Local State Transition of Feedback Controlled Quantum Systems with Imperfect Detector Efficiency: Part I: Differential Geometric Analysis for Dynamical Systems with Matrix-Valued States

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.

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

“…Now we can formulate the problem in this paper. We investigate local state transition of the controlled quantum systems under continuous quantum measurement through analyzing equation (6) using the method developed in the first part of this series [1]. This approach is justified because any statement true for equation (6) is also true even if we restrict the variable X to S n .…”

“…To avoid the obstacle and make further analysis possible, the authors developed a theory and techniques of differential geometric analysis for dynamical systems with matrix-valued states in [1]. In this paper we apply the result to the analysis of local state transition of the controlled quantum systems with imperfect detector efficiency.…”

“…A quantum dot system measured by quantum point contact is also described by the SME in [20]. Section 3 of [1] showed that investigation for controlled quantum system under continuous measurement is boiled down to the analysis for the following H n -valued ordinary differential equation:…”

“…We analyze local state transition of controlled quantum systems under continuous quantum measurement using the method developed in [1]. We start this paper with a review of quantum control.…”

“…This paper is a continuation of [1]. We analyze local state transition of controlled quantum systems under continuous quantum measurement using the method developed in [1].…”