Inequalities for quantum divergences and the Audenaert-Datta conjecture

TL;DR

This paper reviews inequalities related to quantum divergences, focusing on the Audenaert-Datta conjecture about the Data Processing Inequality for the lpha-z-Renyi relative entropies, summarizing current progress and methods.

Contribution

It provides a unified review of the problem, its context, and the partial solutions to the Audenaert-Datta conjecture on quantum divergences.

Findings

Summary of known results on the DPI for lpha-z-Renyi entropies

Analysis of methods used in partial solutions

Identification of open problems and future directions

Abstract

Given two density matrices $\rho$ and $\sigma$, there are a number of different expressions that reduce to the $\alpha$-R\'enyi relative entropy of $\rho$ with respect to $\sigma$ in the classical case; i.e., when $\rho$ and $\sigma$ commute. Only those expressions for which the Data Processing Inequality (DPI) is valid are of potential interest as quantum divergences in quantum information theory. Audenaert and Datta have made a conjecture on the validity of the DPI for an interesting family of quantum generalizations of the $\alpha$ - R\'enyi relative entropies, the $\alpha-z$ - R\'enyi relative entropies. They and others have contributed to the partial solution of this conjecture. We review the problem, its context, and the methods that have been used to obtain the results that are known at present, presenting a unified treatment of developments that have unfolded in a number of different papers.