯
¯
u(x)|2 dx − p̄2 .
var(p) =|p − p̄|2 ¯ũ(p)¯ dp =
|
idx
R
R
2¯

(12.5)

Immediately after Heisenberg’s discovery (=“Proposition 12.1”, 1927), Kennard, by asimple calculation, showed the following uncertainty relation:
1

1[var(q)] 2 · [var(p)] 2 ≥~.
2(12.6)(=(12.2))(cf. Lemma 12.13 later). Of course, it is clear that there is a great gap between Heisenberg’s uncertainty relation (12.3) and Kennard’s uncertainty relation (12.6).

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Next we shall introduce the nice idea by Arthurs-Kelly [7], that is, a certain approximate simultaneous measurement of the position q and the momentum p of a particle A
in one dimensional real line R, which has a state function u(x) ( ∈ L2 (R), ∥u∥L2 (R) = 1).
~dNote that the position observable Q( ≡ x) and the momentum observable P ( ≡ idx)

do not commute, that is,³´

QP − P Q = i~ ̸= 0 .

(12.7)

Therefore, any simultaneous measurement of the position observable x and the momentum~dobservable idx
for a particle “A” can not be realized. However, Arthurs-Kelly’s idea is

excellent as follows: We ﬁrst prepare another particle “B” with the state u0 (y) such that:
Z
Z ¯
¯2
h ~d
i
¯
¯
y ¯u0 (y)¯ dy =u0 (y)u0 (y) dy = 0(12.8)
idyRR
2yfor example, u0 (y) = (π~)11/4 exp( − 2~). Further we regard these two particles “A” and“B” as a “particle C” in two dimensional Euclidean space R2 with the state u(x)u0 (y)( ∈ L2 (R2 ), ∥u · u0 ∥L2 (R2 ) = 1). Now consider the self-adjoint operators (x − y) and
~∂
~∂
+ i∂y
in L2 (R2 ), which commute, that is, it holds that:
i∂x

(

~∂
~∂
~∂
~∂
+
)(x − y) = (x − y)(+)
i∂x i∂yi∂x i∂y(12.9)

That is because we can easily calculate:~∂~∂+)(x − y)]f (x, y)i∂x i∂y
~
~∂
~∂
~∂
~
~∂
= f (x, y) + x
f (x, y) − yf (x, y) + x
f (x, y) − f (x, y) − yf (x, y)
i
i∂x
i∂x
i∂y
i
i∂y
~∂
~∂
=[(x − y)(+)]f (x, y).
i∂x i∂y[(~∂~∂Thus the simultaneous measurement of observables (x − y) and i∂x+ i∂y
for a “particle

C” (= “A” + “B” ) can be realized. Moreover, we can easily calculate these expectations
as follows:

ZZ
R2

and

Z ¯
¯2
h
i
¯
¯
u(x)u0 (y) (x − y)u(x)u0 (y) dxdy =x¯u(x)¯ dx(12.10)

R

ZZ

Z
i
i
h ~∂
h ~d
~∂
+
)u(x)u0 (y) dxdy =u(x) dx.
u(x)u0 (y) (u(x)
i∂x i∂yidx
R2
R(12.11)

By the reason that the equalities (12.10)= q̄ and (12.11)= p̄ hold, we may say that

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299(♯) An “approximate simultaneous measurement” of the position observable Q( ≡ x)~dand the momentum observable P ( ≡ idx) can be realized.

Here, the variances varasm (q) and varasm (p) in the approximate simultaneous measurement of the position q and the momentum p of a particle “C” are given respectively
by:
ZZ

¯ ZZ
¯2
h
i
¯
¯
varasm (q) =[(x − y)u(x)u0 (y) dxdy − ¯u(x)u0 (y) (x − y)u(x)u0 (y) dxdy ¯
2
R2
Z R¯
¯2 ¯2
¯2
¯Z ¯
¯2 ¯2 ¯ Z ¯
¯
¯
¯ ¯
¯
¯
¯ ¯
¯
¯
(12.12)=x¯u(x)¯ dx¯ + ¯¯yu0 (y)¯ dy ¯
¯xu(x)¯ dx − ¯
i2

R

R

R

and
Z ¯
¯2
¯Z
h ~d
i ¯2 ¯ Z
h ~d
i
¯ ~d
¯
¯
¯
¯
varasm (p) =u(x)¯ dx − ¯u(x)u(x) dx¯ + ¯u(y)u0 (y) dy.
¯
idx
idy
R idx
R
R
(12.13)Hence, we can get, by the arithmetic-geometric inequality and the well-known uncertainty relation (Robertson uncertainty relation, cf. Lemma 12.13 later), the following
simultaneous uncertainty relation;[varasm (q)]1/2 · [varasm (p)]1/2
"Z
#
" Z
#
¯
¯2
¯Z ¯
¯2 ¯2 1/4¯¯
¯2 ¯ 1/4
¯
¯
¯
¯
¯ ¯
¯
¯
¯ ¯
=2
x¯u(x)¯ dx¯× ¯¯xu(x)¯ dx − ¯¯yu0 (y)¯ dy ¯
R

"Z
×

R

R

#
" Z
#
¯ ~d
¯2
¯
¯Z
h ~d
i ¯2 1/4
h ~di ¯2 1/4
¯
¯
¯
¯
¯
¯
u(x)¯ dx − ¯u(x)u(x) dx¯× ¯
u0 (y)u0 (y) dy ¯
¯
idx
idx
idy
R
R
R

≥~.

(12.14)This is Arthurs-Kelly’s idea. We believe that Arthurs-Kelly’s discovery (12.14) is the ﬁrst
great step to the understanding of Heisenberg’s uncertainty relation.

12.3

Approximate simultaneous measurement

Since our main purpose in this chapter is to describe Proposition 12.1 in terms ofmathematics and further to prove it, we must clarify the ambiguous words (i.e., “approximate simultaneous”, “error”) in Proposition 12.1. For this, we prepare several deﬁnitionsin this section.

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According to the well-known spectral representation theorem (cf. [92]), there is abijective correspondence of a crisp observable (Rn , BRn , E) in B(H) to an n-tuple (A1 ,R..., An ) of commutative (unbounded) self-adjoint operators on H such that Ai = Rn λi
E(dλ1 ...dλn ). That is,(A1 , A2 , ..., An )(commutative self-adjoint operators on H)←→
R
Ai = Rn λi E(dλ1 ...dλn )(Rn , BRn , E)

(12.15)(crisp observable in B(V ))In particular, we frequently identify a crisp observable (R, BR , E) in B(H) with a (un³ R´
bounded) self-adjoint operator A = R λ E(dλ) on H.
³
´
W∗
W∗
Note that Proclaim 1 (9.9) or, Axiom 1 (9.11) says as follows:[♯] Let O ≡ (Rn , BRn , F ) be an observable in B(H). And consider a measurement
MB(H) (O ≡ (Rn , BRn , F ), S [ρu ] ), where ρu = |u⟩⟨u|. When we take a measurement
MB(H) (O ≡ (Rn , BRn , F ), S [ρu ] ), the probability that the measured value λ( ∈ Rn )belongs to a set Ξ ( ∈ BRn ) is given by⟨u, F (Ξ)u⟩H³´= tr[ρu F (Ξ)] .

