Quantum inference as an under-determined linear problem and the collective of accessible states

A detailed mathematical analysis of the collective of accessible micro-states compatible with a given set of a priori fixed expectation values (generalized ensemble) is undertaken. To such an effect, the problem of quantum inference is framed as an under-determined linear problem. The analysis is based upon the splitting of the pertinent statistical distribution (the density matrix one is trying to infer) into two orthogonal components. One of the components happens to coincide with the minimum norm solution of the linear problem posed by the fulfillment of the a priori conditions on the given expectation values. The remaining component belongs to the Null space of the expectation values transformation.