Anyway, to add to the confusion: I just got a nice email from Giampiero Campa, who in week294 had pointed me to the fascinating papers on control theory by Jan Willems. Control theory is the art of getting open systems — systems that interact with their environment — to behave in ways you want.

Since complex systems like ecosystems or the entire Earth are best understood as made of many interacting open systems, and/or being open systems themselves, I think ideas from control theory could become very important in understanding the Earth and how our actions affect it. But I’m also fascinated by control theory because of how it combines standard ideas in physics with new ideas that are best expressed using category theory — a branch of math I happen to know and like. (See week296 and subsequent issues for more on this.) And quantum control theory — the art of getting quantum systems to do what you want — is the sort of thing people here at the CQT may find interesting.

In short, control theory seems like a promising meeting-place for some of my disparate interests. Not necessarily the most important thing for ‘saving the planet’, by any means! But the kind of thing I can’t resist thinking about.

In his email, Campa pointed me to two new papers on this subject:

• Anthony M. Bloch, Roger W. Brockett, and Chitra Rangan, Finite controllability of infinite-dimensional quantum systems, IEEE Transactions on Automatic Control 55 (August 2010), 1797-1805.

• Matthew James and John E. Gough, Quantum dissipative systems and feedback control design by interconnection, IEEE Transactions on Automatic Control 55 (August 2010), 1806-1821.

The second one is related to the ideas of Jan Willems:

Abstract: The purpose of this paper is to extend J.C. Willems’ theory of dissipative systems to open quantum systems described by quantum noise models. This theory, which combines ideas from quantum physics and control theory, provides useful methods for analysis and design of dissipative quantum systems. We describe the interaction of the plant and a class of external systems, called exosystems, in terms of feedback networks of interconnected open quantum systems. Our results include an infinitesimal characterization of the dissipation property, which generalizes the well-known Positive Real and Bounded Real Lemmas, and is used to study some properties of quantum dissipative systems. We also show how to formulate control design problems using network models for open quantum systems, which implements Willems’ “control by interconnection” for open quantum systems. This control design formulation includes, for example, standard problems of stabilization, regulation, and robust control.

I don’t have anything intelligent to say about these papers yet. Does anyone out know if ideas from quantum control theory have been used to tackle the problems that decoherence causes in quantum computation? The second article makes me wonder about this:

In the physics literature, methods have been developed to model energy loss and decoherence (loss of quantum coherence) arising from the interaction of a system with an environment. These models may be expressed using tools which include completely positive maps, Lindblad generators, and master equations. In the 1980s it became apparent that a wide range of open quantum systems, such as those found in quantum optics, could be described within a new unitary framework of quantum stochastic differential equations, where quantum noise is used to represent the influence of large heat baths and boson fields (which includes optical and phonon fields). Completely positive maps, Lindblad generators, and master equations are obtained by taking expectations.

Quantum noise models cover a wide range of situations involving light and matter. In this paper, we use quantum noise models for boson fields, as occur in quantum optics, mesoscopic superconducting circuits, and nanomechanical systems, although many of the ideas could be extended to other contexts. Quantum noise models can be used to describe an optical cavity, which consists of a pair of mirrors (one of which is partially transmitting) supporting a trapped mode of light. This cavity mode may interact with a free external optical field through the partially transmitting mirror. The external field consists of two components: the input field, which is the field before it has interacted with the cavity mode, and the output field, being the field after interaction. The output field may carry away energy, and in this way the cavity system dissipates energy. This quantum system is in some ways analogous to the RLC circuit discussed above, which stores electromagnetic energy in the inductor and capacitor, but loses energy as heat through the resistor. The cavity also stores electromagnetic energy, quantized as photons, and these may be lost to the external field…