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arXiv:1010.1335 · Journal of Mathematical Physics

Upper continuity bounds on the relative q-entropy for q > 1

Alexey E. Rastegin

Published Jun 1, 2011• 15 citations
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Abstract

Generalized entropies and relative entropies are the subject of active research. Similar to the standard relative entropy, the relative q-entropy is generally unbounded for q > 1. Upper bounds on the quantum relative q-entropy in terms of norm distances between its arguments are obtained in finite-dimensional context. These bounds characterize a continuity property in the sense of Fannes.

DOI
10.1063/1.3600535

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Generalized entropies and relative entropies are the subject of active research. Similar to the standard relative entropy, the relative q-entropy is generally unbounded for q > 1. Upper bounds on the quantum relative q-entropy in terms of norm distances between its arguments are obtained in finite-dimensional context. These bounds characterize a continuity property in the sense of Fannes.
The quantum relative entropy is frequently used as a distance, or distinguishability measure between two quantum states. In this article we study the relation between this measure and a number of other measures used for that purpose, including the trace norm distance. More specifically, we derive lower and upper bounds on the relative entropy in terms of various distance measures for the difference of the states based on unitarily invariant norms. The upper bounds can be considered as statements of continuity of the relative entropy distance in the sense of Fannes. We employ methods from optimisation theory to obtain bounds that are as sharp as possible.
The quantum relative entropy is frequently used as a distance measure between two quantum states, and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory. We have derived many such inequalities in previous work. The present paper is a follow-up on this, and provides a sharp upper bound on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned. The result obtained here is more general than the corresponding one from our previous work. As a corollary, we obtain a sharp upper bound on the regularised relative entropy introduced by Lendi, Farhadmotamed, and van Wonderen.
The relative entropy S(rho∣∣sigma)S(\\rho||\\sigma)S(rho∣∣sigma) is frequently used as a distance, or distinguishability measure between two states rho\\rhorho and sigma\\sigmasigma. In this paper we study the relation between this measure and a number of other measures used for that purpose, including the trace norm distance. More precisely, we derive lower and upper bounds on the relative entropy S(rho∣∣sigma)S(\\rho||\\sigma)S(rho∣∣sigma) in terms of various distance measures for rho−sigma\\rho-\\sigmarho−sigma based on unitarily invariant norms. The upper bounds can be considered as statements of continuity of the relative entropy distance in the sense of Fannes. We employ optimisation theory to obtain bounds that are as sharp as possible.
We show that Frenkel’s integral representation of the quantum relative entropy provides a natural framework to derive continuity bounds for quantum information measures. Our main general result is a dimension-independent semi-continuity relation for the quantum relative entropy with respect to the first argument. Using it, we obtain a number of results: (1) a tight continuity relation for the conditional entropy in the case where the two states have equal marginals on the conditioning system, resolving a conjecture by Wilde in this special case; (2) a stronger version of the Fannes–Audenaert inequality on quantum entropy; (3) better estimates on the quantum capacity of approximately degradable channels; (4) an improved continuity relation for the entanglement cost; (5) general upper bounds on asymptotic transformation rates in infinite-dimensional entanglement theory; and (6) a proof of a conjecture due to Christandl, Ferrara, and Lancien on the continuity of ’filtered’ relative entropy distances.