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arXiv:0708.3147

Controllability of Quantum Systems on the Lie Group SU(1,1)

Jianwu Wu, Chunwen Li, Jing Zhang, Tzyh‐Jong Tarn

Published Aug 23, 2007• 0 citations
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Abstract

This paper examines the controllability for quantum control systems with SU(1,1) dynamical symmetry, namely, the ability to use some electromagnetic field to redirect the quantum system toward a desired evolution. The problem is formalized as the control of a right invariant bilinear system evolving on the Lie group SU(1,1) of two dimensional special pseudo-unitary matrices. It is proved that the elliptic condition of the total Hamiltonian is both sufficient and necessary for the controllability. Conditions are also given for small time local controllability and strong controllability. The results obtained are also valid for the control systems on the Lie groups SO(2,1) and SL(2,R).

DOI
10.48550/arxiv.0708.3147

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This paper examines the controllability for quantum control systems with SU(1,1) dynamical symmetry, namely, the ability to use some electromagnetic field to redirect the quantum system toward a desired evolution. The problem is formalized as the control of a right invariant bilinear system evolving on the Lie group SU(1,1) of two dimensional special pseudo-unitary matrices. It is proved that the elliptic condition of the total Hamiltonian is both sufficient and necessary for the controllability. Conditions are also given for small time local controllability and strong controllability. The results obtained are also valid for the control systems on the Lie groups SO(2,1) and SL(2,R).
Bilinear systems, matrix systems and right-invariant systems are distinguished first. Then controllability theorems of such three different systems are summarized, the key points focus on the necessary and sufficient controllability conditions for right-invariant systems, and the relation with bilinear systems. Controllability theorems of these three different systems are especially emphasized by using Lie groups and Lie algebra. Different controllability theorems of quantum systems using similar method are analyzed in detail. Corresponding relations of controllability between quantum systems with bilinear systems are pointed out by means of comparison, and each type of controllability applied to different situations are also given. Relationships among different notions of controllability of quantum systems are established.
We develop some general results on the properties of the reachable sets for right invariant bilinear systems with state varying on compact Lie groups. The main results consist of a characterization of the set of states reachable in arbitrary time from the identity of the group. This, under suitable assumptions, is proved to be a Lie subgroup of the underlying Lie group. We apply these results to the analysis of the controllability of particles with spin. For these systems we also obtain estimates of the time after which every state is reachable from the identity. The results are motivated by the problem of controlling a two-level quantum system in implementations of quantum computers.
In this paper, we develop some general results on the properties of the reachable sets for right invariant bilinear systems with state varying on compact Lie groups. The main results consist of a characterization of the set of states reachable in arbitrary time from the identity of the group. This, under suitable assumptions, is proved to be a Lie subgroup of the underlying Lie group. We apply these results to the analysis of the controllability of particles with spin. For these systems we also obtain estimates of the time after which every state is reachable from the identity. The results are motivated by the problem of controlling a two-level quantum system in implementations of quantum computers.
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.
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.