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We analyze the asymptotic behavior of discrete-time, Markovian quantum systems with respect to a subspace of interest. Global asymptotic stability of subspaces is relevant to quantum information processing, in particular for initializing the system in pure states or subspace codes. We provide a linear-algebraic characterization of the dynamical properties leading to invariance and attractivity of a given quantum subspace. We then construct a design algorithm for discrete-time feedback control that allows to stabilize a target subspace, proving that if the control problem is feasible, then the algorithm returns an effective control choice. In order to prove this result, a canonical QR matrix decomposition is derived, and also used to establish the control scheme potential for the simulation of open-system dynamics.

We study feedback-controlled, discrete-time quantum Markovian dynamics focusing on pure-state stabilization problem. Assuming that the system is unitarily controllable, and accessible via a given quantum measurement, we explicitly construct a choice of control actions conditioned on the measurement outcome that globally stabilizes the target state for the averaged dynamics. A key step in deriving this result is the definition of a canonical QR decomposition for complex matrices.

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.