 Solution

For an ensemble $\mathcal{E}=\{p_i,{\rho }_i\}$, write its channel-output Holevo quantity asTake ensembles ${\mathcal{E}}_1=\{p_i,{\rho }_i\}$ and ${\mathcal{E}}_2=\{q_j,{\sigma }_j\}$ independently, giving the product ensemble $\{p_i{q}_j,{\rho }_i\otimes {\sigma }_j\}$. Its average output factors:The Von Neumann entropy of a tensor product is the sum of its entropies. Applying this both to the average and to every labelled output givesChoose each ensemble within $\epsilon$ of its supremum. The supremum for the joint channel includes this product ensemble, so its value is at least ${\chi }^{\ast }({\Phi }_1)+{\chi }^{\ast }({\Phi }_2)-2\epsilon$. Let $\epsilon \downarrow 0$. The superadditivity of Holevo capacity follows:This argument does not restrict the joint supremum to product ensembles and does not require the individual suprema to be attained.