Quantum indirect estimation theory and joint estimate of all moments of two incompatible observables

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.