# On the discrete Wigner function for SU(N)

## TL;DR

This paper develops a rigorous framework for defining a discrete Wigner function for finite-dimensional quantum systems, establishing a mapping between SU(N) generators and phase space operators, with applications to three-level systems.

## Contribution

It introduces a mathematically sound method to construct discrete Wigner functions for SU(N) groups, enabling analysis of finite-dimensional quantum states.

## Key findings

- Derived a general discrete Wigner function for SU(3)
- Applied the framework to a three-level quantum system
- Discussed extensions to other phase space quasiprobability functions

## Abstract

We present a self-consistent theoretical framework for finite-dimensional discrete phase spaces that leads us to establish a well-grounded mapping scheme between Schwinger unitary operators and generators of the special unitary group $\mathrm{SU(N)}$. This general mathematical construction provides a sound pathway to the formulation of a genuinely discrete Wigner function for arbitrary quantum systems described by finite-dimensional state vector spaces. To illustrate our results, we obtain a general discrete Wigner function for the group $\mathrm{SU(3)}$ and apply this to the study of a particular three-level system. Moreover, we also discuss possible extensions to the discrete Husimi and Glauber-Sudarshan functions, as well as future investigations on multipartite quantum states.