Weak values of a quantum observable and the cross-Wigner distribution
aa r X i v : . [ qu a n t - ph ] S e p WEAK VALUES OF A QUANTUMOBSERVABLE AND THE CROSS-WIGNERDISTRIBUTION
Maurice A. de GossonUniversity of ViennaFaculty of Mathematics, NuHAGA-1090 ViennaSerge M. de GossonSwedish Social Insurance AgencyDepartment for Analysis and Forecasts103 51 StockholmJuly 28, 2018
Abstract
We study the weak values of a quantum observable from the point ofview of the Wigner formalism. The main actor is here the cross-Wignertransform of two functions, which is in disguise the cross-ambiguityfunction familiar from radar theory and time-frequency analysis. Itallows us to express weak values using a complex probability distribu-tion. We suggest that our approach seems to confirm that the weakvalue of an observable is, as conjectured by several authors, due to theinterference of two wavefunctions, one coming from the past, and theother from the future.
PACS:
Keywords: weak values, weak measurements, Wigner distribution, inter-ference
We study in the present Letter the notion of weak measurement introducedby Aharonov and Albert [2], Bergmann, and Lebowitz in [1, 2, 3, 4, 5, 6]1rom the point of view of the Wigner phase space formalism. This will allowus to discuss the claim made by these authors that the weak value can beseen as the interference of two wavepackets, one going forward in time andthe other backwards in time.Let us briefly recall the difference between an ideal (also called strong, orvon Neumann) measurement, and a weak measurement. Let b A be a (quan-tum) observable, realized as an essentially self-adjoint operator; we assumefor simplicity that b A has a eigenvalues a , a , ... with corresponding orthog-onal eigenfunctions ψ , ψ , ... . In an ideal measurement the expectationvalue of b A in a pre-selected state ψ is h b A i ψ = h ψ | b A | ψ ih ψ | ψ i ; (1)if the sequence of eigenvalues lies in some interval [ a min , a max ] then we willhave a min ≤ h b A i ψ ≤ a max . In fact, if one performs the ideal measurementthe outcome will always be one of the eigenvalues λ j , and the probabil-ity of this outcome is | λ j | / || ψ j || where λ j is the coefficient of ψ j in theFourier expansion ψ = P j λ j ψ j . Moreover the system will be left in thestate ψ j after the ideal measurement yielding the value a j . The situationis very different for weak measurements. As is explained in Ritchie et al.[25] (also see Berry and Shukla [9], Steinberg [27]), in a weak measurementthe eigenvalues are not fully resolved and the system is left in a superposi-tion of the unresolved states. If an appropriate post-selection is made, thissuperposition can interfere to produce a measurement result which can besignificantly outside the range of the eigenvalues of the observable b A . Thepost-selection can then be accomplished by making an ideal measurementof some other observable b B and selecting one particular outcome. Thus,the post-selected state | φ i is an eigenstate of b B which can be expressed asa linear combination of the eigenstates of b A (we note that, conversely, anideal measurement can be expressed as a convex sum of weak values: seeHosoya and Shikano [21]). If h φ | ψ i 6 = 0 (and if φ , ψ are square integrable)the weak value of b A is then the complex number h b A i φ,ψ weak = h φ | b A | ψ ih φ | ψ i . (2)We will show that this weak value can be expressed in terms of thecross-Wigner transform W ( φ, ψ )( x, p ) = (cid:0) π ~ (cid:1) N Z R N e − i ~ py φ ∗ ( x + y ) ψ ( x − y ) dy (3)2f the pair ( φ, ψ ) whose physical interpretation is that of an interferenceterm in the Wigner distribution of the sum φ + ψ ; we mention that theimportance of these interference terms have been emphasized and studiedby Zurek [29] in the context of the sub-Planckian structures in phase space.The cross-Wigner transform is a very important object being intensivelystudied in the harmonic analysis literature and in time-frequency analysis;see e.g. Cohen [11], Folland [14] Gr¨ochenig [17], Hlawatsch and Flandrin[20]. Notice that W ( φ, ψ )( x, p ) reduces to the familiar Wigner distribution(Hillery et al. [19], Littlejohn [22] when φ = ψ .We will not address here the ontological debates arising around the prob-lem of “Elements of Reality” (see Cohen and Hiley [12, 13]); these questionsare difficult and have led to profound philosophical controversies. Notation.