(12.16)Ã!

ihTherefore, the expectation E MB(H) (O, S [ρu ] )

³

h

i´n
≡ E(i) MB(H) (O, S [ρu ] )of thei=1measured value obtained by the measurement MB(H) (O ≡ (Rn , BRn , F ), S [ρu ] ) is given
by
E
Z

(i)

h

i
MB(H) (O, S [ρu ] )λi ⟨u, F (dλ1 · · · dλn )u⟩H=i = 1, 2, ..., n,(12.17)Rnhwhere ρu = |u⟩⟨u|. Further, its variance var MB(H) (O, S [ρu ] )!
i´n
S [ρu ] )is given by

i

Ã

³
h
≡ var(i) MB(H) (O,

i=1

h
i
var(i) MB(H) (O, S [ρu ] )
Z ¯
h
i¯2
¯
¯
(i)
λ
−
E
=
M
(O,
S
)
¯ i
B(H)
[ρu ] ¯ ⟨u, F (dλ1 · · · dλn )u⟩H
n
ZR
¯2
¯Z
¯
¯
2
λi ⟨u, F (dλ1 · · · dλn )u⟩H ¯=|λi | ⟨u, F (dλ1 · · · dλn )u⟩H − ¯RnRn

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301(i = 1, 2, ..., n).
We begin with the following deﬁnition.
Deﬁnition 12.3. Let H be a Hilbert space with the inner product ⟨·, ·⟩H .
b tnsr = (K, s, (X, F, Fb)) is called a “tensor observable” (or precisely,(1). A triplet O
H⊗K“tensor represented observable”) in B(H ⊗ K), if it satisﬁes the following conditions (i)and (ii):(i) K is a Hilbert space and s is an element in K such that ∥s∥ = 1,(ii) (X, F, Fb) is a crisp observable in B(H ⊗ K), where H ⊗ K is a tensor Hilbert spacewith the inner product ⟨·, ·⟩H⊗K .
b tnsr = (K, s, (X, F,(2). Let (X, F, F ) be any observable in B(H). A tensor observable O
H⊗KbF )) is called a realization of the observable (X, F, F ) in tensor Hilbert space H ⊗ K, if itholds that⟨u ⊗ s, Fb(Ξ)(u ⊗ s)⟩H⊗K = ⟨u, F (Ξ)u⟩H(∀u ∈ H, ∀Ξ ∈ F).

(12.20)¥The following proposition is essential to our argument.
Proposition 12.4. [Holevo [34]]. Let (X, F, F ) be an observable in B(H). Then, thereb tnsr = (K, s, (X, F, Fb)) that is the realization of (X, F, F ),exists a tensor observable O
H⊗Kthat is, it holds that⟨u ⊗ s, Fb(Ξ)(u ⊗ s)⟩H⊗K = ⟨u, F (Ξ)u⟩H(u ∈ H, Ξ ∈ F).

(12.21)

Conversely any crisp observable (X, F, Fb) in B(H ⊗ K) and any s( ∈ K, ∥s∥K = 1) giverise to the unique observable (X, F, F ) in B(H) satisfying (12.21).
¥
We shall use the following notations.
³ R´
Notation 12.5. [Domain]. Let A = R λ EA (dλ), the spectral representation of A
be a (unbounded) self-adjoint operator on H. Then, we deﬁne the Dom(A), the domainof A, byZ
Dom(A) := {u ∈ H :|λ|2 ⟨u, EA (dλ)u⟩ < ∞}.
R

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CHAPTER 12. HEISENBERG’S UNCERTAINTY RELATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/

b tnsr = (K, s, (Rn , BRn , Fb)) be an observable and a tensor
Let O ≡ (Rn , BRn , F ) and O
H⊗K
observable in B(H) and in B(H ⊗ K) respectively. Then, we deﬁne thatmar[O]mar(k) := (R, BR , [F ](k) )(it will be called the kth marginal observable of O) ,where
[F ]mar· · × R} ×Ξ × R· · × R} )(k) (Ξ) := F ( R| × ·{z| × ·{zk − 1 times(∀Ξ ∈ BR )n − k times

Further, deﬁne that
Z
³
´
mar
mar
Dom([O](k) ) ≡ Dom([F ](k) ) := {u ∈ H :|λk |2 ⟨u, F (dλ1 ...dλn )u⟩ < ∞},
n
R
Z
³
´
mar
mar
b
b
Dom([O](k) ) ≡ Dom([F ](k) ) := {v̂ ∈ H ⊗ K :|λk |2 ⟨v̂, Fb(dλ1 ...dλn )v̂⟩H⊗K < ∞},Rn³´
b tnsr ]mar ) ≡ Dom⊗s ([Fb]mar )Dom⊗s ([OH⊗K (k)(k)Z:= {u ∈ H :|λk |2 ⟨u ⊗ s, Fb(dλ1 ...dλn )(u ⊗ s)⟩H⊗K < ∞},(12.22)Rnb marbwhere Dom([O]mar(k) ) (or Dom([O](k) )) is called the k-th domain of O (or O).
¥
Now we have the following main deﬁnition.
Deﬁnition 12.6. [Approximate simultaneous observable]. Let A1 , ..., An be (unbounded)ASO

self-adjoint operators in H. An observable O[Al ]nl=1 ≡ (Rn , BRn , F ) in B(H) is called theapproximate simultaneous observable of A1 , ..., An , if it satisﬁes the following conditions

ASO(i) (domain condition) for each i (= 1, 2, ..., n), Dom([O[Al ]nl=1 ]mar(i) ) ∩ Dom(Ai ) is densein H(ii) (unbias condition) for each i (= 1, 2, ..., n),Z⟨u, Ai u⟩ =λ⟨u, [F ]mar(i) (dλ)u⟩,RASO(u ∈ Dom([O[Al ]nl=1 ]mar(i) ) ∩ Dom(Ai )). (12.23)¥
ASO

Remark 12.7. [1]. As seen later (cf. Lemma 12.14(iii)), it holds that Dom([O[Al ]nl=1 ]mar
(i) )
ASO

ASO

⊆ Dom(Ai ) holds. Thus, Dom([O[Al ]nl=1 ]mar
]mar )(i) ) ∩ Dom(Ai ) = Dom([O[Al ]nl=1 (i)

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12.3. APPROXIMATE SIMULTANEOUS MEASUREMENT303

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/[2]. There is a very reason to assume the following condition (iii) or (iv) instead of the³´
above (i). (iii) and (iv) are stronger than (i), more precisely, (iv) =⇒ (iii) =⇒ (i).
(iii) (self-adjointness) for each i (= 1, 2, ..., n), Ai is essentially self-adjoint on
ASO

Dom([O[Al ]nl=1 ]mar(i) ) ∩ Dom(Ai ),or³(iv) (commutative condition) for each i (= 1, 2, ..., n), Ai´
ASO= R λ Ei (dλ) and [O[Al ]nl=1 ]mar(i)Rcommute.
Although each of (i), (iii) and (iv) has merit and demerit respectively, the physical meaningof the (iv) is the clearest.

