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arXiv:1703.05244 · Letters in Mathematical Physics

Maps on positive operators preserving Rényi type relative entropies and maximal f-divergences

Marcell Gaál, Gergő Nagy

Published Oct 26, 2017• 2 citations
arXivHTMLPDF
DOI
10.1007/s11005-017-1021-4

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The concept of classical fff-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the R\'enyi divergences. Various quantum versions of this concept, and more narrowly, the concept of R\'enyi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard fff-divergences), Matsumoto's maximal fff-divergences, measured fff-divergences, and sandwiched and α\alphaα-zzz-R\'enyi divergences. In this paper we give a systematic overview of the various concepts of quantum fff-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum fff-divergence by a quantum operation. In particular, we compare the standard and the maximal fff-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum fff-divergences are strictly different for non-commuting operators unless fff is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard fff-divergences. We also study the monotonicity of the α\alphaα-zzz-R\'enyi divergences under the special class of bistochastic maps that leave one of the arguments of the R\'enyi divergence invariant, and determine domains of the parameters α,z\alpha,zα,z where monotonicity holds, and where the preservation of the α\alphaα-zzz-R\'enyi divergence implies the reversibility of the quantum operation.
The concept of classical [Formula: see text]-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the Rényi divergences. Various quantum versions of this concept, and more narrowly, the concept of Rényi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz’ quasi-entropies (standard [Formula: see text]-divergences), Matsumoto’s maximal [Formula: see text]-divergences, measured [Formula: see text]-divergences, and sandwiched and [Formula: see text]-[Formula: see text]-Rényi divergences. In this paper, we give a systematic overview of the various concepts of quantum [Formula: see text]-divergences, with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum [Formula: see text]-divergence by a quantum operation. In particular, we compare the standard and the maximal [Formula: see text]-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum [Formula: see text]-divergences are strictly different for non-commuting operators unless [Formula: see text] is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard [Formula: see text]-divergences. We also study the monotonicity of the [Formula: see text]-[Formula: see text]-Rényi divergences under the special class of bistochastic maps that leave one of the arguments of the Rényi divergence invariant, and determine domains of the parameters [Formula: see text] where monotonicity holds, and where the preservation of the [Formula: see text]-[Formula: see text]-Rényi divergence implies the reversibility of the quantum operation.
We consider the reversibility problem for quantum operations (stochastic maps) in connection with the preservation of various quantum divergences, including two different versions of quantum fff-divergences, sandwiched R\'enyi divergences, and α\alphaα-zzz-R\'enyi relative entropies.