by $\langle\Delta S(\rho) \rangle$. We show that for inefficient measurements this becomes $$M(I\! :\! J) \leq \langle\Delta S(\rho) \rangle - \sum_i P(i) \langle \Delta S(\rho_i)\rangle ,$$ where $\langle \Delta S(\rho_i)\rangle$ is the average entropy reduction which would have resulted if the initial state had been $\rho_i$, and as above $\rho = \sum_i P(i)\rho_i$.

The second is the fundamental property that, for all quantum operations, the average reduction in von Neumann entropy is concave in the initial state $\rho$. That is $$\langle \Delta S(\rho) \rangle \geq \sum_i P(i) \langle \Delta S(\rho_i) \rangle .$$

Finally, in the third part of this paper, we use the above result to show that for measurements which are uniform in their sensitivity across state-space (that is, measurements which are unitarily covariant), the amount which one learns about the final state always increases with the initial uncertainty, where this uncertainty is characterized by [*majorization*]{}. This is a quantum version of the much simpler classical result (which we also show) that the mutual information always increases with the initial uncertainty for classical measurements which are permutation symmetric. In addition we show that, for unitarily covariant measurements, the mutual information for pure-state ensembles also has this property. One can 