Geometric approach to the discrete Wigner function

TL;DR

This paper explores a geometric method for constructing the discrete Wigner function using finite field-based rotation and displacement operators, addressing algebraic issues and providing explicit formulas for different characteristics.

Contribution

It introduces a geometric approach to the discrete Wigner function, clarifies the algebraic origin of its non-uniqueness, and derives explicit expressions for odd and even characteristics.

Findings

Explicit formulas for Wigner kernels in odd and even cases

Analysis of algebraic origin of non-uniqueness

Clarification of geometric construction of the Wigner function

Abstract

We analyze the Wigner function constructed on the basis of the discrete rotation and displacement operators labeled with elements of the underlying finite field. We separately discuss the case of odd and even characteristics and analyze the algebraic origin of the non uniqueness of the representation of the Wigner function. Explicit expressions for the Wigner kernel are given in both cases.