**Unitary Control of Quantum Systems in Finite and Infinite Dimensions**

Referent: Prof. Dr. M. Keyl, Freie Universität Berlin

Veranstalter: Karl-Hermann Neeb

Abstract:

Quantum control theory studies the dynamics of quantum systems which can be manipulated, e.g., by external controls like electro-magnetic fields. Mathematically, one of the central topics in this framework is the decision problem of controllability: Can we express an arbitrary unitary on the system’s Hilbert space as a time-evolution operator of the given dynamics with appropriate controls?

In finite dimensions this question can be completely answered in a Lie-algebraic framework. Infinite dimensions, on the other hand, trigger more challenging mathematics and require methods from operator analysis and (extensions of) infinite dimensional Lie theory.

This talk will provide an introduction into this topic, and an overview on some of its central questions and results. In finite dimensions we show in particular how a system Lie algebra can be associated to a quantum control system, which leads to an easy condition for deciding controllabiliy: the celebrated Lie algebra rank conditon.

The second part of the talk shows under which assumptions the finite dimensional results can be translated more or less directly to infinite dimensions — now using infinite dimensional algebras and groups, and the strong instead of the norm topology. Beyond these assumptions a number of interesting, new open questions arise, which we will briefly sketch in an outlook.