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arXiv:quant-ph/0106115

The Lie algebra Structure and Nonlinear Controllability of Spin Systems

Francesca Albertini, Domenico D’Alessandro

Published Jun 20, 2001• 7 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0106115

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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.
We consider a network of n spin 12 systems which are pairwise interacting via Ising interaction and are controlled by the same electro-magnetic control field. Such a system presents symmetries since the Hamiltonian is unchanged if we permute two spins. This prevents full (operator) controllability, in that not every unitary evolution can be obtained. We prove however that controllability is verified if we restrict ourselves to unitary evolutions which preserve the above permutation invariance. For low dimensional cases, n = 2 and n = 3, we provide an analysis of the Lie group of available evolutions and give explicit control laws to transfer between two arbitrary permutation invariant states. This class of states includes highly entangled states such as Greenberger-Horne-Zeilinger (GHZ) states and W states, which are of interest in quantum information.
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.
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.
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.