Complete conditions for legitimate Wigner distributions

TL;DR

This paper establishes a comprehensive set of conditions to determine if a phase-space function is a valid Wigner distribution for quantum states, with applications to entanglement detection in continuous-variable systems.

Contribution

It introduces a hierarchy of complete criteria for verifying the physical realizability of Wigner distributions using normally-ordered expansions.

Findings

Derived a hierarchy of conditions for Wigner distribution validity

Showed elliptical distributions can be diagonalized easily in this framework

Connected the formulation to bipartite entanglement detection

Abstract

Given a real-valued phase-space function, it is a nontrivial task to determine whether it corresponds to a Wigner distribution for a physically acceptable quantum state. This topic has been of fundamental interest for long, and in a modern application, it can be related to the problem of entanglement detection for multi-mode cases. In this paper, we present a hierarchy of complete conditions for a physically realizable Wigner distribution. Our derivation is based on the normally-ordered expansion, in terms of annihilation andcreation operators, of the quasi-density operator corresponding to the phase-space function in question. As a by-product, it is shown that the phase-space distributions with elliptical symmetry can be readily diagonalized in our representation, facilitating the test of physical realizability. We also illustrate how the current formulation can be connected to the detection of bipartite entanglement for continuous variables.