  We consider the problem of determining the weights of a quantum ensemble.
That is to say, given a quantum system that is in a set of possible known
states according to an unknown probability law, we give strategies to estimate
the individual probabilities, weights, or mixing proportions. Such strategies
can be used to estimate the frequencies at which different independent signals
are emitted by a source. They can also be used to estimate the weights of
particular terms in a canonical decomposition of a quantum channel. The quality
of these strategies is quantified by a covariance-type error matrix. According
with this cost function, we give optimal strategies in both the single-shot and
multiple-copy scenarios. The latter is also analyzed in the asymptotic limit of
large number of copies. We give closed expressions of the error matrix for
two-component quantum mixtures of qubit systems. The Fisher information plays
an unusual role in the problem at hand, providing exact expressions of the
minimum covariance matrix for any number of copies.
