ICRR,
Kashiwa
7/5, 2012

Uncertainty Relation and Weak Value 	Akio	
  Hosoya	
  	
  	
  	
  (Tokyo	
  Tech)	  	
  1. Introduction
Uncertainty Relation, Heisenberg vs Ozawa
2. Experimental Verification of Ozawa’s inequality
by Hasegawa et al.
3. A formal theory of (Weak) Value
Main theorem☆ The weak value is a “conditional value” of an observable☆ Derivation of Born’s rule
4. Summary
arXiv1104.1873
J.Phys.A44:415303,2011
with Minoru Koga

1.	
  Uncertainty	
  Principle	
	
  	
  In	
  the	
  early	
  20th	  century	  Bohr,	
  Heisenberg,	
  Schroedinger	
  and	
  other	
  people	
  hotly	
  discussed	
  possible	
  moDon	  of	
  electron	
  in	
  an	
  atom,	
  when	  it	
  makes	
  a	
  transiDon	
  between	  two	
  quantum	
  states.	
  	
  
	
  	
  Burkhardt	
  Drude, one	  of	  Heisenberg’s	
  classmates	
  even	  proposed	
  a	
  gamma-­‐ray	
  microscope	
  to	
  seLle	  the	
  issue.	
  

	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  	
  W.	
  Heisenberg,	
  “DER	  TEIL	  UND	  DAS	
  GANZE”(1969)	
  	
  	
  	
  

“Newton”(2012)	

ε(q)	
  η(p)	
  >	
  	
  	
  	
  	
  h/4π	This motivated Heisenberg to examine the gedanken experiment
for the error ε(q) in the measurement of the position q of anelectron and the disturbance η(p) of its momentum p by themeasurement of q. He concluded that we cannot define
any “trajectory of an electron”.
W. Heisenberg: Z.Phys.43,172 (1927)

This inequality should not be confused with the
Kennerd-Robertson inequality (1928),

σ(q) σ (p) ≥h/4π (#)
where σ(q) is the standard deviation of
the initial wavefunction before measurement.

Unfortunately many textbooks mixed up these two.
(Dirac mentioned only # in his textbook but carefully.)	

Ozawa reformulated Heisenberg’s uncertainty principle, on thebasis of rigorous measurement theory of the Completely
Positive (CP) map.

ε(A) η(B) +ε(A)σ(B)+σ (A) η(B)≥|<ψ|[A,B]|ψ>|/2
where ε(A) is the error in the measurement of A, η(B) is thedisturbance of B by the measurement of A.
σ (A) and σ(B) are the quantum fluctuations of A and B in thestate |ψ> i.e., the standard deviations.

M.Ozawa: Phys.Rev. A67,042105 (2003)	

(sketch of proof)
The quantum measurement theory based on Complete
Positive Map shows all physical measurement can be
described by an unitary transformation U defined in
the Hilbert space H= Hsystem⊗Hdetector of the systemand detector.

|ψ>	

system	detector	
|ξ>	

M. Ozawa, Ann. Phys. (N.Y.), 311, 350–416 (2004).
See also Chuang and Nielsen textbook	CP map
A density operator ρ is called a positive operator, since
the eigen values are non-negative because of the probabilistic
interpretation. Any map which sends positive operators topositive operators is called a positive map. Obviously
physical operation Λ to a state has to be a positive map.
It is natural to further demand that Λ is completely positive in thesense that(Λ⊗1)(ρ⊗σ) ≥ 0

for all σ.

The map Λ should be positive even if there are unaffected
experimental apparatuses outside your own laboratory.

This seemingly obvious requirement is actually verystrong and has been shown to be equivalent to(1) Kraus representation

ρ	
  	
  	
  Λ	
  (ρ)	
  	
  =	
  ∑	
  	
  i	
  Ai	
  ρA*i	
  	
  	
  ,	
  
with	
  	
  ∑	
  i	
  A	
  *	
  i	
  Ai=1.	
  The	
  probability	
  of	
  the	
  measurement	
  outcome	
  i	
  is	
  	
  

	
  	
  	
  	
  	
  	
  pi	
  =	
  	
  Tr[ρEi],	
  	
  	
  	
  
	
  	
  	
  	
  	
  	
  Ei:=A	
  *	
  i	
  Ai	
  	
  	
  	
  PosiDve	
  Operator	
  Valued	
  Measure	
  (POVM)	
  	  (2) Existence of the unitary operator in themeasurement model of the system and detector suchthat Ai=<ξ|U|i>. 	

