92

数理解析研究所講究録
第1186巻 2001年 92-105

Measurements of Nondegenerate Observables
MASANAO OZAWA( 小澤正直 )School of Informatics and Sciences, Nagoya University, Nagoya 464‐8601, Japan

1.

Introduction

Every measuring apparatus inputs the state \rho of the measured system and outputs

the classical output x and the state \rho_{\{X=x} } of the measured system conditional upon

the outcome x=x . In the conventional approach, the probability distribution of theclassical output x and the output state \rho_{\{x=x\}} are determined by the spectral projec‐

tions of the measured observable by the Born statistical formula and the projection
postulate, respectively. This description has been a very familiar principle in quan‐tum mechanics, but is much more restrictive than what quantum mechanics allows.
In the modern measurement theory, the problem has been inve\overline{s}tigated as to what is
the most general description of measurement allowed by quantum mechanics. This
paper investigates the problem of the determination of all the possible measurementsof observables with nondegenerate spectrum and shows that the following conditions
are equivalent for measurements of nondegenerate observables: (i) The joint proba‐

bility distribution of the outcomes of successive measurements depends affinely on

the initial state. (ii) The apparatus has an indirect measurement model. (iii) The
state change is described by a positive superoperator valued measure. (iv) The statechange is described by a completely positive superoperator valued measure. (v) The

family of output states is a Borel family of density operators independent of the
input state and can be arbitrarily chosen by the choice of the apparatus.
2.

Measurement schemes

Let \mathcal{H} be a separable Hilbert space. A density operator on \mathcal{H} is a positive operator

on \mathcal{H} with unit trace . The state space of \mathcal{H} is the set S(\mathcal{H}) of density operators on \mathcal{H} .
In what follows, we shall give a general mathematical formulation for the statisticalproperties of measuring apparatuses. For heuristics, we shall consider a measuringapparatus which measures the quantum system described by the Hilbert space \mathcal{H} .
Every measuring apparatus has the output variable that gives the outcome on eachmeasurement. We assume that the output variable takes values in a standard Borel

space which is specified by each measuring apparatus. We shall denote by A(x) the
measuring apparatus with the output variable x taking values in a standard Borel
space A with the Borel a‐field B(\Lambda) . The statistical property of the apparatus A(x)93consists of the output distribution Pr\{x\in\triangle||\rho\} and the state reduction \rho\vdash\Rightarrow\rho\{x=x\} .
The output distribution Pr\{x\in\triangle||\rho\} describes the probability distribution of theoutput variable x when the input state is \rho\in S(\mathcal{H}) , where \triangle\in B(\Lambda) . The state
reduction \rho\mapsto\rho\{bx=x\} describes the state change from the input state \rho to the output

state \rho\{x=x\} , when the measurement leads to the output x=x . Since the output
variable can have a continuous probability distribution, the output state \rho_{\{x=x\}} isdetermined up to probability one with respect to the output distribution. The state

reduction determines the collective state reduction \rho\mapsto\rho\{x\in\Delta\} that describes theoutput state \rho\{x\in\Delta\} given that the output of the measurement is in a Borel set\triangle .

The collective state reduction is naturally related to the state reduction by the

integral formula\rho_{\{x\in\Delta\} =\frac{1}{Pr\{x\in\triangle| \rho\}}\int_{\Delta} \rho_{\{X=x}\}Pr\{x\in dX||\rho\}.

In order to justify the integral we need to require that the function \rho\mapsto\rho\{x=x\} isa density operator valued Borel function on A. Formal description of the statisticalproperties of measuring apparatuses will be given as follows.
Let A be a standard Borel space [1] with ‐field B(\Lambda) . A Borel family of statesfor (\mathcal{H}, \Lambda) is a family \{\rho_{x}\}_{x\in\Lambda} of density operators on \mathcal{H} such that the function\sigmax\mapsto Tr[A\rho_{x}] is a Borel function on A for every A\in \mathcal{L}(\mathcal{H}) , where \mathcal{L}(\mathcal{H}) stands

for the space of bounded operators on \mathcal{H} . Since \mathcal{H} is separable, every Borel familyof states is Bochner integrable with respect to every probability measure on A [2].
Denote by B(\Lambda, S(\mathcal{H})) the space of Borel families of states for (\mathcal{H}, \Lambda) . The state
space of A is the set S(\Lambda) of probability measures on B(\Lambda) . A measurement schemefor (\mathcal{H}, \Lambda) is the pair (P, Q) of a function \mathcal{P} from S(\mathcal{H}) to S(\Lambda) and a function 2
from S(\mathcal{H}) to B(\Lambda, S(\mathcal{H})) . The function /p is called the output probability scheme,

and 2 is called the state reduction scheme. Two measurement schemes (\mathcal{P}, Q) and\mathcal{P}=P' and they
(P', Q\prime) are said to be equivalent, in symbols (\mathcal{P}, Q)\cong(P', Q') , ifdiffer only on a null set of the probability measure \mathcal{P}\rho , i.e.,(\mathcal{P}\rho)\{x\in\Lambda|(Q\rho)(_{X)\neq(}QJ\rho)(x)\}=0,

for all \rho\in S(\mathcal{H}) .