(Continued on Remark 12.12.)[3]. Also, see the condition (i) in Example 11.5. This condition is equivalent to
´
³
ASO
∩
Dom(A
)
.
• the formula (12.23) holds on a dense set ∩ni=1 Dom([O[Al ]nl=1 ]mari(i)¥
Deﬁnition 12.8. [Approximate simultaneous tensor observable]. Let A1 , ..., An be (unb ASTnO = (K, s, (Rn , BRn , Fb))bounded) self-adjoint operators in H. A tensor observable O[Al ]l=1

b ASTnO = (K, s,
is called an approximate simultaneous tensor observable of A1 , ..., An , if O
[Al ]l=1nb(R , BRn , F )) satisﬁes the following conditions:b ASTnO ]mar ) ∩ Dom(Ai ) is(i) (domain condition) for each i (= 1, 2, ..., n), Dom⊗s ([O
[Al ]
(i)
l=1

dense in H(ii) (unbias condition) for each i (= 1, 2, ..., n),Z⟨u, Ai u⟩ =λi ⟨u ⊗ s, Fb(dλ1 · · · dλn )(u ⊗ s)⟩(12.24)Rn

b ASTnO ]mar ) ∩ Dom(Ai ), ).
(u ∈ Dom⊗s ([O[Al ]l=1 (i)¥ASO
b ASTnO is characterized by the following proposition.
The relation between O[Al ]nl=1 and O[Al ]l=1

Proposition 12.9. Let A1 , ..., An be (unbounded) self-adjoint operators in H.

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CHAPTER 12. HEISENBERG’S UNCERTAINTY RELATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/b ASTnO ≡ (K, s, (Rn , BRn , Fb) be an approximate simultaneous tensor observable(i) Let O
[Al ]
l=1

of A1 , ..., An in H. Then, there exists an approximate simultaneous observable
ASO
b ASTnO is a realization of OASOn .
O n ≡ (Rn , BRn , F ) such as O
[Al ]l=1[Al ]l=1[Al ]l=1

ASO(ii) Let O[Al ]nl=1 ≡ (Rn , BRn , F ) be an approximate simultaneous observable of A1 , ..., An
b ASTnO
in H. Then, there exists a approximate simultaneous tensor observable O
ASO≡ (K, s, (R , BRn , Fb) such as it is a realization of O[Al ]nl=1 .

[Al ]l=1nASO(iii) Let O[Al ]nl=1 ≡ (Rn , BRn , F ) be an approximate simultaneous observable of A1 , ..., An
b ASTnO ≡ (K, s, (Rn , BRn , Fb) be an approximate simultaneous tensor
in H. Let O[Al ]l=1

b ASTnO is a realization of OASOn .
observable of A1 , ..., An in H. And assume that O[Al ]l=1[Al ]
l=1

Then, for each i ( = 1, 2, ..., n),b AST O ]mar ) ⊆ Dom(Ai ).
Dom([O[Al ]nl=1 ]mar(i) ) = Dom⊗s ([O[Al ]nl=1 (i)ASO(12.25)

Proof. The statement (i) is trivial. Also the statement (ii) and the equality “=” in(12.25) immediately follow from Proposition 12.4. Also, the inclusion “⊆” in (12.25) isproved in Lemma 12.14(iii) later.
Deﬁnition 12.10. [Uncertainty] Let A1 , ..., An be (unbounded) self-adjoint operators on
a Hilbert space H.
ASO[I]. Let O[Al ]nl=1 = (Rn , BRn , F ) ) be an approximate simultaneous observable of A1 , . . . , An .
µ¶n
ASOof O[Al ]nl=1 for a state u (∥u∥H = 1) is(i). Then, the uncertainty ∆OASOn (Ai , u)
[Al ]
l=1

i=1

deﬁned byZ∆OASOn (Ai , u) =
[Al ]
l=1

Rn

Z
λ2i ⟨u, F (dλ1 · · · dλn )u⟩ −λ2 ⟨u, Ai (dλ)u⟩(12.26)R(u ∈ H such that ∥u∥ = 1 ),where (12.26) should be interpreted that ∆OASOn (Ai , u) = ∞ for u ∈/ Dom([F ]mar(i) ) (cf.
[Al ]
l=1
³
Dom([F ]mar“∆OASOn (Ai , u) ≥ 0” will be shown in Theorem(i) ) ⊆ Dom(Ai ) in (12.25)).
[Al ]
l=1
´
12.15 later.
ASO(ii). Also the i-th variance var(i) [O[Al ]nl=1 , u] is deﬁned by

ASO
var(i) [O[Al ]nl=1 , u] =Z
|λi − ⟨u, Ai u⟩|2 ⟨u, F (dλ1 · · · dλn )u⟩H
Rn

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12.3. APPROXIMATE SIMULTANEOUS MEASUREMENTFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/

305(i = 1, 2, ..., n),b ASTnO = (K, s, (Rn , BRn , Fb) ) be an approximate simultaneous tensor observable[II] Let O
[Al ]
l=1

of A1 , . . . , An .

µ(i). Then, the uncertainty¶n∆O
b ASTnO (Ai , u)
[Al ]
l=1

deﬁned byZ∆O
b ASTnO (Ai , u) =
[Al ]
l=1

Rn

b ASTnO for a state u (∥u∥H = 1) is
of O
[Al ]
l=1

i=1

λ2i ⟨u ⊗ s, Fb(dλ1 · · · dλn )(u ⊗ s)⟩ −Z
λ2 ⟨u, Ai (dλ)u⟩(12.28)R(u ∈ H such that ∥u∥ = 1 ),where (12.28) should be interpreted that ∆O/ Dom⊗s ([Fb]mar
b ASTnO (Ai , u) = ∞ for u ∈
(i) )
[Al ]
l=1
³
(cf. Dom⊗s ([Fb]mar“∆O
b ASTnO (Ai , u) ≥ 0” will be shown in(i) ) ⊆ Dom(Ai ) in (12.25)).
[Al ]
l=1
´
Theorem 12.15 later.
b ASTnO , u] is deﬁned by(ii). Also the i-th variance var(i) [O[Al ]l=1

b ASTnO , u] =var(i) [O[Al ]l=1Z|λi − ⟨u, Ai u⟩|2 ⟨u ⊗ s, Fb(dλ1 · · · dλn )(u ⊗ s)⟩H⊗K(12.29)Rn(i = 1, 2, ..., n).
¥
Proposition 12.11. Let A1 , ..., An be (unbounded) self-adjoint operators on a Hilbertb ASTnO = (K, s, (Rn , BRn , Fb)) is a realization of OASOn =space H. Assume that O[Ai ]i=1[Ai ]i=1(Rn , BRn , F ). Let u ∈ H (∥u∥H = 1). Then it holds that
∆OASOn (Ai , u) = ∆O
b ASTnO (Ai , u)
[Al ]
l=1