Let A be an observable acting on Hsystem we are goingto measure. Introduce a detector observable M on Hdetector
We look at the difference of the time-evolved pointer variable
M and the physical observable A before measurement,

E(A):=U+ (1⊗M)U-A⊗1.
Define the error ε(A) in the measurement of A by M as

ε(A)2 =<ψ|⊗<ξ|E(A) 2|ψ>⊗|ξ>

The change of another observable B by the measurementof A is 	
D(B):=U+ (B⊗1)U-B⊗1
so that the disturbance η(B) is defined as

η(B) 2 = <ψ|⊗<ξ| D(B)2|ψ>⊗|ξ>0=U+ [B⊗1,1⊗M] U=[U+ (B⊗1) U, U+ (1⊗M) U=[D(B)+B⊗1,E(A)+A⊗1]=[D(B), E(A)]+[D(B), A⊗1]+[B⊗1, E(A)]+[B⊗1, A⊗1]
Then	
[D(B), E(A)]+[D(B), A⊗1]+[B⊗1, E(A)]=[A⊗1, B⊗1]	
By the triangular inequality, we have	|| [D(B), E(A)] ||+||[D(B), A⊗1]||+|| [B⊗1, E(A)]||≥ || [A⊗1, B⊗1] ||where ||#||=|<ψ|⊗<ξ| #|ψ>⊗|ξ>|

The Kennerd-Robertson inequality implies
ε(A) η(B) ≥ |<ψ|⊗<ξ| [D(B), E(A)] |ψ>⊗|ξ>|/2
and so on, which we use three times to obtain
Ozawa’s inequality,
ε(A) η(B) +ε(A)σ(B)+σ (A) η(B)≥|<ψ|[A,B]|ψ>|/2

Recently, Ozawa’s inequality is experimentally verified
by Hasegawa’s group in Wien in the neutron spin case.
J.Erhart et.al: Nature Phys. 8 185 (2012)

I would like to call your attention to the fact that the error
and the disturbance are experimentally accessible if
Ozawa’s definitions are accepted.
However, there are some criticism on the definition of error
and disturbance.
Here I will argue that the definition of error and disturbanceactually leads to a deeper understanding of quantum
mechanics that the “true value” of an observable is the weakvalue.

Preparatories
Look at the expression for the error
ε(A)2 = <ψ|⊗<ξ|(U+ (1⊗M)U-A⊗1) 2|ψ>⊗|ξ>= <ψ|⊗<ξ|(U+ (1⊗M2) U |ψ>⊗|ξ>- <ψ|⊗<ξ|U+ (1⊗M)UA⊗1 |ψ>⊗|ξ>-c.c.
+<ψ|A2|ψ>,Let

ΣmUm|ψ>⊗|m>,

U |ψ>⊗|ξ>=

where M|m>=m|m>

<ψ|⊗<ξ|(U+ (1⊗M2) U |ψ>⊗|ξ>=Σm<ψ|U+mm2Um|ψ>=<ψ|OA2|ψ><ψ|⊗<ξ|U+ (1⊗M)UA⊗1 |ψ>⊗|ξ>=Σm<ψ|U+mmUmA|ψ>=<ψ|OAA|ψ>For the projection measurement Um, we write

OA:=ΣmmUm

and use OA⊗1 as a substitute of 1⊗M of the pointer.

