  Knowing and guessing, these are two essential epistemological pillars in the
theory of quantum-mechanical measurement. As formulated quantum mechanics is a
statistical theory. In general, a priori unknown states can be completely
determined only when measurements on infinite ensembles of identically prepared
quantum systems are performed. But how one can estimate (guess) quantum state
when just incomplete data are available (known)? What is the most reliable
estimation based on a given measured data? What is the optimal measurement
providing only a finite number of identically prepared quantum objects are
available? These are some of the questions we address. We present several
schemes for a reconstruction of states of quantum systems from measured data:
(1) We show how the maximum entropy (MaxEnt) principle can be efficiently used
for an estimation of quantum states on incomplete observation levels. (2) We
show how Bayesian inference can be used for reconstruction of quantum states
when only a finite number of identically prepared systems are measured. (3) We
describe the optimal generalized measurement of a finite number of identically
prepared quantum systems which results in the estimation of a quantum state
with the highest fidelity.
