Quantum Hypothesis Testing and the Operational Interpretation of the Quantum Rényi Relative Entropies

Abstract

We show that the new quantum extension of Rényi’s α-relative entropies, introduced recently by Müller-Lennert et al. (J Math Phys 54:122203, 2013) and Wilde et al. (Commun Math Phys 331(2):593–622, 2014), have an operational interpretation in the strong converse problem of quantum hypothesis testing. Together with related results for the direct part of quantum hypothesis testing, known as the quantum Hoeffding bound, our result suggests that the operationally relevant definition of the quantum Rényi relative entropies depends on the parameter α: for α < 1, the right choice seems to be the traditional definition ${{D_\alpha^{(old)}} (\rho \| \sigma) :=\frac{1}{\alpha-1} \,\,{\rm log\,\,Tr}\,\, \rho^{\alpha} \sigma^{1-\alpha}}$, whereas for α > 1 the right choice is the newly introduced version ${D_\alpha^{(new)}} (\rho \| \sigma) := \frac{1}{\alpha-1}\,{\rm log\,\,Tr}\,\big(\sigma^{\frac{1-\alpha}{2 \alpha}}\rho \sigma^{\frac{1-\alpha}{2 \alpha}}\big)^{\alpha}$.On the way to proving our main result, we show that the new Rényi α-relative entropies are asymptotically attainable by measurements for α > 1. From this, we obtain a new simple proof for their monotonicity under completely positive trace-preserving maps.