# The Reconstruction Problem and Weak Quantum Values

**ABSTRACT** Quantum Mechanical weak values are an interference effect measured by the

cross-Wigner transform W({\phi},{\psi}) of the post-and preselected states,

leading to a complex quasi-distribution {\rho}_{{\phi},{\psi}}(x,p) on phase

space. We show that the knowledge of {\rho}_{{\phi},{\psi}}(z) and of one of

the two functions {\phi},{\psi} unambiguously determines the other, thus

generalizing a recent reconstruction result of Lundeen and his collaborators.

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**ABSTRACT:**A quantum state contains the maximal amount of information available for a given quantum system. In this paper we use weak-value expressions to reconstruct quantum states of continuous-variable systems in the quantum optical domain. The role played by postselecting measured data will be particularly emphasized in the proposed setup, which is based on an interferometer just using simple homodyne detection.Physical Review A 11/2012; 86(5). · 2.99 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the notion of weak value of a quantum observable introduced by Aharonov from the point of view of the Born-Jordan quantization scheme. While both quantizations agree for observables of the type "kinetic energy plus potential" they lead to different weak values for general quantum observables. Weak measurements could thus provide an experimental test for the determination of the correct quantization scheme.12/2012;

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The Reconstruction Problem and Weak Quantum

Values

Maurice A. de Gosson

University of Vienna

Faculty of Mathematics, NuHAG

A-1090 Vienna

Serge M. de Gosson

Swedish Social Insurance Agency

Department for Analysis and Forecasts

S-103 51 Stockholm

February 5, 2012

Abstract

Quantum Mechanical weak values are an interference effect mea-

sured by the cross-Wigner transform W(φ,ψ) of the post-and pres-

elected states, leading to a complex quasi-distribution ρφ,ψ(x,p) on

phase space. We show that the knowledge of ρφ,ψ(z) and of one of the

two functions φ,ψ unambiguously determines the other, thus general-

izing a recent reconstruction result of Lundeen and his collaborators.

1 Introduction

In 1958 W. Pauli [22] mentions the problem of the reconstruction of a quan-

tum state knowing its position and momentum; this conjecture was later

disproved; see H. Reichenbach’s book [23]; also Corbett [8] for a review of

the Pauli problem. However, in [19] Lundeen and his coworkers consider

the following experiment: a weak measurement of position is performed on

a particle; thereafter a strong measurement of momentum is made. This

allows them to effectively reconstruct the wavefunction ψ(x) pointwise by

scanning through all values of x. The aim of the present paper is to show

that, more generally, the wavefunction can be reconstructed from the knowl-

edge of a complex probability distribution ρφ,ψ(x,p) on phase space, related

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to the notion of weak values, and expressed in the terms of the cross-Wigner

transform W(φ,ψ) of the pre- and postselected states. Our proof is based on

the fact that the knowledge of W(φ,ψ) and of one of the two functions φ,ψ

allows to determine uniquely and unambiguously (in infinitely many ways)

the other function. This is of course in strong contrast with the case of the

usual Wigner distribution Wψ = W(ψ,ψ) whose knowledge only determines

ψ up to a factor with modulus one.

In what follows the position and momentum vectors are x = (x1,...,xn)

and p = (p1,...,pn), respectively; px = p1x1+···+pnxnis their scalar prod-

uct. We write dnx = dx1dx2···dxn, and all the integrations are performed

over n-dimensional space Rn. The ?-Fourier transform of a function ψ is

defined by

?

Fψ(p) =

1

2π?

?n?

e−i

?pxψ(x)dnx.

2 Weak Values and the Cross-Wigner Transform

Let? A be a quantum observable associated to a function A by the Weyl

Aharonov and coworkers [1, 2, 3, 4, 5] have introduced and studied the notion

of weak measurement and the weak value of an observable (also see [16]).

The weak value of? A with respect to the pair (φ,ψ) is the complex number

?? A?φ,ψ

According to

?? A?φ,ψ

When φ = ψ this is the usual average value ?? A?ψof? A in the state ψ.

[17] and Parks and Gray [21] for concise and clear expositions): viewing

|ψ? as a preselected state and ?φ| as a postselected state, if we couple a

measuring device whose pointer has position coordinate x to the system and

subsequently measure that coordinate then the mean value of the pointer is

correspondence: ? A

Weyl

←→ A [9, 10, 18] and φ,ψ two non-orthogonal states.

weak=?φ|? A|ψ?

