Reverse test and quantum analogue of classical fidelity and generalized fidelity

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}(ρ||σ) is D^{R}(ρ||σ):=trρlnσ^{1/2}ρ^{-1}σ^{1/2} and D(ρ||σ):= trρ(lnρ-lnσ), respectively. In the latter setting, we prove uniqueness of quantum relative entropy, that is, D^{Q}(ρ||σ) should equal a constant multiple of D(ρ||σ). In the analysis, we define and use reverse test and asymptotic reverse test, which are natural inverse of hypothesis test.

arXiv commentA new proof of joint convexity of $D^R$ is added. Also, some technical correction, Title changed

References in corpus (3)

• Distinguishability and Accessible Information in Quantum Theory
• Reverse Test and Characterization of Quantum Relative Entropy
• Reverse estimation theory, Complementality between RLD and SLD, and monotone distances

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• A fidelity measure for quantum states based on the matrix geometric mean
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