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The proceedings of the IEEE Conference on Control Applications/The proceedings of the ... IEEE Conference on Control Applications

Accessibility analysis for controlled quantum systems under continuous quantum measurement

Tomotake Sasaki, Shinji Hara, Koji Tsumura

Published Sep 1, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.1109/cca.2010.5611164

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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.
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.
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.
We study quantum system dynamics driven by continuous laser fields under the measurement process. In order to take into account the system transition due to the measurement, we define the superoperator which eliminates the coherence relevant to the measured quantum states. We clarify that the dynamics of the measured states is frozen in the frequent measurement limit, while the space spanned by unmeasured states is isolated from the original system. We also derive the effective Liouvillian which governs incoherent population dynamics under the condition, in which measurements are frequently applied. We apply the formulation to two-level and Lambda-type three-level systems and clarify how the quantum measurements hinder the coherent population dynamics driven by the continuous laser fields in practical examples. Analysis on the laser field amplitude dependency of the final distribution in the t-->infinity limit suggests the possibility of the measurement-assisted quantum control.
The ability to navigate and control the state space of a quantum system is fundamental to quantum computing, quantum information processing, and advanced spectroscopy. This paper explores a rigorous mathematical framework for quantifying state space accessibility by leveraging the theory of operator commutation relations, particularly within the context of Lie algebras. We demonstrate that the span of iterated commutators of system Hamiltonians and control operators forms a dynamical Lie algebra, whose dimension and structure directly dictate the reachable states and unitary transformations. For closed quantum systems, the Lie algebra rank condition provides a powerful criterion for complete controllability, asserting that if the dynamical Lie algebra spans the entire space of anti-Hermitian operators, then any state can be reached from any other, and any unitary operation can be implemented. For open quantum systems, while environmental interactions introduce decoherence and dissipation, the Lie algebraic approach can still offer insights into the attainable subspaces and the limits of control, often involving Lie semigroups and Lie wedges. We present methods to analyze the accessibility of both pure and mixed states, illustrating how the algebraic properties encoded in commutation relations provide a precise measure of the manipulability of quantum systems. The implications extend to the design of robust quantum control protocols, optimized state preparation, and the development of error-resilient quantum technologies.