THE PROPER FORMULA FOR RELATIVE ENTROPY AND ITS ASYMPTOTICS IN QUANTUM PROBABILITY

Umegaki's relative entropy S(omega, phi) = Tr D-omega(log D-omega-log D-phi) (of states omega and phi with density operators D-omega and D-phi, respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegaki's one. As a byproduct, the inequality Tr A log AB greater-than-or-equal-to Tr A(log A + log B) is obtained for positive definite matrices A and B.