Quantum Control: Approach based on Scattering Theory for Non-commutative Markov Chains and Multivariate Operator Theory

Modern control theory has frequently used concepts and results from abstract mathematics. The aim of this proposal is to explore genuinely non-commutativeversions with a view toward direct applications to the emergent discipline of quantum control. Some elements of open-loop and closed-loop control have already been developed, and recently there has been much interest in fully quantum (or, coherent) control models. Experimental advances mean that physicists have anunprecedented ability to manipulate quantum mechanical systems, and from the technological point of view there is currently much interest in deriving atheory of quantum engineering as the foundation for a much anticipated quantum technological revolution.The proposed research is on the theoretical side and aims to enlarge the toolkit of mathematical methods available to control quantum systems. The classical theory of control has many deep and productive connections with disciplines of mathematical analysis. Complex analysis is indispensable in the use of Laplace transforms and transfer functions for controlled systems. Based on that the theory of state space models and robust control profited from ideas in operator theory. But there are difficulties to adapt these methods to the world of noncommutative mathematics needed for quantum control. There is partial success in specific models but an integrated, first-principles discipline of quantum control is still missing.A basic idea of this proposal is to make use of recent developments in multivariate operator theory. While in classical operator theory a single operator is analysed, in multivariate operator theory the joint action of a family of operators is studied. These operators may not commute with each other. Nevertheless there are analogues to classical results in complex analysis such as the idea of multi-analytic operators. In fact, many of the operator results which are relevant for classical control theory can be extended to this setting. We propose to develop these tools with applications to quantum control. Scattering theory for non-commutative Markov chains is a theory about open quantum systems with many connections to operator theory. Recently the wave operator occurring in this theory has been rewritten as a multi-analytic operator. On the other hand it is possible to interpret this theory as a version of open-loop control, for example it has been successfully applied to the preparation of states in a micromaser interacting with a stream of atoms. Hence it is very natural to start here to develop the methods of multivariate operator theory as applied to the problems in quantum control.Once the bridge between quantum control and multivariate operator theory is understood in the specific directions described above we speculate that a considerable amount of related and deep mathematics becomes available for engineering applications. In the later parts of the project this also includes the more advanced methods of robust control.The investigators Dr Gohm and Professor Gough are the members of the Quantum Control Research Group at Aberystwyth University, set up in 2007. Both have a strong background in mathematical physics and they share the conviction that building bridges between the demands of an applied discipline such as quantum control and ideas in pure mathematics such as multivariate operator theory is as interesting as it is useful.