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 D^{ Q}({ \rho}||{ \sigma}) is D^{ R}({ \rho}||{ \sigma}):=tr{ \rho}ln{ \sigma}^{ 1/2}{ \rho}^{ -1}{ \sigma}^{ 1/2} and D({ \rho}||{ \sigma}):= tr{ \rho}(ln{ \rho}-ln{ \sigma}), respectively. In the latter setting, we prove uniqueness of quantum relative entropy, that is, D^{ Q}({ \rho}||{ \sigma}) should equal a constant multiple of D({ \rho}||{ \sigma}). In the analysis, we define and use reverse test and asymptotic reverse test, which are natural inverse of hypothesis test.
Categories: quant-ph math-ph math.MP