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Reverse test and quantum analogue of classical fidelity and generalized fidelity

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Preprint
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arXiv
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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 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.

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