Superadditivity of Holevo capacity (${\chi }^{\ast }({\Phi }_1\otimes {\Phi }_2)\ge {\chi }^{\ast }({\Phi }_1)+{\chi }^{\ast }({\Phi }_2)$)

Take independent ensembles for two quantum channels and their Cartesian product ensemble. The Von Neumann entropy of a tensor product is the sum of the two entropies, both for its average output and for each labelled output. Thus its Holevo quantity is the sum of the two individual quantities. Taking ensembles arbitrarily close to each supremum proves the displayed inequality without requiring maximizing ensembles to exist. It does not assert additivity for arbitrary channels.