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arXiv:0812.1830 · Physical Review A

Inadequacy of a classical interpretation of quantum projective measurements via Wigner functions

Amir Kalev, A. Mann, Pier A. Mello, M. Revzen

Published Jan 29, 2009• 10 citations
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Abstract

We study the possibility of giving a classical interpretation to quantum projective measurements for a particle described by a pure Gaussian state whose Wigner function is non-negative. We analyze the case of a projective measurement which gives rise to a proper Wigner function---i.e., taking on, as its values, the eigenvalues of the projector. We find that, despite having this property, this kind of projector produces a state whose Wigner function ceases to be non-negative and hence precludes its interpretation as a classical probability density. We also study the general case in which the projected state has a non-negative Wigner function, but then we find that the Wigner function of the projector is not a proper one. Thus, we conclude that a non-negative Wigner function is inadequate to serve as a hidden-variable model for quantum processes in which projective measurements take place.

DOI
10.1103/physreva.79.014104

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We study the possibility of giving a classical interpretation to quantum projective measurements for a particle described by a pure Gaussian state whose Wigner function is non-negative. We analyze the case of a projective measurement which gives rise to a proper Wigner function---i.e., taking on, as its values, the eigenvalues of the projector. We find that, despite having this property, this kind of projector produces a state whose Wigner function ceases to be non-negative and hence precludes its interpretation as a classical probability density. We also study the general case in which the projected state has a non-negative Wigner function, but then we find that the Wigner function of the projector is not a proper one. Thus, we conclude that a non-negative Wigner function is inadequate to serve as a hidden-variable model for quantum processes in which projective measurements take place.
In this paper we give a partial answer to the problem: When does an initially non-negative Wigner function remain non-negative under the effect of the time evolution? We show that, for pure states, this is the case for linear systems only; to prove this we use the fact that the Wigner function is non-negative if and only if the wavefunction is Gaussian. We also prove that the Green’s solution of the evolution equation of the Wigner function, which in the framework of probability theory corresponds to the conditional probability density, takes on negative values. We utilize a theorem, about moments, borrowed from Pawula. We conclude that the Wigner phase-space formulation of quantum mechanics cannot receive a genuine probabilistic interpretation.
We study the Wigner function for a quantum system with a discrete, infinite-dimensional Hilbert space, such as a spinless particle moving on a one-dimensional infinite lattice. We discuss the peculiarities of this scenario and of the associated phase-space construction, propose a meaningful definition of the Wigner function in this case and characterize the set of pure states for which it is non-negative. We propose a measure of non-classicality for states in this system, which is consistent with the continuum limit. The prescriptions introduced here are illustrated by applying them to localized and Gaussian states and to their superpositions.
One of the central foundational questions of physics is to identify what makes a system quantum as opposed to classical. One seminal notion of classicality of a quantum system is the existence of a non-contextual hidden variable model as introduced in the early work by Bell, Kochen and Specker. In quantum optics, the non-negativity of the Wigner function is a ubiquitous notion of classicality. In this work we establish an equivalence between these two concepts. In particular, we show that any non-contextual hidden variable model for Gaussian quantum optics has an alternative non-negative Wigner function description. Conversely, it was known that the Wigner representation provides a non-negative non-contextual description of Gaussian quantum optics. It follows that contextuality and Wigner negativity are equivalent notions of non-classicality and equivalent resources for this quantum subtheory. In particular, both contextuality and Wigner negativity are necessary for a computational speed-up of quantum Gaussian optics. At the technical level, our result holds true for any subfamily of Gaussian measurements that include homodyne measurements, i.e., measurements of standard quadrature observables.
Recently, quantum models of carrier transport in semiconductors have been proposed on the basis of the Wigner formulation of quantum mechanics. Such models are demanded by the semiconductor industry due to the fast transition from microelectronics to nanoelectronics. However, the Wigner function cannot be interpreted as a quantum analog of a distribution function, since it is not a probability density. In this work we show how this difficulty could be overcome by introducing a non standard quantum theory, based on the concept of Positive-Operator-Valued (POV) measures [3–5,7].Such a theory is demanded by the semiconductor industry due to the fast transition from microelectronics to nanoelectronics. The Wigner function cannot be strictly interpreted as a quantum analog of a distribution function, because it is not a probability density. In this communication we show that this difficulty can be overcome by introducing a non standard quantum theory, based on the concept of Positive-Operator-Valued (POV) measures rather than on Projector-Valued (PV) measures. The introduction of this more general class of mathematical objects accounts for the stochastic effect of measure errors in the effective evaluation of observables described by PV measures [1, 2]. Following Prugovecki [3], we show that, on an appropriately defined stochastic phase space, it is possible to introduce a “quantum distribution function” whose marginals give the desired macroscopic quantities, and which does not present the disadvantages of the Wigner function. We present the resulting quantum Boltzmann equations for an ensemble of free carriers, and discuss extensions to the case of carriers in a semiconductor.
The Wigner function of quantum systems is an effective instrument for constructing an approximate classical description of the systems for which the classical approximation is possible. In recent years, the Wigner function formalism has been applied as well to seek indications of specific quantum properties of quantum systems leading to the impossibility of constructing the classical approximation. Used most of all as such an indication is the existence of negative values in the Wigner function for specific states of the quantum system being studied. The existence of such values itself prejudices the probabilistic interpretation of the Wigner function, though for an arbitrary observable depending jointly on the coordinate and the momentum of the quantum system just the Wigner function provides an effective instrument for calculating the average value and the other statistical characteristics. In this paper, a probabilistic interpretation of the Wigner function based on coordinating a theoretical probabilistic definition of the probability density, with restrictions to a physically small domain of phase space due to the uncertainty principle, is proposed.
Understanding what distinguishes quantum mechanics from classical mechanics is crucial for quantum information processing applications. In this work, we consider two notions of non-classicality for quantum systems, negativity of the Wigner function and contextuality for Pauli measurements. We prove that these two notions are equivalent for multi-qudit systems with odd local dimension. For a single qudit, the equivalence breaks down. We show that there exist single qudit states that admit a non-contextual hidden variable model description and whose Wigner functions are negative.