alpha-z-relative Renyi entropies

Abstract: We consider a two-parameter family of R\'enyi relative entropies $D_{\alpha,z}(\rho||\sigma)$ that are quantum generalisations of the classical R\'enyi divergence $D_{\alpha}(p||q)$. This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum R\'enyi divergences, as well as the quantum R\'enyi relative entropies. All its members satisfy the quantum generalizations of R\'enyi's axioms for a divergence. We consider the range of the parameters $\alpha,z$ 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.