Inequalities for Tsallis relative entropy and generalized skew information

Abstract

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.

Keywords:

• Tsallis relative entropy
• trace inequality
• skew information
• correlation measure and positive operators

AMS Subject Classifications:

• 15A45
• 47A63
• 94A17

Acknowledgement

The author was supported in part by the Japanese Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists (B), 20740067.