In this work we address the problems of state stabilization and convergence dynamics for finite-dimensional Quantum Dynamical Semigroups generators. We do so by building on the powerful notions of invariant and attractive quantum subsystems. We first recall some recent results due to Francesco Ticozzi and Lorenza Viola providing explicit algebraic characterizations of invariant and attractive subsystems. Then we present a new result on convergence dynamics.
In this work we address the problems of state stabilization and convergence dynamics for finite-dimensional Quantum Dynamical Semigroups generators. We do so by building on the powerful notions of invariant and attractive quantum subsystems. We first recall some recent results due to Francesco Ticozzi and Lorenza Viola providing explicit algebraic characterizations of invariant and attractive subsystems. Then we present a new result on convergence dynamics.
We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics accurately describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway for faithfully representing quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model parameter and initial state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov’s stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular,
We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway to faithfully represent quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular, explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained.
We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway to faithfully represent quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model-parameter and initial-state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov's stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular, explicit results for the synthesis of stabilizing semigroups and noiseless subspaces in finite-dimensional Markovian systems are obtained.
We propose a general framework for investigating a large class of stabilization problems in Markovian quantum systems. Building on the notions of invariant and attractive quantum subsystem, we characterize attractive subspaces by exploring the structure of the invariant sets for the dynamics. Our general analysis results are exploited to assess the ability of open-loop Hamiltonian and output-feedback control strategies to synthesize Markovian generators which stabilize a target subsystem, subspace, or pure-state. In particular, we provide an algebraic characterization of the manifold of stabilizable pure states in arbitrary finite-dimensional Markovian systems, that leads to a constructive strategy for designing the relevant controllers. Implications for stabilization of entangled pure states are addressed by example.
We propose a general theoretical framework that is suitable to study a wide class of stabilization problems for quantum Markovian dynamical systems. Building on systemtheoretic ideas, we propose definitions of invariant and attractive quantum subsystem, characterize Markovian invariance properties, and provide sufficient conditions for attraction. The general framework and results are illustrated by addressing the potential of output-feedback Markovian control strategies for quantum pure state-stabilization. In particular, constructive results for the synthesis of stabilizing semigroups in arbitrary finite-dimensional Markovian systems are established. I. BACKGROUND AND MOTIVATIONS Stabilization problems are of central relevance for many quantum control applications, ranging from state preparation of quantum-optical and nano-mechanical systems to generation of noise-protected realizations of quantum information in realistic devices [1]. Dynamical systems undergoing Markovian evolution [2], [3] are both widely relevant from a physical standpoint and present distinctive control challenges – preventing, in particular, open-loop quantum-engineering and stabilization methods based on dynamical decoupling to be viable [4], [5]. However, we show here how a wide class of stabilization problems can be effectively treated in a general framework, provided by attractive quantum subsystems. After introducing the main ideas and definitions along with some general results, we shall explore their application to pure-state preparation problems for Markovian output-feedback control. We refer to the forthcoming journal version of the present paper [6] for detailed proofs we shall omit or merely sketch in the following sections. Consider a separable Hilbert space H over the complex field C. Let B(H) represent the set of linear bounded operators on H, H(H) denoting the real subspace of Hermitian operators, with I, O being the identity and the zero operator, respectively. In the standard statistical formulation of quantum mechanics [7], [8], the dimension of the Hilbert space H associated with the quantum system of interest, Q, is determined by the physics of the problem. In what follows, we consider finite-dimensional systems, i.e. dim(H) 0 being described by a TracePreserving, Completely-Positive (TPCP) map Tt(·) [12], [1]. A differential equation for the density operator of I may be derived provided that a forward composition law holds: Definition 1 (QDS): A quantum dynamical semigroup is a one-parameter family of TPCP maps {Tt(·), t ≥ 0} that satisfies: (i) T0 = I, (ii) Tt ◦ Ts = Tt+s, ∀t, s > 0, (iii) trace(Tt(ρ)X) is a continuous function of t, ∀ρ ∈ D(HI), ∀X ∈ B(HI). Due to the trace and positivity preserving assumptions, a QDS is a semigroup of contractions. It has been proved [10], [13] that the Hille-Yoshida generator for a QDS exists and can be cast in the following canonical form:
The speed of convergence of controlled dissipative dynamics to the desired state or set of states is critical in many tasks in experimental physics and quantum information processing. Here we focus on quantum Markovian master equations and, by exploiting the linear character of the dynamical generator, we derive an alternative characterization of the semigroups stabilizing a target subspace, as well as the asymptotic speed of convergence to the desired set. Our analysis highlights the effect of the Hamiltonian parameters (which can be considered as our controls) on the convergence speed, and suggests two basic principles for their design.