Coding Theorems for Compound Problems via Quantum Rényi Divergences

We show two-sided bounds between the traditional quantum Rényi divergences and the new notion of Rényi divergences introduced recently in Müller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, arXiv:1306.1586. The bounds imply that the two versions can be used interchangeably near α = 1, and hence one can benefit from the best properties of both when proving coding theorems in the case of asymptotically vanishing error. We illustrate this by giving short and simple proofs of the quantum Stein’s lemma with composite null-hypothesis, universal source compression, and the achievability part of the classical capacity of compound quantum channels. Apart from the above interchangeability, we benefit from a weak quasi-concavity property of the new Rényi divergences that we also establish here.