Adaptation of the Alicki-Fannes-Winter method for the set of states with bounded energy and its use

Abstract

We describe a modification of the Alicki-Fannes-Winter method (used for proving uniform continuity of functions on the set of quantum states). It allows to show uniform continuity on the set of states with bounded energy of any approximately affine function having limited growth with increasing energy. Some applications in quantum information theory are considered. In particular, the uniform finite-dimensional approximation theorem for the Holevo capacity of energy constrained channels is proved.