Translations and reflections on the torus: Identities for discrete Wigner functions and transforms

TL;DR

This paper explores the mathematical structure of discrete Wigner functions on a torus, introducing new identities and extending operators to a doubled phase space for better understanding of quantum state representations.

Contribution

It introduces new identities for products of pure state Wigner and chord functions and extends operators to a doubled phase space for superoperator representation.

Findings

Derived identities involving inverse phase space participation ratio

Established correlations of states with their translations

Constructed propagator of Wigner functions from the Weyl representation

Abstract

A finite Hilbert space can be associated to a periodic phase space, that is, a torus. A finite subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure state Wigner and chord functions, involving, for instance the inverse phase space participation ratio and correlations of a state with its translate. New identities are derived involving {\it transition} Wigner or chord functions of transition operators $\ket{\psi_1}\bra{\psi_2}$. Extension of the reflection and translation operators to a doubled torus phase space leads to the representation of superoperators and so to the construction of the propagator of Wigner functions from the Weyl representation of the evolution operator.