Uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy

TL;DR

This paper establishes tight continuity bounds for key information measures of quantum channels, considering input dimension and energy constraints, with special focus on multi-mode quantum oscillators, enhancing understanding of channel capacity stability.

Contribution

It provides new tight continuity bounds for quantum channel capacities based on input dimension and energy, extending previous bounds and focusing on infinite-dimensional systems.

Findings

Derived bounds for output conditional mutual information.

Estimated variation of the Holevo quantity under channel and ensemble changes.

Established tight bounds for capacities of energy-constrained infinite-dimensional channels.

Abstract

We obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension (if it is finite) and on the input energy bound (if the input dimension is infinite). We pay a special attention to the case of a multi-mode quantum oscillator as an input system. First we prove continuity bounds for the output conditional mutual information for a single channel and for $n$ copies of a channel. Then we obtain estimates for variation of the output Holevo quantity with respect to simultaneous variations of a channel and of an input ensemble. As a result tight and close-to-tight continuity bounds for basic capacities of quantum channels depending on the input dimension are obtained. They complement the Leung-Smith continuity bounds depending on the output dimension. Finally, we obtain tight and close-to-tight continuity bounds for basic capacities of infinite-dimensional energy-constrained channels with respect to the energy-constrained Bures distance generating the strong convergence of quantum channels.