An essay submitted in the competition for a Smith-Knight or Rayleigh-Knight prize. This essay combines my work on uniform continuity bounds (arXiv:1707.04249) with that on local continuity bounds (arXiv:1706.02212), and unifies the notation. Abstract: “Majorization is a pre-order of vectors, giving a sense in which one vector can be said to be more disordered than another. Two given vectors, however, may be incomparable. This concept has been extended
to quantum mechanical states, to particular success in the theory of entanglement manipulation. Here, we investigate the majorization pre-order over the ε-ball of quantum states defined by the trace distance. We find, suprisingly, that the ε-ball admits a maximal and minimal state in majorization order, which are comparable to every other state in the set. This property depends in particular on the geometry described by the trace distance. We apply these minimal and maximal states to find explicit formulas for “smooothed” single-partite entropies, briefly consider implications for state estimation via the MaxEnt principle, and establish local continuity bounds for a broad class of functions. Next, we investigate further properties of the minimal state in majorization order, and in particular of the map taking the center of the ε-ball to the minimal state in this ball. We find that this map satisfies a so-called semigroup property, which allows us to derive continuity bounds which are uniform over the set of states from our local results. The construction of the minimal state is motivated by a more general result we derive via Fermat’s rule of convex optimization, providing a first step to extending our results to functions of bipartite states, such as the conditional entropy.