Asymptotic statistics of repeated indirect quantum measurements

This research project analyzes the asymptotic behaviour of a quantum system subject to a sequence of indirect measurements. These quantum measurements give rise to a stochastic process, called quantum trajectory, which describes the state of the system after each measurement. Using martingale techniques we will prove that this quantum trajectory converges non-deterministically to one of the minimal invariant subspaces determined by the quantum channel, which is a linear map that describes the mean evolution of the state. The probability of convergence to each subspace depends on the initial state of the system. The convergence can be steered towards a chosen target subspace, modifying the dynamics with a feedback control scheme properly designed using Lyapunov techniques and graph-theoretic ideas, generalizing the control scheme preseneted in [2]. Preparation of quantum states in a target sub- space finds one of its applications in cooling techniques and in state preparation in quantum information. The other focus of this research project is on the derivation of some statistical asymptotic laws (Law of Large Numbers - Central Limit Theroem - Law of Iterated Logarithms) for the stochastic process describing the measurement outcomes, without requiring any ergodicity assumption on the quantum channel, and thus generalizing the results obtained in [3]. These statistical asymptotic laws can be used for solving estimation problems like process tomography. This research project puts together probability theory and control theory in order to prove asymptotic results on quantum stochastic processes and in order to design a feedback control scheme that is able to prepare a quantum system in a precise target subspace. A rigorous mathematical treatment is employed in deriving results having important applications in quantum engineering problems, like information encoding or parameter estimation.