Sandwiched R\'enyi Divergence Satisfies Data Processing Inequality

TL;DR

This paper proves that the sandwiched quantum Rényi divergence satisfies the data processing inequality for all α>1 and demonstrates super-additivity of the α-Holevo information, advancing understanding of quantum information measures.

Contribution

It establishes the data processing inequality for sandwiched Rényi divergence for all α>1 and shows super-additivity of α-Holevo information, using advanced mathematical tools.

Findings

Sandwiched Rényi divergence satisfies data processing inequality for all α>1.

α-Holevo information is super-additive.

Results are derived using Hölder's inequality, Riesz-Thorin theorem, and complex interpolation.

Abstract

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.