We focus on quantum systems subject to external interactions (laser, magnetic fields) taken as controls, which are contaminated with non perturbative noise. The measured observables come thus in the form of probability laws; we ask the following question: is it possible, from the knowledge of these probability laws, to recover the free and interaction (dipole) Hamiltonians ? We see that the theoretical answer is positive (provided some assumptions on the controllability of the quantum system hold); then we explore numerical approaches which exploit particular metrics on the space of probability laws.