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arXiv:quant-ph/0503218 · Journal of Mathematical Physics

Continuity bounds on the quantum relative entropy

Koenraad M. R. Audenaert, Jens Eisert

Published Oct 1, 2005• 83 citations
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Abstract

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.

DOI
10.1063/1.2044667

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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.