Tight continuity bounds for the quantum conditional mutual information, for the Holevo quantity and for capacities of quantum channels

TL;DR

This paper establishes tight continuity bounds for quantum information measures like the conditional mutual information and Holevo quantity, with applications to quantum channel capacities, enhancing understanding of their stability under various conditions.

Contribution

It provides new tight continuity bounds for the Holevo quantity and quantum capacities, refining previous bounds and analyzing continuity under different metrics and energy constraints.

Findings

Tight continuity bounds for quantum conditional mutual information.

Fannes' and Winter's type bounds for the Holevo quantity.

Refined bounds for quantum channel capacities.

Abstract

We start with 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 nonequivalent metrics on the set of discrete ensembles of quantum states. We show that the Holevo quantity is continuous on the set of all ensembles of m states with respect to all the 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 for discrete and continuous ensembles are found. Winter's type tight continuity bound for the Holevo quantity under constraint on the average energy 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).