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Recently, a new notion of quantum Re&#x0301;nyi divergences has been introduced by Mu&#x0308;ller-Lennert, Dupuis, Szehr, Fehr, and Tomamichel and Wilde, Winter, and Yang, which found a number of applications in strong converse theorems. Here, we show that these new Re&#x0301;nyi 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 Re&#x0301;nyi divergences. The novelty of our proofs is that the composite/compound coding theorems can be almost directly obtained from the single-shot error bounds, essentially with the same effort as for the case of simple null-hypothesis/single source/single channel.

Coding Theorems for Compound Problems via Quantum Rényi Divergences