On the Non-Uniqueness of Statistical Ensembles Defining a Density Operator and a Class of Mixed Quantum States with Integrable Wigner Distribution

TL;DR

This paper explores the non-uniqueness of statistical ensembles in quantum mechanics, introduces Feichtinger states based on modulation spaces, and extends Jaynes' results on the non-uniqueness of density operators.

Contribution

It introduces Feichtinger states defined via modulation spaces and extends Jaynes' non-uniqueness result to these states and convex sums of Wigner transforms.

Findings

Feichtinger states satisfy certain marginal properties of Wigner distributions.

Extended Jaynes' non-uniqueness theorem to a broader class of quantum states.

Provided new insights into the structure of mixed quantum states and their ensembles.

Abstract

It is standard to assume that the Wigner distribution of a mixed quantum state consisting of square-integrable functions is a quasi-probability distribution, that is that its integral is one and that the marginal properties are satisfied. However this is in general not true. We introduce a class of quantum states for which this property is satisfied, these states are dubbed "Feichtinger states" because they are defined in terms of a class of functional spaces (modulation spaces) introduced in the 1980's by H. Feichtinger. The properties of these states are studied, which gives us the opportunity to prove an extension to the general case of a result of Jaynes on the non-uniqueness of the statistical ensemble generating a density operator. As a bonus we obtain a result for convex sums of Wigner transforms.