Reverse Test and Characterization of Quantum Relative Entropy

The aim of the present paper is to give axiomatic characterization of quantum relative entropy utilizing resource conversion scenario. We consider two sets of axioms: non-asymptotic and asymptotic. In the former setting, we prove that the upperbound and the lowerbund of $\mathrm{D}^{Q}(\rho||\sigma) $ is $\mathrm{D}^{R}(\rho||\sigma) :=\mathrm{tr}% \,\rho\ln\sqrt{\rho}\sigma^{-1}\sqrt{\rho}$ and $\mathrm{D}(\rho||\sigma) :=$ $\mathrm{tr}\,\rho(\ln\rho-\ln\sigma) $, respectively. In the latter setting, we prove uniqueness of quantum relative entropy, that is, $\mathrm{D}^{Q}(\rho||\sigma) $ should equal a constant multiple of $\mathrm{D}(\rho||\sigma) $. In the analysis, we define and use reverse test and asymptotic reverse test, which are natural inverse of hypothesis test.