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

We show that the new quantum extension of Renyi's alpha-relative entropies, introduced recently by Muller-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 Renyi relative entropies depends on the parameter alpha: for alpha < 1, the right choice seems to be the traditional definition D-alpha((old)) (rho parallel to sigma) := 1/alpha-1 log Tr rho(alpha)sigma(1-alpha), whereas for alpha > 1 the right choice is the newly introduced version D-alpha((new)) (rho parallel to sigma) := 1/alpha-1 log Tr (sigma 1-alpha/2 alpha rho sigma 1-alpha/2 alpha)(alpha). On the way to proving our main result, we show that the new Renyi alpha-relative entropies are asymptotically attainable by measurements for alpha > 1. From this, we obtain a new simple proof for their monotonicity under completely positive trace-preserving maps.