The set of equivalence classes of measurement schemes for (\mathcal{H}, \Lambda) is denoted by\mathcal{H} is a\mathcal{M}(\mathcal{H}, \Lambda) and we define At ( \mathcal{H})=\bigcup_{\Lambda}\mathcal{M}(\mathcal{H}, \Lambda) . A measurement theory for
A
M
A
\mat
h
cal
{
M
}
(
\
mat
h
cal{H}) . An.
pair (A, M) consisting of a nonempty set and a function from toelement of A is called an apparatus. Hivery apparatus has its distinctive output vari‐able. We denote the apparatuses with output variables x, y,
by A(x), A(y), ,A(x)=A(y)x=yrespectively. We assume that
if and only if. The image of\ldots\ldotsA(x) by M is denoted by M(x) instead of M(A(x)) for simplicity, so that M(x)denotes the equivalence class of the measurement scheme corresponding to the ap‐

paratus A(x) . The output variablexor the apparatus A(x) is called \Lambda ‐valued if

M(x)\in \mathcal{M}(\mathcal{H}, \Lambda) . For any A(x)\in A , the equivalence class M(x) of the measure‐ment scheme is called the statistical property of A(x) . We shall denote by (P_{X}, Qx)

94a representative of M(x) ; in this case, we shall also write M(x)=[P_{X}, Qx] . The
function P_{x} is called the output probability of A(x) and Q_{X} the state reduction ofA(x) which is determined uniquely up to the output probability one. Two appara‐tuses A(x) and A(y) are said to be statistically equivalent, in symbols A(x)\cong A(y) ,if they have the same statistical property, i.e., M(x)--M(y) . Two measurement
theories (A, M) and (A', M') are said to be consistent if M(x)=M'(x) for all
A(x)\in A\cap A' . Two consistent measurement theories (A, M) and (A', M') are said
to be statistically equivalent, in symbols (A, M)\cong(A', M\prime) , if they have the same
set of the statistical equivalence classes of apparatuses, i.e., M(A)=M'(A') .
Suppose that a measurement theory (A, M) is given. Let A(x)\in A and M(x)=[\mathcal{P}_{X}, Q_{x}] . The probability distribution of the output variable x in the state \rho is definedbyPr\{x\in\triangle||\rho\}=(^{p_{X}}\rho)(\triangle)
for all \rho\in S(\mathcal{H}) and \triangle\in B(\Lambda) . This probability distribution is called the output
distribution of A(x) in \rho . The output states \{\rho_{\{xx}=\}\}_{x\in\Lambda} of A(x) in \rho is defined by\rho_{\{x=x}\}=(QX\rho)(x)for all \rho\in S(\mathcal{H}) and x\in\Lambda . This is defined uniquely up to the output probability
one.

Let \Lambda_{1},\ldots, \Lambda_{n} be standard Borel spaces. For j=1,

\ldots,n, let A(x_{j}) be a \Lambda_{j^{-}valued apparatus. A successive measurement in the input state \rho is a sequence ofmeasurements using A(x_{1}),A(x_{n}) such that the input state of the apparatus
A(x_{1}) is \rho and the input state of the apparatus A(X_{j+1}) is the output state of then-1 . The joint probability distribution of theapparatus A(x_{j}) for j=1,
outcomes of the successive measurements using A(x_{1}),A(x_{n}) in the input state\rho is naturally defined recursively by\ldots,\ldots,\ldots,Pr\{x_{1}\in\triangle_{1}, \ldots, x_{n}\in\triangle_{n}||\rho\}=

I_{\triangle_{1} ^{p_{r} \{x_{2}\in\triangle_{2}, \ldots, x_{n}\in\triangle_{n} | \rho\{x_{1}=x_{1}\}\}pr\{X_{1}\in dx_{1}| \rho\}(2.1)

for \triangle_{1}\in \mathcal{B}(\Lambda_{1}), \ldots, \triangle_{n}\in B(\Lambda_{n}) .

Now, we consider the following requirement for a measurement theory (A, M) :Mixing law of the n ‐joint probability distributions (nMLPD) : For any
A(x_{n}) of apparatuses with values in \Lambda_{1},\Lambda_{n)} respectively, if
sequence A(x_{1}),the input state \rho is the mixture of \rho_{1} and \rho_{2} such that \rho=\alpha\rho_{1}+(1-\alpha)\rho_{2} with\ldots,\ldots,0<\alpha<1 then, we havePr\{x_{1}\in\triangle_{1}, \ldots, X_{n}\in\triangle n||\rho\}\alpha Pr\{x_{1}\in\triangle_{1}, \ldots, x_{n}\in\triangle_{n}||\rho_{1}\}+(1-\alpha)Pr\{X_{1}\in\triangle_{1}, \ldots, x_{n}\in\triangle_{n}||\rho_{2}\}=for all \rho\in S(\mathcal{H}) and \triangle_{1}\in B(\Lambda_{1}),\ldots, \triangle_{n}\in B(\Lambda_{n}) .

(2.2)

95

This requirement is justified as follows. If the system is in the state \rho_{1} withprobability \alpha and in the state \rho_{2} with probability 1-\alpha then the joint probability
should be the mixture of the joint probabilities in the right hand side. On the other
hand, the state of the system is described by the density operator \rho and the joint
probability is also expressed as in the left hand side, so that the above equality
should hold.

It can be seen by induction that the relationPr\{x_{1}\in\triangle_{1}, \ldots, x_{n-1}\in\triangle_{n-1}, X_{n} \in\Lambda_{n}||\rho\}=Pr\{x_{1}\in\triangle_{1}, \ldots, x_{n-1}\in\triangle_{n -1}||\rho\}holds generally, so that the nMLPD implies the (n-1)MLpD .

