Quantum $f$-divergences via Nussbaum-Szkoła Distributions and Applications to $f$-divergence Inequalities

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-Szkoła 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-Szkoła distributions are used, is also provided in this article.

Text overlap with arXiv:2203.01964v2. That article is divided into two different articles. The present article is the first one in this series and contains two new sections including a review section. Accepted for publication at Rev. Math. Phys. 27 pages

References in corpus (9)

• The Quantum Chernoff Bound
• Quantum state discrimination and its applications
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• The Converse Part of The Theorem for Quantum Hoeffding Bound
• Quantum correlations and distinguishability of quantum states
• Large deviations and Chernoff bound for certain correlated states on a spin chain
• Quantum state discrimination bounds for finite sample size
• Hypothesis testing for Gaussian states on bosonic lattices