Attractive Quantum Subsystems and Feedback-Stabilization Problems

摘要: We propose a general theoretical framework that is suitable to study wide class of stabilization problems for quantum Markovian dynamical systems. Building on systemtheoretic ideas, we definitions invariant and attractive subsystem, characterize invariance properties, provide sufficient conditions attraction. The results are illustrated by addressing the potential output-feedback control strategies pure state-stabilization. In particular, constructive synthesis stabilizing semigroups in arbitrary finite-dimensional systems established. I. BACKGROUND AND MOTIVATIONS Stabilization central relevance many applications, ranging from state preparation quantum-optical nano-mechanical generation noise-protected realizations information realistic devices [1]. Dynamical undergoing evolution [2], [3] both widely relevant physical standpoint present distinctive challenges – preventing, open-loop quantum-engineering methods based decoupling be viable [4], [5]. However, show here how can effectively treated framework, provided subsystems. After introducing main ideas along with some results, shall explore their application pure-state control. refer forthcoming journal version paper [6] detailed proofs omit or merely sketch following sections. Consider separable Hilbert space H over complex field C. Let B(H) represent set linear bounded operators H, H(H) denoting real subspace Hermitian operators, I, O being identity zero operator, respectively. standard statistical formulation mechanics [7], [8], dimension associated system interest, Q, determined physics problem. what follows, consider systems, i.e. dim(H) 0 described TracePreserving, Completely-Positive (TPCP) map Tt(·) [12], A differential equation density operator I may derived forward composition law holds: Definition 1 (QDS): semigroup one-parameter family TPCP maps {Tt(·), t ≥ 0} satisfies: (i) T0 = (ii) Tt ◦ Ts Tt+s, ∀t, s > 0, (iii) trace(Tt(ρ)X) continuous function t, ∀ρ ∈ D(HI), ∀X B(HI). Due trace positivity preserving assumptions, QDS contractions. It has been proved [10], [13] Hille-Yoshida generator exists cast canonical form: