In the framework of bilinear control of the Schrödinger equation with bounded control operators, it has been proved that the reachable set has a dense complemement in ${\cal S}\cap {\cal H}^{2}$ . Hence, in this setting, exact quantum control in infinite dimensions is not possible. On the other hand it is known that there is a simple choice of operators which, when applied to an arbitrary state, generate dense orbits in Hilbert space. Compatibility of these two results is established in this paper and, in particular, it is proved that the closure of the reachable set of bilinear control by bounded operators is dense in ${\cal S}\cap {\cal H}^{2}$ . The requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group.In the framework of bilinear control of the Schrödinger equation with bounded control operators, it has been proved that the reachable set has a dense complemement in ${\cal S}\cap {\cal H}^{2}$. Hence, in this setting, exact quantum control in infinite dimensions is not possible. On the other hand it is known that there is a simple choice of operators which, when applied to an arbitrary state, generate dense orbits in Hilbert space. Compatibility of these two results is established in this paper and, in particular, it is proved that the closure of the reachable set of bilinear control by bounded operators is dense in ${\cal S}\cap {\cal H}^{2}$. The requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group.

arXiv:1004.3447Journal of Physics A Mathematical and TheoreticalFeb 25, 20115 citations

Quantum control in infinite dimensions and Banach-Lie algebras

Michael Keyl

In finite dimensions, controllability of bilinear quantum control systems can be decided quite easily in terms of the "Lie algebra rank condition" (LARC), such that only the systems Lie algebra has to be determined from a set of generators. In this paper we study how this idea can be lifted to infinite dimensions. To this end we look at control systems on an infinite dimensional Hilbert space which are given by an unbounded drift Hamiltonian H0and bounded control Hamiltonians H1,...,HN. The drift H0is assumed to have empty continuous spectrum. We use recurrence methods and the theory of Abelian von Neumann algebras to develop a scheme, which allows us to use an approximate version of LARC, in order to check approximate controllability of the control sysPtem in question.In finite dimensions, controllability of bilinear quantum control systems can be decided quite easily in terms of the "Lie algebra rank condition" (LARC), such that only the systems Lie algebra has to be determined from a set of generators. In this paper we study how this idea can be lifted to infinite dimensions. To this end we look at control systems on an infinite dimensional Hilbert space which are given by an unbounded drift Hamiltonian H0and bounded control Hamiltonians H1,...,HN. The drift H0is assumed to have empty continuous spectrum. We use recurrence methods and the theory of Abelian von Neumann algebras to develop a scheme, which allows us to use an approximate version of LARC, in order to check approximate controllability of the control sysPtem in question.

IEEE Conference on Decision and Control (CDC)Dec 1, 20193 citations

Advances in Quantum Control Theory

Jeffrey L. Krause

APSMar 1, 20010 citations

Directions in the Theory of Quantum Control

Domenico D’Alessandro

Springer eBooksAug 10, 20077 citations

In the framework of bilinear control of the Schrödinger equation with bounded control operators, it has been proved that the reachable set has a dense complemement in

{\cal S}\cap {\cal H}^{2}

. Hence, in this setting, exact quantum control in infinite dimensions is not possible. On the other hand it is known that there is a simple choice of operators which, when applied to an arbitrary state, generate dense orbits in Hilbert space. Compatibility of these two results is established in this paper and, in particular, it is proved that the closure of the reachable set of bilinear control by bounded operators is dense in

{\cal S}\cap {\cal H}^{2}

. The requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group.