Thenε(A)2 =<ψ|(OA- A) 2|ψ>.
Similarly

η(B)2 =Σm<ψ|([Um,B] 2|ψ>.
Hasegawa et al. measured the error and disturbancefor A=σx and B=σy. OA is a linear combination of the Paulimatrices.
We parameterize it as OA=cosφ σx+sinφ σy	How can we measure ε(A) and η (B)?
ε(A)2 =<ψ|(OA- A) 2|ψ>=<ψ|OA2 +A2 – OAA-AOA|ψ>=2+<ψ|OA |ψ>+<Aψ|OA |Aψ>-<(1+A)ψ|OA |(1+A)ψ>Each term can be evaluated by a sequential
spin measurement. For example,<ψ|OA |ψ>=[(I+++ I+-)-(I-++ I--)]/[I+++ I-++I+-+ I--]
I-+= # of neutrons when the 1st polarizer selects
down-spin and the 2nd one up-spin.

|Aψ>=|σxψ> is the 180 degree rotated state of |ψ>and|(1+A)ψ>is the +1state along the x-axis.
All the terms can be evaluated by sequential spinmeasurements with appropriate rotation of thequantization axis. 	

ε2(A)=<(U(1⊗M)U+-A⊗1) 2>
η2 (B)=<(U(B⊗1)U+-B⊗1) 2>
where M is the meter operator to measure A and U is theunitary operator for the system and detector.
<….> means the expectation value w.r.t. the initiallyprepared state |ψ>⊗|ξ> of the system and detector.
Hereafter we concentrate on ε2(A). 	

長谷川さん	Therefore, they concluded that Ozawa’s versionof uncertainty relation has been veryfied, while theoriginal version is violated.
Since then there have been debates on the definitionof error and disturbance.
e.g. ,Yu Watanabe (Ph.D thesis,Univ.Tokyo)
For the purpose of finding the fundamental limit oferror/disturbance relation for all possible
measurements Ozawa’s definition seems mostnatural. Even more, it opens up a new conceptof quantum mechanics ---weak value--.

Very recently Steinberg announced that his groupalso verified Ozawa’s inequality in an optical set-up
by using measuring the weak value.
A. Steinberg (Vaexjo talk, June 2012) 	

The hint can be found in the paper by Lund and Wiseman
New J. Phys. 12 093011 (2010) quoted in the paper by
Ozawa and Hasegawa et al. Ozawa found the same
formua much earlier
They gave an expression for the error/disturbance in termsof the weak values
ε2(A)=<(U(1⊗M)U+-A⊗1) 2>=Σδa(δa) 2Pwv(a+δa)η2(B)=<(U(B⊗1)U+-B⊗1) 2>=Σ

2P (b+δb)
(δb)
δb
wv

Here
Pwv(δa):=Σa<(U(1⊗Π(a+δa))U+(Π(a) ⊗1)>is the weak probability, where Π(a)=|a><a| is the
projection operator to the eigen state |a> of theoperator A.
Note that in general<AB>=Σω<AΠ(ω)B>=Σω<AΠ(ω)B>=Σωλ*ω (A) λω (B)P(ω)
can be expressed in terms of the weak values
which can be measured with an arbitrary acuracy .

The error ε(A) can also be rewritten as

ε2(A)=Σ ω,mP（ω,m)|Am-λωm (A)|2	
  (Hofmann, different from Lund Weisman’s)P（ω,m)=|<ω|Km|Ψ>|2	
  is the probability to obtain the post-selected state<ω|Km according to the standard Born’s rule.
Am is the measured value of A defined by
Am|m>=M|m>.	
  

λωm (A):=	
  <ω|KmA|Ψ>/<ω|Km|Ψ>	
  where Km is the Kraus operator: Km=	  <0|U|m> 	
  Note that the error can be interpreted as theGaussian mean of the difference of the measured value
Am and the “true value” of A, λωm (A).
One can instantly notice thatλωm (A):=	
  <ω|KmA|Ψ>/<ω|Km|Ψ>	
  is the weak value proposed by Aharonov and his
collaborators.	
  	