?

?φ|ψ?

, ?φ|ψ? ?= 0.(1)

weak=A(x,p)ρφ,ψ(x,p)dnxdnp(2)

The physical interpretation of the weak value is the following (see Jozsa

?? x? = g Re?? A?φ,ψ

weak

if the coupling interaction is the standard von Neumann interaction Hamil-

tonian? H = g? Ap. In addition the mean of the pointer momentum is given

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by

?? p? = 2g

?v Im?? A?φ,ψ

weak.

(This needs sufficiently weak interaction, see the discussions in Duck et al.

[12] and in Parks et al. [20]).

In a recent work [11] we have shown that the weak value can be calculated

by averaging A over the complex phase space function

ρφ,ψ(x,p) =W(φ,ψ)(x,p)

?φ|ψ?

(3)

where

W(φ,ψ)(x,p) =?

1

2π?

?n?

e−i

?pyφ∗(x +1

2y)ψ(x −1

2y)dny (4)

is the cross-Wigner transform (we are conjugating φ and not ψ in order

to be consistent with the bra-ket notation; in most mathematical texts the

transform defined by (4) would be denoted by W(ψ,φ)). The function ρφ,ψ

satisfies the marginal conditions

?

and hence in particular

?

The function ρφ,ψcan thus be viewed as a complex quasi-probability density

on phase space; its real and imaginary parts moreover satisfy

?

The appearance of the cross-Wigner function is characteristic of interfer-

ence phenomena, and suggests the following interpretation of weak values1

[2, 3, 4]. Assume that we measure an observable? A an initial time tinand that

state); similarly at a final time tfina measurement of another observable

?B yields |φ(tfin)? = |?B = b? (the post-selected state). Choose now some

nity, and has led to an ongoing epistemological debate on the back-action of future on the

past.

ρφ,ψ(x,p)dnp =φ∗(x)ψ(x)

?φ|ψ?

,

?

ρφ,ψ(x,p)dnx =[Fφ(p)]∗Fψ(p)

?φ|ψ?

; (5)

ρφ,ψ(x,p)dnxdnp = 1.

Reρφ,ψ(x,p)dnxdnp = 1 ,

?

Imρφ,ψ(x,p)dnxdnp = 1

a non-degenerate eigenvalue was found: |ψ(tin)? = |? A = a? (the pre-selected

1But this interpretation is far from being unanimously shared in the scientific commu-

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intermediate time t : tin< t < tfin. Following the time-symmetric approach

to quantum mechanics, at this intermediate time the system is described by

the two wavefunctions

ψ = ψt= UHin

t,tinψ(tin) , φ = φt= UHfin

t,tfinφ(tfin) (6)

where UH

? H is the quantum Hamiltonian). Notice that φttravels backwards in time

system under consideration is in the state |ψt?? = UH

Wigner distribution Wψt?; at any time t??> t the system is in the state

|φt??? = UH

is the superposition |ψt? + |φt? of both states, and the Wigner distribution

t,t? = e−i? H(t−t?)/?is the Schrödinger unitary evolution operator if

since t < tfin. The situation is thus the following: at any time t?< t the

t?,tin|ψ(tin)? and has

t??,tfin|φ(tfin)? and has Wigner distribution Wφt?. But at time t it

W(φt+ ψt) = W(φt+ ψt,φt+ ψt)

of this state is

W(φt+ ψt) = Wφt+ Wψt+ 2ReW(φt,ψt). (7)

This equality shows the emergence at time t of the interference term 2ReW(φt,ψt) =

2ReW(φ,ψ), signalling a strong interaction between the states |ψt? = |ψ?

and |φt? = |φ?.

3 The Reconstruction Problem

3.1 An example

As briefly explained in the Introduction Lundeen and his coworkers [19] con-

sider the following experiment: a weak measurement of position is performed

on a particle; this amounts to applying the projection operator?Πx= |x??x|

strong measurement of momentum is made, yielding a value p0; the result

of the weak measurement is thus

to the pre-selected state |ψ?, which yields?Πx|ψ? = ψ(x)|x?. Thereafter a

??Πx?φ,ψ

weak=e

i

?p0xψ(x)

Fψ(p0)

.(8)

This allows the reconstruction of the function ψ from ??Πx?φ,ψ

ψ(x) = ke−i

weak:

?p0x??Πx?φ,ψ

weak, k = Fψ(p0). (9)

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Let us retrieve this result using the Wigner formalism developed above.

