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Lecture Note

Open Quantum Systems

Volker Bach


These notes give an informal introduction to open quantum systems. Starting from Schrödinger’s equation, we are lead to density matrices and then to states which are positive, normalized linear functionals on observable algebras. We then go through the notions of KMS states and C*-, W*-, and quantum dynamical systems and mathematically formulate the property of return to equilibrium, which we then demonstrate to hold true for a spin-boson model. Finally, we give an application to decoherence of qubits of a quantum computer.

These notes are based on articles whose precise reference is appended. (Particularly useful were Pillet’s article:

S. Attal, A. Joye, and C.-A. Pillet, editors. Open Quantum Systems I: The Hamiltonian Approach, volume 1880 of Lecture Notes in Mathematics. Springer-Verlag, 2006.

M. Mück. Thermal Relaxation for Particle Systems in Interaction with several bosonic Heat Reservoirs. PhD thesis, Johannes Gutenberg-Universität Mainz, July 2004. ISBN 3-8334-1866-4.)

It is a pleasure to thank Klaus Sibold and Manfred Salmhofer for organizing a most interesting and pleasant workshop at the Institute for Theoretical Physics of Leipzig University in February 2008 and Kay-Uwe Giering for excellent typesetting.