Rényi relative entropies and noncommutative $L_p$-spaces II

Abstract

We study an extension of the sandwiched Rényi relative entropies for normal positive functionals on a von Neumann algebra, for parameter values $\alpha \in [1/2,1)$. This work is intended as a continuation of [A. Jenčová, Ann. Henri Poincaré 19, 2513-2542, 2018], where the values $\alpha >1$ were studied. We use the Araki-Masuda divergences of [Berta et al., Ann. Henri Poincaré 9, 1843-1867, 2018] and treat them in the framework of Kosaki's noncommutative $L_p$-spaces. Using the variational formula, recently obtained by F. Hiai, for $\alpha \in [1/2,1)$, we prove the data processing inequality with respect to positive trace preserving maps and show that for $\alpha \in (1/2,1)$, equality characterizes sufficiency (reversibility) for any 2-positive trace preserving map.

A continuation of arXiv:1604.08462, rewritten an updated. Comments welcome

References in corpus (4)

• Rényi divergences as weighted non-commutative vector valued $L_p$-spaces
• Optimized quantum f-divergences and data processing
• Rényi relative entropies and noncommutative $L_p$-spaces
• Quantum $f$-divergences in von Neumann algebras I. Standard $f$-divergences

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• Geometric relative entropies and barycentric Rényi divergences
• Quantum Rényi divergences and the strong converse exponent of state discrimination in operator algebras
• Proof of Renyi QNEC for free fermions
• Trace- and improved data processing inequalities for von Neumann algebras
• Geometric conditions for saturating the data processing inequality
• Revisiting the equality conditions of the data processing inequality for the sandwiched Rényi divergence
• Uhlmann's theorem for relative entropies
• The exponential Orlicz space in quantum information geometry
• Asymptotic Equipartition Theorems in von Neumann algebras