An observable of \mathcal{H} is a self‐adjoint operator (densely defined) on \mathcal{H} . We denoteby E^{A} the spectral measure of an observable A . According to the Born statisticalformula, we say that an R‐valued apparatus A(x) measures an observable A if

Pr\{x\in\triangle||\rho\}=Tr[E^{A}(\triangle)\rho](2.3)

for all \rho\in S(\mathcal{H}) and \triangle\in B(R) , where R stands for the real number field. A
measurement theory (A, M) is called nonsuperselective if for any observable A there
is at least one apparatus measuring A .
A \Lambda ‐valued observable of \mathcal{H} is a projection valued measure E from B(\Lambda) to \mathcal{L}(\mathcal{H})such that E(\Lambda)=I ; A‐valued observables are also called spectral measures, forwhich we refer the reader to [3]. The Born statistical formula is generalized asfollows. We say that a A‐valued apparatus A(x) measures a \Lambda ‐valued observable E
if

(2.4)

Pr\{x\in\triangle||\rho\}=Tr[E(\triangle)\rho]

for all \rho\in S(\mathcal{H}) and \triangle\in B(\Lambda) . According to spectral theory, the observables A of\mathcal{H} are in one- t_{o^{-}one} correspondence with the R ‐valued observables E^{A} for (\mathcal{H}, R) .

From (2.3) and (2.4), an R‐valued apparatus measures an observable A if and only
if it measures R‐valued observable E^{A} . Given a finite sequence (A_{1}, \cdots, A_{n}) of mu‐
tually commuting observables, the simultaneous measurement of those observables
is considered as a measurement of R^{n} ‐valued observable E^{(A_{1},\ldots,A_{n})} such that

E^{(A,\ldots,A_{n})}1(\triangle 1\cross\cdots\cross\triangle)n=E^{A_{1}} (\triangle)\cdots EAn(\triangle_{n})(2.5)\triangle_{n}\in B(R) .
A probability operator valued measure (POVM) for (\dot{\mathcal{H} , \Lambda) is a positive operator
valued measure F from \mathcal{B}(\Lambda) to \mathcal{L}(\mathcal{H}) such that F(A)=I ; for general theory werefer the reader to [4]. Since the correspondence \triangle\mapsto Tr[F(\triangle)\rho] is a probability
measure, the Born statistical formula can be generalized formally to POVMs as
folIows. We say that a A‐valued apparatus A(x) measures a POVM F for (\mathcal{H}, \Lambda) if

for any \triangle_{1},\ldots,.

Pr\{x\in\triangle||\rho\}=Tr[F(\triangle)\rho]

(2.6)

96

for all \rho\in S(\mathcal{H}) and \triangle\in B(\Lambda) . Conventional measurement theory is devoted
to measurements of observables but modern theory extends the notion of measure‐ments to measurements of POVMs [5, 6, 7, 8, 9]. We shall describe in the following
requirement the essential feature of the modern approach.

Existence of probability operator valued measures (EPOVM): For any
apparatus A(X)_{l} there exists a POVM F_{x} uniquely such that A(x) measures F_{x} .
The EPOVM is justified by the following theorem proved in [7].
Theorem 2.1. For any measurement theory.(A, M) , the EPOVM is equivalent

to the IMLPD.

3.

Collective measurement schemes

In order to provide an alternative definition of measurement schemes, we call a pair(\mathcal{P}, \mathcal{R}) as a collective measurement scheme if \mathcal{P} is a function from S(\mathcal{H}) to S(\Lambda)
and \prime\mathcal{R} is a function from B(\Lambda)\cross S(\mathcal{H}) to S(\mathcal{H}) satisfying\sum_{n}(\mathcal{P}\rho)(\triangle_{n})\mathcal{R}(\triangle_{n}, \rho)= R(\Lambda, \rho)(3.1)

for any countable Borel partition \{\triangle_{1}, \triangle_{2}, \ldots\} of A and \rho\in S(\mathcal{H}) , where the sum
is convergent in the trace norm. The function \mathcal{R} is called the collective reduction
scheme. Two collective measurement schemes (\mathcal{P}, \mathcal{R}) and (\mathcal{P}', \mathcal{R}\prime) are said to beequivalent, in symbols (P,\mathcal{R})\cong(P',R') , if \mathcal{P}=P' and R(\triangle, \rho)=\mathcal{R}'(\triangle, \rho) for all\triangle\in B(\Lambda) with (P\rho)(\triangle)>0 .
Theorem 3.1. The relation( \mathcal{P}\rho)(\triangle)\mathcal{R}(\triangle, \rho)=\int_{\Delta}(Q\rho) (X)d(P\rho)(X)(3.2)

where \triangle\in B(\Lambda) and \rho\in S(\mathcal{H})_{f} sets up a one-t_{0} ‐one correspondence between theequivalence classes of measurement schemes (P, Q) and the equivalence classes ofcollective measurement schemes (\mathcal{P}, \mathcal{R}) .

Let (\mathcal{P}, Q) be a measurement scheme for (\mathcal{H}, \Lambda) . The collective measurement

scheme (P, \mathcal{R}) defined by (3.2) up to equivalence is called the collective measurement
scheme induced by (\mathcal{P}, Q) and the function \prime\mathcal{R} is called the collective reduction scheme
induced by (P, Q) .

