Fundamental properties of Tsallis relative entropy

TL;DR

This paper explores fundamental properties of Tsallis relative entropy in classical and quantum systems, including inequalities, monotonicity, and convexity, extending existing concepts and proving new inequalities.

Contribution

It provides a parametric extension of trace inequalities, proves monotonicity without invertibility assumptions, and introduces generalized Tsallis relative entropy with subadditivity and Peierls-Bogoliubov inequality.

Findings

Extended trace inequality for quantum Tsallis relative entropy

Monotonicity of quantum Tsallis relative entropy without invertibility

Subadditivity and generalized Peierls-Bogoliubov inequality for Tsallis entropy

Abstract

Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied. As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz. The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible. The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity. Moreover, the generalized Peierls-Bogoliubov inequality is also proven.