  We introduce the quantum indirect estimation theory, which provides a general
framework to address the problem of which ensemble averages can be estimated by
means of an available set of measuring apparatuses, e. g. estimate the ensemble
average of an observable by measuring other observable. A main ingredient in
this approach is that of informationally complete (infocomplete in short)
measurements, which allow to estimate the ensemble average of any arbitrary
system operator, as for quantum tomography. This naturally leads to the more
stringent concept of AB-informationally complete measurements, by which one can
estimate jointly all the moments of two incompatible observables A and B. After
analyzing all general properties of such measurements, we address the problem
of their optimality, and we completely solve the case of qubits, showing that a
sigma_x sigma_y-infocomplete measurement is less noisy than any infocomplete
one. We will also discuss the relation between the concept of
AB-infocompleteness and the notion of joint measurement of observables A and B.