(12.30)

[Al ]
l=1

and
b ASTnO , u].
var(i) [O[Al ]nl=1 , u] = var(i) [O[Al ]l=1
ASO(12.31)

Proof. This immediately follows from Deﬁnition 12.10.
Remark 12.12. [Continued from Remark 12.7]. Again note that, if the commutative condition (iv) in Remark 12.7 is assumed in the Deﬁnition 12.10, we can deﬁne
³
´
ASO
ASO
∆ MB(H) (Ai × [O[Al ]nl=1 ]mar,S(ρ)), the distance between Ai and [O[Al ]nl=1 ]maru(i)(i) , cf. Deﬁnition 11.1. And further we see that³´
ASO mar∆ MB(H) (Ai × [O[Al ]nl=1 ](i) , S(ρu )) =(“error” deﬁned in Deﬁnition 11.1)∆OASOn (Ai , u)
[Al ]
l=1

(“uncertainty” deﬁned in Deﬁnition 12.10)

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Thus, in this case, the physical meaning of “uncertainty” is clear.

¥

12.4

LemmasIn this section, we shall prepare some Lemmas.
Lemma 12.13. [Robertson’s uncertainty relation]. Let A1 and A2 be any symmetric
operators on a Hilbert space H. Then, it holds thathi1/2 hi1/2 1 D
E D
E∥A1 u∥2 − |⟨u, A1 u⟩|2
· ∥A2 u∥2 − |⟨u, A2 u⟩|2
≥ | A1 u, A2 u − A2 u, A1 u |2(12.33)for all u ∈ Dom(A1 ) ∩ Dom(A2 ) .
Proof. Using Schwartz inequality, we see|⟨A1 u, A2 u⟩ − ⟨A2 u, A1 u⟩|D
E DE=| A1 u − ⟨u, A1 u⟩u, A2 u − ⟨u, A2 u⟩u − A2 u − ⟨u, A2 u⟩u, A1 u − ⟨u, A1 u⟩u |hi1/2 hi1/2≤2 ∥A1 u∥2 − |⟨u, A1 u⟩|2
· ∥A2 u∥2 − |⟨u, A2 u⟩|2.
(12.34)

Lemma 12.14. Let A1 , · · · , An be any (unbounded) self-adjoint operators in a Hilbert
space H. Let (K, s, (Rn , BRn , Fb) be an approximate simultaneous tensor observable
³
´
R
R
mar
bk =
b
b
for A1 , · · · , An . Put A
λ
F
(dλ
dλ
·
·
·
dλ
)
≡
λ[
F
]
=
(i)
(dλ)
(k =
1
2
n
Rn k
R
1, 2, ..., n). Then, the following equalities (i) ∼ (iii) hold(i)bi (u ⊗ s)⟩ =⟨v, Ak u⟩ = ⟨v ⊗ s, AZ

λk ⟨v ⊗ s, Fb(dλ1 dλ2 · · · dλn )(u ⊗ s)⟩

R2bk ) and all v ∈ H (k = 1, 2, ..., n),
for all u ∈ Dom⊗s (A(ii)
Zλi λj ⟨u ⊗ s, Fb(dλ1 dλ2 · · · dλn )(u ⊗ s)⟩Rnbi (u ⊗ s), A
bj (u ⊗ s)⟩= ⟨A

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12.4. LEMMAS307

For Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/bi − Ai ⊗ I)(u ⊗ s), (Abj − Aj ⊗ I)(u ⊗ s)⟩= ⟨Ai u, Aj u⟩ + ⟨(A(12.36)bi ) ∩ Dom⊗s (Abj ),
for all i ̸= j and all u ∈ Dom⊗s (A(iii)
Z|λk |2 ⟨u ⊗ s, Fb(dλ1 dλ2 · · · dλn )(u ⊗ s)⟩R2

bk (u ⊗ s)∥2 = ∥Ak u∥2 + ∥(A
bk − Ak ⊗ I)(u ⊗ s)∥2 ≥ ∥Ak u∥2= ∥A(12.37)bk ) (k = 1, 2, ..., n). Thus, it holds that, for each i ( = 1, 2, ..., n),
for all u ∈ Doms (AASO
b AST O ]mar ) ⊆ Dom(Ai ).
Dom([O[Al ]nl=1 ]mar(i) ) = Dom⊗s ([O[Al ]nl=1 (i)(12.38)bk ).
Proof. First we prove (i). Fix k ∈ {1, 2}. We can see that, for any v, u ∈ Dom⊗s (A⟨v, Ak u⟩
1= {⟨(v + u), Ak (v + u)⟩ − ⟨(v − u), Ak (v − u)⟩4− i⟨(v + iu), Ak (v + iu)⟩ + i⟨(v − iu), Ak (v − iu)⟩}1bk ((v + u) ⊗ s)⟩ − ⟨(v − u) ⊗ s, A
bk ((v − u) ⊗ s)⟩= {⟨(v + u) ⊗ s, A
4bk ((v + iu) ⊗ s)⟩ + i⟨(v − iu) ⊗ s, A
bk ((v − iu) ⊗ s)⟩}− i⟨(v + iu) ⊗ s, A
bk (u ⊗ s)⟩= ⟨v ⊗ s, AZZ= ⟨v ⊗ s,
λk Fbk (dλ1 dλ2 · · · dλn )(u ⊗ s)⟩ =Rn

λk ⟨v ⊗ s, Fbk (dλ1 dλ2 · · · dλn )(u ⊗ s)⟩.

