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IEEE Conference on Decision and Control (CDC)

The Lie algebra structure of spin systems and their controllability properties

Francesca Albertini, Domenico D’Alessandro

Published Nov 13, 2002• 0 citations
Publisher

Abstract

In this paper, we provide a complete analysis of the Lie algebra structure of a system of n interacting spin 1/2 particles with different gyromagnetic ratios in an electro-magnetic field. We relate the structure of this Lie algebra to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. We prove that for these systems all the controllability notions, including the possibility of driving the state or the evolution operator of the system, are equivalent. We also give a necessary and sufficient condition for controllability in terms of the properties of the above described graph. We provide extensions to the case of possibly equal gyromagnetic ratios.

DOI
10.1109/.2001.980658

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In this paper, we provide a complete analysis of the Lie algebra structure of a system of n interacting spin 1/2 particles with different gyromagnetic ratios in an electro-magnetic field. We relate the structure of this Lie algebra to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. We prove that for these systems all the controllability notions, including the possibility of driving the state or the evolution operator of the system, are equivalent. We also give a necessary and sufficient condition for controllability in terms of the properties of the above described graph. We provide extensions to the case of possibly equal gyromagnetic ratios.
In this paper, we provide a complete analysis of the Lie algebra structure of a system of n interacting spin 1/2 particles with different gyromagnetic ratios in an electro-magnetic field. We relate the structure of this Lie algebra to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. We prove that for these systems all the controllability notions, including the possibility of driving the state or the evolution operator of the system, are equivalent. We also give a necessary and sufficient condition for controllability in terms of the properties of the above described graph. We provide extensions to the case of possibly equal gyromagnetic ratios.
In this paper, we study the controllability properties and the Lie algebra structure of networks of particles with spin immersed in an electro-magnetic field. We relate the Lie algebra structure to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. For networks with different gyromagnetic ratios, we provide a necessary and sufficient condition of controllability in terms of the properties of the above mentioned graph and describe the Lie algebra structure in every case. For these systems all the controllability notions, including the possibility of driving the evolution operator and/or the state, are equivalent. For general networks (with possibly equal gyromagnetic ratios), we give a sufficient condition of controllability. A general form of interaction among the particles is assumed which includes both Ising and Heisenberg models as special cases. Assuming Heisenberg interaction we provide an analysis of low dimensional cases (number of particles less then or equal to three) which include necessary and sufficient controllability conditions as well as a study of their Lie algebra structure. This also, provides an example of quantum mechanical systems where controllability of the state is verified while controllability of the evolution operator is not.
Controllability of dynamical systems is a fundamental problem for both classical control as well as quantum control.This paper focuses on investigating the controllability Lie algebra of single-spin and two spin quantum systems.Firstly,based on the equivalent controllability conditions of quantum systems,some general necessary and sufficient conditions related to the basis coefficients are obtained by computing Lie brackets of Hamiltonian operators for single-spin quantum systems.Secondly,the matrix basis of Lie algebra su(4) is generated by Cartan decomposition.Using the matrix basis and employing the controllability theorem of quantum system,a calculation method of multiple Lie brackets for the Hamiltonian operators and a controllability criterion of quantum systems are also put forward in this article.
In this paper, we present a controllability analysis of the quantum Ising periodic chain of n spin 1/2 particles where the interpolating parameter between the two Hamiltonians plays the role of the control. A fundamental result in the control theory of quantum systems states that the set of achievable evolutions is (dense in) the Lie group corresponding to the Lie algebra generated by the Hamiltonians of the system. Such a dynamical Lie algebra therefore characterizes all the state transitions available for a given system. For the Ising spin periodic chain we characterize such a dynamical Lie algebra and therefore the set of all reachable states. In particular, we prove that the dynamical Lie algebra is a (3n-1)-dimensional Lie sub-algebra of su(2^n) which is a direct sum of a two dimensional center and a (3n-3)-dimensional semisimple Lie subalgebra. This in turn is the direct sum of n-1 Lie algebras isomorphic to su(2) parametrized by the eigenvalues of a fixed matrix. We display the basis for each of these Lie subalgebras. Therefore the problem of control for the Ising spin periodic chain is, modulo the two dimensional center, a problem of simultaneous control of n-1 spin 1/2 particles. In the process of proving this result, we develop some tools which are of general interest for the controllability analysis of quantum systems with symmetry.
A quantum mechanical system S is indirectly controlled when the control affects an ancillary system A and the evolution of S is modified through the interaction with A only. A study of indirect controllability gives a description of the set of states that can be obtained for S with this scheme. In this paper, we study the indirect controllability of quantum systems in the finite dimensional case. After discussing the relevant definitions, we give a general necessary condition for controllability in Lie algebraic terms. We present a detailed treatment of the case where both systems, S and A, are two-dimensional (qubits). In particular, we characterize the dynamical Lie algebra associated with S + A, extending previous results, and prove that complete controllability of S + A and an appropriate notion of indirect controllability are equivalent properties for this system. We also prove several further indirect controllability properties for the system of two qubits, and illustrate the role of the Lie algebraic analysis in the study of reachable states.
In this paper, we define three different notions of controllability for quantum mechanical systems involving the possibility of driving the evolution operator as well as the state of the system. By using general results on transitivity of transformation groups on spheres we establish the connections among these different notions of controllability. Motivated by the physical model of multilevel quantum systems, we also study the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove for the system of two interacting spin 1/2 particles the negative result that not every state transfer can be obtained in arbitrary time.