Markov triplets on CCR-algebras

Jenčová, Anna; Petz, Dénes; Pitrik, József

doi:10.1007/bf03549824

Cited by 7 publications

(5 citation statements)

References 12 publications

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“…As it turns out, the preservation of S f is in general strictly stronger than the preservation of S f , i.e., in general the preservation of S f does not imply the reversibility of the quantum operation as in Definition 3.16. This can be seen in various ways. In [34,Remark 5.4], an example from [44] was used to show states ̺, σ and a CPTP map Φ such that Φ is not reversible on {̺, σ}, but…”

Section: The Relation Of the Preservation Conditionsmentioning

confidence: 99%

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Different quantum f-divergences and the reversibility of quantum operations

2017

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The concept of classical f -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 f -divergences), Matsumoto's maximal f -divergences, measured f -divergences, and sandwiched and α-z-Rényi divergences.In this paper we give a systematic overview of the various concepts of quantum fdivergences, with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum f -divergence by a quantum operation. In particular, we compare the standard and the maximal f -divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum f -divergences are strictly different for non-commuting operators unless f is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard f -divergences.We also study the monotonicity of the α-z-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 α, z where monotonicity holds, and where the preservation of the α-z-Rényi divergence implies the reversibility of the quantum operation.

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Section: The Relation Of the Preservation Conditionsmentioning

confidence: 99%

“…By Example 4.2 and (c) of Theorem 3.34, this latter condition implies that S f (Φ(̺) Φ(σ)) = S f (̺ σ) for every operator convex function f on (0, +∞); yet reversibility does not hold. The example from [44] is rather involved; below we give a much simpler one, in Example 4.8.…”

Section: The Relation Of the Preservation Conditionsmentioning

confidence: 99%

Different quantum f-divergences and the reversibility of quantum operations

2017

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“…Preservation of the f -divergence by a stochastic map is not automatic in this case; however, it is not sufficient for the reversibility of the map, either. Indeed, it was shown in Example 2.2 of [28] that there exists a positive definite operator D 123 on a tripartite Hilbert space…”

Section: Corollary Assume That Suppmentioning

confidence: 99%

QUANTUM f-DIVERGENCES AND ERROR CORRECTION

et al. 2011

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Quantum f -divergences are a quantum generalization of the classical notion of fdivergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f -divergences; special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f -divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f -divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f -divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.

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“…The case of equality is studied in the paper [5] and the general properties of entropy are in the book [8].…”

Section: S(α(a)) := −Tr α(A) Log α(A)mentioning

confidence: 99%