## A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality.(English)Zbl 1218.81025

The convexity conjecture on Wigner-Yanase-Dyson skew information was successfully solved by E. H. Lieb [Adv. Math. 11, 267–288 (1973; Zbl 0267.46055)] with an affirmative answer.
The present paper studies the convexity problem for a generalized relative entropy $$J_p(K,A,B)$$ introduced by the authors. The quantity $$J_p(K,A,A)$$ becomes Wigner-Yanase-Dyson skew information up to a constant factor, where $$K$$ is a selfadjoint operator. $$J_p(I,A,I)$$ is also equivalent to quantum Tsallis entropy up to a constant factor.
Therefore it may be important to study the mathematical properties of the quantity $$J_p(K,A,B)$$ in general from the viewpoint of entropy theory.
The main results in the present paper are the joint convexity with respect to $$A$$ and $$B$$, and monotonicity on the partial trace. In addition, the equality condition of the above properties (trace inequalities) are studied.

### MSC:

 81P45 Quantum information, communication, networks (quantum-theoretic aspects) 82B10 Quantum equilibrium statistical mechanics (general) 47N50 Applications of operator theory in the physical sciences 47A63 Linear operator inequalities 15A45 Miscellaneous inequalities involving matrices 94A17 Measures of information, entropy 81P15 Quantum measurement theory, state operations, state preparations

Zbl 0267.46055
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