On asymptotic continuity of functions of quantum states

A useful kind of continuity of quantum states functions in asymptotic regime is so-called asymptotic continuity. In this paper we provide general tools for checking if a function possesses this property. First we prove equivalence of asymptotic continuity with so-called it robustness under admixture. This allows us to show that relative entropy distance from a convex set including maximally mixed state is asymptotically continuous. Subsequently, we consider it arrowing - a way of building a new function out of a given one. The procedure originates from constructions of intrinsic information and entanglement of formation. We show that arrowing preserves asymptotic continuity for a class of functions (so-called subextensive ones). The result is illustrated by means of several examples.