We will work with systems having N degrees of freedom. Po-sition (resp. momentum) variables are denoted x = ( x , ..., x N ) (resp. p =( p , ..., p N )); they are vectors in R N . The corresponding phase space vari-able is z = ( x, p ); it is a vector in phase space R N . We will endow the phasespace with the standard symplectic form σ ( z, z ′ ) = px ′ − p ′ x . When integrat-ing we will use, where appropriate, the volume elements dx = dx · · · dx N , dp = dp · · · dp N , dz = dpdx . The unitary ~ -Fourier transform of a function ψ in L ( R N ) is defined by F ψ ( p ) = (cid:0) π ~ (cid:1) N/ Z R N e − i ~ py ψ ( y ) dy. The cross-Wigner transform (3) satisfies the “marginal properties” Z R N W ( φ, ψ )( z ) dp = φ ( x ) ∗ ψ ( x ) (4)and Z R N W ( φ, ψ )( z ) dx = F φ ( p ) ∗ F ψ ( p ) . (5)It follows from the equality (4) that Z R N W ( φ, ψ )( z ) dz = h φ | ψ i . (6)3or h φ | ψ i 6 = 0 we define ρ φ,ψ ( z ) = W ( φ, ψ )( z ) h φ | ψ i . (7)Note the conjugation relation ρ φ,ψ ( z ) ∗ = ρ ψ,φ ( z ); also ρ λφ,λψ ( z ) = ρ φ,ψ ( z )for every complex λ = 0 hence the function ρ φ,ψ only depends on the states | ψ i and | φ i . In view of Eqn. (6) we have Z R N ρ φ,ψ ( z ) dz = 1 (8)hence ρ φ,ψ can be viewed as a complex probability distribution with respectto which the average of the classical observable A is calculated; also, Eqn.(8) implies that Z R N Re ρ φ,ψ ( z ) dz = 1 , Z R N Im ρ φ,ψ ( z ) dz = 0 (9)so that Re ρ φ,ψ can be viewed as a quasi-distribution, in the same way as theusual Wigner transform. When ψ = φ then Im ρ ψ,ψ = 0 and Re ρ ψ,ψ = W ψ .Observe that it immediately follows from Eqns. (7) and (4), (5) that themarginals distributions of ρ φ,ψ are given by Z R N ρ φ,ψ ( z ) dp = φ ∗ ( x ) ψ ( x ) h φ | ψ i , Z R N ρ φ,ψ ( z ) dx = [ F φ ( p )] ∗ F ψ ( p ) h φ | ψ i ; (10)note that anyone of these equalities allows by integrating in the conjugatevariable to recover the normalization condition (8).We point out that the consideration of complex probability densitieshas per se nothing unusual; such complex probabilities have been used inthe context of stochastic processes (see Zak [28]), signal theory (multipathfading channels, see Chayawan [10]) and they also appear in the study ofnon-Hermitian quantum mechanics (see Barkay and Moiseyev [8]).We claim that: Theorem 1
Let A be a classical observable and b A its Weyl quantization;we have h b A i φ,ψ weak = Z R N A ( z ) ρ φ,ψ ( z ) dz. (11)The reader familiar with the Weyl–Wigner–Moyal formalism (de Gosson[15, 16], Littlejohn [22]) will have noticed that when φ = ψ formula (11)reduces to the well-known relation h ψ | b A | ψ i = Z R N A ( z ) W ψ ( z ) dz h b A i ψ = h ψ | b A | ψ i / h ψ | ψ i . We will studythe relative importance of these values when φ and ψ are coherent states inSubsection 2.3 To prove formula (11) it is sufficient, in view of definition (7) of ρ φ,ψ ( z ), toshow that h φ | b A | ψ i = Z R N W ( φ, ψ )( z ) A ( z ) dz. (12)To prove the latter we could perform a direct calculation staring from theright-hand side, inserting the expression (3) of W ( φ, ψ )( z ) and making var-ious changes of variables. We prefer to give a more elegant proof which hassome conceptual advantages. The first step consists in observing that thecross-Wigner transform can be expressed in terms of the Grossmann–Royer[18, 26] operator b T GR ( z ) φ ( x ) = e i ~ p ( x − x ) φ (2 x − x ) (13)(also see de Gosson [16], Chapter 9). A simple calculation shows that wehave W ( φ, ψ )( z ) = (cid:0) π ~ (cid:1) N h b T GR ( z ) φ | ψ i (14)and that the Weyl quantization b A of the observable is given by b Aψ ( x ) = (cid:0) π ~ (cid:1) N Z R N A ( z ) b T GR ( z ) ψ ( x ) dz . (15)Using the latter we have h φ | b A | ψ i = (cid:0) π ~ (cid:1) N Z R N A ( z ) h φ | b T GR ( z ) ψ i dz ; (16)we next observe that b T GR ( z ) is both unitary and involutive (i.e. b T GR ( z ) = b T GR ( z ) − ) and hence h φ | b T GR ( z ) ψ i = h b T GR ( z ) − φ | ψ i = h b T GR ( z ) φ | ψ i (17)so that (16) can be rewritten h φ | b A | ψ i = (cid:0) π ~ (cid:1) N Z R N A ( z ) h b T GR ( z ) φ | ψ i dz = Z R N A ( z ) W ( φ, ψ )( z ) dz which was to be proven. 