Coding theorems for compound problems via quantum Rényi divergences

Recently, a new notion of quantum Rényi divergences has been introduced by Müller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J.Math.Phys. 54:122203, (2013), and Wilde, Winter, Yang, Commun.Math.Phys. 331:593--622, (2014), that has found a number of applications in strong converse theorems. Here we show that these new Rényi divergences are also useful tools to obtain coding theorems in the direct domain of various problems. We demonstrate this by giving new and considerably simplified proofs for the achievability parts of Stein's lemma with composite null hypothesis, universal state compression, and the classical capacity of compound classical-quantum channels, based on single-shot error bounds already available in the literature, and simple properties of the quantum Rényi divergences. The novelty of our proofs is that the composite/compound coding theorems can be almost directly obtained from the single-shot error bounds, with essentially the same effort as for the case of simple null-hypothesis/single source/single channel.