  
Y.	
  Aharonov,	
  D.	
  Z.	
  Albert,	
  and	  L.	
  Vaidman,	  	
  Phys.	
  Rev.	
  LeL.	
  60,	
  1351	
  (1988).	
  	
  There is a deep connection between thevalue of physical quantity before measurement
and the weak value.
I will show this in a somewhat axiomatic way assuming(1) Linearity: λ(αA+βB)=αλ(A)+βλ(B ) α,β∈ R(2) Product rule when restricted to the Abelian subalgebra:λ(ST )=λ(S) λ(T )~classical theory
for all S, T∈Vmax(3) The prepared state is |Ψ>	  nothing	
  else;	  λ(|Ψ><Ψ|)=1,
λ(|Ψ⊥><Ψ⊥|)=０ , for all |Ψ⊥> s.t.	
  <Ψ⊥|Ψ>=0	  (4)The expectation value Ex[A] and the variance Var[A]be independent of the choice of CONS Ω={<ω|} ω,i.e.,
Vmax∈V(N).

２. Formal theory of value of an observable
2.1 Context (finite dimension)
Let V(N) be a set of Abelian sub-algebras of allobservables N .There may be many choices of the
sub-algebra V1,V2,V3 ….. ∈ V(N).
Choose Vmax∈V(N). We call Vmax as a context. The
idea is that the mutually commutable set of observables{P,Q,R,….} define a set of simultaneous eigenvectors of
P,Q,R,….{<ω|}, which corresponds to the resultant
states after the projective measurements of P,Q,R,…. .
★ Inspired by Doering’s lecture 2010 at Nagoya	As Bohr frequently emphasized, the description ofquantum experiment has to be classical. The word“classical” may be paraphrased by the eigenvaluesof mutually commuting observables {P,Q,R,….}in the maximal subalgebra Vmax
The way of description (=context) of experiments
is characterized by the choice of Vmax.
We are going to define the value of an observable A
in the state |ψ> in the context Vmax, i.e.,. .{<ω|}.

Corresponding to the choice of the Abelian sub-algebra
V1,V2,V3 …. ∈V(N), we have a collection oforthonormal basis {<ω|}1, {<ω|}2, {<ω|}3 , ……
We can think of the collection of the values of an observable
A in the state |ψ> in the context V1, V2,V3 …..i.e.,{<ω|}1, {<ω|}2, {<ω|}3…
We fix a maximal Abelian subalgebra Vmax∈V(N)for the moment of discussion and therefore thecontext Ω:={<ω|} ω . We shall find an expression for thevalue of an observable A- λ(A) ∈C, complex number.

	2.2 Main Theorem
We demand that the “value” λ(A)∈C of an observable A∈N
satisfies the following properties:(1) Linearity:λ(αA+βB)=αλ(A)+βλ(B ) α,β∈ R(2) Product rule when restricted to the Abelian subalgebra:λ(ST )=λ(S) λ(T )~classical theory
for all S, T∈Vmax(3) The prepared state is |Ψ>	  nothing	
  else;	  λ(|Ψ><Ψ|)=1,
λ(|Ψ⊥><Ψ⊥|)=０ , for all |Ψ⊥> s.t.	
  <Ψ⊥|Ψ>=0	
  The above reqirements via Riez’s theorem lead toλ(A)=Tr[WA]/Tr[W] ---(1) λ(1)=1 --- (3)with
W=a|Ψ><ω|+b|ω><Ψ|	  ---(2)(3)	
  	
  	
  
where <ω|	  is a simultaneous eigenvector of Vmax.

The formal classical probability theory a laKolmogorov presupposes the probability
measure P(ω) and λω (A) the value of aphysical quantity A for an event ω.
The expectation value Ex[A] and the variance
Var[A] are given by
Ex[A]=ΣωP(ω) λω (A) ,with Ex[1]=1,
Var[A]=ΣωP(ω)| λω (A)-Ex[A]|2
We adopt these expressions also in quantum
mechanics. Note that P(ω) is independent of A.

Correspondence with the classical probability theory.
ω∈Ω: event (<ω¦∈Vmax∈V(N) )	
dP(ω): probability measure (independent of A) (P(<ω¦) )	
hA(ω): a random variable (real) (λω (A): complex)	
Expectation value and variance:	
Ex(A)	  :=	  ∫ dP(ω) hA(ω)	
Var(A)	
  :=	
  ∫ dP(ω) ¦hA(ω)-Ex(A)¦2	
  （４）We demand the expectation value Ex[A]and the variance Var[A] be independent of the choiceof CONS Ω={<ω|} ω,i.e., Vmax∈V(N).
According to the central limit theorem, the distributionof values of observable A approaches the normal(Gaussian ) distribution characterized by its mean
Ex[A] and the variance Var[A].
The requirement (４) demands that the distribution
should be independent of how we measure A but
depends only on the prepared state |Ψ>. 	

