arXiv:1112.5773v2 [math-ph] 5 Feb 2012
The Reconstruction Problem and Weak Quantum
Maurice A. de Gosson
University of Vienna
Faculty of Mathematics, NuHAG
Serge M. de Gosson
Swedish Social Insurance Agency
Department for Analysis and Forecasts
S-103 51 Stockholm
February 7, 2012
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.
In 1958 W. Pauli  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 ; also Corbett  for a review of
the Pauli problem. However, in  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
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
2Weak Values and the Cross-Wigner Transform
Let? A be a quantum observable associated to a function A by the Weyl
←→ A [9, 10, 18] and φ,ψ two non-orthogonal states.
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
). The weak value of? A with respect to the pair (φ,ψ) is the complex
When φ = ψ this is the usual average value ?? A?ψof? A in the state ψ.
 and Parks and Gray  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
, ?φ|ψ? ?= 0.
The physical interpretation of the weak value is the following (see Jozsa
?? x? = g Re?? A?φ,ψ
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
?? p? = 2g
(This needs sufficiently weak interaction, see the discussions in Duck et al.
 and in Parks et al. ).
In a recent work  we have shown that the weak value can be calculated
by averaging A over the complex phase space function
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
ρφ,ψ(x,p)dnxdnp = 1.
The function ρφ,ψcan thus be viewed as a complex quasi-probability density
on phase space; its real and imaginary parts moreover satisfy
Reρφ,ψ(x,p)dnxdnp = 1 ,
Imρφ,ψ(x,p)dnxdnp = 1
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
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-
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
? 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 Schr¨ odinger 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).
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?.
3The Reconstruction Problem
As briefly explained in the Introduction Lundeen and his coworkers  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
This allows the reconstruction of the function ψ from ??Πx?φ,ψ
ψ(x) = ke−i
weak, k = Fψ(p0).
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).
Choosing for the post-selected state φ the normalized momentum wavefunc-
φp0(x) = (2π?)−n/2e
sion is given by
which –after the change of variables x′= x +1
2y and integrating– becomes
?(p−p0)xFψ(2p − p0).
Taking into account the fact that ?φp0|ψ? = Fψ(p0) this yields
?(p−p0)xFψ(2p − p0)
In view of Eqn.
Πx(x′,p′) = δ(x′− x) and hence
(10) the classical observable Πx
←→?Πx is given by
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 .
We note that the difficulties inherent to the theory of measurement of
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)
(see , Chapter 9). Using the following well-known formula (“Moyal iden-
tity”, , Chapter 9):
We are going to show that, more generally, a quantum state ψ can always be
It is in
addition a unitary self-adjoint operator:?TGR(z0)∗=?TGR(z0)−1=?TGR(z0).
by the simple formula
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
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
Moyal’s identity (14) we see that W(φ,ψ) is square integrable. In view of
the Cauchy–Schwarz inequality we have
|W(φ,ψ)(z0)||?TGR(z0)γ(x)|d2nz0≤ ||W(φ,ψ)||L2||γ||L2 < +∞
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
??TGR(z)γ|θ? = (π?)nW(γ,θ)(z)
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(x) = 2n?φ|ψ?
4Relation 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 , Binz
and Pods ), and related to the Wigner transform by a symplectic Fourier
A(φ,ψ)(z) = FσW(φ,ψ)(z) = FW(φ,ψ)(Jz)
where F is here the usual Fourier transform on R2nand J =
is given by the analytical expression
The symplectic Fourier transform being an involution we can rewrite the
definition (3) of ρφ,ψ(x,p) in the form
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
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