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arXiv · 1010.1030

Reverse Test and Characterization of Quantum Relative Entropy

Keiji Matsumoto
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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 $\mathrm{D}^{Q}(ρ||σ) $ is $\mathrm{D}^{R}(ρ||σ) :=\mathrm{tr}% \,ρ\ln\sqrtρσ^{-1}\sqrtρ$ and $\mathrm{D}(ρ||σ) :=$ $\mathrm{tr}\,ρ(\lnρ-\lnσ) $, respectively. In the latter setting, we prove uniqueness of quantum relative entropy, that is, $\mathrm{D}^{Q}(ρ||σ) $ should equal a constant multiple of $\mathrm{D}(ρ||σ) $. In the analysis, we define and use reverse test and asymptotic reverse test, which are natural inverse of hypothesis test.

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Keiji Matsumoto. 2010-10-05. Reverse Test and Characterization of Quantum Relative Entropy. https://arxiv.org/abs/1010.1030

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