arXiv:quant-ph/0610146 · Journal of Physics A Mathematical and Theoretical

A sharp continuity estimate for the von Neumann entropy

Koenraad M. R. Audenaert

Published Jun 27, 2007• 284 citations

Abstract

We derive an inequality relating the entropy difference between two quantum states to their trace norm distance, sharpening a well-known inequality due to Fannes. In our inequality, equality can be attained for every prescribed value of the trace norm distance.

DOI: 10.1088/1751-8113/40/28/s18

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arXiv:quant-ph/0610146 · Journal of Physics A Mathematical and Theoretical

A sharp continuity estimate for the von Neumann entropy

Koenraad M. R. Audenaert

Published Jun 27, 2007• 284 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1751-8113/40/28/s18

Related papers

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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.

We establish a tight upper bound for the difference in von Neumann entropies between two quantum states,

\rho _{1}

and

\rho _{2}

. 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

(\rho _{1} - \rho _{2})

. 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.