Permutation Symmetry Determines the Discrete Wigner Function

TL;DR

This paper establishes that the permutation symmetry of the discrete Wigner function is equivalent to the symmetry group being a unitary 2-design, uniquely characterizing the Wigner function in odd prime power dimensions and linking it to Clifford covariance.

Contribution

It proves the equivalence between permutation symmetry and unitary 2-designs for the discrete Wigner function, identifying conditions for its uniqueness in odd prime power dimensions.

Findings

Permutation symmetry corresponds to unitary 2-designs in discrete Wigner functions.

Unique discrete Wigner function is characterized by Clifford covariance in odd prime power dimensions.

No Clifford covariant Wigner function exists in even prime power dimensions.

Abstract

The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2-design. Such a highly symmetric representation can only appear in odd prime power dimensions besides dimensions 2 and 8. It suffices to single out a unique discrete Wigner function among all possible quasiprobability representations. In the course of our study, we show that this discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension.