α-z-Renyi relative entropies

We consider a two-parameter family of Renyi relative entropies D-alpha,D-z(rho parallel to sigma) that are quantum generalisations of the classical Renyi divergence D-alpha(p parallel to q). This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum Renyi divergences, as well as the quantum Renyi relative entropies. All its members satisfy the quantum generalizations of Renyi's axioms for a divergence. We consider the range of the parameters a, z for which the data-processing inequality holds. We also investigate a variety of limiting cases for the two parameters, obtaining explicit formulas for each one of them. (C) 2015 AIP Publishing LLC.