Journal Menu

- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Advances in Mathematical Physics

Volume 2011 (2011), Article ID 625978, 16 pages

http://dx.doi.org/10.1155/2011/625978

Research Article

## Quantum Dynamical Semigroups and Decoherence

^{1}Faculty of Physics, University of Bielefeld, Universitätsstraβe 25, 33615 Bielefeld, Germany^{2}Bundesamt für Strahlenschutz (Federal Office for Radiation Protection), Willy-Brandt-Straße 5, 38226 Salzgitter, Germany

Received 22 June 2011; Accepted 31 August 2011

Academic Editor: Christian Maes

Copyright © 2011 Mario Hellmich. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- R. Haag,
*Local Quantum Physics*, Springer, Berlin, Germany, 2nd edition, 1996. - F. Strocchi,
*An Introduction to the Mathematical Structure of Quantum Mechanics*, World Scientific, Hackensack, NJ, USA, 2005. - P. Blanchard and R. Olkiewicz, “Decoherence induced transition from quantum to classical dynamics,”
*Reviews in Mathematical Physics*, vol. 15, no. 3, pp. 217–243, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Olkiewicz, “Environment-induced superselection rules in Markovian regime,”
*Communications in Mathematical Physics*, vol. 208, no. 1, pp. 245–265, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. P. Landsman, “When champions meet: rethinking the Bohr-Einstein debate,”
*Studies in History and Philosophy of Science B*, vol. 37, no. 1, pp. 212–242, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - N. P. Landsman, “Between classical and quantum,” in
*Handbook of the Philosophy of Science: Philosophy of Physics*, J. Butterfield and J. Earman, Eds., North Holland, Amsterdam, The Netherlands, 2007. - E. B. Davies,
*Quantum Theory of Open Systems*, Academic Press, London, UK, 1976. - H.-P. Breuer and F. Petruccione,
*The Theory of Open Quantum Systems*, Oxford University Press, New York, NY, USA, 2002. - K. J. Engel and R. Nagel,
*One-Parameter Semigroups for Linear Evolution Equations*, Springer, New York, NY, USA, 2000. - O. Bratteli and D. W. Robinson,
*Operator Algebras and Quantum-Statistical Mechanics I*, Springer, New York, NY, USA, 2nd edition, 1987. - K. Jacobs, “Fastperiodizitätseigenschaften allgemeiner Halbgruppen in Banach-Räumen,”
*Mathematische Zeitschrift*, vol. 67, pp. 83–92, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. De Leeuw and I. Glicksberg, “Almost periodic compactifications,”
*Bulletin of the American Mathematical Society*, vol. 65, pp. 134–139, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. De Leeuw and I. Glicksberg, “Applications of almost periodic compactifications,”
*Acta Mathematica*, vol. 105, pp. 63–97, 1961. View at Publisher · View at Google Scholar - J. van Neerven,
*The Asymptotic Behaviour of Semigroups of Linear Operators*, Birkhäuser, Basel, Switzerland, 1996. - W. Arendt and C. J. K. Batty, “Tauberian theorems and stability of one-parameter semigroups,”
*Transactions of the American Mathematical Society*, vol. 306, no. 2, pp. 837–852, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. I. Lyubich and Q. P. Vû, “Asymptotic stability of linear differential equations in Banach spaces,”
*Studia Mathematica*, vol. 88, no. 1, pp. 37–42, 1988. View at Zentralblatt MATH - B. Blackadar,
*Operator Algebras*, Springer, Berlin, Germany, 2006. - B. Kümmerer and R. Nagel, “Mean ergodic semigroups on ${W}^{\ast}$-algebras,”
*Acta Scientiarum Mathematicarum*, vol. 41, no. 1-2, pp. 151–159, 1979. - A. Bátkai, U. Groh, C. Kunszenti-Kovács, and M. Schreiber, “Decomposition of operator semigroups on ${W}^{\ast}$-algebras,” http://arxiv.org/abs/1106.0287v1. View at Publisher · View at Google Scholar
- D. E. Evans, “Irreducible quantum dynamical semigroups,”
*Communications in Mathematical Physics*, vol. 54, no. 3, pp. 293–297, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Tomiyama, “On the projection of norm one in ${W}^{\ast}$-algebras,”
*Proceedings of the Japan Academy*, vol. 33, pp. 125–129, 1957. - D. W. Robinson, “Strongly positive semigroups and faithful invariant states,”
*Communications in Mathematical Physics*, vol. 85, no. 1, pp. 129–142, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Frigerio, “Stationary states of quantum dynamical semigroups,”
*Communications in Mathematical Physics*, vol. 63, no. 3, pp. 269–276, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Hellmich,
*Decoherence in infinite quantum systems*, Ph.D. thesis, University of Bielefeld, Bielefeld, Germany, 2009. - R. Carbone, E. Sasso, and V. Umanità, “Decoherence for positive semigroups on ${M}_{2}(\u2102)$,”
*Journal of Mathematical Physics*, vol. 52, no. 3, Article ID 032202, 2011. - A. Dhahri, F. Fagnola, and R. Rebolledo, “The decoherence-free subalgebra of a quantum Markov semigroup with unbounded generator,”
*Infinite Dimensional Analysis, Quantum Probability and Related Topics*, vol. 13, no. 3, pp. 413–433, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Rebolledo, “Decoherence of quantum Markov semigroups,”
*Annales de l'Institut Henri Poincar é*, vol. 41, no. 3, pp. 349–373, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Rebolledo, “A view on decoherence via master equations,”
*Open Systems and Information Dynamics*, vol. 12, no. 1, pp. 37–54, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Alicki, “Controlled quantum open systems,”
*Lecture Notes in Physics*, vol. 622, pp. 121–139, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Tomiyama, “On the projection of norm one in ${W}^{\ast}$-algebras II,”
*The Tôhoku Mathematical Journal*, vol. 10, pp. 125–129, 1958. - P. Ługiewicz and R. Olkiewicz, “Classical properties of infinite quantum open systems,”
*Communications in Mathematical Physics*, vol. 239, no. 1-2, pp. 241–259, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Christensen, “Generators of semigroups of completely positive maps,”
*Communications in Mathematical Physics*, vol. 62, no. 2, pp. 167–171, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Connes, “Classification of injective factors,”
*Annals of Mathematics*, vol. 104, no. 1, pp. 73–115, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Lindblad, “On the generators of quantum dynamical semigroups,”
*Communications in Mathematical Physics*, vol. 48, no. 2, pp. 119–130, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Spohn, “An algebraic condition for the approach to equilibrium of an open
*N*-level system,”*Letters in Mathematical Physics*, vol. 2, no. 1, pp. 33–38, 1977. - A. Frigerio and M. Verri, “Long-time asymptotic properties of dynamical semigroups on ${W}^{\ast}$-algebras,”
*Mathematische Zeitschrift*, vol. 180, no. 2, pp. 275–286, 1982. View at Publisher · View at Google Scholar