Spectra of phase point operators in odd prime dimensions and the extended Clifford group

TL;DR

This paper investigates how the Extended Clifford group influences the spectral classification of phase point operators in odd prime dimensions, using symplectic vector spaces over finite fields to enhance understanding of discrete Wigner distributions.

Contribution

It introduces a detailed analysis of the Extended Clifford group's role in classifying spectra of phase point operators and derives conjugacy classes of ${\rm ESL}(2, \mathbb{F}_N)$.

Findings

Classification of spectra of phase point operators in odd prime dimensions

Derivation of conjugacy classes of ${\rm ESL}(2, \mathbb{F}_N)$

Insights into the structure of discrete phase spaces

Abstract

We analyse the role of the Extended Clifford group in classifying the spectra of phase point operators within the framework laid out by Gibbons et al for setting up Wigner distributions on discrete phase spaces based on finite fields. To do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field. Auxiliary results include a derivation of the conjugacy classes of ${\rm ESL}(2, \mathbb{F}_N)$.