Reverse Test and Characterization of Quantum Relative Entropy

Abstract: 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.