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SICE Journal of Control Measurement and System Integration

Local State Transition of Feedback Controlled Quantum Systems with Imperfect Detector Efficiency: Part II: Accessibility Analysis for Quantum Systems

Tomotake Sasaki, Shinji Hara, Koji Tsumura

Published Nov 1, 2010• 1 citation
Publisher

Abstract

In this paper, we analyze local state transition of controlled quantum systems under continuous quantum measurement, which is described by a matrix-valued nonlinear stochastic differential equation. To this end, we utilize the method of differential geometric analysis for systems with matrix-valued states developed in the first part of this series. The method provides us a direct and efficient way of analysis with a clear perspective. We study local state transition of the controlled quantum systems with imperfect detector efficiency which has not been studied enough in previous works. The controlled quantum system with perfect detector efficiency is also investigated as a special case. Sufficient conditions for the measurement operator and the control Hamiltonian are derived, under which the local state transition is quite limited. We also show that the conditions are satisfied in many typical situations. The results reveal fundamental nature of the controlled quantum systems under continuous quantum measurement.

DOI
10.9746/jcmsi.3.417

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In this paper, we analyze local state transition of controlled quantum systems under continuous quantum measurement, which is described by a matrix-valued nonlinear stochastic differential equation. To this end, we utilize the method of differential geometric analysis for systems with matrix-valued states developed in the first part of this series. The method provides us a direct and efficient way of analysis with a clear perspective. We study local state transition of the controlled quantum systems with imperfect detector efficiency which has not been studied enough in previous works. The controlled quantum system with perfect detector efficiency is also investigated as a special case. Sufficient conditions for the measurement operator and the control Hamiltonian are derived, under which the local state transition is quite limited. We also show that the conditions are satisfied in many typical situations. The results reveal fundamental nature of the controlled quantum systems under continuous quantum measurement.
We consider a new class of dynamical systems whose state is represented by a Hermitian matrix motivated by treating quantum control problems. We develop theory and techniques of differential geometric analysis for dynamical systems in that class, where a Lie product of matrix functions is introduced and plays an important role. We provide a simple and coordinate-free calculation method for the Lie product of matrix functions which enables efficient differential geometric analysis. The result of this paper will be used in a follow-up paper on analysis of local state transition of controlled quantum systems under continuous quantum measurement with imperfect detector efficiency.
We investigate controlled quantum systems under continuous quantum measurement utilizing the theory and techniques of differential geometric analysis for dynamical systems with affine state space. At first, by employing the notion of the strong accessibility distribution, we analyze the local state transition in detail and show that it is quite limited under conditions on the setting of measurement and actuator. Furthermore, we show an explicit form of the accessible set under control with certain conditions.
Quantum mechanical systems exhibit an inherently probabilistic nature upon measurement which excludes in principle the singular direct observability continual case. Quantum theory of time continuous measurements and quantum prediction theory, developed by the author on the basis of an independent-increment model for quantum noise and nondemolition causality principle in the 80's, solves this problem allowing continual quantum predictions and reducing many quantum information problems like problems of quantum feedback control to the classical stochastic ones. Using explicit indirect observation models for diffusive and counting measurements we derive quantum filtering (prediction) equations to describe the stochastic evolution of the open quantum system under the continuous partial observation. Working in parallel with classical indeterministic control theory, we show the Markov Bellman equations for optimal feedback control of the a posteriori stochastic quantum states conditioned upon these two kinds of measurements. The resulting filtering and Bellman equation for the diffusive observation is then applied to the explicitly solvable quantum linear-quadratic-Gaussian (LQG) problem which emphasizes many similarities and differences with the corresponding classical nonlinear filtering and control problems and demonstrates microduality between quantum filtering and classical control.
Knowledge about to what extent quantum dynamical systems can be steered by coherent controls is indispensable for future developments in quantum technology. The purpose of this paper is to analyze such controllability aspects for finite dimensional bilinear quantum control systems. We use a unified approach based on Lie-algebraic methods from nonlinear control theory to revisit known and to establish new results for closed and open quantum systems. In particular, we provide a simplified characterization of different notions of controllability for closed quantum systems described by the Liouville-von Neumann equation. We derive new necessary and sufficient conditions for accessibility of open quantum systems modelled by the Lindblad-Kossakowski master equation. To this end, we exploit a well-studied topic of differential geometry, namely the classification of all matrix Lie-groups which act transitively on the Grassmann manifold or the punctured Euclidean space. For the special case of coupled spin-1/2 systems, we obtain a remarkably simple characterization of accessibility. These accessibility results correct and refine previous statements in the quantum control literature.
This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence--contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement--estimation--feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.