Sandwiched Rényi divergence satisfies data processing inequality

Abstract

Sandwiched (quantum) α-Rényi divergence has been recently defined in the independent works of Wilde et al. [“Strong converse for the classical capacity of entanglement-breaking channels,” preprint arXiv:1306.1586 (2013)] and Müller-Lennert et al. [“On quantum Rényi entropies: a new definition, some properties and several conjectures,” preprint arXiv:1306.3142v1 (2013)]. This new quantum divergence has already found applications in quantum information theory. Here we further investigate properties of this new quantum divergence. In particular, we show that sandwiched α-Rényi divergence satisfies the data processing inequality for all values of α > 1. Moreover we prove that α-Holevo information, a variant of Holevo information defined in terms of sandwiched α-Rényi divergence, is super-additive. Our results are based on Hölder's inequality, the Riesz-Thorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.