The above requirement uniquely determines both
W=|Ψ><ω|	  and	
  therefore the “value” coincides with Aharonov’s weak valueλω (A)=Tr[WA]/Tr[W] =	
  <ω|A|Ψ>/<ω|Ψ>,	
  	  (	  i.e.,	  b=0)
Aharonov, Albert and Vaidman,

and the measure,P(ω)=¦<ω|Ψ>|2	
  and therefore we have “derived” the Born formulafor the expectation value and the variance
Ex[A]=<Ψ|A|Ψ>	
  
	
  
2
Var[A]=<Ψ|(A-­‐Ex[A]) |Ψ>.

PRL 60,1351

Outline of proof	
Introducing the lagrange multiplier µ to ensure thecompleteness relation, we demand the variation of
the lagrangian L[<x|,µ] w.r.t. <x| and µ vanish for all
Observable A
L[<x|,µ]=Ex[A]-µ(Σx<Ψ|x><x|A|Ψ>-­‐<Ψ|A|Ψ>)=ΣxP(x) λx (A)-µ(Σx<Ψ|x><x|A|Ψ>-­‐<Ψ|A|Ψ>)where λx (A)=Tr[WA]/Tr[W]
W=a|Ψ><x|+b|x><Ψ|Idea	  of	
  the	
  proof:	
  if	
  P(ω)=¦<Ψ|ω>|2	
  Ex[A]=ΣωP(ω) λω (A)	  =Σω	  ¦<Ψ|ω>|2[<ω|A|Ψ>/<ω|Ψ>]	
  =	
  Σω<Ψ|ω><ω|A|Ψ>	
  =<Ψ|A|Ψ>	
  Var[A]=ΣωP(ω) |λω (A)|2	
  =Σω	  ¦<Ψ|ω>|2|<ω|A|Ψ>/<ω|Ψ>) |2	  =	
  Σω<Ψ|A|ω><ω|A|Ψ>	
  =<Ψ|A2|Ψ>	
  do	
  not	
  depend	
  on	
  {<ω|}	
  i.e.,	
  the	
  choice	
  of	
  Vmax∈V(N).

Note	
  that	  	  	
  P(ω)=¦<Ψ|ω>|p	
  would	
  not	
  work	  for	
  p≠2!	
  	
  The	
  key	  is	
  the	
  completeness	
  relaDon	
  Σω|ω><ω|=1.	
  
c.f.	
  	
  	
  	
  	
  J.Phys. A;43 025304 (2010) with Shikano	  	  The weak value λω	
  (A) is a fundamental quantity inquantum mechanics which is experimentally accessible
only slightly disturbing the prepared statee.g., by the weak measurements with an arbitrary
accuracy in principle. For me, at least, it is almost a“physical reality” in the sense of EPR.
The probability measure P(ω)=|<ω|Ψ>|2	
  is
not an axiom any more but a consequence of quantum
mechanics and the probability theory.
λω (A) is interpreted as a value of A in the context
of the pre-selected state |Ψ>	  and the post-selected states{<ω|} of the intended projective measurements of a
maximal set of commuting observables Vmax∈V(N).

