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Further results on generalized conditional entropies

Alexey E. Rastegin

Published Jan 1, 2015• 9 citations
Publisher

Abstract

We further examine some properties of the conditional Rényi and Tsallis–Havrda–Charvát (THC) entropies. Such properties are interesting from the viewpoint of applications in studying protocols of quantum information science and foundations of quantum mechanics. In particular, we consider properties of the conditional Rényi and THC entropies with respect to conditioning on more. We also exemplify that the desired property can be violated with the conditional min-entropy. Applications of such results to the TCH entropy rate are considered. Connections between generalized conditional entropies and error probability are examined. Several relations between various conditional entropies are obtained. It is shown that such relations can be used for bounding the conditional Rényi and TCH entropies.

DOI
10.1051/ita/2014029

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We further examine some properties of the conditional Rényi and Tsallis–Havrda–Charvát (THC) entropies. Such properties are interesting from the viewpoint of applications in studying protocols of quantum information science and foundations of quantum mechanics. In particular, we consider properties of the conditional Rényi and THC entropies with respect to conditioning on more. We also exemplify that the desired property can be violated with the conditional min-entropy. Applications of such results to the TCH entropy rate are considered. Connections between generalized conditional entropies and error probability are examined. Several relations between various conditional entropies are obtained. It is shown that such relations can be used for bounding the conditional Rényi and TCH entropies.
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The Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.
The R\'enyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of R\'enyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of R\'enyi entropies and the corresponding relative entropies that contains the min-entropy, collision entropy, von Neumann entropy as well as the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities and a limited duality relation. We conclude that the treatment of these entropies is technically challenging and requires sophisticated tools from linear algebra. We share several conjectured properties which we were unable to prove.
Entropies are fundamental measures of uncertainty with central importance in information theory and statistics and applications across all the quantitative sciences. Under a natural set of operational axioms, the most general form of entropy is captured by the family of Rényi entropies, parameterized by a real number ααα. Conditional entropy extends the notion of entropy by quantifying uncertainty from the viewpoint of an observer with access to potentially correlated side information. However, despite their significance and the emergence of various useful definitions, a complete characterisation of measures of conditional entropy that satisfy a natural set of operational axioms has remained elusive. In this work, we provide a complete characterisation of conditional entropy, defined through a set of axioms that are essential for any operationally meaningful definition: additivity for independent random variables, invariance under relabeling, and monotonicity under conditional mixing channels. We show that the most general conditional entropy is a generalized convex combination of conditional entropies defined as exponential averages of Rényi entropies of the conditioned distribution, parameterized by a real number and a probability measure on the positive reals.