5 .3 The case of coherent states Suppose that both wavefunctions are normalized coherent states concen-trated near z = ( x , p ) and − z at time t in , that is we choose θ and ψ = ψ z where θ ( x ) = (cid:0) π ~ (cid:1) N/ b T ( z ) e − ~ | x | , ψ ( x ) = (cid:0) π ~ (cid:1) N/ b T ( − z ) e − ~ | x | ; (18)where b T ( z ) = e − i ~ ( x b p − p b x ) is the Heisenberg–Weyl operator. These statesare minimum uncertainty states (they saturate the Heisenberg inequalities∆ x j ∆ p j ≥ ~ ). A standard calculation of Gaussian integrals shows that thescalar product of these states is h θ | ψ i = e − ~ | z | . (19)Let us calculate W ( φ, ψ ). Using the translation formula (see de Gosson [16]) W ( b T ( α ) φ, b T ( β ) ψ )( z ) = e − i ~ χ αβ ( z ) W ( φ, ψ )( z − ( α + β )) (20)where χ αβ is the phase function defined by χ αβ ( z ) = σ ( z, α − β ) + σ ( α, β ) (21)( σ the standard symplectic form). We thus have W ( φ, ψ )( z ) = e i ~ σ ( z,z ) W ( ξ , ξ )( z )where σ ( z, z ) = px − p x and ξ ( x ) = ( π ~ ) − N/ e −| x | / ~ is the standardfiducial coherent state (Littlejohn [22]). Now, W ( ξ , ξ ) = W ξ , the Wignerdistribution of ξ , which is given by W ξ ( z ) = (cid:0) π ~ (cid:1) N e − ~ | z | , | z | = | x | + | p | (22)(de Gosson [15, 16], Littlejohn [22]). We thus conclude that W ( φ, ψ )( z ) = (cid:0) π ~ (cid:1) N e i ~ σ ( z,z ) e − ~ | z | . (23)Using the scalar product formula (19) we see that the complex probabilitydistribution ρ φ,ψ is given by ρ φ,ψ ( z ) = (cid:0) π ~ (cid:1) N e i ~ σ ( z,z ) e ~ | z | e − ~ | z | . (24)6his formula shows that ρ α,β ( z ) has an oscillatory behavior which is sharplypeaked near the origin. We notice that since | ρ φ,ψ ( z ) | ≤ (cid:0) π ~ (cid:1) N e ~ | z | e − ~ | z | the weak value h b A i φ,ψ weak satisfies |h b A i φ,ψ weak | ≤ Z R N | ρ φ,ψ ( z ) || A ( z ) | dz = (cid:0) π ~ (cid:1) N e ~ | z | Z R N e − ~ | z | | A ( z ) | dz ≤ (cid:0) π ~ (cid:1) N e ~ | z | sup | A ( z ) | Z R N e − ~ | z | dz. The integral in the third line is easy to evaluate; its value is ( π ~ ) N hencewe have the estimate |h b A i φ,ψ weak | ≤ e ~ | z | sup | A ( z ) | . (25)This inequality shows that even if the observable A is small, the weak valuecan a priory take very large values provided that the phase space distancebetween both wavepackets φ, ψ is large; this is in strong contrast with whathappens for the individual states | φ i and | ψ i , for which lead to the estimates |h b A i φ | ≤ sup | A ( z ) | , |h b A i ψ | ≤ sup | A ( z ) | ;the relative phase space localization of these states does not play any rolein these inequalities. We will shortly discuss non-trivial extensions of thesuperposition considered above in the discussion below. Let us apply the phase space formalism to a discussion of the situation ini-tially considered in [3, 4] where at a time t in an observable b A is measured anda non-degenerate eigenvalue was found: | ψ ( t in ) i = | b A = a i (the pre-selectedstate); similarly at a later time t fin a measurement of another observable b B yields | φ ( t fin ) i = | b B = b i (the post-selected state). Let t be some intermedi-ate time: t in < t < t fin . Following the time-symmetric approach to quantummechanics (see the review in [6]), at this intermediate time the system isdescribed by the two wavefunctions ψ t = U Ht,t in ψ ( t in ) , φ t = U Ht,t fin φ ( t fin ) (26)7here U Ht,t ′ = e − i b H ( t − t ′ ) / ~ is the Schr¨odinger unitary evolution operator( b H the quantum Hamiltonian). Notice that φ t travels backwards in timesince t < t fin . The situation is thus the following: at any time t ′ < t thesystem under consideration is in the state | ψ t ′ i = U Ht ′ ,t in | ψ ( t in ) i and hasWigner distribution W ψ t ′ ; at any time t ′′ > t the system is in the state | φ t ′′ i = U Ht ′′ ,t fin | φ ( t in ) i and has Wigner distribution W φ t ′ . But at time t it isthe superposition | ψ t i + | φ t i of both states, and the Wigner distribution ofthis cat-like state is W ( φ t + ψ t ) = W φ t + W ψ t + 2 Re W ( φ t , ψ t ) . (27)This equality shows the abrupt emergence at time t –and only at that time!