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On a canonical QR decomposition and feedback control of discrete-time quantum dynamics

Francesco Ticozzi, Saverio Bolognani

Published Jan 1, 2010• 2 citations
OpenAlex

Abstract

We study feedback-controlled, discrete-time quan- tum Markovian dynamics focusing on pure-state stabilization problem. Assuming that the system is unitarily controllable, and accessible via a given quantum measurement, we explic- itly 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. I. INTRODUCTION

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We study feedback-controlled, discrete-time quan- tum Markovian dynamics focusing on pure-state stabilization problem. Assuming that the system is unitarily controllable, and accessible via a given quantum measurement, we explic- itly 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. I. INTRODUCTION
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.
We analyze the asymptotic behavior of discretetime, 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 linearalgebraic 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 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.
In this paper, we present several refinements of the results in [1] concerning feedback stabilization of invariant subspaces of quantum trajectories. First, we consider a more realistic dynamics that accounts for potential measurement imperfections. Second, the strong controllability assumption can be replaced by a simpler and more natural condition whenever the channel is primitive on its minimal invariant subspaces. Finally, we show that the control strategy can be based on the estimated trajectory (starting from an estimated initial state) while still ensuring exponential stabilization towards the target subspace at the same rate, thereby demonstrating the robustness of the approach.
Abstract. The fundamental mathematical definitions of the controlled feedback Markov dynamics of quantum-mechanical systems are introduced with regard to the dynamical reduction and filtering of quantum states in the course of quantum measurement in either discrete or continuous real time. The concept of sufficient coordinates for the description of a posteriori quantum states in a given class is introduced and it is proved that they form a classical Markov process with values in either state operators or state vector space. The general problem of optimal control of a quantum-mechanical system is discussed and the corresponding Bellman equation in the space of sufficient coordinates is derived. The results are illustrated in the example of control of the semigroup dynamics of a quantum system that is instantaneously observed at discrete times and evolves between measurement times according to the Schrődinger equation. 1.
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,