Let (A, M) be a measurement theory satisfying the IMLPD. For any A(x)\in A ,define \mathcal{R}_{x} to be the collective reduction scheme induced by (P_{x}, QX) . The functions\mathcal{R}_{x} is called the collective reduction of the apparatus A(x) . The collective output
states \{\rho\{_{X\in\Delta}\}\}_{\Delta\beta}\epsilon(\Lambda) of A(x) in \rho is defined by\rho\{x\in\Delta\}=\mathcal{R}X(\triangle, \rho)

97

for all \rho\in S(\mathcal{H}) and \triangle\in B(\Lambda) .

From the IMLPD, for any \triangle, \Delta'\in B(\Lambda) and \rho\in S(\mathcal{H}) , we havePr\{x\in\triangle, y\in\triangle^{J}||\rho\}=\int_{\Delta}Tr[F_{y}(\triangle\prime)\rho\{x=x\}]Pr\{x\in dx||\rho\}[F_{y}( \triangle\prime)\int_{\Delta}\rho\{x=x\}Pr\{X\in dx||\rho\}]

=

Tr

=

Pr\{y\in\triangle J||\rho_{\{x\in}\triangle\}\}^{p\{_{X}\in\triangle|}r|\rho\}.

Thus, the collective output state is characterized by

Pr\{x\in\triangle, y\in\triangle^{J}||\rho\}=Pr\{y\in\triangle^{J} | \rho_{\{X\in\triangle}\}\}pr\{X\in\triangle||\rho\} ,(3.3)

Pr\{y\in\triangle;| \rho_{\{x}\in\Delta\}\}=^{p} r\{y\in\triangle\prime|x\in\triangle||\rho\} ,

(3.4)

or

where the right hand side is the conditional distribution defined by

Pr\{y\in\triangle\prime|x\in\triangle|\rho\}=\frac{Pr\{_{X\in\triangle, \triangle}y\in|J\rho\} {Pr\{x\in\triangle|\rho\} .

(3.5)

Similarly, in the measurement theory satisfying (n-1)MLpD we have the fol‐lowing relationPr\{x_{1}\in\triangle, \ldots,x_{n}\in\triangle_{n}||\rho\}
=

(3.6)

Pr\{x_{n}\in\triangle_{n}||\rho fx_{1\in\Delta}1,\ldots,Xn-1\in\Delta n-1\}\} \cross\cdots\cross Pr\{_{X_{2} \in\triangle_{2}| \rho\{x_{1}\epsilon\Delta_{1}\}\}Pr\{_{X}1 \in\triangle_{1}| \rho\}(3.7)
for all \triangle_{i}\in B(\Lambda_{j}) and j=1, \cdots, n , where\rho\{x_{1}\in\Delta 1,\ldots,x_{n-}1\epsilon\triangle_{n-}1\}=(\cdots(\rho_{\{ \triangle_{1} X_{1}\in)\cdots)\{Xn-1\in\Delta n-1\}
4.

.

Davies‐Lewis postulate

In what follows, we shall introduce some mathematical terminology independent ofparticular measurement theory. A superoperator for \mathcal{H} is a bounded linear transfor‐
mation on the space \tau c(\mathcal{H}) of trace class operators on \mathcal{H} . A dual superoperator for\mathcal{H} is an ultraweakly continuous linear transformation on the space \mathcal{L}(\mathcal{H}) of bounded
operators on \mathcal{H} . The dual of a superoperator L is the dual superoperator L^{*} definedby \langle L^{*}A, \rho\rangle=\langle A, L\rho\rangle for all A\in \mathcal{L}(\mathcal{H}) and \rho\in\tau c(\mathcal{H}) , where \langle \cdot, \cdot) stands for the
duality pairing defined by \langle A, \rho)=Tr[A\rho] for all A\in \mathcal{L}(\mathcal{H}) and \rho\in\tau c(\mathcal{H}) . The
correspondence L\mapsto L^{*} is a one-t_{0} ‐one correspondence between the superoperatorsand the dual superoperators. A superoperator or a dual superoperator is called pos‐
itive if it maps positive operators to positive operators. We denote by \mathcal{L}(\tau c(\mathcal{H}) the
space of superoperators and \mathcal{P}(\tau c(\mathcal{H}) the space of positive superoperators. Positive

98

contractive superoperators are called operations [10]. From Corollary 3.2.6 of [11], apositive superoperator T is an operation if and only if 0\leq T^{*}(I)\leq I , or equivalently
0\leq Tr[T(\rho)]\leq 1 for all \rho\in S(\mathcal{H}) [ 6 , Lemma 2.2.2].
A positive superoperator valued (PSV) measure is a mapping \mathcal{I} from B(\Lambda) to\mathcal{P}(\tau c(\mathcal{H}) such that if \{\triangle_{1}, \triangle_{2}, \ldots\} is a countable Borel partition of \Lambda , then wehave\mathcal{I}(\Lambda)\rho=\sum_{n}\mathcal{I}(\triangle_{n})\rho
for any \rho\in\tau c(\mathcal{H}) , where the sum is convergent in the trace norm. The PSV measure\mathcal{I} is said to be normalized if it satisfies the further condition

Tr[\mathcal{I}(\Lambda)\rho]=Tr[\rho]
for any \rho\in\tau c(\mathcal{H}) . Normalized PSV measures are called instruments [12, 6] forshort. Accordingly, we will not use “instrument” for a synonym for “apparatus”
but eventually it will turn out that an “instrument” corresponds to the statisticalequivalence class of an “apparatus”.
Let \mathcal{I} be an instrument for (\Lambda, \mathcal{H}) . Then, the function \triangle\mapsto Tr[\mathcal{I}(\triangle)] is a
probability measure on B(\Lambda) for all \rho\in S(\mathcal{H}) . The relation

X(\triangle)=\mathcal{I}(\triangle)*I(4.1)

for all \triangle\in B(\Lambda) , defines a POVM for (\mathcal{H}, \Lambda) , called the POVM of \mathcal{I} . The relation
T=\mathcal{I}(\Lambda) defines a trace preserving operation, called the total operation of \mathcal{I} .
A measurement theory (A, M) is said to satisfy the Davies‐Lewis postulate if itsatisfies the follows postulate.

