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arXiv:2409.19206 · Physical Review Research

Particle approximations of Wigner distributions for n arbitrary observables

Ralph Sabbagh, Olga Movilla Miangolarra, Hamid Hezari, Tryphon T. Georgiou

Published Jan 27, 2025• 0 citations

Abstract

A class of signed joint probability measures for n arbitrary quantum observables is derived and studied based on quasicharacteristic functions with symmetrized operator orderings of Margenau-Hill type. It is shown that the Wigner distribution associated with these observables can be rigorously approximated by such measures. These measures are given by affine combinations of Dirac delta distributions supported over the finite spectral range of the quantum observables and give the correct probability marginals when coarse-grained along any principal axis. We specialize to bivariate quasiprobability distributions for the spin measurements of spin- 1 / 2 particles and derive their closed-form expressions. As a side result, we point out a connection between the convergence of these particle approximations and the Mehler-Heine theorem. Finally, we interpret the supports of these quasiprobability distributions in terms of repeated thought experiments.

DOI: 10.1103/physrevresearch.7.013102

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A class of signed joint probability measures for n arbitrary quantum observables is derived and studied based on quasicharacteristic functions with symmetrized operator orderings of Margenau-Hill type. It is shown that the Wigner distribution associated with these observables can be rigorously approximated by such measures. These measures are given by affine combinations of Dirac delta distributions supported over the finite spectral range of the quantum observables and give the correct probability marginals when coarse-grained along any principal axis. We specialize to bivariate quasiprobability distributions for the spin measurements of spin- 1 / 2 particles and derive their closed-form expressions. As a side result, we point out a connection between the convergence of these particle approximations and the Mehler-Heine theorem. Finally, we interpret the supports of these quasiprobability distributions in terms of repeated thought experiments.

We study a generalization of the Wigner function to arbitrary tuples of hermitian operators. We show that for any collection of hermitian operators A1...An , and any quantum state there is a unique joint distribution on R^n, with the property that the marginals of all linear combinations of the operators coincide with their quantum counterpart. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution, because for position and momentum this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.

We study a generalization of the Wigner function to arbitrary tuples of Hermitian operators. We show that for any collection of Hermitian operators A1, …, An and any quantum state, there is a unique joint distribution on Rn with the property that the marginals of all linear combinations of the Ak coincide with their quantum counterparts. In other words, we consider the inverse Radon transform of the exact quantum probability distributions of all linear combinations. We call it the Wigner distribution because for position and momentum, this property defines the standard Wigner function. We discuss the application to finite dimensional systems, establish many basic properties, and illustrate these by examples. The properties include the support, the location of singularities, positivity, the behavior under symmetry groups, and informational completeness.

Marginal properties of arbitrary s-ordered quasiprobability distribution fucntions are investigated. The positivity of marginal distributions for all quasiprobabilities, interpolating between the Wigner function and the Q representation of the density operator (s\ensuremath{\le}0), is proven. We also show that, on the contrary, marginal distributions of the remaining quasiprobabilities, interpolating between the Wigner function and the P representation (s>0), can take on negative values. General formulas for the marginal distributions are given, and their relations to the actual quantum-mechanical probability distributions for position and momentum are established. Our results provide an interesting generalization of some recent results concerning marginal properties of the P representation [G. S. Agarwal, Opt. Commun. 95, 109 (1993)].

We study a generalization of the Wigner function to arbitrary tuples of hermitian operators, which is a distribution uniquely characterized by the property that the marginals for all linear combinations of the given operators agree with the quantum mechanical distributions. Its role as a joint quasi-probability distribution is underlined by the property that its support always lies in the set of expectation value tuples of the operators. We characterize the set of singularities and positivity, and provide some basic examples.

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