Non-negative Wigner functions for orbital angular momentum states

I Rigas,Z Hradil,J Řeháček,L L Sánchez-Soto,A B Klimov

doi:10.1103/physreva.81.012101

Abstract

The Wigner function of a pure continuous-variable quantum state is non-negative if and only if the state is Gaussian. Here we show that for the canonical pair angle and angular momentum, the only pure states with non-negative Wigner functions are the eigenstates of the angular momentum. Some implications of this surprising result are discussed.

Highlights

The Wigner function of a pure continuous-variable quantum state is non-negative if and only if the state is Gaussian
The characterization of quantum states that are classical, in the sense of giving rise to non-negative Wigner functions, is a topic of undoubted interest. It was proven in a classical paper by Hudson [6] that the only states that have non-negative Wigner functions are Gaussian states [8, 9]
The classification of states with nonnegative Wigner functions is an amazing problem that has been solved quite recently by Paz and coworkers [12, 13] and Gross [14, 15], so that the role of Gaussian states is taken on by stabilizer states. These are the only states that can be simulated efficiently in classical computers [16]. Between these two cases, we have the interesting situation of canonical pairs, such as the angle and orbital angular momentum (OAM), for which one variable is continuous while the other one is discrete [17]

Summary

Introduction

The characterization of quantum states that are classical, in the sense of giving rise to non-negative Wigner functions, is a topic of undoubted interest. It was proven in a classical paper by Hudson [6] (later generalized by Soto and Claverie [7] to multipartite systems) that the only states that have non-negative Wigner functions are Gaussian states [8, 9]. The goal of this work is precisely to determine the pure states of these OAM-carrying systems for which the Wigner function is non-negative, filling in this way a long overdue gap.

Objectives

Results

Conclusion