Davies‐Lewis postulate: For any apparatus A(x)_{f} there is a normalized PS V
measure \mathcal{I}_{x} satisfying the following relations for any \rho\in S(\mathcal{H}) and Borel set \triangle\in
B(\Lambda) :(DL1)Pr\{x\in\triangle||\rho\}=Tr[\mathcal{I}_{X}(\triangle)\rho](DL2)\rho_{\ x\in\Delta\} =\frac{\mathcal{I}_{x}(\triangle)\rho} {Tr[\mathcal{I}_{X}(\triangle)\rho]}.

.

It should be noted that the original Davies‐Lewis formulation [12] does not men‐tion the existence of state reduction Q_{X} so that their theory is formulated for theapparatuses associated with the collective measurement scheme. However, we have

proved in Theorem 3.1 the formulation based on the measurement scheme as thepr imitive notion is equivalent to the formulation based on the collective measure‐-ment scheme as the primitive notion. Thus, the above presentation of Davies‐Lewis
postulate is fully equivalent to their original formulation.
From Theorem 3.1, the normalized PSV measure \mathcal{I}_{x} determines the output state\rho_{t^{x=}x}\} uniquely up to equivalence by

99( DL3)\int_{\Delta}\rho_{\{_{X=}x}\}Tr[d\mathcal{I}_{X}(X)\rho]=\mathcal{I}_{x} (\triangle)\rho.

From (DL1) , the Davies‐Lewis postulate implies IMLPD. Although the Davies‐
Lewis description of measurement is quite general, it is not clear by itself whether
it is general enough to allow all the possible measurements. The following theorem
shows indeed it is the case.

Theorem 4.1. For any nonsuperselective measurement theory, the Davies‐

Lewis postulate is equivalent to the

2MLPD .

A measurement theory (A, M) is called a statistical measurement theory if it isnonsuperselective and satisfies

2MLPD .

Theorem 4.2. Every statistical measurement theory satisfies

positive integer n .

nMLPD forall

In the rest of this paper, we shall take it for granted that every measurement

theory consistent with quantum mechanics should be a statistical measurement the‐
ory

5.

.

Measurements of observables

In the preceding section, we have shown that in any statistical measurement theory
the statistical equivalence class of an apparatus is represented by an instrument
which determines the statistical property of the apparatus, although it depends on

the particular measurement theory whether a given instrument (in mathematics)

has the corresponding apparatus (in physics).
In what follows, we shall explore mathematical properties of instruments (nor‐malized PSV measures) independent of particular measurement theory. An instru‐
ment \mathcal{I} for (\Lambda, \mathcal{H}) is said to be decomposable if \mathcal{I}(\triangle)^{*}A=X(\triangle)T^{*}(A) for all\triangle\in B(\Lambda) and A\in \mathcal{L}(\mathcal{H}) , where X is the POVM of \mathcal{I} and T the total opera‐tion. For a given A‐valued observable E for (\Lambda, \mathcal{H}) , an instrument \mathcal{I} is said to be
E ‐compatible if the POVM of \mathcal{I} is E. i.e., \mathcal{I}^{*}(\triangle)I=E(\triangle)\} for all\triangle\in B(\Lambda) ; suchan instrument is also called observable measuring. For an observable A , an instru‐
ment is called A ‐compatible if it is E^{A} ‐compatible. The following theorem showsin particular that every observable measuring instrument is decomposable (see [9,Proposition 4.3] for the case of completely positive instruments).
Theorem 5.1. Let E be a \Lambda ‐valued observable of \mathcal{H} . Let \mathcal{I} be an E‐compatible
instrument and T its total operation. Then, we have the following statements.
(i) For any \triangle\in B(\Lambda) and \rho\in\tau c(\mathcal{H}) , we have\mathcal{I}(\triangle)\rho=\tau[E(\triangle)\rho]=T[\rho E(\triangle)]= T[E(\triangle)\rho E(\triangle)] .

(5.1)(ii) For any \triangle\in B(\Lambda) and B\in \mathcal{L}(\mathcal{H}) , we have\mathcal{I}(\triangle)^{*}B=E(\triangle)T^{*}(B)=T^{*}(B)E(\triangle)= E(\triangle)\tau*(B)E(\triangle) .

(5.2)

100

It follows from (5.2) that the range of \tau* is included in the commutant of therange of E , i.e., T^{*}[\mathcal{L}(\mathcal{H})]\subseteq E[B(\Lambda)]' . An operation L is called E ‐compatible if
L^{*}[\mathcal{L}(\mathcal{H})]\subseteq E[\mathcal{B}(\Lambda)]' . For an observable A , the operation L is called A‐compatible
if it is E^{A} ‐compatible.

All the E ‐compatible instruments are determined as follows.
Theorem 5.2. Let E be an \Lambda ‐valued observable of \mathcal{H} . The relation(5.3)\mathcal{I}(\triangle)\rho=^{\tau[E(\triangle})\rho]

for all \triangle\in B(\Lambda) and \rho\in\tau c(\mathcal{H}) sets up a one‐to‐one correspondence between the
E ‐compatible instruments \mathcal{I} and the E ‐compatible trace preserving operations T .