Since an event ω	  is identified with a consequence <ω|ofmeasurement	
  of	
  Vmax	  , P(ω)=|<ω|Ψ>|2	
  is interpretedas 	
  its relative frequency.
From the formula:
Ex[A]=ΣωP(ω) λω (A)	  λω	  (|a><a|) can be interpreted as a conditional
probability to reach <ω|	  via	
  |a><a| with the initiallyprepared	
  state	  |Ψ>.	
  In	
  general	
  λω may be complex. 	
  	  	  J.Phys. A;43 025304 (2010) with Shikano	  	  Going back to the original motivation of the value of anobservable before measurement （“真値”）
we just show an example:	  ξ(t):=<x|X(t)|Ψ>/<x|Ψ>,	
  where	  X(t)	
  ,0≤t≤T	
  is	
  the	  posiDon	
  operator	
  of	
  a	
  
parDcle.	
  <x| is the eigen state of X(T) withthe eigen value x.	
  	
t=T	

ξ(t)	
	

x	

We can ask the following counter-factual question.
We are in a certain initial state andknow the value of X as x by measuring X of 
at t=T.
What the value of X(t) would be before T ?
We can answer in an experimentally verifiable way.

A remark:☆ The weak value is almost a wave function itself
For the case that the momentum is conserved,
η(t):=<x|P(t)|Ψ>/<x|Ψ>=iΨ’(x)/Ψ(x)See the experiment which shows a trajectory of photon
by weak measurements of its transverse momentum“Direct measurement of the quantum wavefunction” 	
by Lundeen et al. (Recent Issue of Nature),Lundeen
et al. 	

There have been a hot debate in Japan over the “true value”
of an observable before the post-selection.
Some people are still reluctant to accept the assertion thatsuch a thing exists and actually coincides with the weak value.
One of the reason may be that the weak value depends
on the choice of the post-selected state by an observer.
This does not fit the “objectivity” of the “truth”.
I would say that the weak value is the subjective valueof objective physical quantity. In contrast the expectation value
is an objective value.
Note that we cannot assign objective value for all theobjective physical quantities as Kochen and Specker proved. 	

Kochen-Specker theorem (’67) (c.f. Zeilinger’s recent paper)
We cannot assign a value of all physical quantities
independently of how we measure it for dim(H)≥3.
Example by Mermin (4 dim)σx⊗1

1⊗σx σx⊗σx

1	

1⊗σz

σz⊗1

1	

σz⊗σz

σx⊗σz σz⊗σx σy⊗σy
1

1

1	

cannot assign
the eigenvalues±1 consistently
Eigenvalues are
only non-contextual
values-1	
Mermin ‘90	The utmost important thing is that the “true value” ofan observable before measurement can be experimentally
verified by weak measurements.
The value of observable certainly exists before
measurement. The measurement only discoversit.

Mach-Zehnder interferometer
A photon injected from A goes out of E only by interference	

BS	A	

mirror	

E	

Double MZ interferometerbeam-splitter
1
A
B

C

mirror

2

E
F

3

D

D’

Mitchison, Jozsa and Popescu arXiv 0706.1508

Click!

We can see

A is pre-selected

photon would pass E(interference)

photon should have passed(which way)

D is post-selected

We can know which-way, whileinterference effect is manifested!
(Aharanov)	

Verfication by a weak mesaurement
Insert a tilted slide glassin the path C
The optical axis will shift.	
（similar for E）	
1
A
B

C

2

E
shift	F

3
D’

D

Click!

4.

Summary of the 1st partI reviewed the uncertainty relation by Ozawa and its
recent verification by Wien group led by Hasegawa on theemphasis,(1) The mathematical background (CP map)
(2) how the original mathematical expression
have been made operational for the actual experiment.
I think both are very innovative! Furthermore this leads
to an important concept of the weak value.

Summary of the 2nd part
Combining quantum mechanics and the formal
probability theory we have shown that the context
dependent (subjective) value of (objective )observable A
is the weak valueλω	
  (A) :=	  <ω|A|Ψ>/<ω|Ψ>	
  and the probability measure is given by
Born’s rule:
P(ω)=|<ω|Ψ>|2,	  where |Ψ>	  is	  the	  	
  iniDal	
  state	  and	
  <ω|	  is	
  the	  post	
  selected	
  state.	
  The value of observable appears only after
measurement not before in the Copenhagen
interpretation.
However,
Carl Friedrich von Weizsäcker denied that
the Copenhagen interpretation asserted:"What cannot be observed does not exist".
He suggested instead that the Copenhagen
interpretation follows the principle:"What is observed certainly exists;
about what is not observed we are still free
to make suitable assumptions.

Thank you for your attention!