Wigner distributions for n arbitrary observables

TL;DR

This paper generalizes the Wigner function to arbitrary sets of hermitian operators, establishing a unique joint distribution that reproduces all linear combinations' quantum distributions, with applications to finite-dimensional systems.

Contribution

It introduces a novel generalization of the Wigner distribution for any hermitian operators, extending its properties and applications beyond position and momentum.

Findings

Existence of a unique joint distribution for any hermitian operators

Properties include support, singularities, positivity, and symmetry behavior

Application to finite-dimensional quantum systems

Abstract

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.