From the above theorem, in every statistical measurement theory we have thefollowing: For any apparatus A(x) measuring a \Lambda ‐valued observable E , there is an
E ‐compatible trace preserving operation T such that the statistical property of A(x)is represented as follows.

output distribution:collective output state:
6.

Pr\{x\in\triangle||\rho\}=Tr[E(\triangle)\rho](5.4)\underline{T[E(\triangle)\rho]}(5.5)\rho\{x\in\Delta\}=Tr[E(\triangle)\rho]Measurements of nondegenerate observables

Let E be a A‐valued observable for \mathcal{H} . We say that E is nondegenerate if thecommutant of E is abelian. The spectral measure E^{A} of an observable A of \mathcal{H}is nondegenerate if and only if A has nondegenerate spectral. Two Borel families\{\rho_{x}\}_{x\in\Lambda} and \{\rho_{x}'\}x\epsilon\Lambda of density operators are said to be E ‐equivalent if they differ
only on an E ‐null set, i.e.,E\{x\in\Lambda|\rho_{x}\neq\rho_{x}'\}=0Theorem 6.1. Let
Bochner integral formulaE.

be a nondegenerate \Lambda ‐valued observable of \mathcal{H} .

T \rho=\int_{\Lambda}\rho_{x}Tr[\rho dE(X)]

The

(6.1)

for all \rho\in\tau c(\mathcal{H}) sets up a one- t_{o^{-}one} correspondence between the E‐compatible
trace preserving operations T and the E ‐equivalence classes of the Borel families\{\rho_{x}\}_{x\epsilon\Lambda} of density operators indexed by A.

Let us consider a statistical measurement theory (A, M(x)) . Let \rho_{\{X=x} } be theoutput state for the input state \rho of an apparatus A(x) measuring a A‐valued ob‐
servable E . Then, the apparatus A(x) has the instrument \mathcal{I}_{x} and the E‐compatible

101

operation T satisfying (5.3) and (5.5). By Theorem 6.1 there is a Borel family\{\rho_{x}\}_{x\in\Lambda} of density operators satisfying (6.1), so that we have

\int_{\Delta}\rho_{\{x=x}\}Tr[d\mathcal{I}X(X)\rho]=\int_{\Delta}\rho_{x} Tr[d\mathcal{I}_{x}(X)\rho]
for all \triangle\in B(\Lambda) . It follows that the output state for the output

x=x\rho_{\{x=x\}}=\rho_{x}is given by(6.2)

up to the output probability one.
From the above theorem, in any statistical measurement theory we conclude thefollowing: For any apparatus A(x) measuring an \Lambda ‐valued observable E , there is a
Borel family \{\rho_{x}\}_{x\in\Lambda} of density operators uniquely up to E ‐equivalence such thatthe statistical property of A(x) is represented as follows.

output distribution:
output state:Pr\{x\in\triangle||\rho\}=Tr[E(\triangle)\rho]\rho_{\{x=x\}}=\rho_{x}

(6.3)
(6.4)

It follows that the problem of determining all the possible quantum state re‐ductions arising in the measurement of a nondegenerate A‐valued observable E is
reduced to the problem as to what Borel family \{\rho_{x}\} of states can be obtained

from the measurement of E . In order to obtain the answer to this question, in thenext section we shall consider indirect measurement models and ask what familiesof states can be obtained from those models.
7.

Indirect measurement models

An indirect measurement model for (\Lambda,\mathcal{H}) is defined to be a 4‐tuple (\mathcal{K}, \sigma, U, E)

consisting of a separable Hilbert space \mathcal{K} , a density operator \sigma on \mathcal{K} , a unitary
operator U on \mathcal{H}\otimes \mathcal{K} , and a A‐valued observable E of \mathcal{K} . For a given indirect
measurement model (\mathcal{K}, \sigma, U, E) the relation\mathcal{I}(\triangle)\rho=Tr_{\mathcal{K}}[(I\otimes E(\triangle)) U(\rho\otimes\sigma)U^{*}] ,(7.1)

where \triangle\in B(\Lambda) and \rho\in\tau c(\mathcal{H}) , defines an instrument \mathcal{I} for (\Lambda, \mathcal{H}) , which is called
the instrument determined by (\mathcal{K}, \sigma, U, E) .
Let S be the quantum system described by the Hilbert space \mathcal{H} . If an apparatus
A(x) for (\mathcal{H}, \Lambda) has the the indirect measurement model (\mathcal{K}, \sigma, U, E) , the process ofmeasurement on S using the apparatus A(x) is described as follows [13, 14] . The
measurement is carried out by the interaction between the measured system S andthe apparatus A(x) for the finite time interval from the time t to the time t+\triangle t .
We assume that the measured system is free from the apparatus A (x) after the time
t+\triangle t . The time t is called the time of measurement. The time t+\triangle t is called thetime just after measurement. We define the probe system of the apparatus A(x) to102

be the smallest subsystem P of the apparatus A(x) such that the composite system
S+P is isolated from t to t+\triangle t . We assume that the probe P is a quantum systemdescribed by the Hilbert space \mathcal{K} .
At the time t , the system S is supposed to be in the input state \rho and the probe
P is in the fixed state \sigma . The time evolution of the composite system S+P fromt to t+\triangle t is described by the unitary operator U on \mathcal{H}\otimes \mathcal{K} . Hence, at the time
t+\triangle t the composite system S+P is in the state U(\rho\otimes\sigma)U^{*} . The outcome ofthis measurement is obtained by the measurement of the A‐valued observable E of
P at the time t+\triangle t locally at the system P. Thus, the output distribution of theapparatus A(x) is given by

Pr\{x\in\triangle||\rho\}=Tr[(I\otimes E(\triangle))U(\rho\otimes\sigma)U^{*}](7.2)

for all \triangle\in B(\Lambda) . The apparatus A(e) to measure E is supposed to be a part of theapparatus A(x) comprising of the subsequent stages later than P , so that we may
assume A(x)=P+A(e) and that the output variables x and e might correspondto the same macroscopic variable, though we need to distinguish them by different
symbols.

