Theorem 8.10 in Chapter 8 of Theory of Quantum Information asserts that the Holevo capacity of a quantum channel (between density operators on $\mathbb{C}^d$) can be achieved by an ensemble consisting of ${d^2}$ pure states.
The proof for general ensembles is a nice application of the conditions under which the Holevo information is convex (Proposition 5.48) along with Proposition 2.52 which is essentially a consequence of Catheodory's theorem and the space of density operators having real vector space dimension $d^2-1$ and therefore we can always pick the ensemble to have at most $d^2$ states.
I can also see that any value achieved by an ensemble of mixed states can be achieved by one of consisting only of pure states. My problem is that I can't see why this optimal pure state ensemble also has the $d^2$ bound on the number of states in the ensemble. Any insight into this argument would be appreciated.
