# Tight continuity bounds for the quantum conditional mutual information, for the Holevo quantity and for capacities of a channel

Research paper by M. E. Shirokov

Indexed on: 16 Jul '16Published on: 16 Jul '16Published in: Quantum Physics

#### Abstract

First we consider Fannes' type and Winter's type tight continuity bounds for the quantum conditional mutual information and their specifications for states of special types. Then we analyse continuity of the Holevo quantity with respect to two nonequivalent metrics on the set of ensembles of quantum states. We show that the Holevo quantity is continuous on the set of all ensembles of \$m\$ states with respect to both metrics if either \$m\$ or the dimension of underlying Hilbert space is finite and obtain Fannes' type tight continuity bounds for the Holevo quantity in this case. In general case conditions for local continuity of the Holevo quantity and their corollaries (preserving local continuity under quantum channels, stability with respect to perturbation of states) are considered. Winter's type tight continuity bound for the Holevo quantity under the energy constraint on the average state of ensembles is obtained and applied to the system of quantum oscillators. The above results are used to obtain tight and close-to-tight continuity bounds for basic capacities of finite-dimensional channels (refining the Leung-Smith continuity bounds) and for classical capacities of infinite-dimensional channels with energy constraints.