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World Congress on Engineering

A mathematical review of the generalized entropies and their matrix trace inequalities.

Shigeru Furuichi

Published Jan 1, 2007• 3 citations
OpenAlex

Abstract

We review the properties of the generalized entropies in our previous papers in the following way. (1)A generalized Fannes’ inequality is shown by the axiomatically characterized Tsallis entropy. (2)The maximum entropy principles in nonextensive statistical physics are revisited as an application of the Tsallis relative entropy defined for the nonnegative matrices in the framework of matrix analysis. (3)A variational expression for Tsallis relative entropy is derived and some related inequalities are studied. Matrix trace inequalities, Tsallis relative entropy, Fannes’ inequality, variational expression and maximum entropy principle

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We review the properties of the generalized entropies in our previous papers in the following way. (1)A generalized Fannes’ inequality is shown by the axiomatically characterized Tsallis entropy. (2)The maximum entropy principles in nonextensive statistical physics are revisited as an application of the Tsallis relative entropy defined for the nonnegative matrices in the framework of matrix analysis. (3)A variational expression for Tsallis relative entropy is derived and some related inequalities are studied. Matrix trace inequalities, Tsallis relative entropy, Fannes’ inequality, variational expression and maximum entropy principle
We review the mathematical properties of the generalized entropies, based on our previous papers. This review is an extended version of our article [9]. The Tsallis entropy defined for a quantum state is uniquely derived from three axioms. A generalized Fannes’ inequality is shown. The maximum entropy principle in Tsallis statistics are revisited as an application of the Tsallis relative entropy defined for nonnegative matrices in the framework of matrix analysis. A variational expression for Tsallis relative entropy is derived and some related inequalities are studied. Finally, an application to qunatum information theory is discussed.
Quantum entropy and skew information play important roles in quantum information science. They are defined by traces of positive operators, hence trace inequalities often have important roles to develop the mathematical theory in quantum information science. In this article, we study some properties for information quantities in quantum systems through trace inequalities. Especially, we give upper bounds and lower bounds on the Tsallis relative entropy, which is a one-parameter extension of the relative entropy in quantum systems. In addition, we compare the known bounds and the new bounds, for both upper and lower bounds, respectively. We also give an inequality for generalized skew information by introducing a generalized correlation measure.
Quantum entropy and skew information play important roles in quantum information science. They are defined by the trace of the positive operators so that the trace inequalities often have important roles to develop the mathematical theory in quantum information science. In this paper, we study some properties for information quantities in quantum system through trace inequalities. Especially, we give upper bounds and lower bounds of Tsallis relative entropy, which is a one-parameter extension of the relative entropy in quantum system. In addition, we compare the known bounds and the new bounds, for both upper and lower bounds, respectively. We also give an inequality for generalized skew information by introducing a generalized correlation measure.