matrix $\rho=\sum_{i}P(i)|\psi_{i}\rangle\langle \psi_{i}|$.

We first show this for UC measurements generated from a single operator. Writing the mutual information in the reverse form one has $$\begin{aligned}
     M(I\! :\! J)) & = & H[P(U)] - \sum_i P_i H[P(U|i)] ,
\label{forIc}\end{aligned}$$ where $U$ is the continuum index for the measurement operators (and thus the measurement outcomes) which are $A_U = UAU^\dagger$ for some appropriately normalized $A$. Naturally all this means is that $P(U|i)$ is a function of $U$, where $U$ ranges over all unitaries. Since the measurement is unitarily covariant, $H[P(U|i)]$ is the same for all initial states $|\psi_{i}\rangle$, and therefore the second term is the same for all initial ensembles. Thus $M$ depends only on the first term $H[P(U)] = H[\mbox{Tr}[UA^\dagger AU^\dagger\rho]]$, which depends only on $\rho$, and is invariant under all unitary transformations of $\rho$. Thus $M$ depends only on the eigenvalues of $\rho$. Since the mutual information for a mixture of measurements is merely a function of the respective mutual informations for each measurement (in particular it is a linear combination of them), the result holds for all UC measurements.

Since $M$ depends only on $\rho$, in establishing the Schur concavity of $M$ with respect to $\rho$, we need 