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 ${\mathrm{D}}^R(\rho ||\sigma ):=\mathrm{tr}$ 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.

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• Min- and Max- Relative Entropies and a New Entanglement Monotone
• An Information-Spectrum Approach to Classical and Quantum Hypothesis Testing for Simple Hypotheses
• On metric of quantum channel spaces
• Reverse estimation theory, Complementality between RLD and SLD, and monotone distances

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