A Family of Weyl-Wigner Transforms for Discrete Variables Defined in a Finite-Dimensional Hilbert Space

TL;DR

This paper introduces a family of Weyl-Wigner transforms for discrete finite-dimensional quantum systems, highlighting a special parameter value where properties mirror those of continuous-variable systems, with a brief geometric perspective.

Contribution

It defines a parametric family of Weyl-Wigner transforms for finite prime-dimensional Hilbert spaces and identifies the unique parameter value that aligns their properties with continuous-variable cases.

Findings

Only one parameter value yields properties similar to continuous variables

A geometric interpretation of the transforms is proposed

The transforms are defined specifically for prime-dimensional Hilbert spaces

Abstract

We study the Weyl-Wigner transform in the case of discrete variables defined in a Hilbert space of finite prime-number dimensionality $N$. We define a family of Weyl-Wigner transforms as function of a phase parameter. We show that it is only for a specific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-dimensional Hilbert space. A geometrical interpretation is briefly discussed.