In order to determine the output state of A(x) , suppose that at the time t+\triangle t ,the above measurement on S using A(x) is followed immediately by the measurementof an observable B of S using another apparatus A(b) . Then, the joint probability

distribution of the outcomes x and b of this successive measurement coincides withthe joint probability distribution of the outcomes e and b of the E ‐measurement on
P and the B ‐measurement on S in U(\rho\otimes\sigma)U^{*} , i.e.,Pr\{x\in\triangle, b\in\triangle'| \rho\}=Pr\{e\in\triangle, b\in\triangle^{J} | U(\rho\otimes\sigma)U^{*}\} .

(7.3)

Since the E ‐measurement is carried out locally at the system P , thejoint probability
distribution of their outcomes is given by

Pr\{e\in\triangle, b\in\triangle^{J}||U(\rho\otimes\sigma)U^{*}\}==

Tr[(EB(\triangle\prime)\otimes E(\triangle))U(\rho\otimes\sigma)U^{*}]
Tr[E^{B}(\triangle\prime)Tr\kappa[(I\otimes E(\triangle))U(\rho\otimes\sigma)U^ {*}]] ,(7.4)

where Tr_{\mathcal{K} stands for the partial trace over \mathcal{K} . For the detailed justification of theabove formula, we refer the reader to [15, 16, 17, 14]. From (3.3), the (collective)

output state \rho\{X\in\triangle\} satisfies

Pr\{X\in\triangle, b\in\triangle^{J}||\rho\}=Tr[E^{B}(\triangle J)p_{r} \{x\in\triangle||\rho\}\rho\{X\in\triangle\}] .

(7.5)

From (7.2)-(7.5) , we determine the output state as\rho_{\{x\in\Delta\}}=\underline{Tr_{\mathcal{K} [(I\otimes E(\triangle) U(\rho\otimes\sigma)U^{*}]}Tr [(I\otimes E(\triangle))U(\rho\otimes\sigma)U^{*}]

(7.6)

103

We have, thus, demonstrated that if the apparatus A(x) has the indirect mea‐
surement model (\mathcal{K}, \sigma, U, E) , then A(x) has the instrument \mathcal{I}_{x} defined by\mathcal{I}_{x}(\triangle)\rho=Tr_{\mathcal{K}}[(I\otimes E(\triangle)) U(\rho\otimes\sigma)U^{*}] ,(7.7)

where \triangle\in B(\Lambda) and \rho\in\tau c(\mathcal{H}) , so that the statistical property of A(x) is describedby \mathcal{I}_{x} with relations (DL1) and (DL2) . The above instrument \mathcal{I}_{x} is called the

instrument of A(x) and is actually the instrument \mathcal{I} of (\mathcal{K}, \sigma, U, E) defined by (7.1).
Now, we consider the following hypothesis.

Indirect measurability hypothesis: For any indirect measurement model(\mathcal{K}, \sigma, U, E) , there is an apparatus A(x) with the instrument \mathcal{I}_{X} defined by (7.7).
In general, an instrument \mathcal{I} is said to be realized by an indirect measurement

model (\mathcal{K}, \sigma, U, E) if (7.1) holds for any \rho\in\tau c(\mathcal{H}) . In this case, the instrument\mathcal{I} is called

unitarily realizable. Under the indirect measurability hypothesis, every
unitarily realizable instrument represents the statistical property of an apparatus.
In the sequel, a statistical measurement theory (A, M) is called a standard mea‐
surement theory, if it satisfies the indirect measurability hypothesis. It is natural
to consider that any standard measurement theory is consistent with the standard
formulation of quantum mechanics by the following argument. In the standard for‐mulation of quantum mechanics, the system P can be prepared in any \sigma with given
finite accuracy and the unitary operator U can be realized as a time evolution withgiven finite accuracy in principle, otherwise a certain superselection rule would be
required. Thus, it suffices to show that the A‐valued observable can be measured
by an apparatus. We have already argue that the statistical measurement theory
is consistent with the standard formulation of quantum mechanics. Thus, for any
observable A we have an apparatus A(a) to measure A . Since A is a standard Borel
space, there is a Borel subset \Lambda' of R Borel isomorphic with A. It follows that there
is a Borel function A arrow R and an observable A such that

E(\triangle)=E^{A}(f^{-1}(\triangle))

for all \triangle\in B(\Lambda) . Thus, the apparatus A(z) measuring E is constructed by anapparatus A(x) measuring A with output processing z=f(x) .
8.

Complete positivity

Let D=\tau c(\mathcal{H}) or D=\mathcal{L}(\mathcal{H}) . A linear transformation L on D is called completely
A_{n}\in D and vectors
positive (CP) iff for any finite sequences of operators A_{1},\xi_{1},\xi_{n}\in \mathcal{H} we have\ldots,\ldots,\sum_{ij}\langle\xi_{i}|L(A_{i}*A_{j})|\xi j\rangle\geq 0.

