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arXiv:1304.0409 · Journal of Mathematical Physics

On the asymmetry of the relative entropy

Koenraad M. R. Audenaert

Published Jul 1, 2013• 11 citations
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Abstract

The quantum relative entropy S(ρ||σ) is a widely used dissimilarity measure between quantum states, but it has the peculiarity of being asymmetric in its arguments. We quantify the amount of asymmetry by providing a sharp upper bound in terms of two parameters: the trace norm distance between the two states, and the smallest of the smallest eigenvalues of both states. The bound is essentially the asymmetry between two binary distributions governed by these two parameters.

DOI
10.1063/1.4811856

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The quantum relative entropy S(ρ||σ) is a widely used dissimilarity measure between quantum states, but it has the peculiarity of being asymmetric in its arguments. We quantify the amount of asymmetry by providing a sharp upper bound in terms of two parameters: the trace norm distance between the two states, and the smallest of the smallest eigenvalues of both states. The bound is essentially the asymmetry between two binary distributions governed by these two parameters.
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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.
Fermionic Gaussian states are a fundamental tool in many-body physics, faithfully representing non-interacting quantum systems and allowing for efficient numerical simulations. Given a many-body wave function, it is therefore interesting to ask how much it differs from that of a Gaussian state, as quantified by the notion of non-Gaussianity. In this work, we relate measures of non-Gaussianity with the Shannon entropy of the particle-number distribution, coinciding with the particle-number asymmetry for pure states. We derive a lower bound on the relative entropy of non-Gaussianity in terms of the exponential of the Shannon entropy, and study numerically its tightness for large system sizes. Our bound is non-trivial for large values of the asymmetry and relies on the concentration of the particle-number distribution of (mixed) fermionic Gaussian states. Since the Shannon entropy of the particle-number distribution is often efficient to compute or experimentally measure, our results can be viewed as a practical way to lower bound non-Gaussianity, highlighting a non-trivial interplay with particle-number asymmetry.