Reverse Test and Characterization of Quantum Relative Entropy

TL;DR

This paper provides an axiomatic characterization of quantum relative entropy using resource conversion scenarios, establishing bounds and uniqueness results for different asymptotic settings.

Contribution

It introduces reverse test concepts and proves bounds and uniqueness of quantum relative entropy under axiomatic frameworks.

Findings

Upper and lower bounds for quantum relative entropy established.

Uniqueness of quantum relative entropy shown in asymptotic setting.

Reverse test and asymptotic reverse test are introduced as inverse hypothesis tests.

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 $\mathrm{D}^{R}(\rho||\sigma) :=\mathrm{tr}% \,\rho\ln\sqrt{\rho}\sigma^{-1}\sqrt{\rho}$ 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.