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arXiv:2107.10112 · Mathematical Inequalities & Applications

Modulus of continuity of the quantum f-entropy with respect to the trace distance

Iosif F Pinelis

Published Jan 1, 2021• 1 citation
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Abstract

A well-known result due to Fannes is a certain upper bound on the modulus of continuity of the von Neumann entropy with respect to the trace distance between density matrices; this distance is the maximum probability of distinguishing between the corresponding quantum states. Much more recently, Audenaert obtained an exact expression of this modulus of continuity.

DOI
10.7153/mia-2021-24-66

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A well-known result due to Fannes is a certain upper bound on the modulus of continuity of the von Neumann entropy with respect to the trace distance between density matrices; this distance is the maximum probability of distinguishing between the corresponding quantum states. Much more recently, Audenaert obtained an exact expression of this modulus of continuity.
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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 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 establish a tight upper bound for the difference in von Neumann entropies between two quantum states,ρ1\rho _{1}ρ1​andρ2\rho _{2}ρ2​. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator(ρ1−ρ2)(\rho _{1} - \rho _{2})(ρ1​−ρ2​). This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.
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