Sandwiched Rényi Divergence Satisfies Data Processing Inequality

Abstract

Sandwiched (quantum) $\alpha$-Rényi divergence has been recently defined in the independent works of Wilde et al. (arXiv:1306.1586) and Müller-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 $\alpha$-Rényi 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 $\alpha$-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.

13 pages, title changed, fixed typos, results unchanged, to appear in J. Math. Phys

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