only consider one ensemble for each $\rho$. We therefore choose the eigen-ensemble $\{\lambda_i,|\phi_i\rangle\}$, where $\lambda_i$ and $|\phi_i\rangle$ are the eigenvalues and eigenvectors of $\rho$ respectively. We know that the mutual information is concave in the vector of initial ensemble probabilities, and for the ensemble we have chosen, the initial probabilities are the eigenvalues of $\rho$. As a result the mutual information is concave in the eigenvalues of $\rho$. Since $M$ is invariant under unitary transformations, and since unitary transformations include permutations as a subgroup, it is also a symmetric function of the eigenvalues. Thus $M$ is Schur-concave.

conclusion
==========

In using a quantum channel, if there are limitations on the completeness (or alternatively the [*strength*]{}, in the terminology of [@FJ]) or efficiency of the measurements that the receiver can perform, then it is possible to give a bound on the mutual information which is stronger than the Holevo bound. Further, this bound has a very simple form in terms of the Holevo $\chi$ quantity, and the $\chi$ quantities of the ensembles, one of which remains after the measurement is made.

This bound also allows us to obtain a relationship between the mutual information and the average von Neumann entropy reduction 