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ON THE CONTROLLABILITY OF SYSTEMS ON COMPACT LIE GROUPS AND QUANTUM MECHANICAL SYSTEMS

Domenico D ' Alessandro

Published Jan 1, 2000• 4 citations
OpenAlex

Abstract

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.

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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.
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 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.
For a right-invariant system on a compact Lie group G , I present two methods to design a control to drive the state from the identity to any element of the group. The first method, under appropriate assumptions, achieves exact control to the target but requires estimation of the ‘size’ of a neighborhood of the identity in G and solution of a nonlinear algebraic equation. The second method does not involve any mathematical difficulty and obtains control to a desired target with arbitrary accuracy . A third method is then given combining the main ideas of the previous methods. This is also very simple in its formulation and turns out to be generically more efficient as illustrated by one of the examples I consider. The methods described in the paper provide arbitrary constructive control for any right-invariant system on a compact Lie group. In particular, the results can be applied to the coherent control of general multilevel quantum systems to an arbitrary target.
In this paper we consider the (state) reachability and controllability problems of special two-level quantum systems, the so-called quantum bits via externally applied electro-magnetic field. The system is described by special a bilinear right-invariant model whose state varies on the Lie group of 2 x 2 special unitary matrices. We show that if two or more independent controls are used, then every state can be achieved in arbitrary time using finite energy. The mathematical construction is motivated by the demand of manipulating (or logically operating on) the state of quantum bits, and the results provide some insight into the feasibility of realizing given operations in quantum computers. The paper has a special actuality: both computer science and quantum mechanics are closely related to the famous Hungarian mathematician John von Neumann, and this year there is the 50 th Anniversary of his early and sorrowful death.
In this paper, we define four different notions of controllability of physical interest for multilevel quantum mechanical systems. These notions involve the possibility of driving the evolution operator as well as the state of the system. We establish the connections among these different notions as well as methods to verify controllability. The paper also contains results on 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 that, for the system of two interacting spin 1/2 particles, not every state transfer can be obtained in arbitrary small time.