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arXiv:1112.5773 · Journal of Physics A Mathematical and Theoretical

The reconstruction problem and weak quantum values

Maurice A. de Gosson, Serge M. de Gosson

Published Mar 1, 2012• 12 citations
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Abstract

Quantum mechanical weak values are an interference effect measured by the cross-Wigner transform of the post- and pre-selected states, leading to a complex quasi-distribution ρ ϕ, ψ ( x , p ) on phase space. We show that the knowledge of ρ ϕ, ψ ( z ) and of one of the two functions ϕ, ψ unambiguously determines the other, thus generalizing a recent reconstruction result of Lundeen and his collaborators.

DOI
10.1088/1751-8113/45/11/115305

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Quantum mechanical weak values are an interference effect measured by the cross-Wigner transform of the post- and pre-selected states, leading to a complex quasi-distribution ρ ϕ, ψ ( x , p ) on phase space. We show that the knowledge of ρ ϕ, ψ ( z ) and of one of the two functions ϕ, ψ unambiguously determines the other, thus generalizing a recent reconstruction result of Lundeen and his collaborators.
Closely associated with the notion of weak value is the problem of reconstructing the post-selected state: this is the so-called reconstruction problem. We show that the reconstruction problem can be solved by inversion of the cross-Wigner transform, using an ancillary state. We thereafter show, using the multidimensional Hardy uncertainty principle, that maximally concentrated cross-Wigner transforms corresponds to the case where a weak measurement reduces to an ordinary von Neumann measurement.
We generalize the concept of the weak value of a quantum observable to cover arbitrary real positive operator measures. We show that the definition is operationally meaningful in the sense that it can be understood within the quantum theory of sequential measurements. In particular, we show that the weak value can be obtained from a single measurement scheme. We then present a detailed analysis of the recent experiment [J. S. Lundeen et al., Nature (London) 474, 188 (2011)] concerning the reconstruction of the state of a photon using weak measurements. We compare their method with the reconstruction method through informationally complete phase space measurements. In particular, we show that unlike with phase space measurements, the reconstruction of a completely unknown state is not always possible using the method of weak measurements.
The two-state vector formalism provides a complete description of pre- and post-selected quantum systems and has uncovered a host of new quantum phenomena which were previously hidden. The most important feature is that any weak coupling to a pre- and post-selected system is effectively a coupling to a "weak value" which is given by a simple expression depending on the two-state vector. In particular, weak values, are the outcomes of so called "weak measurements" which have recently become a very powerful tool for ultra-sensitive measurements. Using weak values, I will show that the classical limit of quantum mechanics is a far more complicated issue; it is in fact dramatically more involved and it requires a complete revision of all our intuitions. The revised intuitions can then serve as a guide to finding novel quantum effects.
A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an ``anomalous weak value'', is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures.
We demonstrate that the Fourier transform of a function leads to interference effects in phase space which allow the reconstruction of the essential properties of the original function, even when we restrict the inverse Fourier transform to the phase factor rather than the complete Fourier transform. We analyze this phase-only reconstruction, known from signal processing, in quantum mechanics for the elementary example of a superposition of two real-valued Gaussians. We employ the Wigner function as well as techniques of semiclassical quantum mechanics to show that their key features, such as their separation and their width, are indeed encoded in the phase of the Fourier transform. For this example, the amplitude-only reconstruction performing the inverse Fourier transform of the absolute value of the Fourier transform fails.
The quantum mechanical weak value Aw=⟨ϕ∣A∣ψ⟩/⟨ϕ∣ψ⟩A_w=\left\langle ϕ|A|ψ\right \rangle / \left\langle ϕ| ψ\right\rangleAw​=⟨ϕ∣A∣ψ⟩/⟨ϕ∣ψ⟩ of an observable AAA is a measurable quantity associated with an observable AAA and pre- and post-selected states ∣ψ⟩,∣ϕ⟩\vertψ\rangle, \vert ϕ\rangle∣ψ⟩,∣ϕ⟩. Much has been discussed about the meaning and metrological uses of anomalous weak values, lying outside of the range of eigenvalues of AAA. We present a simple proof that anomalous weak values require that the (possibly mixed) pre- and post- selection states have coherence in the eigenbasis of AAA. We also present conditions under which anomalous AwA_wAw​ are witnesses of generalized contextuality, dispensing with the operational weak measurement set-up.