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Publications of the Research Institute for Mathematical Sciences

From Quasi-Entropy to Various Quantum Information Quantities

Fumio Hiai, Dėnes Petz

Published Sep 19, 2012• 37 citations
Publisher

Abstract

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.

DOI
10.2977/prims/79

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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.
The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.
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.
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 concept of classical [Formula: see text]-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the Rényi divergences. Various quantum versions of this concept, and more narrowly, the concept of Rényi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz’ quasi-entropies (standard [Formula: see text]-divergences), Matsumoto’s maximal [Formula: see text]-divergences, measured [Formula: see text]-divergences, and sandwiched and [Formula: see text]-[Formula: see text]-Rényi divergences. In this paper, we give a systematic overview of the various concepts of quantum [Formula: see text]-divergences, with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum [Formula: see text]-divergence by a quantum operation. In particular, we compare the standard and the maximal [Formula: see text]-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum [Formula: see text]-divergences are strictly different for non-commuting operators unless [Formula: see text] is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard [Formula: see text]-divergences. We also study the monotonicity of the [Formula: see text]-[Formula: see text]-Rényi divergences under the special class of bistochastic maps that leave one of the arguments of the Rényi divergence invariant, and determine domains of the parameters [Formula: see text] where monotonicity holds, and where the preservation of the [Formula: see text]-[Formula: see text]-Rényi divergence implies the reversibility of the quantum operation.