Equality conditions of Data Processing Inequality for $\alpha$-$z$ R\'enyi relative entropies

TL;DR

This paper characterizes the algebraic conditions under which the Data Processing Inequality holds with equality for the two-parameter family of quantum Re9nyi relative entropies, extending previous results.

Contribution

It provides necessary and sufficient algebraic conditions for equality in DPI for all parameter ranges of - and z-parameters, generalizing prior work.

Findings

Necessary and sufficient conditions for DPI equality across parameters

Generalization of previous DPI equality results

Strengthening of algebraic criteria for quantum Re9nyi entropies

Abstract

The $\alpha$-$z$ R\'enyi relative entropies are a two-parameter family of R\'enyi relative entropies that are quantum generalizations of the classical $\alpha$-R\'enyi relative entropies. In \cite{zhang20CFL} we decided the full range of $(\alpha,z)$ for which the Data Processing Inequality (DPI) is valid. In this paper we give algebraic conditions for the equality in DPI. For the full range of parameters $(\alpha,z)$, we give necessary conditions and sufficient conditions. For most parameters we give equivalent conditions. This generalizes and strengthens the results of Leditzky, Rouz{\'e} and Datta in \cite{LRD17DPI}.