Quantum $f$-divergences in von Neumann algebras I. Standard $f$-divergences

Abstract

We make a systematic study of standard $f$-divergences in general von Neumann algebras. An important ingredient of our study is to extend Kosaki's variational expression of the relative entropy to an arbitary standard $f$-divergence, from which most of the important properties of standard $f$-divergences follow immediately. In a similar manner we give a comprehensive exposition on the Rényi divergence in von Neumann algebra. Some results on relative hamiltonians formerly studied by Araki and Donald are improved as a by-product.

33 pages

References in corpus (3)

• Different quantum f-divergences and the reversibility of quantum operations
• Rényi divergences as weighted non-commutative vector valued $L_p$-spaces
• Rényi relative entropies and noncommutative $L_p$-spaces

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• The strong converse exponent of discriminating infinite-dimensional quantum states
• On the error exponents of binary state discrimination with composite hypotheses
• Recoverability of quantum channels via hypothesis testing
• Geometric relative entropies and barycentric Rényi divergences
• State convertibility in the von Neumann algebra framework
• Quantum Rényi divergences and the strong converse exponent of state discrimination in operator algebras
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• Quantum $f$-divergences via Nussbaum-Szkoła Distributions and Applications to $f$-divergence Inequalities
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• Quantum teleportation in the commuting operator framework
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