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arXiv:quant-ph/9704040 · Journal of Physics A Mathematical and General

Asymptotics of quantum relative entropy from a representation theoretical viewpoint

Masahito Hayashi

Published Apr 10, 2001• 47 citations
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Abstract

In this paper it is proved that the quantum relative entropy D (ρ||σ) can be asymptotically attained by the relative entropy of probabilities given by a certain sequence of positive-operator-valued measures (POVMs). The sequence of POVMs depends on σ, but is independent of the choice of ρ.
DOI
10.1088/0305-4470/34/16/309

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Recently, an interesting quantity called the quantum Rényi divergence (or 'sandwiched' Rényi relative entropy) was defined for pairs of positive semi-definite operators ρ and σ. It depends on a parameter α and acts as a parent quantity for other relative entropies which have important operational significance in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Rényi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Rényi entropy is obtainable from the quantum Rényi divergence only if ρ and σ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived—the quantum Rényi divergence with parameter α ⩾ 1/2, and the α-relative Rényi entropy with α ∈ [0, 1).
We investigate the non-Markovian behavior in open quantum systems from an information-theoretic perspective. Our main tool is the max-relative entropy, which quantifies the maximum probability with which a state ρ can appear in a convex decomposition of a state σ . This operational interpretation provides a new view for the non-Markovian process. We also find that max-relative entropy can be the witness and measure of non-Markovian processes. As applications, some examples are also given and compared with other measures in this paper.