Sandwiched R\'enyi Divergence Satisfies Data Processing Inequality

Sandwiched (quantum) $\alpha$-R\'enyi divergence has been recently defined in the independent works of Wilde et al. (arXiv:1306.1586) and M\"uller-Lennert et al (arXiv:1306.3142v1). 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 $\alpha$-R\'enyi divergence satisfies the data processing inequality for all values of $\alpha> 1$. Moreover we prove that $\alpha$-Holevo information, a variant of Holevo information defined in terms of sandwiched $\alpha$-R\'enyi divergence, is super-additive. Our results are based on H\"older's inequality, the Riesz-Thorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.