Rn

(12.39)bk ) is dense in H, we see that
Since Dom⊗s (AZbbk (dλ1 dλ2 · · · dλn )(u ⊗ s)⟩⟨v, Ak u⟩ = ⟨v ⊗ s, Ak (u ⊗ s)⟩ =λk ⟨v ⊗ s, A

(12.40)Rnbk ) and all v ∈ H. This completes the proof of (i).
for all u ∈ Dom⊗s (ANext, we prove (ii). Without loss of generality, we put i = 1 and j = 2. Let u be anyb1 ) ∩ Dom⊗s (Ab2 ). Then, we see, by the above (i), thatelement in Dom⊗s (AZ

λ1 λ2 ⟨u ⊗ s, Fb(dλ1 dλ2 · · · dλn )(u ⊗ s)⟩ZZb=⟨
λ1 F (dλ1 dλ2 · · · dλn )(u ⊗ s),
λ2 Fb(dλ1 dλ2 · · · dλn )(u ⊗ s)⟩
Rn

Rn

Rn

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CHAPTER 12. HEISENBERG’S UNCERTAINTY RELATIONFor Clear Printing, See http://www.keio-up.co.jp/kup/mfomt/
b1 (u ⊗ s), A
b2 (u ⊗ s)⟩= ⟨A
b1 − A1 ⊗ I)(u ⊗ s) + (A1 u ⊗ s), (Ab2 − A2 ⊗ I)(u ⊗ s) + (A2 u ⊗ s)⟩= ⟨(A
b1 − A1 ⊗ I)(u ⊗ s), (Ab2 − A2 ⊗ I)(u ⊗ s)⟩= ⟨(A
b1 − A1 ⊗ I)(u ⊗ s), A2 u ⊗ s⟩+ ⟨(A
b2 − A2 ⊗ I)(u ⊗ s)⟩ + ⟨A1 u ⊗ s, A2 u ⊗ s⟩
+ ⟨A1 u ⊗ s, (Ab1 − A1 ⊗ I)(u ⊗ s), (Ab2 − A2 ⊗ I)(u ⊗ s)⟩= ⟨(A
b1 (u ⊗ s), A2 u ⊗ s⟩ − ⟨A1 u, A2 u⟩+ ⟨A
b2 (u ⊗ s)⟩ − ⟨A1 u, A2 u⟩ + ⟨A1 u, Au ⟩+ ⟨A1 u ⊗ s, A

b1 − A1 ⊗ I)(u ⊗ s), (Ab2 − A2 ⊗ I)(u ⊗ s)⟩ − ⟨A1 u, A2 u⟩
= ⟨(A
Z
Z
+
λ2 ⟨A1 u ⊗ s, Fb(dλ1 dλ2 )(u ⊗ s)⟩ +λ1 ⟨Fb(dλ1 dλ2 )(u ⊗ s), A2 u ⊗ s⟩Rn

Rn

b1 − A1 ⊗ I)(u ⊗ s), (Ab2 − A2 ⊗ I)(u ⊗ s)⟩.
= ⟨A1 u, A2 u⟩ + ⟨(A(12.41)Hence, the proof of (ii) is completed. Also, the proof of (12.37) is carried out just ina similar way. Lastly, we can easily see that (12.37) implies (12.38) since we see thatASO
b ASTnO ]mar ) in (12.25).
Dom([O n ]mar ) = Dom⊗s ([O[Al ]l=1 (i)[Al ]l=1 (i)Now we have the following theorem, which is one of our main results.
b ASTnO = (K, s, (Rn , BRn , Fb) ) be a realization of an approxiTheorem 12.15. Let O[Al ]l=1

ASO
bi =mate simultaneous tensor observable O[Al ]nl=1 = (Rn , BRn , Fb ) of A1 , . . . , An . Put A
R
λ[Fb]mar(i) (dλ). Then, we see that
R
Z
Z
2
mar
∆O
λ ⟨u, [F ](i) (dλ)u⟩ −λ2 ⟨u, Ai (dλ)u⟩b ASTnO (Ai , u) =∆OASOn (Ai , u) =
[Al ]
l=1

[Al ]
l=1

R

R

bi (u ⊗ s)∥2 − ∥Ai u∥2=∥Abi − Ai ⊗ I)(u ⊗ s)∥2=∥(A

(12.42)(∀u ∈ H such that ∥u∥ = 1(12.43)

Proof. It immediately follows from Lemma 12.14.

12.5

Existence theorem

Now we shall mention the following theorem, which assures the existence of an approximate simultaneous tensor observable of arbitrary observables A1 , ..., An . For two
observables A1 and A2 , the similar theorem was proved by P. Busch, et al. [15, 14].

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12.5. EXISTENCE THEOREM

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Theorem 12.16. [Cf.[36]] Let A1 , ..., An be (unbounded) self-adjoint operators on aP
Hilbert space H. Let a1 , ..., an be any positive numbers such that ni=1 (1 + ai 2 )−1 = 1.
Then, we see,b ASTnO ≡ (K, s, (Rn , BRn ,(i) there exists an approximate simultaneous tensor observable O[Al ]l=1bF )) of A1 , ..., An such that:b AST O mar∆Ob ASTnO (Ai , u) = ai ∥Ai u∥ (u ∈ Dom⊗s ([O[Al ]n ](i) ) i = 1, 2, ..., n).
l=1(12.44)[Al ]l=1

and equivalently,ASO(ii) there exists an approximate simultaneous observable O[Al ]nl=1 ≡ (Rn , BRn , F ) of
A1 , ..., An such that:ASO∆OASOn (Ai , u) = ai ∥Ai u∥ (u ∈ Dom([O[Al ]nl=1 ]mar(i) ) i = 1, 2, ..., n).

(12.45)[Al ]l=1

Proof. By Proposition 12.11, it suﬃces to prove (i). Put K = Cn = {z = (z1 , ..., zn ) :zi ∈ C (i = 1, 2, ..., n)}, i.e., the n-dimensional Hilbert space with the norm ∥z∥n =[Σni=1 |zi |2 ]1/2 . Put e1 = (1, 0, ..., 0), e2 = (0, 1, 0, ..., 0) , ...., en = (0, 0, ..., 1) ∈ Cn . Put
s = e1 . And put Pi : Cn → Cn , (i = 1, 2, ..., n), a projection such that Pi ei = ei , Pi ek =0(k ̸= i), that is, Pi = |ei ⟩⟨ei |. Put bi = (1 + ai 2 )1/2 and Bi = bi 2 Ai (i = 1, 2, ..., n).
Consider the spectral representations
ZZ
Ai =λEAi (dλ), Bi =λEBi (dλ),
R

Z
0=

R

and

Z
Pi =
R

λE0 (dλ)

in H

R

Z
n
λEPCi (dλ),

nλEIC (dλ)

I=in Cn .

Rb : H ⊗ Cn → H ⊗ Cn
Note that EAi (d(λ/bi 2 )) = EBi (dλ). Deﬁne the unitary operator U
b = I ⊗ U where a unitary operator U on Cn satisﬁes that U e1 = Σn ei /bi . And
by U
i=1

b b ) in B(H ⊗ Cn ) by
deﬁne the crisp observable (R, BR , EAib b (dξ) = Ub ∗ [EB (dξ) ⊗ Pi + E0 (dξ) ⊗ (I − Pi )]U
b
E
i
Ai

(i = 1, 2, ..., n).

