Inequalities for quantum divergences and the Audenaert-Datta conjecture

Abstract

Given two density matrices $\rho$ and $\sigma$, there are a number of different expressions that reduce to the $\alpha$-Rényi 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 $\alpha$ - Rényi relative entropies, the $\alpha -z$ - Rényi 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.

arXiv comment23 pages

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• Geometric relative entropies and barycentric Rényi divergences
• Gradient flow structure and exponential decay of the sandwiched Rényi divergence for primitive Lindblad equations with GNS-detailed balance
• Interpolation of quasi noncommutative $L_p$-spaces
• Generalized Entropic Quantum Speed Limits
• Geometric properties for a class of deformed trace functions
• Generalizing Lieb's Concavity Theorem via Operator Interpolation
• Reverse Holder, Minkowski, And Hanner Inequalities For Matrices
• Log-majorizations between quasi-geometric type means for matrices
• Some convexity and monotonicity results of trace functionals
• Variational Representations related to Quantum Rényi Relative Entropies
• Conditions for Large-Sample Majorization of Pairs of Flat States in Terms of $\alpha$-z Relative Entropies