of $A$ and $Q$ after the von Neumann measurement is $$\sigma^{(AQ)} = |kj\rangle \langle kj|^{(A)} \otimes \frac{A_{kj} \rho^{(Q)} A_{kj}^\dagger}
                                                              {P(k,j)} .$$ where $|kj\rangle$ is the state of $A$ selected by the von Neumann measurement. The second fact is that the state which results from discarding all information about the measurement outcomes $k$ and $j$ can be obtained by performing a unitary operation between $A$ and another system $E$ which perfectly correlates the states $|kj\rangle$ of $A$ with orthogonal states of $E$, and then tracing out $E$. The final key fact we require is a result proven by SWW [@SWW], which is that the Holevo $\chi$ quantity is non-increasing under partial trace. That is, if we have two quantum systems $A$ and $B$, and an ensemble of states $\rho_i^{(AB)}$ with associated probabilities $P_i$, then $$\begin{aligned}
   \chi^{(A)} & =    & S(\rho^{(A)}) - \sum_i S(\rho_i^{(A)})  \nonumber \\
              & \leq & S(\rho^{(AB)}) - \sum_i S(\rho_i^{(AB)}) = \chi^{(AB)} ,\end{aligned}$$ where $\rho_i^{(A)} = \mbox{Tr}_B[\rho_i^{(AB)}]$. To prove this result SWW use strong subadditivity. [@subadd]

We now encode information in system $Q$ using the ensemble $\varepsilon$, and consider the joint system which consists of the three systems $Q$, $A$, $E$ and a forth system $M$, with dimension $N_1$. 