Abstract

We employ majorization theory to obtain a powerful tool for deriving simple and universal proofs of continuity bounds for various entropies which are relevant in classical and quantum information theory. In obtaining this, we first derive a more general result which may be of independent interest: a necessary and sufficient condition under which a state maximizes a concave, continuous, Gateaux-differentiable function in an epsilon-ball in trace distance. Examples of such a function include the von Neumann entropy, Renyi entropies, and the conditional entropy. In particular, by introducing a notion of majorization flow, we prove that the alpha-Rényi entropy is Lipschitz continuous, for alpha > 1, thus resolving an open problem and providing a substantial improvement over previously known bounds. We also discuss some challenging open questions. This is joint work with Eric Hanson. Note that no prior knowledge in quantum information theory is required to understand the talk.