Wigner distributions for finite state systems without redundant phase point operators

TL;DR

This paper develops a uniform approach to defining Wigner distributions for finite quantum systems using an N×N phase space grid, avoiding redundancies and analyzing differences between odd and even N cases.

Contribution

It introduces a phase space formulation for finite quantum systems that avoids redundant phase points, providing a unified framework for both odd and even N.

Findings

Odd N allows full marginals implementation.

Even N permits only restricted marginals.

Multiple valid definitions of Wigner distributions for even N.

Abstract

We set up Wigner distributions for $N$ state quantum systems following a Dirac inspired approach. In contrast to much of the work on this case, requiring a $2N\times 2N$ phase space, particularly when $N$ is even, our approach is uniformly based on an $N\times N $ phase space grid and thereby avoids the necessity of having to invoke a `quadrupled' phase space and hence the attendant redundance. Both $N$ odd and even cases are analysed in detail and it is found that there are striking differences between the two. While the $N$ odd case permits full implementation of the marginals property, the even case does so only in a restricted sense. This has the consequence that in the even case one is led to several equally good definitions of the Wigner distributions as opposed to the odd case where the choice turns out to be unique.