IEEE Conference on Decision and Control (CDC)

Notions of controllability for quantum mechanical systems

Francesca Albertini, Deanna M. D’Alessandro

Published Nov 13, 2002• 3 citations

Abstract

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.

DOI: 10.1109/.2001.981126

Related papers

Notions of controllability for quantum mechanical systems

Francesca Albertini, Deanna M. D’Alessandro

IEEE Conference on Decision and Control (CDC)Jul 10, 200335 citations

Notions of controllability for quantum mechanical systems

Francesca Albertini, Domenico D’Alessandro

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.

arXiv:quant-ph/0106128Jun 23, 20010 citations

Notions of controllability for bilinear multilevel quantum systems

Francesca Albertini, Domenico D’Alessandro

In this note, 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.

IEEE Transactions on Automatic ControlAug 1, 2003224 citations

On the controllability of systems on compact Lie groups and quantum mechanical systems

Domenico D’Alessandro

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.

IEEE Conference on Decision and Control (CDC)Nov 11, 20026 citations

Finite Controllability of Infinite-Dimensional Quantum Systems

Anthony M. Bloch, Roger W. Brockett, Chitra Rangan

Quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates, and their linear superpositions. For some quantum systems, such as spin systems, the quantum evolution equation (the Schrödinger equation) is finite-dimensional and old results on controllability of systems defined on on Lie groups and quotient spaces provide most of what is needed insofar as controllability of non-dissipative systems is concerned. However, in an infinite-dimensional setting, controlling the evolution of quantum systems often presents difficulties, both conceptual and technical. In this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues. In particular, we analyze controllability for infinite-dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre-defined subspaces. This result, which we call the Finite Controllability Theorem, provides a set of sufficient conditions for controllability in an infinite-dimensional setting. We consider specific physical systems that are of interest for quantum computing, and provide insights into the types of quantum operations (gates) that may be developed.

arXiv:quant-ph/0608075IEEE Transactions on Automatic ControlMar 2, 201070 citations

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

Domenico D ' Alessandro

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.

Jan 1, 20004 citations