As an introduction to quantum probability this school pursues similar goals as the school on Quantum Independent Increment Processes that took place in March 2003. But the content of the lectures will be largely disjoint from those of the previous school.
The lectures are intended to be accessible to graduate students in mathematics and theoretical physics having no previous experience in this field, the necessary prerequisites from classical probability, functional analysis and quantum physics will be part of the first week. The second week will present recent progress in quantum potential theory and its applications.
School QPT2007
c/o Prof. Michael Schürmann
Institut für Mathematik und Informatik
Ernst-Moritz-Arndt Universität
Jahnstr. 15a
D-17487 Greifswald, Germany
Phone: ++49 (0) 38 34 - 86 46 33
Fax: ++49 (0) 38 34 - 86 46 39
Potential Theory in Classical Probability
Lecturer: Nicolas Privault
Lecture Notes (PDF)
Syllabus:
- Classical potential theory
- Markovian setting
- Markov processes
- Dirichlet forms
- Probabilistic interpretations
- Martin boundaries
Quantum Probabilistic Methods in Operator Space Theory
Lecturer: Quanhua Xu
Abstract (PDF)
Lecture Notes (PDF, version from August 28, 2007)
Syllabus:
- First part: An introduction to operator spaces, including completely bounded maps, column and row Hilbertian spaces and Pisier's operator Hilbertian spaces
- Second part: Noncommutative Khintchine inequalities and martingale inequalities. Their application to operator spaces.
Applications of Controlled Quantum Processes in Quantum Optics
Lecturer: Luc Bouten
Abstract (PDF)
Lecture Notes (PDF, version from October 31, 2007
Syllabus:
- The spectral theorem, constructing classical Brownian motion on the Fock space
- Conditional expectations, quantum Kallianpur-Striebel formula
- QSDE's as encountered in quantum optics, formulation of the filtering problem
- Derivation of the (controlled) quantum filtering equation
- Dynamical programming, separation and optimal control of a two-level atom (qubit) with a laser
Noncommutative Dirichlet Forms
Lecturers: Fabio Cipriani
Abstract (PDF)
Lecture Notes (PDF, version from 10 January 2008)
Syllabus:
- Lecture 1: General theory
- Lecture 2: Dirichlet forms and differential calculus on C*-algebras
- Lecture 3: Potential theoretic and dynamical characterizations of Riemannian manifolds with positive curvature
- Lecture 4: Construction of dissipative dynamics for quantum statistical systems
- Lecture 5: Dirichlet forms and noncommutative geometry
Random Walks on Noncommutative Spaces
Lecturer: Philippe Biane
Abstract (PDF)
Lecture Notes (PDF, version from May 23, 2007)
Syllabus:
- random walks on duals of groups and on quantum groups
- boundary theory
- relation to classical processes
- processes with free increments
Quantum Walks
Lecturer: Norio Konno
Keywords: limit theorem, absorption problem, localization, uniform mixing property, quantum probabilistic approach.
Abstract (PDF)
Lecture Notes (PDF, version from November 26, 2007)
