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

We describe a modification of the Alicki-Fannes-Winter method which allows to prove uniform continuity on the set of quantum states with bounded energy of any locally almost affine function having limited growth with increasing energy. Some applications in quantum information theory are considered. The asymptotic continuity of the relative entropy of entanglement and of its regularization under the energy constraint on one subsystem is proved. Channel-independent continuity bounds for the quantum mutual information at the output of a local channel and for the output Holevo quantity under the input energy constraint are obtained.