Toeplitz Density Operators and their Separability Properties

TL;DR

This paper investigates the properties of Toeplitz density operators, especially their separability, when their symbols are in specific functional spaces, with a focus on the Gaussian case, linking operator theory to quantum states.

Contribution

It explores the separability properties of Toeplitz density operators with symbols in functional spaces, extending understanding of their quantum mechanical implications.

Findings

Analysis of Toeplitz operators with symbols in the Feichtinger algebra

Discussion of separability properties in the Gaussian case

Connection between Toeplitz operators and quantum state representations

Abstract

Toeplitz operators (also called localization operators) are a generalization of the well-known anti-Wick pseudodifferential operators studied by Berezin and Shubin. When a Toeplitz operator is positive semi-definite and has trace one we call it a density Toeplitz operator. Such operators represent physical states in quantum mechanics. In the present paper we study several aspects of Toeplitz operators when their symbols belong to some well-known functional spaces (e.g. the Feichtinger algebra) and discuss (tentatively) their separability properties with an emphasis on the Gaussian case.