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arXiv:0712.2881 · International Journal of Mathematics

FROM QUASI-ENTROPY TO SKEW INFORMATION

D�nes Petz, Viktor Szabó

Published Nov 1, 2009• 14 citations
arXivHTMLPDF

Abstract

This paper gives an overview about particular quasi-entropies, generalized quantum covariances, quantum Fisher informations, skew informations and their relations. The point is the dependence on operator monotone functions. It is proven that a skew information is the Hessian of a quasi-entropy. The skew information and some inequalities are extended to a von Neumann algebra setting.

DOI
10.1142/s0129167x09005832

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This paper gives an overview about particular quasi-entropies, generalized quantum covariances, quantum Fisher informations, skew informations and their relations. The point is the dependence on operator monotone functions. It is proven that a skew information is the Hessian of a quasi-entropy. The skew information and some inequalities are extended to a von Neumann algebra setting.
Csiszár’s ƒ-divergence of two probability distributions was extended to the quantum case by the author in 1985. In the quantum setting, positive semidefinite matrices are in the place of probability distributions and the quantum generalization is called quasi-entropy, which is related to some other important concepts as covariance, quadratic costs, Fisher information, Cram´er-Rao inequality and uncertainty relation. It is remarkable that in the quantum case theoretically there are several Fisher information and variances. Fisher information are obtained as the Hessian of a quasi-entropy. A conjecture about the scalar curvature of a Fisher information geometry is explained. The described subjects are overviewed in details in the matrix setting. The von Neumann algebra approach is also discussed for uncertainty relation.
The subject of this paper is a mathematical transition from the Fisher information of classical statistics to the matrix formalism of quantum theory. If the monotonicity is the main requirement, then there are several quantum versions parametrized by a function. In physical applications the minimal is the most popular. There is a one-to-one correspondence between Fisher informations (called also monotone metrics) and abstract covariances. The skew information and the chi-square-divergence are treated here as particular cases.
We consider a quantum quasi-relative entropy [Formula: see text] for an operator [Formula: see text] and an operator convex function [Formula: see text]. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the [Formula: see text]-divergences (i.e. [Formula: see text]). We also provide an error term for a class of operator inequalities, that generalizes operator strong subadditivity inequality. We apply those results to demonstrate explicit bounds for the logarithmic function, that leads to the quantum relative entropy, and the power function, which gives, in particular, a Wigner–Yanase–Dyson skew information. In particular, we provide the remainder terms for the strong subadditivity inequality, operator strong subadditivity inequality, WYD-type inequalities, and the Cauchy–Schwartz inequality.
The subject is the applications of the use of quasi-entropy in finite dimensional spaces to many important quantities in quantum information. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the f -divergence, monotone metrics, covariance and the \chi^2 -divergence are the most important particular cases. The extension of monotone metrics to those with two parameters is a new concept. Monotone metrics are also characterized by their joint convexity property.
Quantum coherence is one of the characteristic features of quantum mechanics and underpins many quantum mysteries. To eliminate the influence of the reference basis on the coherence of a quantum state and uncover its intrinsic properties, it is common to study coherence by averaging over different reference bases. Using the metric-adjusted skew information, we explore three natural approaches to average coherence of a state: average over all orthonormal bases, average over all elements of an operator orthonormal base, and average over a complete family of mutually unbiased bases. We establish the equivalence among these three types of average coherence and interpret the unified average coherence as the coherence of a quantum state relative to a depolarizing channel. Additionally, we employ the unified average coherence to introduce a notion of quantum fff entropy (where fff is an operator monotone function associated with the metric-adjusted skew information) and demonstrate that quantum fff entropy possesses properties analogous to the ubiquitous von Neumann entropy. Furthermore, we illuminate some connections between quantum fff entropy and quasientropy, and compare fff entropy with von Neumann entropy, R\'enyi entropy, and Tsallis entropy for some typical states.