(12.46)b b , ..., Eb b commute, we can deﬁne a crisp observable (Rn , BRn , Eb b) in B(H ⊗Cn )Since E
A1
An
A
such that:b b(dξ1 dξ2 ...dξn ) =
E
A

n
Y
i=1

b b (dξi ).
E
Ai

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b tnsr = (Cn , e1 , (Rn , Bn , Eb b)) is an apNow, we shall show that the tensor observable O
H⊗KA
R
bi = n ξi E
b b (dξ1 dξ2 ...dξn )proximate simultaneous tensor observable of A1 , ..., An . Put ARA(i = 1, ..., n). Then we see that,Zb b(dξ1 dξ2 ...dξn )(u ⊗ e1 )⟩|ξi |2 ⟨u ⊗ e1 , E
AnZRb b (dξi )(u ⊗ e1 )⟩=|ξi |2 ⟨u ⊗ e1 , E
Ai
R
Z
³
´
=
|ξi |2 ⟨u ⊗ e1 , [(I ⊗ U ∗ ) EBi (dξi ) ⊗ Pi + E0 (dξi ) ⊗ (I − Pi ) (I ⊗ U )](u ⊗ e1 )⟩ZR|ξ|2 ⟨u, EBi (dξ)u⟩ · ⟨e1 , U ∗ Pi U e1 ⟩
=
ZR
ej
ek
=
|ξ|2 ⟨u, EBi (dξ)u⟩ · ⟨Σnj=1 , Pi Σnk=1 , ⟩
bj
b
R
Z
Z k
= |bi |−2|λ|2 ⟨u, EBi ((dλ)u⟩ = |bi |2|λ|2 ⟨u, EAi (dλ)u⟩.
(12.48)RRbi ) = Dom(Ai ) (where s = e1 ). Similarly we see
Hence, Dom⊗s (AZ
b b(dξ1 dξ2 ...dξn )(u ⊗ e1 )⟩ξi ⟨u ⊗ e1 , E
A
n
R
Z
Z
−2
= |bi |
λ⟨u, EBi ((dλ)u⟩ =λ⟨u, EAi (dλ)u⟩.
R(12.49)R

b ASTnO satisﬁes the condition (ii) in Deﬁnition 12.6. Also, noting that I(dλ) =Thus, O[Al ]l=1

I(1 ∈ dλ), = 0(1 ∈/ dλ), we also see that, for each i (i = 1, 2, ..., n) and Ξk ∈ B,
b b (Ξ1 ) · (EA (Ξ2 ) ⊗ I)
E
i
Ai
³
´
∗
= (I ⊗ U ) EBi (Ξ1 ) ⊗ Pi + E0 (Ξ1 ) ⊗ (I − Pi ) (I ⊗ U )(EAi (Ξ2 ) ⊗ I)³´
= (EAi (Ξ2 ) ⊗ I)(I ⊗ U ∗ ) EBi (Ξ1 ) ⊗ Pi + E0 (Ξ1 ) ⊗ (I − Pi ) (I ⊗ U )b b (Ξ1 ).
= (EAi (Ξ2 ) ⊗ I) · E
Ai(12.50)RRbi and Ai ⊗ I commute since A
bi =
So, A
ξ
E
(dξ)
and
A
⊗
I
=
ξ (EAi (dξ) ⊗ I).
b
i
A
R
R
i
bi − Ai ⊗ I on Dom(Abi ) ∩ Dom(Ai ⊗ I) has the unique self-adjoint extension
Hence, Abi − Ai ⊗ I], which has the spectral representation[AZbb b (dξ1 )(EA (dξ2 ) ⊗ I).
[Ai − Ai ⊗ I] =(ξ1 − ξ2 )Ei
Ai(12.51)

R2

Then, we see thatbi − Ai ⊗ I](u ⊗ e1 )∥2∥[A

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Z

=
ZR

2

|ξ1 − ξ2 |2 ⟨u ⊗ e1 , EAbi (dξ1 )(EAi (dξ2 ) ⊗ I)(u ⊗ e1 )⟩|ξ|2 ⟨u ⊗ e1 , EAbi (dξ1 )(u ⊗ e1 )⟩RZ−2
ξ1 ξ2 ⟨u ⊗ e1 , EAbi (dξ1 )(EAi (dξ2 ) ⊗ I)(u ⊗ e1 )⟩R2Z+|ξ2 |2 ⟨u ⊗ e1 , (EAi (dξ2 ) ⊗ I)(u ⊗ e1 )⟩RZ
2= (|bi | − 2 + 1)
|ξ|2 ⟨u, EAi (dξ)u⟩=R= |ai | ∥Ai u∥ ,22

(12.53)bi − Ai ⊗ I]) = Dom(Ai ) (where s = e1 ) and ∆ b AST O (Ai , u)which implies that Dom⊗s ([A
O
n
[Al ]
l=1

= ai ∥Ai u∥. Therefore, the proof of theorem is completed.

Remark 12.17. In the above proof, the following statements were also proved:bi and Ai ⊗ I commute, so A
bi − Ai ⊗ I on Dom(Abi ) ∩ Dom(Ai ⊗ I) has a unique(i) Abi − Ai ⊗ I] (i = 1, 2),
self-adjoint extension [Abi ) = Dom⊗s ([Abi − Ai ⊗ I]) = Dom(Ai ) (i = 1, 2).
(ii) Dom⊗s (AThus the commutative condition (iv) in Remark 12.7 is satisﬁed.
¥

12.6

Uncertainty relations

Now we propose the following theorem, which is our main result in this chapter.
We believe that this theorem is the ﬁnal version of Heisenberg’s uncertainty relation
concerning measurement errors.
Theorem 12.18. [Heisenberg’s uncertainty relation, cf. [36, 67]]. Let A1 and A2 be any(unbounded) self-adjoint operators on a Hilbert space H. Then, we see,b AST2 O ≡ (K, s, (R2 , BR2 , Fb))(i) for any approximate simultaneous tensor observable O
[Al ]
l=1

of A1 and A2 , the following inequality holds:
1∆O
b AST O (A1 , u) · ∆O
b AST O (A2 , u) ≥ |⟨A1 u, A2 u⟩ − ⟨A2 u, A1 u⟩|
2[Al ]2[Al ]2l=1
l=1

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for all u ∈ H such that ∥u∥ = 1, where the left hand side of (12.54) is deﬁned by∞ if ∆Ob AST O (Ai , u) = ∞ for some i,[Al ]2l=1

and equivalently,ASO(ii) for any approximate simultaneous observable O[Al ]2l=1 ≡ (R2 , BR2 , F ) of A1 and A2 ,the following inequality holds:
1∆OASO2 (A1 , u) · ∆OASO2 (A2 , u) ≥ |⟨A1 u, A2 u⟩ − ⟨A2 u, A1 u⟩|
[Al ]
[Al ]
2
l=1
l=1

