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تسجيل مستخدم جديدQuantum $f$-divergences in von Neumann algebras II. Maximal $f$-divergences
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Fumio Hiai
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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 Haagerups $L^1$-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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We make a systematic study of standard $f$-divergences in general von Neumann algebras. An important ingredient of our study is to extend Kosakis variational expression of the relative entropy to an arbitary standard $f$-divergence, from which most o
f the important properties of standard $f$-divergences follow immediately. In a similar manner we give a comprehensive exposition on the Renyi divergence in von Neumann algebra. Some results on relative hamiltonians formerly studied by Araki and Donald are improved as a by-product.
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r quantum key distribution. This lemma was originally proven in finite dimension, with a direct extension to type I von Neumann algebras. Here we show that the use of modular theory allows to give more transparent meaning to the objects constructed by the lemma, and to prove it for general von Neumann algebras. This yields immediate generalizations of e.g. [arXiv:1510.04682].
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