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

TL;DR

This paper modifies the Alicki-Fannes-Winter method to establish uniform continuity of certain quantum information measures on states with bounded energy, with applications to entanglement and channel capacities.

Contribution

It introduces a new approach to prove continuity bounds for quantum information quantities under energy constraints, extending previous methods.

Findings

Proves asymptotic continuity of the relative entropy of entanglement.

Establishes energy-constrained continuity bounds for quantum mutual information.

Demonstrates channel-independent bounds for output Holevo quantity.

Abstract

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.