α-z-Rényi relative entropies

We consider a two-parameter family of Rényi relative entropies Dα,z(ρ ∥ σ) that are quantum generalisations of the classical Rényi divergence Dα(p ∥ q). This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum Rényi divergences, as well as the quantum Rényi relative entropies. All its members satisfy the quantum generalizations of Rényi’s axioms for a divergence. We consider the range of the parameters α, 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.