  We consider the statistics of overlaps between a mixed state and its image
under random unitary transformations. Choosing the transformations from the
unitary group with its invariant (Haar) measure, the distribution of overlaps
depends only on the eigenvalues of the mixed state. This allows one to estimate
these eigenvalues from the overlap statistics. In the first part of this work,
we present explicit results for qutrits, including a discussion of the expected
uncertainties in the eigenvalue estimation. In the second part, we assume that
the set of available unitary transformations is restricted to $SO(3)$, realized
as Wigner $D$-matrices. In that case, the overlap statistics does not depend
only on the eigenvalues, but also on the eigenstates of the mixed state under
scrutiny. The overlap distribution then shows a complicated pattern, which may
be considered as a fingerprint of the mixed state. When using random
transformations from the unitary group, the eigenvalues can be determined quite
simply from the lower and the upper limit of the overlap statistics. This may
still be possible in the $SO(3)$ case, but only at the expense of a finite
systematic uncertainty.
