Wigner distributions for finite-state systems without redundant phase-point operators

Abstract

We set up Wigner distributions for N-state quantum systems following a Dirac-inspired approach. In contrast to much of the work in this study, requiring a 2N × 2N phase space, particularly when N is even, our approach is uniformly based on an N × 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 marginal 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.