(12.55)for all u ∈ H such that ∥u∥ = 1, where the left hand side of (12.55) is deﬁned by∞ if ∆OASO2 (Ai , u) = ∞ for some i.
[Al ]
l=1

R
bi =
Proof. By Proposition 12.11, it suﬃces to prove (i). Put A
λ Fe(dλ1 dλ2 )R2 ibi ) for some i, we see, by the(i = 1, 2). Let u ∈ D(A1 ) ∩ D(A2 ). If u ∈/ Dom⊗s (Adeﬁnition of the uncertainty, that ∆O
b AST O (Ai , u) = ∞, so (12.55) clearly holds. Hence,[Al ]2l=1b1 ) ∩ Dom⊗s (Ab2 ). Let u be any element in
it is suﬃcient to prove (12.55) for u ∈ Dom⊗s (Ab1 ) ∩ Dom⊗s (Ab2 ). We see, by the part (ii) of Lemma 12.14, thatu ∈ Dom⊗s (Ab1 − A1 ⊗ I)(u ⊗ s), (Ab2 − A2 ⊗ I)(u ⊗ s)⟩⟨A1 u, A2 u⟩ + ⟨(AZ=
λ1 λ2 ⟨u ⊗ s, Fe(dλ1 dλ2 )(u ⊗ s)⟩R2b2 − A2 ⊗ I)(u ⊗ s), (Ab1 − A1 ⊗ I)(u ⊗ s)⟩= ⟨A2 u, A1 u⟩ + ⟨(A(12.56)from which, we get, by Schwarz inequality, that
1|⟨A1 u, A2 u⟩ − ⟨A2 u, A1 u⟩|2
1 bb= |⟨(A
1 − A1 ⊗ I)(u ⊗ s), (A2 − A2 ⊗ I)(u ⊗ s)⟩2b2 − A2 ⊗ I)(u ⊗ s), (Ab1 − A1 ⊗ I)(u ⊗ s)⟩|− ⟨(A
b1 − A1 ⊗ I)(u ⊗ s)∥ · ∥(Ab2 − A2 ⊗ I)(u ⊗ s)∥.
≤ ∥(A

(12.57)

Hence (by Theorem 12.15), the proof is completed.
The following theorem was ﬁrst discovered by Arthurs and Goodman [6]. However we
did not know their discovery in the preparation of [36].
Theorem 12.19. [Approximate simultaneous uncertainty relation, cf [6]]. Let A1 andA2 be any (unbounded) self-adjoint operators on a Hilbert space H. Then, we see,

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313

b AST2 O = (K, s, (R2 , BR2 , Fb),(i) for any approximate simultaneous tensor observable O
[Al ]
l=1

) of (A1 , A2 ), the following inequality holds:
b AST2 O , u]1 )1/2 · (var[Ob AST2 O , u]2 )1/2 ≥ |⟨A1 u, A2 u⟩ − ⟨A2 u, A1 u⟩|(var[O
[Al ]
[Al ]
l=1

l=1

(12.58)for all u ∈ H such that ∥u∥ = 1, where the left hand side of (12.58) is deﬁned by ∞b AST2 O , u]mar = ∞ for some i, also the right hand side of (12.58) is deﬁned
if var[O(i)[Al ]l=1

by ∞ if u ∈/ Dom(A1 ) ∩ Dom(A2 ),and equivalentlyASO(ii) for any approximate simultaneous observable O[Al ]2l=1 = (R2 , BR2 , Fb ) of (A1 , A2 ),the following inequality holds:
ASO

ASO

(var[O[Al ]2l=1 , u]1 )1/2 · (var[O[Al ]2l=1 , u]2 )1/2 ≥ |⟨A1 u, A2 u⟩ − ⟨A2 u, A1 u⟩|(12.59)for all u ∈ H such that ∥u∥ = 1, where the left hand side of (12.59) is deﬁned by ∞b AST2 O , u]mar = ∞ for some i, also the right hand side of (12.59) is deﬁned
if var[O(i)[Al ]l=1

by ∞ if u ∈/ Dom(A1 ) ∩ Dom(A2 ).
R
bi =Proof. By Proposition 12.11, it suﬃces to prove (i). Put A
λ Fb(dλ1 dλ2 )R2 ibi ) for some i, we see, by the deﬁnition of the variance, that(i = 1, 2). If u ∈/ Dom⊗s (Ab AST2 O , u]mar = ∞, so, (12.58) clearly holds. Hence, it is suﬃcient to prove (12.58)
var[O
(i)
[Al ]l=1b1 ) ∩ Dom⊗s (Ab2 ). Let u be any element in Dom⊗s (Ab1 ) ∩
in the case that u ∈ Dom⊗s (Ab2 ). Then, we see, by (iii) in Lemma 12.14, that
Dom⊗s (Ab AST2 O , u]mar = ∥Abi (u ⊗ s)∥2 − |⟨u ⊗ s, Abi (u ⊗ s)⟩|2
var[O
(i)
[Al ]
l=1

(12.60)bi − Ai ⊗ I)(u ⊗ s)∥2 − |⟨u, Ai u⟩|2=∥Ai u∥2 + ∥(Abi − Ai ⊗ I)(u ⊗ s)∥ (i = 1, 2),≤2(∥Ai u∥2 − |⟨u, Ai u⟩|2 )1/2 · ∥(A(12.61)

therefore, by Lemma 12.13 and Theorem 12.18 we get,b AST2 O , u]1 · var[Ob AST2 O , u]2
var[O
[Al ]
[Al ]
l=1

l=1

≥ 4(∥A1 u∥ − |⟨u, A1 u⟩| )
2

2 1/2

· (∥A2 u∥2 − |⟨u, A2 u⟩|2 )1/2

b1 − A1 ⊗ I)(u ⊗ s)∥ · ∥(Ab2 − A2 ⊗ I)(u ⊗ s)∥· ∥(A≥ |⟨A1 u, A2 u⟩ − ⟨A2 u, A1 u⟩|2 .

