On accumulated Cohen's class distributions and mixed-state localization operators

TL;DR

This paper extends the concept of accumulated spectrograms to accumulated Cohen class distributions for mixed-state localization operators, using quantum harmonic analysis to analyze their eigenvalues, eigenvectors, and phase space properties.

Contribution

It introduces the accumulated Cohen class distribution for mixed-state localization operators, generalizing previous spectrogram-based results through quantum harmonic analysis techniques.

Findings

Extended results from accumulated spectrograms to Cohen class distributions.

Developed a framework for analyzing eigenvalues and eigenvectors of mixed-state localization operators.

Provided tools for phase space analysis of time-frequency distributions.

Abstract

Recently we introduced mixed-state localization operators associated to a density operator and a (compact) domain in phase space. We continue the investigations of their eigenvalues and eigenvectors. Our main focus is the definition of a time-frequency distribution which is based on the Cohen class distribution associated to the density operator and the eigenvectors of the mixed-state localization operator. This time-frequency distribution is called the accumulated Cohen class distribution. If the trace class operator is a rank-one operator, then the mixed-state localization operators and the accumulated Cohen class distribution reduce to Daubechies' localization operators and the accumulated spectrogram. We extend all the results about the accumulated spectrogram to the accumulated Cohen class distribution. The techniques used in the case of spectrograms cannot be adapted to other distributions in Cohen's class since they rely on the reproducing kernel property of the short-time Fourier transform. Our approach is based on quantum harmonic analysis on phase space which also provides the tools and notions to introduce the analogues of the accumulated spectrogram for mixed-state localization operators; the accumulated Cohen's class distributions.