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Generalized quantum entropies: a definition and some properties

Steeve Zozor, G. M. Bosyk, Federico Holik, M. Portesi, Pedro W. Lamberti

Published Nov 3, 2016• 0 citations
OpenAlex

Abstract

In this presentation, a family of quantum entropies inspired by the clasical (h,φ)- entropies proposed by Salicrú et al. (more precisely, inspired by the Csiszár’s divergen- cies) will be introduced. The proposed family includes several well-known entropies such that the von Neumann entropy or quantum versions of the Rényi’s and Havrda- Charvát-Daróczy-Tsallis’s entropies, among many others. The main properties of the proposed quantum (h, φ )-entropies lie on the the fundamental concept of majorization. The behavior of these entropies when a quantum state is subject to some quantum op- erations (unitary, measurement) will be characterized, and its behavior when dealing with composite systems as well. Some potential applications in detection of entan- glement will be exposed. Finally, we will present possible definitions of associate measures such that conditional generalized quantum entropies.

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In this presentation, a family of quantum entropies inspired by the clasical (h,φ)- entropies proposed by Salicrú et al. (more precisely, inspired by the Csiszár’s divergen- cies) will be introduced. The proposed family includes several well-known entropies such that the von Neumann entropy or quantum versions of the Rényi’s and Havrda- Charvát-Daróczy-Tsallis’s entropies, among many others. The main properties of the proposed quantum (h, φ )-entropies lie on the the fundamental concept of majorization. The behavior of these entropies when a quantum state is subject to some quantum op- erations (unitary, measurement) will be characterized, and its behavior when dealing with composite systems as well. Some potential applications in detection of entan- glement will be exposed. Finally, we will present possible definitions of associate measures such that conditional generalized quantum entropies.
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.
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.
The theory of majorization is a powerful mathematical tool which naturally arises in the quantum theory as a consequence of its fundamental connection with unitarity. In this work, we look for majorization relations based on entropic inequalities. We begin by introducing the theory of majorization. We then give some notions of quantum mechanics, like the density matrix and some algebra, before presenting the concept of entropy which is closely related to the theory of majorization. We define in the process the Shannon, von Neumann and Renyi entropies, along with some of their properties and applications. We then define quantum entanglement and give some criteria for separability, which turn out to be applications of the concepts of majorization and entropy in the quantum theory. We present some new majorization relations similar to already existing entropic inequalities, and exhibit their advantages and limitations. The rest of the report is dedicated to the study of the field of Gaussian quantum information. We present the theory, before introducing other majorization relations, which turn out to be useful because they allow one to know if it is possible to transform one quantum state into another when there exists majorization relations between them.
This paper introduces a generalized definition of quantum information entropy, aiming to provide a more comprehensive framework for quantifying information in complex quantum systems. The core of this work lies in extending the conventional concept of entropy, traditionally applied to classical information, to the realm of quantum mechanics. We explore various types of quantum entropy, including von Neumann entropy, relative entropy, and a newly proposed generalized entropy, denoted as *Eg*. The definition of *Eg* is based on a novel approach that considers the entanglement structure and correlations within quantum states. We establish relationships between these different entropy measures, demonstrating their interconnectedness within the broader context of quantum information theory. The paper further outlines a mathematical formalism for calculating *Eg*, providing a tool for analyzing and characterizing quantum information. Our generalized entropy provides a more flexible and potentially more accurate measure of quantum information, particularly in scenarios involving complex quantum correlations and multi-partite quantum systems. ---
We review the concept of majorization and its relation with generalized information measures. Majorization theory provides an elegant framework for comparing two probability distributions, leading to a rigorous concept of disorder which is more stringent than that based on the Shannon entropy. Nevertheless, it is shown that it can be fully captured through general entropic inequalities based on generalized entropic forms. A brief review of generalized entropies is also provided. As illustration, we discuss the majorization properties of generalized thermal distributions derived from generalized entropies, and identify rigorous mixing parameters. We also describe majorization in quantum systems. We discuss in particular its capability for providing a disorder based criterion for the detection of quantum entanglement, which is stronger than that based on the von Neumann entropy and leads to a generalized entropic separability criterion.