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Hence, the proof is completed.
Now we have the following corollary.3

Corollary 12.20. [Uncertainty relations concerning a pair of conjugate observables]. Let
A1 and A2 be a pair of conjugate observables in a Hilbert space H.
ASO(i: cf. [7]) There exists an approximate simultaneous observable O[Al ]2l=1 = (R2 , BR2 .F) of A1 and A2 . Thus, we can take an approximate simultaneous measurement
ASO

MB(H) (O[Al ]2l=1 , S [|u⟩⟨u|] ).
(ii: cf. [36]) For any positive number ϵ and any k(= 1, 2), there exists an approximate
ASO

simultaneous observable O[Al ]2l=1 = (R2 , BR2 .F ) of A1 and A2 such that:∆OASO2 (Ak , u) ≤ ϵ∥Ak u∥H
[Al ]
l=1

(∀u ∈ H such that ∥u∥ = 1),(iii: cf. [36, 67]) (Heisenberg’s uncertainty relation) However the following inequality
holds∆OASO2 (A1 , u) · ∆OASO2 (A2 , u) ≥ ~/2
[Al ]
l=1

for all u ∈ H(12.63)[Al ]l=1(∥u∥H = 1),(iv: cf. [6]) The following inequalities hold: (approximate simultaneous uncertainty
relation)
ASO

ASO(var[O[Al ]2l=1 , u]1 )1/2 · (var[O[Al ]2l=1 , u]2 )1/2 ≥ ~for all u ∈ H(12.64)(∥u∥H = 1).

Proof. Note that ⟨A1 u, A2 u⟩ − ⟨A2 u, A1 u⟩ = i~ (u ∈ Dom(A1 ) ∩ Dom(A2 ), ∥u∥H
= 1). Then, the above assertions (i) and (ii) are consequences of Theorem 12.16. Also,
the above assertions (iii) and (iv) are respectively consequences of Theorem 12.18 andTheorem 12.19.

3

There are other uncertainty relations, For the recent variants, see [68].

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12.7

315

EPR-experiment and Heisenberg’s uncertainty
relation

Now we have the complete form of Heisenberg’s uncertainty relation as Corollary 12.20.
To be compared with Corollary 12.20, we should note that the conventional Heisenberg’s
uncertainty relation (= Proposition 12.1) is ambiguous. Wrong conclusions are sometimes
derived from the ambiguous statement (= Proposition 12.1). For example, in some booksof physics, it is concluded that EPR-experiment (Einstein, Podolosky and Rosen [22])contradicts with Heisenberg’s uncertainty relation. That is,(I) Heisenberg’s uncertainty relation says that the position and the momentum of aparticle can not be measured simultaneously and exactly.
On the other hand,(II) EPR-experiment says that the position and the momentum of a certain “particle” canbe measured simultaneously and exactly.
Thus someone may conclude that the above (i) and (ii) includes a paradox, and therefore,
EPR-experiment contradicts with Heisenberg’s uncertainty relation. Of course, this is amisunderstanding. This “paradox” was solved in [36]. Now we shall explain the solutionof the paradox.
[Concerning the above (I)]Put H = L2 (Rq ). Consider two-particles system in

H ⊗ H = L2 (R2(q1 ,q2 ) ). In the EPR problem, we, for example, consider the state us³´( ∈ H ⊗ H = L2 (R2(q1 ,q2 ) )) or precisely, |us ⟩⟨us | such that:rus (q1 , q2 ) =1 − 12 (q1 −q2 −a)2 − 12 (q1 +q2 −b)2 iϕ(q1 ,q2 )
8ϵ
e 8σ
·e
2πϵσ

(12.65)where ϵ is assumed to be a suﬃciently small positive number and ϕ(q1 , q2 ) is a real-valued
function. This is the quantum form of EPR-experiment in Remark 12.2(ii). Let A1 :
L2 (R2(q1 ,q2 ) ) → L2 (R2(q1 ,q2 ) ) and A2 : L2 (R2(q1 ,q2 ) ) → L2 (R2(q1 ,q2 ) ) be self-adjoint operatorssuch thatA 1 = q1 ,

A2 =

~∂
.
i∂q1

(12.66)

Then, Corollary 12.20 (i) says that there exists an approximate simultaneous observable
ASO

O[Al ]2l=1 = (R2 , BR2 .F ) of A1 and A2 . Thus we can take an approximate simultaneous

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ASO

measurement MB(H) (O[Al ]2l=1 , S [|us ⟩⟨us |] ). And thus, the following Heisenberg’s uncertainty
relation (= Corollary 12.20 (iii)) holds,∆OASO2 (A1 , us ) · ∆OASO2 (A2 , us ) ≥ ~/2
[Al ]
l=1

[Al ]

(12.67)l=1[Concerning the above (II)] However, it should be noted that, in the above situation
we assume that the state us is known before the measurement. In such a case, we may
b1 : L2 (R2take another measurement as follows: Deﬁne the self-adjoint operators A)→22
b2 : L2 (R2
L (R2(q1 ,q2 ) ) and A(q1 ,q2 ) ) → L (R(q1 ,q2 ) ) such that

(q1 ,q2 )2

b1 = b − q2 ,A

b2 = A2 = ~∂Ai∂q1

(12.68)Note that these operators commute. Therefore,
b1 and A
b2 (for the state us ).
(♯) we can take an exact simultaneous measurement of A
And moreover, we can easily calculate as follows (cf. Deﬁnition 11.1 and Remark 12.12).
³´bb1 us − A1 us ∥∆ MB(H) (A1 × A1 , S(ρus )) = ∥Arh ZZ ¯i1/2
1 − 12 (q1 −q2 −a)2 − 12 (q1 +q2 −b)2 iϕ(q1 ,q2 ) ¯¯2
¯
8σ
8ϵ
=
e
·e
¯ dq1 dq2¯((b − q2 ) − q1 )
2πϵσ
R2
r
i1/2
h ZZ ¯1 − 12 (q1 −q2 −a)2 − 12 (q1 +q2 −b)2 ¯¯2
¯
8ϵ
e 8σ
=
¯ dq1 dq2¯((b − q2 ) − q1 )
2πϵσ
2
√ R
(12.69)
= 2ϵ,
and
³
´
b2 , S(ρus )) = ∥A
b2 us − A2 us ∥ = 0.
∆ MB(H) (A2 × A(12.70)Thus we see³´³´b1 , S(ρus )) · ∆ MB(H) (A2 × A
b2 , S(ρus )) = 0.
∆ MB(H) (A1 × A(12.71)Since ϵ ( > 0) can be taken suﬃciently small, the above measurement (♯) is superior to³
ASO
the approximate simultaneous measurement MB(H) (O[Al ]2l=1 , S [|us ⟩⟨us |] ). Here, S [|us ⟩⟨us |]´
is identiﬁed with S(|us ⟩⟨us |) since |us ⟩⟨us | is a pure state. However it should be again
noted that, the measurement (♯) is made from the knowledge of the state us .

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317[(I) and (II) are consistent, cf. [36] ] The above conclusion (12.71) does not contradicts with Heisenberg’s uncertainty relation (12.67), since the measurement (♯) is not anapproximate simultaneous measurement of A1 and A2 .
¥In the above arguments, note that Theorem 12.19 (approximate simultaneous uncertainty relation) is powerless to solve the paradox (i.e., the paradox between EPRexperiment and Heisenberg’s uncertainty relation). That is because the concept “error” (or“uncertainty” ) is not explicit in Theorem 12.19.

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