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We study quantum channels that are close to another channel with weaklyadditive Holevo information and derive upper bounds on their classicalcapacity. Examples of channels with weakly additive Holevo information areentanglement-breaking channels, unital qubit channels, and Hadamard channels.Related to the method of approximate degradability, we define approximationparameters for each class above that measure how close an arbitrary channel isto satisfying the respective property. This gives us upper bounds on theclassical capacity in terms of functions of the approximation parameters, aswell 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 upperbounds can be computed efficiently using semidefinite programming (SDP). Weexhibit the usefulness of our method with two example channels: a convexmixture of amplitude damping and depolarizing noise, and a composition ofamplitude damping and dephasing noise. For both channels, our bounds performwell in certain regimes of the noise parameters in comparison to a recentlyderived SDP upper bound on the classical capacity. Along the way, we define thenotion of a generalized channel divergence (which includes the diamond distanceas an example), and we prove that for jointly covariant channels thesequantities are maximized by purifications of a state invariant under thecovariance group. This latter result may be of independent interest.

Approaches for approximate additivity of the Holevo information of quantum channels