Measurement-based Feedback Control Of Stochastic Quantum Systems

Quantum control theory plays an important role in the fields of quantum chemistry and quantum information, and has already attained significant success. Quantum measurement is an effective way to acquire the information of the controlled system. When one observable is measured, one will obtain one of the eigenvalues of this observable, and meanwhile the system state will collapse or continuously evolve to the eigenstate of the observable corresponding to that eigenvalue. In the late 1980’s, measurement-based quantum feedback control was proposed. After that, some measurement-based feedback control methods were successively proposed and studied.No matter whether deterministic or stochastic quantum systems, the preparation problem of quantum states has attracted a lot of attention and been widely studied. Based on the quantum measurement theory, only the eigenstates of the measured observable can be prepared. Bell states, the maximally entangled states of two-qubit systems, are powerful resources for quantum communication. Entanglement is one of the key features that distinguish quantum systems from classical ones and has many applications. This paper studies the state preparation problem of the measurement-based stochastic quantum systems including the Bell states of two-qubit systems and the eigenstates of angular momentum operators. The main research content of this dissertation can be listed as follows.1) The origin and development of quantum control are introduced, and the applications of traditional control methods to quantum systems are reviewed. Based on the detailed analysis of the status quo of quantum feedback control, some problems to be solved in this dissertation are given.2) For two-qubit systems, the preparation schemes of a given target Bell state is studied. Two special control strategies are proposed:the switching control strategy based on the state space partition, and the switching control strategy based on the switch between different models. As for the first switching control strategy, the dissertation first chooses an appropriate measured observable and two control channels within the theoretical framework of quantum continuous measurement; then divides the state space into several different regions, and designs a switching control law between these regions; and finally proves the stability of the whole closed-loop system, and verifies the feedback stabilization effect of the switching control strategy via the simulation experiments on a two-qubit system. As for the second switching control strategy, we will first propose a method based on the switch between two different models to achieve the control objective; then, according to the given target Bell state, choose proper control Hamiltonians for the two models, and propose the design method for the switching rules between the two models; and finally prove the stability of the whole switching system in theory, and show the effectiveness of the proposed switching control strategy between two different models via simulation experiments.3) The applications of the switching control strategy to stochastic angular momentum systems will be studied to prepare a given eigenstate of the angular momentum operator (measurement operator). Based on the partition of the state space, the dissertation will give a switching condition of Bang-Bang control law, and prove the global stability of the closed-loop system around the target eigenstate via the Lyapunov stability theory and the stochastic LaSalle invariance principle.