# Representations of two-qubit and ququart states via discrete Wigner functions

## TL;DR

This paper develops a theoretical framework using discrete Wigner functions for two-qubit and ququart states, linking phase space representations with experimental quantum state tomography and quantum correlations.

## Contribution

It introduces a detailed study of discrete SU(2)⊗SU(2) and SU(4) Wigner functions, connecting them with experimental density matrix reconstructions and quantum correlation visualization.

## Key findings

- Established a formal correspondence between different phase space descriptions.
- Analyzed quantum correlation effects in two-qubit X-states.
- Proposed extensions to other distribution functions like Husimi and Glauber-Sudarshan.

## Abstract

By means of a well-grounded mapping scheme linking Schwinger unitary operators and generators of the special unitary group $\mathrm{SU(N)}$, it is possible to establish a self-consistent theoretical framework for finite-dimensional discrete phase spaces which has the discrete $\mathrm{SU(N)}$ Wigner function as a legitimate by-product. In this paper, we apply these results with the aim of putting forth a detailed study on the discrete $\mathrm{SU(2)} \otimes \mathrm{SU(2)}$ and $\mathrm{SU(4)}$ Wigner functions, in straight connection with experiments involving, among other things, the tomographic reconstruction of density matrices related to the two-qubit and ququart states. Next, we establish a formal correspondence between both the descriptions that allows us to visualize the quantum correlation effects of these states in finite-dimensional discrete phase spaces. Moreover, we perform a theoretical investigation on the two-qubit X-states, which combines discrete Wigner functions and their respective marginal distributions in order to obtain a new function responsible for describing qualitatively the quantum correlation effects. To conclude, we also discuss possible extensions to the discrete Husimi and Glauber-Sudarshan distribution functions, as well as future applications on spin chains.