The above condition is equivalent to that L\otimes I maps positive operators in thealgebraic tensor pr.oductD\otimes \mathcal{L}(\mathcal{K}) to positive operators in D\otimes \mathcal{L}(\mathcal{K}) for any Hilbert
space \mathcal{K} . Obviously, every CP superoperators are positive. A superoperator is CP
if and only if its dual superoperator is CP . An instrument \mathcal{I} is called completely
positive (CP) if every operation \mathcal{I}(\triangle) is CP . It can be seen easily from (7.1) that

104

unitarily realizable instruments are CP . Conversely, the following theorem, provedin [18, 9] , asserts that every CP instrument is unitarily realizable.

Theorem 8.1. For any CP instrument \mathcal{I} for (\Lambda, \mathcal{H}) , there is a separable Hilbert
space \mathcal{K}_{f} a unit vector \Phi in \mathcal{K} , a unitary operator U on \mathcal{H}\otimes \mathcal{K} , and a A‐valued
observable E of \mathcal{K} satisfying the relation\mathcal{I}(\triangle)\rho=Tr_{\mathcal{K}}[(I\otimes E(\triangle)) U(\rho\otimes|\Phi\rangle\langle\Phi|)U^{*}]
for all \triangle\in B(\Lambda) and \rho\in\tau c(\mathcal{H}) .

The following theorem shows that the complete positivity of observable measur‐ing instruments is determined by their total operations.
Theorem 8.2. Let E be a \Lambda ‐valued observable. Then_{f} an E ‐compatible instru‐
ment is CP if and only if its total operation is CP .

From the above theorem, in any standard measurement theory we conclude thefollowing statement [9]: The statistical equivalence classes of apparatuses A(x) mea‐suring a \Lambda ‐valued observable E with indirect measurement models are in one-t_{0} ‐onecorrespondence with the E ‐compatible trace preserving CPoperationS_{\rangle} where the sta‐tistical property is represented by (5.4) and (5.5).
For the case of nondegenerate observables, we have the following simple charac‐terizations.

Theorem 8.3. Let E be a nondegenerate \Lambda ‐valued observable. Then_{f} every E‐

compatible operation is completely positive. Every E ‐compatible instrument is com‐pletely positive.

From the above theorem and Theorem 8.1, in the statistical measurement theory

we conclude: Every apparatus measuring a nondegenerate (\Lambda ‐valued) observable isstatistically equivalent to the one having an indirect measurement model.
Every Borel family \{\rho_{x}\} of density operators indexed by A defines an E‐
compatible trace preserving operation by Theorem 6.1, and it is automatically com‐pletely positive so that it is realized by an indirect measurement model. Thus,
we conclude the following: The statistical equivalence classes of apparatuses A(x)measuring a nondegenerate \Lambda ‐valued observable E are in one-t_{0} ‐one correspondencewith the Borel family \{\rho_{x}\} of density operators indexed by \Lambda_{f} where the statisticalproperty is represented by (6.3) and (6.4).
References[1] W. Arveson, An Invitation to C^{*}‐Algebras, Springer, New York, 1976.

[2] E. Hille and R. S. Phillips, Functional Analysis and Semigroups, Coll. Publ.,
Amer. Math. Soc., Providence, R.I., 2nd edition, 1957.

105[3] P. R. Halmos, Introduction to Hilbert Space Theory and the Theory of Spcetral
Multiplicity, Chelsea, New York, 1951.

[4] S. K. Berberian, Notes on Spectral Theory, Van Nostrand, Princeton, NJ, 1966.
[5] C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press,
New York, 1976.

[6] E. B. Davies, Quantum Theory of Open Systems, Academic Press, London,
1976.

[7] M. Ozawa, Optimal measurements for general quantum systems, Reports on
Math. Phys. 18, 11‐28 (1980).

[8] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North‐Holland, Amsterdam, 1982.

[9] M. Ozawa, Quantum measuring processes of continuous observables, J. Math.
Phys. 25, 79‐87 (1984).
[10] R. Haag and D. Kastler, An algebraic approach to quantum field theory, J.
Math. Phys. 5, 848‐861 (1964).

[11] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical
Mechanics I, Springer, New York, 1979.

[12] E. B. Davies and J. T. Lewis, An operational approach to quantum probability,
Commun. Math. Phys. 17, 239‐260 (1970).

[13] M. Ozawa, Measurements of nondegenerate discrete observables, Phys. Rev. A(to appear), [online preprint: LANL quant‐ph/0003033].
[14] M. Ozawa, Operations, disturbance, and simultaneous measurability, [online
preprint: LANL quant‐ph/0005054].

[15] M. Ozawa, Quantum state reduction and the quantum Bayes principle, in
Quantum Communication, Computing, and Measurement edited by O. Hirota,
A. S. Holevo, and C. M. Caves, pages 233‐241, Plenum, New York, 1997.

[16] M. Ozawa, Quantum state reduction: An operational approach, Fortschr.
Phys. 46, 615‐625 (1998).

[17] M. Ozawa, Operational characterization of simultaneous measurements inquantum mechanics, Phys. Lett. A (to appear), [online preprint: LANL quant‐ph/9802039].

[18] M. Ozawa, Conditional expectation and repeated measurement of continuous
quantum observables, in Probability Theory and Mathematical Statistics, editedby K. Itô and J. V. Prohorov, pages 518‐525, Lecture Notes in Math. 1021,
Springer, Berlin, (1983).