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

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