AIP conference proceedings

A review of the mathematical properties of the Tsallis entropies

Shigeru Furuichi, Alexander M. Korsunsky

Published Jan 1, 2008• 3 citations

Abstract

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.

DOI: 10.1063/1.2991343

Related papers

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

Shigeru Furuichi

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

World Congress on EngineeringJan 1, 20073 citations

A generalized Fannes inequality

Shigeru Furuichi, Kenjiro Yanagi, Ken Kuriyama

We axiomatically characterize the Tsallis entropy of a finite quantum system. In addition, we derive a continuity property of Tsallis entropy. This gives a generalization of the Fannes inequality.

arXiv:1001.1390Journal of Inequalities in Pure & Applied MathematicsJan 9, 201011 citations

A generalized Fannes' inequality

茂古市, 研二郎柳, 憲栗山

We axiomatically characterize the Tsallis entropy of a finite quantum system.In addition, we derive a continuity property of Tsallis entropy.This gives a generalization of the Fannes' inequality.

Jan 1, 20070 citations

Inequalities for Tsallis relative entropy and generalized skew information

Shigeru Furuichi

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.

Linear and Multilinear AlgebraSep 23, 201126 citations

On bounds of Tsallis relative entropy and an inequality for generalized skew information

Shigeru Furuichi

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.

arXiv:1008.3441Aug 20, 20101 citation

Generalized ‘Useful’ Relative Information Measures of Type (α, β)

Satish Kumar, Arun Choudhary

In this paper we define some new generalized measures of ‘useful’ relative information and study their particular cases. We also obtain their relations with common entropy measures. Moreover, we define the quantum generalized Tsallis entropy and show that projective measurement does not decrease the quantum generalized Tsallis entropy of a quantum state. Finally, we define generalized quantum Tsallis and Renyi relative entropies and discuss their particular cases.

Open Systems & Information DynamicsMar 1, 20150 citations