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Continuity bounds on the relative entropy

Audenaert, K M R, Jens Eisert

Published Mar 29, 2005• 0 citations
OpenAlex

Abstract

The relative entropy $S(\\rho||\\sigma)$ is frequently used as a distance, or distinguishability measure between two states $\\rho$ and $\\sigma$. 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 $S(\\rho||\\sigma)$ in terms of various distance measures for $\\rho-\\sigma$ 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.

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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.
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