 Solution

The quantity ${\chi }^{\ast }(\Lambda )$ is the one-use Holevo information of the channel: by the Holevo--Schumacher--Westmoreland theorem, its regularization gives the classical capacity, and without entangled inputs across uses it is the asymptotically achievable classical rate using collective output measurements.

If an ensemble contains a mixed ${\rho }_x={\sum }_j{q}_{j|x}|{\psi }_{xj}⟩⟨{\psi }_{xj}|$, refine its label to $(x,j)$ with probability $p_x{q}_{j|x}$. The average channel output is unchanged, while Concavity of Von Neumann entropy givesThus refinement cannot decrease the Holevo quantity, so the maximum may be restricted to pure input states.