  We consider the problem of determining the mixed quantum state of a large but
finite number of identically prepared quantum systems from data obtained in a
sequence of ideal (von Neumann) measurements, each performed on an individual
copy of the system. In contrast to previous approaches, we do not average over
the possible unknown states but work out a ``typical'' probability distribution
on the set of states, as implied by the experimental data. As a consequence,
any measure of knowledge about the unknown state and thus any notion of ``best
strategy'' (i.e. the choice of observables to be measured, and the number of
times they are measured) depend on the unknown state. By learning from
previously obtained data, the experimentalist re-adjusts the observable to be
measured in the next step, eventually approaching an optimal strategy. We
consider two measures of knowledge and exhibit all ``best'' strategies for the
case of a two-dimensional Hilbert space. Finally, we discuss some features of
the problem in higher dimensions and in the infinite dimensional case.
