us with the state $$\sigma^{(QM)} = \sum_j |j\rangle\langle j|^{(M)} \otimes \sum_{k} P(k,j) \sigma_{kj}^{(Q)}$$

After this final unitary, and the partial trace over $A$, the Holevo quantity for the remaining systems $Q$ and $M$, which we will denote by $\chi''^{(QM)}$, satisfies $\chi''^{(QM)} \leq \chi'^{(QAM)} \leq \chi^{(Q)}$. We have gone through the above process using the initial state $\rho$, but we could just as easily have started with any of the initial states, $\rho_i$, in the ensemble, and we will denote the final states which we obtain using the initial state $\rho_i$ as $\sigma_i^{(QM)}$. Calculating $\chi''^{(QM)}$ we have

$$\begin{aligned}
  \chi''^{(QM)} & = & S(\sigma^{(QM)}) - \sum_i P(i) S(\sigma_i^{(QM)}) \nonumber \\ 
              & = & H[J] - \sum_i P(i) H[J|i] \nonumber \\
              &   & + \sum_j P(j) \left[ S(\sigma_j) - \sum_i P(i|j) \sigma_{j|i} \right] \\
          & = & M(J:I) + \sum_j P(j) \chi_j^{(Q)} \leq \chi^{(Q)}.\end{aligned}$$

Rearranging this expression gives the desired result.

Properties of entropy reduction
===============================

We now rewrite the above information bound using the fact that $P(i|j)P(j) =
P(j|i)P(i)$. The result is $$M(I\! :\! J) \leq \langle \Delta S(\rho)\rangle - \sum_i P(i) \langle \Delta S(\rho_i)\rangle
\label{genHall}$$ where $\rho = \sum_i P_i \rho_i$. Ozawa has shown that for efficient measurements $\langle \Delta S(\rho) 