applications have been found for majorization in quantum information theory. [@NielsenLett; @JPV; @NK; @Nielsen; @Chefles; @FJ]

We thus desire to show that for measurements with the specified symmetry, $\langle \Delta S(\sigma)\rangle \geq \langle \Delta S(\rho)\rangle $ whenever $\sigma \prec \rho$ (and similarly for the mutual information). Functions with this property (of which the von Neumann entropy, $S(\rho)$, is one example) are referred to as being [*Schur-concave*]{}. To show that a function is Schur-concave, it is sufficient to show that it is concave, and symmetric in its arguments [@MO; @Bhatia], which in our case are the eigenvalues of the density matrix $\rho$ (if our functions did not depend only on the eigenvalues of $\rho$, then they could not be Schur-concave, since the majorization condition only involves these eigenvalues).

The desired result for classical completely symmetric measurements is now immediate. In the classical case the mutual information is the unique measure of information gain, and $ M(I\! :\! J) = \langle \Delta S(\rho)\rangle$. The mutual information is concave in the initial classical probability vector ${\bf P} =
(P_1,\ldots,P_n)$ (being the vector of the eigenvalues of $\rho$ in our quantum formalism), as is indeed implied by the concavity of $\langle \Delta
S(\rho)\rangle$. Since 