  We address quantum estimation in situations where one has at disposal data
from the measurement of an incomplete set of observables and some a priori
information on the state itself. By expressing the a priori information in
terms of a bias toward a given state the problem may be faced by minimizing the
quantum relative entropy (Kullback entropy) with the constraint of reproducing
the data. We exploit the resulting minimum Kullback entropy principle for the
estimation of a quantum state from the measurement of a single observable,
either from the sole mean value or from the complete probability distribution,
and apply it as a tool for the estimation of weak Hamiltonian processes. Qubit
and harmonic oscillator systems are analyzed in some details.
