The characteristics of limit points of any trajectory of mixed-state quantum systems in the Lyapunov-based control design

For mixed-state quantum systems without degenerate transition, this paper chooses the expectation value of a mechanical quantity as a Lyapunov function, designs the control laws that guarantee the monotonic decreasing of the Lyapunov function, and mainly studies the characteristics of limit points of any system trajectory under the action of control fields. Research results show that for a fully connected system, the limit point of any trajectory is one of its equilibrium states while for an only connected system, the limit points of any trajectory is not isolated. In the latter case, this paper also gives the strict expression of the limit points via the Bloch vector framework of density operators.