Quantum $f$-divergences via Nussbaum-Szko{\l}a Distributions and Applications to $f$-divergence Inequalities

TL;DR

This paper establishes that quantum $f$-divergences can be analyzed through classical $f$-divergences of Nussbaum-Szko{ }a distributions, enabling new inequalities and insights in quantum information theory applicable to finite and infinite-dimensional systems.

Contribution

It demonstrates the equivalence of quantum and classical $f$-divergences via Nussbaum-Szko{ }a distributions, providing a unified framework for studying quantum entropic quantities.

Findings

Quantum $f$-divergence equals classical $f$-divergence of Nussbaum-Szko{ }a distributions.

Derived several quantum $f$-divergence inequalities from classical counterparts.

Results applicable to both finite and infinite-dimensional quantum systems.

Abstract

The main result in this article shows that the quantum $f$-divergence of two states is equal to the classical $f$-divergence of the corresponding Nussbaum-Szko{\l}a distributions. This provides a general framework for studying certain properties of quantum entropic quantities using the corresponding classical entities. The usefulness of the main result is illustrated by obtaining several quantum $f$-divergence inequalities from their classical counterparts. All results presented here are valid in both finite and infinite dimensions and hence can be applied to continuous variable systems as well. A comprehensive review of the instances in the literature where Nussbaum-Szko{\l}a distributions are used, is also provided in this article.