On the quantum Renyi relative entropies and related capacity formulas

We show that the quantum $alpha$-relative entropies with parameter $alphain (0,1)$ can be represented as generalized cutoff rates in the sense of [I. Csiszar, IEEE Trans. Inf. Theory 41, 26-34, (1995)], which provides a direct operational interpretation to the quantum $alpha$-relative entropies. We also show that various generalizations of the Holevo capacity, defined in terms of the $alpha$-relative entropies, coincide for the parameter range $alphain (0,2]$, and show an upper bound on the one-shot epsilon-capacity of a classical-quantum channel in terms of these capacities.