On trace inequalities and their applications to noncommutative communication theory

Certain trace inequalities related to matrix logarithm are shown. These results enable us to give a partial answer of the open problem conjectured by A.S. Holevo. That is, concavity of the auxiliary function which appears in the random coding exponent as the lower bound of the quantum reliability function for general quantum states is proven in the case of 0 ⩽ s ⩽ 1.