# On the Positivity of Trace Class Operators

## TL;DR

This paper investigates the positivity properties of trace-class Weyl operators in quantum mechanics, providing simplified proofs, a phase space characterization, and practical numerical approximations using time-frequency analysis techniques.

## Contribution

It introduces a phase space version of the KLM positivity condition and offers numerically feasible discrete approximations using Gabor frames and Wigner formalism.

## Key findings

- Simplified proofs of known positivity results for trace-class Weyl operators
- A phase space characterization of the KLM positivity condition
- Numerical methods for approximating the KLM condition

## Abstract

The characterization of positivity properties of Weyl operators is a notoriously difficult problem, and not much progress has been made since the pioneering work of Kastler, Loupias, and Miracle-Sole (KLM). In this paper we begin by reviewing and giving simpler proofs of some known results for trace-class Weyl operators; the latter play an essential role in quantum mechanics. We then apply time-frequency analysis techniques to prove a phase space version of the KLM condition; the main tools are Gabor frames and the Wigner formalism. Finally, discrete approximations of the KLM condition, which are tractable numerically, are provided.