Additivity of Holevo capacity (${\chi }^{\ast }(\Lambda \otimes \Gamma )={\chi }^{\ast }(\Lambda )+{\chi }^{\ast }(\Gamma )$)

A channel's Holevo capacity is additive with another channel when their joint capacity equals the sum of the individual capacities. If this holds with every other channel, the channel's tensor powers have capacity $n{\chi }^{\ast }(\Lambda )$ and regularization does not increase its unassisted classical capacity of a quantum channel. Additivity is a special property, not a general rule for quantum channels.

Every entanglement-breaking channel $\mathcal{E}$ has additive Holevo capacity with every finite-dimensional quantum channel $\mathcal{N}$. A measure-and-prepare channel factorization exposes a classical measurement outcome. The quantum mutual information balance identity and data processing for quantum mutual information then bound the joint output Holevo quantity by the sum of the individual capacities, using the conditional input ensemble after a local measurement for the other channel. Independent product ensembles attain the reverse inequality.