Pure state stabilization with discrete-time quantum feedback

Global asymptotic stabilization of quantum pure states is relevant to chemical process control, quantum cooling, state purification, and is crucial to the initialization of quantum information processing algorithms. We provide a linear-algebraic characterization of discrete-time Markovian dynamics leading to invariance and attractivity of a given quantum state. Assuming that the system is unitarily controllable, and accessible via a given quantum measurement, we provide a condition that that characterize the stabilizable target states. We also argue that if the control problem is feasible, then an effective control choice can be explicitly constructed. The result strongly relies on some remarlable properties of a canonical QR decomposition for complex matrices.