Particle approximations of Wigner distributions for n arbitrary observables

TL;DR

This paper introduces a new class of signed joint probability measures to approximate Wigner distributions for multiple quantum observables, providing rigorous methods and explicit formulas, especially for spin-1/2 particles.

Contribution

It develops a novel approach using quasi-characteristic functions and affine combinations of delta distributions to approximate Wigner distributions for arbitrary quantum observables.

Findings

Derived explicit formulas for bivariate quasi-probability distributions of spin-1/2 particles.

Established a connection between convergence of approximations and the Mehler-Heine theorem.

Provided a physical interpretation of the distribution supports through thought experiments.

Abstract

A class of signed joint probability measures for n arbitrary quantum observables is derived and studied based on quasi-characteristic 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 quasi-probability 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 quasi-probability distributions in terms of repeated thought experiments.