Some topics in control of two-level quantum systems

This talk is devoted to some topics in control of two-level quantum systems.
In the first part we will consider coherent control of quantum systems. The goal of control is to maximize the objective functional. A physical example of this problem is control of spin 1/2 system by laser pulse. An important problem is the analysis of traps — local but not global maxima of the objective functional. We will present some theorems about traps for two-level quantum systems.
In the second part we will consider incoherent control of two-level quantum systems using non-selective measurements as control. In addition to the unitary evolution, the system state is transformed under back-action of non-selective measurements of some observables $Q_i$, $1\le i\le n$, at times $t_i$. The control goal is to maximize an objective functional on the set of sequences $Q_1,...,Q_n$. This problem can be used to approximate the quantum Zeno effect. The theorem which gives explicit expressions for the optimal observables will be presented.