Equality conditions of Data Processing Inequality for $\alpha$-$z$ Rényi relative entropies

Abstract

The $\alpha$-$z$ Rényi relative entropies are a two-parameter family of Rényi relative entropies that are quantum generalizations of the classical $\alpha$-Rényi 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{é} and Datta in \cite{LRD17DPI}.

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Cited by in corpus (7)

• Recoverability for optimized quantum $f$-divergences
• Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings
• Generalized Entropic Quantum Speed Limits
• Revisiting the equality conditions of the data processing inequality for the sandwiched Rényi divergence
• Geometric conditions for saturating the data processing inequality
• Rényi-Holevo inequality from $\alpha$-$z$-Rényi relative entropies
• Sufficiency of Rényi divergences