Entropies, majorization flow, and continuity bounds

Abstract

Entropies play a fundamental role in classical and quantum information theory. There are a plethora of quantum entropies and they satisfy a host of interesting properties. The von Neumann entropy, the first quantum entropy to be defined and arguably the most important of them, satisfies a powerful property called strong subadditivity. It was proved by Lieb and Ruskai in 1973 and has turned out to be an essential tool in establishing various important results in quantum information theory. We briefly review this property and its applications. Our main focus, however, is on another desirable property of entropies, namely, continuity. We introduce a notion of majorization flow and demonstrate it to be a powerful tool for deriving simple and universal proofs of continuity bounds for Schur concave functions with a particular emphasis on entropic functions relevant in information theory. In particular, for the case of the $\alpha$-Rényi entropy, whose connections to thermodynamics are discussed in this chapter, majorization flow yields a Lipschitz continuity bound for the case $\alpha >1$, thus resolving an open problem and providing a substantial improvement over previously known bounds.