all operators commute with the density matrix, $\langle
\Delta S(\rho)\rangle$ is only a function of the $\{P_i\}$. From the form of $\langle \Delta S(\rho)\rangle$ we see that a permutation of the elements of ${\bf P}$ is equivalent to a permutation applied to the measurement operators, and since these are invariant under such an operation, $\langle \Delta
S(\rho)\rangle$, and thus $M(I\! :\! J)$, is a symmetric function of its arguments. Thus $M(I\! :\! J)$ is Schur-concave.

The Schur-concavity of $\langle \Delta S(\rho)\rangle$ for unitarily covariant (UC) quantum measurements is just as immediate. Because of the unitary covariance of the measurement, we see from the form of $\langle \Delta S(\rho)\rangle$ that it is invariant under a unitary transformation of $\rho$. As a result, it only depends upon the eigenvalues of $\rho$. Since the permutations are a subgroup of the unitaries, it is also a symmetric function of its arguments (the eigenvalues), and thus Schur-concave.

We wish finally to show that the mutual information is also Schur-concave in $\rho$ for unitarily covariant measurements on ensembles of pure states. This requires a little more work. First we need to show that once we have fixed a set of encoding states, the mutual information is 