Approaches for approximate additivity of the Holevo information of quantum channels

TL;DR

This paper develops methods to approximate the additivity of Holevo information in quantum channels, providing efficiently computable upper bounds on classical capacity and demonstrating their effectiveness on specific noisy channels.

Contribution

It introduces a framework for bounding quantum channel capacities using approximation parameters related to weak additivity, with efficient SDP-based computation.

Findings

Bounds are tight in certain noise regimes.

Method applies to channels close to entanglement-breaking, unital, and Hadamard channels.

Provides new insights into generalized channel divergences.

Abstract

We study quantum channels that are close to another channel with weakly additive Holevo information and derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above that measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as well as an outer bound on the dynamic capacity region of a quantum channel. Since these parameters are defined in terms of the diamond distance, the upper bounds can be computed efficiently using semidefinite programming (SDP). We exhibit the usefulness of our method with two example channels: a convex mixture of amplitude damping and depolarizing noise, and a composition of amplitude damping and dephasing noise. For both channels, our bounds perform well in certain regimes of the noise parameters in comparison to a recently derived SDP upper bound on the classical capacity. Along the way, we define the notion of a generalized channel divergence (which includes the diamond distance as an example), and we prove that for jointly covariant channels these quantities are maximized by purifications of a state invariant under the covariance group. This latter result may be of independent interest.