Quantum Probability Communications

CONTENTS.

CONTENTS OF QPC XI.

PREFACE for QPC Volumes XI & XII.

These notes consist of an introduction to A. Guichardet's functor which induces structure on the finite power set of a set from structure on the set; a re-examination of I.E. Segal's notion of duality endowing Fock space with (quantum) probabilistic interpretation and algebraic structure; and an exposition of integral-sum kernel operators, as mixed multiple quantum Wiener integrals, extending H. Maassen's original idea and P.-A. Meyer's complement. We shall touch on some primitive cohomology, and deliver some sharp estimates. The notes might serve as an introduction to the new approach to quantum stochastic calculus developed by S. Attal and the author, and also to W. von Waldenfels' recent application of integral-sum kernel operators to physical models.

In this introductory course we sketch the framework of quantum probability in order to discuss open quantum systems, in particular the damped harmonic oscillator.

The following sections are included:

• Introduction

• A comparative description of classical and quantum probability

• The role of tensor products of Hilbert spaces

• Some basic operators on Fock spaces

• From urn model to canonical commutation relations

• Stochastic operators on C*-algebras

• Stinespring's theorem

• Extreme points of the convex set of stochastic operators

• Stinespring's theorem in two steps

• Construction of a quantum Markov process

• The central part of minimal dilation

• One parameter semigroups of stochastic maps on a C*-algebra

• Noncommutative stop times

• Markov process at simple stop times

• Minimal Markov flow at simple stop times

• Strong Markov property of the minimal flow for a general stop time

• Strong Markov property under a smoothness condition

• A quantum version of Dynkin's localization formula

• Acknowledgements

• References

There is an extensive literature on Scattering Theory, the case of the so called “hamiltonian dynamics” being a well known subject for anyone interested in Mathematical Physics. Thus, I want to say it loudly, these lectures do not pretend to teach that subject even though it provided an important motivation for the current research. These pages are aimed mostly at providing an insight on some problems of the theory of “quantum open systems” which are inspired by scattering theory.

The text is divided into five sections. The introductory section contains some well-known results from Scattering Theory just quoted here as motivation for the subsequent sections. Section 1 compares the behaviour of two quantum dynamical semigroups at a large time. Most of the results of this section are new and are published here for the first time. The classification of dynamics is the core of Section 3. These results have been obtained and published as a joint research with Luigi Accardi, Claudio Fernández, Humberto Prado and myself. The ergodic properties of quantum Markov flows and semigroups are studied in Section 3, where I recover some old results of Frigerio, Dang Ngoc and others, introducing a notion of quantum recurrence due to myself. The final section is dedicated to explaining the convergence towards the equilibrium as it emerges from a joint work done with Franco Fagnola and published recently in the journal Infinite Dimensional Analysis and Quantum Probability. It is easy to understand that these notes have much of a collective work which I decided to organise as an expository review for the use of young researchers interested in this field. I am greatly indebted to all the coauthors mentioned before as well as to many other colleagues with whom I have had very fruitful discussions. In this respect it is worth mentioning that I learned about the work of Dang Ngoc thanks to Alain Guichardet. On the other hand, Kalyan Sinha became interested in the subject of contiguity (introduced in Section 1), when he visited our group in October 1998, and we started a research for extending the results presented in the current notes. Finally, I want to express my gratitude to Jan van Casteren, for his careful reading of a previous version and for his suggestions to improve the English text.

The following sections are included:

• Complex Banach algebras

• C*-algebras

• von Neumann algebras

• Modular theory

• Conditional expectations

• References

The following sections are included:

• Introduction

• Notations

• Theory of integration with stop times

• Strong Markov Property

• Stopped Processes

• Pre-S and Post-S algebras

• Remarks

• References

The following sections are included:

• Basic definitions and facts

• Combinatorial aspects of freeness: the concept of cumulants

• Free stochastic calculus

• Reference

We present in this paper a theory of quantum stochastic processes based on the notion of kernels, a notion well established in quantum field theory and introduced to quantum probability by H. Maassen. As we restrict ourselves to continuous kernel processes we shall need only the tools of classical calculus and the proofs become rather easy. For the solution of a linear quantum stochatic differential equation an explicit formula is obtained. Under the usual conditions the solution can be extended to a strongly continuous unitary cocycle. It is indicated how the white noise equation can be obtained as a limit of the coloured noise equation.

BIBLIOGRAPHY FOR QPC XI & XII.