Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement which excludes in principle the singular direct observability continual case. Quantum theory of time continuous measurements and quantum prediction theory, developed by the author on the basis of an independent-increment model for quantum noise and nondemolition causality principle in the 80's, solves this problem allowing continual quantum predictions and reducing many quantum information problems like problems of quantum feedback control to the classical stochastic ones. Using explicit indirect observation models for diffusive and counting measurements we derive quantum filtering (prediction) equations to describe the stochastic evolution of the open quantum system under the continuous partial observation. Working in parallel with classical indeterministic control theory, we show the Markov Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these two kinds of measurements. The resulting filtering and Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities and differences with the corresponding classical nonlinear filtering and control problems and demonstrates microduality between quantum filtering and classical control.
This thesis is devoted to the study of the quantum trajectories obtained from the theory of continuous measurement in non relativistic quantum mechanics. New theoretical results as well as examples of applications are presented. On the theoretical front, we study mostly the limit of "strong" measurement where we put forward the emergence of quantum jumps and quantum spikes, two phenomena we characterize in detail. Out of the strong measurement limit, we investigate a method to extract information from a register of qubits optimally. On the application front, we introduce an original method to control quantum systems exploiting only the freedom of changing the measurement intensity and we explain the transition between a ballistic and a diffusive behavior in open quantum random walks; two byproduct of the theoretical study of the strong measurement regime. We further study the problem of semi-classical gravity and show that continuous measurement theory allows to construct a consistent model in the Newtonian regime. We eventually suggest possible extensions of the formalism to a posteriori estimation and hint at generalizations of the results for the strong measurement limit in the wider context of discrete repeated measurements. In the course of our presentation, we emphasize the link with other approaches to the theory of continuous measurement (parallels with stochastic filtering and collapse models in foundations) rather than aim for mathematical rigor.