Transcript Weak measurement with Decoherence Akio Hosoya (Tokyo Tech

竹原
６/６,’11
Born’s Rule and “Value” of an
Observable before
Measurement
Akio Hosoya
1. Introduction
2. A formal theory of (Weak) Value
Born’s rule
3. Main theorem
4. Summary
arXiv:1104.1873
with Minoru Koga
１. Introduction
It seems that the right quantity to discuss in
quantum cosmology is not the probability for
anything but the contextual value of observables.
Example:
Suppose the initial state of the universe is the
one given by Hawking and that we know that
the present state of the universe is such and such
(anthropic? ) .
What would be the value of
physical observables in-between?
Fundamental Problem:
The value of observable appears only after
measurement not before in the Copenhagen
interpretation.
However, there is a revisionist
like me
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.
“Non-contextual values” of observables are not possible by
the Kochen-Specker theorem (‘67)
We cannot assign a value of physical quantity
independently of how we measure it for dim(H)≥3.
Example by Mermin (4 dim)
σx1
1σz
1σx σxσx
σz1
σzσz
σxσz σzσz σyσy
1
1
1
1
1
cannot assign
the eigenvalues
±1 consistently
Eigenvalues are
only non-contextual
values
-1
Mermin ‘90
Therefore the “value” should depend on context,
i.e., how to measure it.
The context is specified by the maximal set of commuting
observables Vmax as explained later in detail.
The non-commutativity of observables V(N) in
quantum mechanics makes different choices of
Vmax V’max∈V(N) non-commutable and therefore of
different context.
I propose the weak value advocated by Aharonov as a
candidate of “contex dependent value of A “.
2
History of Born’s Rule: P(ω)=|<ω|Ψ>|
The Born rule was formulated by Born in a 1926 paper.
In this paper, Born solves the Schrödinger equation for
a scattering problem and, inspired by Einstein's
work on the photoelectric effect, concluded in a
footnote, that the Born rule gives the only possible
interpretation of the solution.
In 1954, together with Walter Bothe, Born was
awarded the Nobel Prize in Physics for this and other
work.
1) Anmerkung bei tier Korrektur: Genauere Uberlegung zeigt,
dab die Wahrscheinlichkeit dem Quadrat der Φ proportional ist.
However, what is “probability”?
There have been many debates over the meaning
of probability.
(1) frequency of events [coin tossing]
(2) expectation [rain forcast,Laplace]
subjective interpretation [Beysian]
…….
But we do not have a consensus yet.
It seems that at present we are content with the
axiomatic theory of probability theory by Kolmogorov
without talking about its meaning.
We believe the combination of quantum mechanics
and axiomatic probability theory reveals the
meaning of probability on the basis of measurement.
Note that the context Ω is fixed once and for all
in classical theory.
Ex(A) :=  dP(ω) hA(ω)
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)
according to Kolmogorov.
２. Formal theory of value of an observable
2.1 Quantum context (finite dimension)
Let V(N) be a set of Abelian sub-algebras of all
observables 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,…. .
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 of
orthonormal 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 the
context Ω:={<ω|} ω . We shall find an expression for the
value 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(A+B)=λ(A)+λ(B ) c.f. von Neumann, Bell…..
(2) Product rule when restricted to the Abelian subalgebra:
λ(ST )=λ(S) λ(T )
close to classical theory
for all S, T∈Vmax
(3) Specification of which state we are definitely living in |Ψ>
λ(|Ψ⊥><Ψ⊥|)=０ , for all |Ψ⊥> s.t. <Ψ⊥|Ψ>=0
The above reqirements lead to
λ(A)=Tr[WA]/Tr[W] ---(1) λ(1)=1 --- (2)
with
W=a|Ψ><ω|+b|ω><Ψ| ---(2)(3)
where <ω| is a simultaneous eigenvector of Vmax.
Note that for S, T∈Vmax
<ω|S=<ω|s, <ω|T=<ω|t
so that λ(ST )=λ(S) λ(T ) holds.
The product rule (2) implies
W=|α><ω|+|ω><β|+ ΣωCnm|ωn><ωm| (♯)
where <ωm|ω>=0, while the condition (3) implies
W=|Ψ><q|+|r><Ψ|
(♭)
Putting ♯ and ♭together we arrive at
W=a|Ψ><ω|+b|ω><Ψ|
( ♮)
The formal classical probability theory a la
Kolmogorov presupposes the probability
measure P(ω) and λω (A) the value of a
physical 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)
Var[A]=ΣωP(ω)| λω (A)|2
We adopt these expressions also in quantum
mechanics.
（４）We demand 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).
According to the central limit theorem, the distribution
of 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.
The above requirement uniquely determines
W=|Ψ><ω|
and therefore the “value”
λω (A)=Tr[WA]/Tr[W] = <ω|A|Ψ>/<ω|Ψ>, ( i.e., b=0)
and the measure,
2
P(ω)=|<ω|Ψ>|
and therefore we have “derived” the Born formula
for the expectation value and the variance
Ex[A]=<Ψ|A|Ψ>
2
Var[A]=<Ψ|A |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>|4 would not work!
The key is the completeness relation Σω|ω><ω|=1.
c.f.
J.Phys. A;43 025304 (2010) with Shikano
Introducing the lagrange multiplier μ to ensure the
completeness relation, we demand the variation of
the “action” L[<ω|,μ] w.r.t. <ω| and μ vanish for all
observable A
L[<ω|,μ]=Ex[A]-μ(Σω<Ψ|ω><ω|A|Ψ>-<Ψ|A|Ψ>)
=ΣωP(ω) λω (A)-μ(Σω<Ψ|ω><ω|A|Ψ>-<Ψ|A|Ψ>)
where λω (A)=Tr[WA]/Tr[W] and W=a|Ψ><ω|+b|ω><Ψ|
δL/δ<ω|
=∂P(ω)/∂<ω| λω (A) +P(ω) ∂λω (A)/∂<ω|-μA|Ψ>=0
and a similar equation for Var[A] lead to
both
2
P(ω)=|<ω|Ψ>|
W=|Ψ><ω|
５. Summary
Combining quantum mechanics and the formal
probability theory we have shown that the context
dependent value of observable A is the weak value
λω (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 initial state and <ω| is the
post selected state, which is inferred by the values
of all the elements of Vmax∈V(N)
i.e., the context of how we intend to measure A.
λω (A) is experimentally accessible at any time by the weak
measurements. 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).
Going back to the original motivation of
the value of an observable before measurement
we just show an example:
ξ(t):=<x|X(t)|Ψ>/<x|Ψ>,
where X(t) ,0≤t≤T is the position operator of a
particle. <x| is the eigen state of X(T) with
the eigen value x.
t=T
ξ(t)
x
We can ask the following counter-factual question.
We are in a certain initial state and
know 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.
Two remarks:
☆ Counter-factual statement:
If A were true, B would hold. A☐B
Caution: the transitive law does not hold.
☆ Recently Englert and Spindel applied the weak value
to the back action problem of the Hawking radiation.
(2010 Arxiv) by analyzing
G  [g] = 8πG[<out| T |in>/<out|in>]