Obviously, the operator? A is here the projector?Πxwhose analytical expres-

?Πxψ(y) = ψ(x)δ(x − y).

tion

φp0(x) = (2π?)−n/2e

sion is given by

(10)

Choosing for the post-selected state φ the normalized momentum wavefunc-

i

?p0x

we have

W(ψ,φp0)(x,p) =?

1

2π?

?3n/2?

?3n/2e

e−i

?pyψ∗(x +1

?

2y)e

i

?p0(x−1

2y)dny

=?

1

2π?

i

?p0x

e−i

?(p+1

2p0)yψ∗(x +1

2y)dny

which —after the change of variables x?= x +1

2y and integrating— becomes

W(ψ,φp0)(x,p) =?1

π?

?ne

2i

?(p−p0)xFψ(2p − p0).

Taking into account the fact that ?φp0|ψ? = Fψ(p0) this yields

?1

ρψ,p0(x,p) =

π?

?n

e

2i

?(p−p0)xFψ(2p − p0)

Fψ(p0)

. (11)

In view of Eqn.

Πx(x?,p?) = δ(x?− x) and hence

(10) the classical observable Πx

Weyl

←→?Πx is given by

??Πx?φp0,ψ

weak=

?

?

ρψ,p0(x?,p?)δ(x?− x)dnp?dnx?

=ρψ,p0(x,p?)dnp?

=φ∗

p0(x)ψ(x)

?φp0|ψ?

where the last equality is a consequence of the first marginal distribution

property (5); this is precisely the expression (8) of the weak value of?Πx.

continuous conjugate variables (such as position and momentum) are real,

and far from being resolved. For a mathematically rigorous approach, using

the properties of the metaplectic group, see Weigert and Wilkinson [25].

We note that the difficulties inherent to the theory of measurement of

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3.2 A general reconstruction formula

Formula (8) shows that we can reconstruct the whole wavefunction ψ by

scanning the weak measurements of the projection operator?Πxthrough x.

reconstructed from the knowledge of the complex quasi-distribution ρφp,ψ,

or, equivalently from the knowledge of the cross-interference term W(φ,ψ).

Let us begin by introducing some notation. Let z = (x,p) and z0= (x0,p0)

be an arbitrary phase space point ,and define the Grossmann—Royer operator

[14, 24]?TGR(z0) by

?TGR(z0)ψ(x) = e

It is, up to the complex exponential factor in front of ψ(2x0−x) a reflection

operator, in fact?TGR(z0)?TGR(z0) =?Id (the identity operator).

A remarkable fact is that the cross-Wigner transform is related to?TGR(z)

W(φ,ψ)(z) =?1

(see [10], Chapter 9). Using the following well-known formula (“Moyal iden-

tity”, [10], Chapter 9):

?

and the Grossmann—Royer formalism we prove that the knowledge of the

cross-Wigner transform W(φ,ψ), and of that one of the two functions φ,ψ

uniquely determines the other. Moreover, this function can be written in

terms of an arbitrary square-integrable auxiliary function γ.

Proposition. Let (φ,γ) be a pair of square integrable functions such that

?γ|φ? ?= 0. We have

2n

?ψ|γ?

ψ(x) =

?φ|γ?

Proof. We begin with a preliminary remark: both integrals in the formulas

above are absolutely convergent. In fact, taking φ = φ?and ψ = ψ?in

We are going to show that, more generally, a quantum state ψ can always be

2i

?p0(x−x0)ψ(2x0− x). (12)

It is in

addition a unitary self-adjoint operator:?TGR(z0)∗=?TGR(z0)−1=?TGR(z0).

by the simple formula

π?

?n??TGR(z)φ|ψ?(13)

W(φ,ψ)∗(z)W(φ?,ψ?)(z)d2nz =?

1

2π?

?n?φ|φ???ψ|ψ??(14)

φ(x) =

?

?

W(φ,ψ)(z0)?TGR(z0)γ(x)d2nz0

W(φ,ψ)∗(z0)?TGR(z0)γ(x)d2nz0.

