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arXiv · 1807.03118

Quantum $f$-divergences in von Neumann algebras II. Maximal $f$-divergences

Fumio Hiai
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Abstract

As a continuation of the paper [20] on standard $f$-divergences, we make a systematic study of maximal $f$-divergences in general von Neumann algebras. For maximal $f$-divergences, apart from their definition based on Haagerup's $L^1$-space, we present the general integral expression and the variational expression in terms of reverse tests. From these definition and expressions we prove important properties of maximal $f$-divergences, for instance, the monotonicity inequality, the joint convexity, the lower semicontinuity, and the martingale convergence. The inequality between the standard and the maximal $f$-divergences is also given.

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Fumio Hiai. 2018-07-09. Quantum $f$-divergences in von Neumann algebras II. Maximal $f$-divergences. https://doi.org/10.1063/1.5051427

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