Statistical Inference in Open Quantum Systems

This thesis concerns the statistical analysis of open quantum systems subject to an external and non-stationary perturbation. In the first paper, a generalization of the explicit-duration hidden Markov models (EDHMM) which takes into account the presence of sparse data is presented. Introducing a kernel estimator in the estimation procedure increases the accuracy of the estimates, and thus allows one to obtain a more reliable information about the evolution of the unobservable system. A generalization of the Viterbi algorithm to EDHMM is developed. In the second paper, we develop a Markov Chain Monte Carlo (MCMC) procedure for estimating the EDHMM. We improve the flexibility of our formulation by adopting a Bayesian model selection procedure which allows one to avoid a direct specification of the number of states of the hidden chain. Motivated by the presence of sparsity, we make use of a non-parametric estimator to obtain more accurate estimates of the model parameters. The formulation presented turns out to be straightforward to implement, robust against the underflow problem and provides accurate estimates of the parameters. In the third paper, an extension of the Cramér-Rao inequality for quantum discrete parameter models is derived. The latter are models in which the parameter space is restricted to a finite set of points. In some estimation problems indeed, theory provides us with additional information that allow us to restrict the parameter space to a finite set of points. The extension presented sets the ultimate accuracy of an estimator, and determines a discrete counterpart of the quantum Fisher information. This is particularly useful in many experiments in which the parameters can assume only few different values: for example, the direction which the magnetic field points to. We also provide an illustration related to a quantum optics problem.