–of the interference term 2 Re W ( φ t , ψ t ), signalling a strong interaction be-tween the states | ψ t i and | φ t i . Such an interaction is due to the wavelikenature of quantum mechanics, and is absent from classical mechanics. Theappearance of interference terms described by the cross-Wigner transform iswell-known and considered as an asset in time-frequency analysis (e.g. radartheory, see Cohen [11], Auslander and Tolimieri [7]). It seems therefore thatour approach could well open new perspectives in the topic of weak measure-ments and values, by importing robust techniques from these Sciences (it isa fact, due mainly to historical and technical reasons, that the mathematicaltechniques related to the Wigner formalism have grown faster and are moresophisticated in signal theory and time-frequency analysis than they are inquantum mechanics, so a feedback seems to be more than welcome!).How the weak values are related to sub-Planckian scales would also beinteresting to investigate; the discussion in Zurek [29], and especially the re-sults in Nicacio et al. [24] could certainly be useful in this context. These au-thors consider superpositions of an arbitrary number of Gaussian states, andstudy their motion under the action of arbitrary Hamiltonian flows. Theyshow that the interference terms coming from the cross-Wigner transformsare always hyperbolic and survive the action of a thermal reservoir. Whilethey mainly have in mind semiclassical dynamics, their approach could beimplemented in the context of weak values. It is actually to a large extentsufficient to study the case of coherent states as in Subsection 2.3, becausethese states form an overcomplete set in the square-integrable functions. Infact, choosing an adequate lattice Λ of points z in phase space the func-tions b T ( z ) ξ ( ξ ( x ) = ( π ~ ) − N/ e −| x | / ~ ) form a Gabor frame (Gr¨ochenig[17]) allowing to write an arbitrary pure state as a linear superposition ofthe states b T ( z ) ξ . The net contribution of all cross-Wigner transforms ofpairs ( b T ( z ) ξ , b T ( z ) ξ ) with z = z is then the total interference leading8o weak values (in [29] Zurek considers a “compass state” consisting of fourterms b T ( z ) ξ , of which he studies interference effects at the sub-Planckianscale; it would be interesting to interpret his results in terms of weak values).There is another aspect of the theory of weak values we have not men-tioned at all, if only because of lack of space and time. It is the possibility ofreconstructing wave functions from weak values, as initiated in Lundeen etal. [23]. It turns out that the Wigner approach sketched in this Letter leadsto useful formulas. For instance, on proves the following inversion formula(de Gosson [16], § η be an arbitrary square integrable functionsuch that h φ | γ i 6 = 0; then ψ ( x ) = 2 N h φ | γ i Z R N W ( φ, ψ ) h b T GR ( z ) ψ | γ i dz . (28)We can reconstruct ψ from the knowledge of the weak value provided thatwe know h φ | γ i . This inversion formula together with the notion of mutu-ally unbiased bases (MUB) could certainly play an important role in thereconstruction problem. Acknowledgements . The first author (MdG) has been supported by theEU FET Open grant UNLocX (255931). Both authors wish to thank BasilHiley (Birkbeck) for useful remarks and helpful criticism. We are also happyto thank Hans Feichtinger for having pointed out several misprints.
References [1] Y. Aharonov, P. G. Bergmann, J. Lebowitz. Time Symmetry in theQuantum Process of Measurement. Phys. Rev. B 134, B1410–B1416(1964)[2] Y. Aharonov, D. Z. Albert, L. Vaidman. How the Result of a Measure-ment of a Component of the Spin of a Spin- δδ