continuum (as in the case of UC measurements) by replacing the relevant sums with integrals.

Now we need to note some further points about UC measurements: A UC measurement may be generated by taking all unitary transformations of any single operator $A$, and dividing them by a common normalization factor. The resulting measurement operators are thus $A_U \propto UAU^\dagger$, where $U$ ranges over all unitaries. The normalization for the $A_U$ comes from $\int U A^\dagger A U^\dagger d\mu(U) = \mbox{Tr}[A^\dagger A] I$ where $d\mu(U)$ is the (unitarily invariant) Haar measure [@Ucov; @Jones] over unitaries.

It is not hard to show that all UC measurements can be obtained by [*mixing*]{} different UC measurements, each generated by a different operator. (Mixing a set of measurements means assigning to each a probability, and then making one measurement from the set at random based on these probabilities [@mixing]).

Next, we need to show that for all UC measurements the mutual information depends only on the eigenvalues of the ensemble density matrix, and we state this as the following lemma.

The mutual information for a UC measurement on a pure-state ensemble, $\varepsilon = \{P(i),|\psi_i\rangle\}$ depends on the ensemble only through the eigenvalues of the density 