the Holevo quantity for the ensemble, $\varepsilon_j$, that remains (from the point of view of the receiver) once the measurement has been made, so that the receiver has learned the outcome $j$, but not the value of $k$. Here the receiver’s overall final state is $$\sigma_{j} = \frac{\sum_{k} A_{kj} \rho A_{kj}^\dagger}{P(j)}  
              = \sum_{ik} P(i,k|j) \sigma_{kj|i} ,$$ where $P(i,k|j)$ is the probability for both $i$ and outcome $k$ given $j$, and $\sigma_{kj|i}$ is the final state that results given the initial state $\rho_i$, and both outcomes $j$ and $k$. The remaining ensemble $\varepsilon_j
= \{P(i|j), \sigma_{j|i}\}$, where $$\sigma_{j|i} = \sum_{k} P(k|j,i) \sigma_{kj|i}  
                = \frac{\sum_{k} A_{kj} \rho_i A_{kj}^\dagger}{P(j|i)} ,$$ and where $P(k|j,i)$ is the probability for outcome $k$ given $j$ and the initial state $\rho_i$.

We begin by collecting various key facts. The first is that any efficient measurement on a system $Q$, described by $N = N_1N_2$ operators, $A_{kj}$, ($j=1,\ldots , N_1$ and $k=1,\ldots , N_2$) can be obtained by bringing up an auxiliary system $A$ of dimension $N$, performing a unitary operation involving $Q$ and $A$, and then making a von Neumann measurement on $A$. [@Krauss; @Schumacher] If the initial state of $Q$ is $\rho^{(Q)}$, then the final joint state 