Wigner function for a particle in an infinite lattice

TL;DR

This paper develops a meaningful definition of the Wigner function for a particle on an infinite lattice, characterizes pure states with non-negative Wigner functions, and introduces a measure of non-classicality consistent with the continuum limit.

Contribution

It proposes a novel phase space construction and a measure of non-classicality for discrete lattice systems, extending the Wigner function framework.

Findings

Characterizes pure states with non-negative Wigner functions.

Applies the framework to localized and Gaussian states.

Introduces a non-classicality measure consistent with the continuum limit.

Abstract

We study the Wigner function for a quantum system with a discrete, infinite dimensional Hilbert space, such as a spinless particle moving on a one dimensional infinite lattice. We discuss the peculiarities of this scenario and of the associated phase space construction, propose a meaningful definition of the Wigner function in this case, and characterize the set of pure states for which it is non-negative. We propose a measure of non-classicality for states in this system which is consistent with the continuum limit. The prescriptions introduced here are illustrated by applying them to localized and Gaussian states, and to their superpositions.