Relative Entropy via Distribution of Observables
Abstract
We obtain formulas for Petz-R\'enyi and Umegaki relative entropy from the idea of distribution of a positive selfadjoint operator. Classical results on R\'enyi and Kullback-Leibler divergences are applied to obtain new results and new proofs for some known results about Petz-R\'enyi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz-R\'enyi -relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory.
Keywords
Cite
@article{arxiv.2203.01964,
title = {Relative Entropy via Distribution of Observables},
author = {George Androulakis and Tiju Cherian John},
journal= {arXiv preprint arXiv:2203.01964},
year = {2023}
}
Comments
Previous version has been divided into two different articles. The first one in this series is `Quantum f-divergences via Nussbaum-Szko{\l}a Distributions and Applications to f-divergence Inequalities' accepted for publication at Rev. Math. Phys. (DOI: https://doi.org/10.1142/S0129055X23600024). The present article is accepted for publication at Infin. Dimens. Anal. Quantum Probab. Relat. Top