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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 I: Differential Geometric Analysis for Dynamical Systems with Matrix-Valued States

Tomotake Sasaki, Shinji Hara, Koji Tsumura

Published Nov 1, 2010• 1 citation
Publisher

Abstract

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.

DOI
10.9746/jcmsi.3.409

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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.
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 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.
Abstract. No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.
No quantum measurement can give full information on the state of a quantum system; hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory. Here we study the properties of controlled quantum filtering equations as classical stochastic differential equations. We then develop methods, using a combination of geometric control and classical probabilistic techniques, for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator.
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.