the mutual information is also concave in the initial state.

The fact that this is true for inefficient measurements, means that once we have made an efficient measurement, no matter what information we throw away regarding the final outcomes (i.e. which outcomes we average over), $\langle \Delta S(\rho)\rangle$ is always greater than the average of the entropy reductions which would have been obtained through measurement in each of the coding states, when we throw away the same information regarding the measurement results.

Information gathering and state-space symmetry
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In this section we show that measurements whose ability to extract information is uniform over the available state-space (that is, does not vary from point to point in the state-space) always extract more information (strictly, never extract less information) the less that is known before the measurement is made. Thus, in this sense, one may regard “the more you know, the less you get” as a fundamental property of measurement. We will show that this is true both for the information obtained regarding the final state (being $\langle \Delta
S(\rho)\rangle$), and the mutual information for a measurement on an ensemble of pure states. We will consider here efficient measurements only; no doubt 