Concepts and Recent Advances in Generalized Information Measures and Statistics

1. Since the introduction of the information measure widely known as Shannon entropy, quantifiers based on information theory and concepts such as entropic forms and statistical complexities have ...
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Majorization and Generalized Entropies

Pp. 100-129 (30)

Norma Canosa and Raul D. Rossignoli

Abstract

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.

Keywords:

Generalized Entropies, Majorization, Generalized Thermal Distributions, Mixing Parameters, Quantum Mixed States, Quantum Entanglement, Separability.

Affiliation:

Departamento de Fisica-IFLP, Universidad Nacional de La Plata, C.C.67, La Plata (1900), Argentina