Tight uniform continuity bound for a family of entropies

We prove a tight uniform continuity bound for a family of entropies which includes the von Neumann entropy, the Tsallis entropy and the $\alpha$-R'enyi entropy, $S_{\alpha }$, for $\alpha \in (0,1)$. We establish necessary and sufficient conditions for equality in the continuity bound and prove that these conditions are the same for every member of the family. Our result builds on recent work in which we constructed a state which was majorized by every state in a neighbourhood ($\epsilon$-ball) of a given state, and thus was the minimal state in majorization order in the $\epsilon$-ball. This minimal state satisfies a particular semigroup property, which we exploit to prove our bound.