celebrated Holevo bound, conjectured by Gordon [@HBoundConj] and Levitin [@Levitin] and proved by Holevo in 1973 [@HBound] gives a bound on the information which may be transmitted from A to B (strictly, the [*mutual information*]{}, $M$, between A and B) when A encodes information in a quantum system using a set of states $\{\rho_i\}$, chosen with probabilities $\{P(i)\}$, and B makes a subsequent measurement upon the system. The Holevo bound is $$M(I\! :\! J) \leq \chi \equiv S(\rho) - \sum_i P(i) S(\rho_i) ,$$ where $\rho = \sum_i P(i) \rho_i$ (and which we will refer to as the [*ensemble state*]{}). We write the mutual information as $M(I\! :\! J)$ to signify that it is the mutual information between the random variables $I$ and $J$, whose values $i$ and $j$ label respectively the encoding used by A, and outcome of the measurement made by B. More recent proofs of the Holevo bound may be found in Refs. [@YO93; @FC94; @SWW] The bound is achieved if and only if the encoding states, $\rho_i$, commute with each other, and the receiver, B, makes a von Neumann measurement in the basis in which they are diagonal. (A von Neumann measurement is one that projects the 