Information about quantum systems is inferred from a sequence of measurements made on it. The maximum likelihood (ML) estimation gives an arguably natural optimum approach in quantum theory. The feasibility is shown on valuable examples including quantum phase or entangled spin states.
The question of what information about a quantum state may be inferred from a sequence of measurements made on it, is addressed. The main result is that maximum-likelihood estimation gives an arguably natural optimum approach in quantum theory. It singles out a particular choice of basis, in which the state is expressed. Various incompatible and incomplete observations made on the ensemble are treated as a complete generalized measurement. Consequently, the maximum likelihood (relative entropy) is the best measure for relating experimental data with theoretical predictions of quantum theory.
Maximum-likelihood estimation is applied to the determination of an unknown quantum measurement. The calibrated measuring apparatus carries out measurements on many different quantum states and the positive operator-valued measure governing the measurement statistics is then inferred from the collected data via the maximum-likelihood principle. In contrast to the procedures based on linear inversion, our approach always provides a physically sensible result. We illustrate the method on the case of the Stern-Gerlach apparatus.
Abstract. The theory of quantum state reconstruction is illustrated here on sev-eral examples taken from modern experimental praxis. Maximum-likelihood estima-tion is applied to experiments on physical systems of increasing complexity, starting with a simple one-dimensional problem of quantum phase estimation, continuing with the absorption and phase neutron tomographies, further discussing quantum tomography of higher-dimensional discrete quantum systems, and closing with the homodyne tomography of an infinite dimensional system- a mode of light. All these experiments nicely demonstrate the utility of present state-of-art techniques for ma-nipulating states of a neutron and internal as well as external states of a photon. Over the past hundred years, quantum theory proved to be an extremely useful tool for describing and understanding nature on a microscopic level. Its usual applications consist in predicting the outcomes of future experimental results given a quantum state describing the physical system.
The maximum likelihood strategy for the estimation of group parameters allows one to derive in a general fashion optimal measurements, optimal signal states, and their relations with other information theoretical quantities. These results provide deep insight into the general structure underlying optimal quantum estimation strategies. The entanglement between representation spaces and multiplicity spaces of the group action appears to be the unique kind of entanglement which is really useful for the optimal estimation of group parameters.