(15)

2n

(16)

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Moyal’s identity (14) we see that W(φ,ψ) is square integrable. In view of

the Cauchy—Schwarz inequality we have

?

for the integrals in (15), (16). We next observe that both formulas (15) and

(16) are equivalent and obtained from each other by swapping φ and ψ and

noting that W(ψ,φ)∗= W(φ,ψ). Let us prove (15). Let us denote by χ(x)

the right hand side of (15):

?

We are going to show that ?χ|θ? = ?φ|θ? for every element θ of the Schwartz

space S(Rn); it will follow that we have χ = φ almost everywhere, which

proves formula (15). We have

?

In view of formula (13) we have

|W(φ,ψ)(z0)||?TGR(z0)γ(x)|d2nz0≤ ||W(φ,ψ)||L2||γ||L2 < +∞

χ(x) =

2n

?ψ|γ?

W(φ,ψ)(z0)?TGR(z0)γ(x)d2nz0.

?χ|θ? =

2n

?ψ|γ?

W(φ,ψ)(z)??TGR(z)γ|θ?d2nz.

??TGR(z)γ|θ? = (π?)nW(γ,θ)(z)

?χ|θ? =(2π?)n

?ψ|γ?

=(2π?)n

?ψ|γ?

Applying Moyal’s identity (14) to the last integral we get

?

and hence ?χ|θ? = ?φ|θ?.

Since the cross-Wigner transform W(φ,ψ) and the weak value ρφ,ψ(x,p)

are equal up to a factor ?φ|ψ? (formula (3)), it follows that we can rewrite

(15) and (16) as

?

ψ(x) = 2n?φ|ψ?∗

?φ|γ?

and hence

?

?

W(φ,ψ)(z)W(γ,θ)(z)d2nz

W(ψ,φ)∗(z)W(γ,θ)(z)d2nz.

W(ψ,φ)∗(z)W(γ,θ)d2nz =?

1

2π?

?n?ψ|γ??φ|θ?

φ(x) = 2n?φ|ψ?

?ψ|γ?

ρφ,ψ(z0)?TGR(z0)γ(x)d2nz0

ρ∗

(17)

?

φ,ψ(z0)(z0)?TGR(z0)γ(x)d2nz0.

7

(18)

Page 8

4 Relation with Time-Frequency Analysis

The Wigner formalism is widely used in time-frequency analysis (TFA), but

there is an interpretational and almost philosophical difference in the view-

points in QM and TFA. While the Wigner formalism is an essential tool

for expressing QM in its phase space version and leads to the Weyl quan-

tization scheme [9, 10, 18], the situation is less clear-cut in TFA where the

appearance of cross-term correlations as W(φ,ψ) is an unwanted artefact;

to eliminate or weaken these interference effects one uses elements of the so-

called Cohen [7, 13] class (which would typically be the Husimi transform

in QM). Everything we have said above can be re-expressed in terms of the

cross-ambiguity function familiar from radar theory (Woodward [26], Binz

and Pods [6]), and related to the Wigner transform by a symplectic Fourier

transform:

A(φ,ψ)(z) = FσW(φ,ψ)(z) = FW(φ,ψ)(Jz)

where F is here the usual Fourier transform on R2nand J =

?0I

0−I

?

. It

is given by the analytical expression

A(φ,ψ)(z) =?

1

2π?

?n?

e−i

?pyφ∗(y +1

2x)ψ(y −1

2x)dny.

The symplectic Fourier transform being an involution we can rewrite the

definition (3) of ρφ,ψ(x,p) in the form

ρφ,ψ(x,p) =FσA(φ,ψ)(x,p)

?φ|ψ?

. (19)

The use of this alternative approach does a priori not bring anything new

from a mathematical point of view, but could perhaps be useful for giving

interpretations of weak values and measurements in terms of the “source-

target” formalism of radar theory.

Acknowledgement 1. The first author (MdG) has been funded by the

Austrian Research Agency FWF (Projektnummer P 23902-N13).

Acknowledgement 2. Both authors wish to express their thanks to the

Referees for a careful reading of our manuscript, and for having pointed out

several inaccuracies and shortcomings. Their comments have substantially

improved the final manuscript.

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Page 9

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