Skip to main content
SpringerLink
Log in

Introduction to Classical and Quantum Markov Semigroups

  • Chapter
  • First Online:
Open Quantum Systems

  • 1178 Accesses

Abstract

We provide a self-contained and fast-paced introduction to the theories of operator semigroups, Markov semigroups and quantum dynamical semigroups. The level is appropriate for well-motivated graduate students who have a background in analysis or probability theory, with the focus on the characterisation of infinitesimal generators for various classes of semigroups. The theorems of Hille–Yosida, Hille–Yosida–Ray, Lumer–Phillips and Gorini–Kossakowski–Sudarshan–Lindblad are all proved, with the necessary technical prerequisites explained in full. Exercises are provided throughout.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. L. Accardi, A. Frigerio, J.T. Lewis, Quantum stochastic processes. Publ. Res. Inst. Math. Sci. 18(1), 97–133 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Alicki, K. Lendi, Quantum Dynamical Semigroups and Applications. Lecture Notes in Physics, vol. 717, 2nd edn. (Springer, Berlin, 2007)

    Google Scholar 

  3. D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edn. (Cambridge University Press, Cambridge, 2009)

    Book  MATH  Google Scholar 

  4. W.B. Arveson, Subalgebras of C ∗-algebras. Acta Math. 123, 141–224 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  5. B.V.R. Bhat, K.R. Parthasarathy, Kolmogorov’s existence theorem for Markov processes in C ∗ algebras. Proc. Indian Acad. Sci. Math. Sci. 104(1), 253–262 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. M.-D. Choi, Positive linear maps on C ∗-algebras. Can. J. Math. 24(3), 520–529 (1972)

    Article  MATH  Google Scholar 

  7. E. Christensen, D.E. Evans, Cohomology of operator algebras and quantum dynamical semigroups. J. Lond. Math. Soc. 20(2), 358–368 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Chruściński, S. Pascazio, A brief history of the GKLS equation. Open Syst. Inf. Dyn. 24(3), 1740001, 20pp. (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. E.B. Davies, Linear Operators and Their Spectra (Cambridge University Press, Cambridge, 2007)

    Book  MATH  Google Scholar 

  10. E.G. Effros, Z.-J. Ruan, Operator Spaces. London Mathematical Society Monographs, vol. 23 (Oxford University Press, Oxford, 2000)

    Google Scholar 

  11. D.E. Evans, Conditionally completely positive maps on operator algebras. Q. J. Math. 28(3), 271–283 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. D.E. Evans, H. Hanche-Olsen, The generators of positive semigroups. J. Funct. Anal. 32, 207–212 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. D.E. Evans, J.T. Lewis, Dilations of Irreversible Evolutions in Algebraic Quantum Theory. Communications of the Dublin Institute for Advanced Studies Series A, vol. 24 (Dublin Institute for Advanced Studies, Dublin, 1977), v+104 pp.

    Google Scholar 

  14. F. Fagnola, Quantum Markov semigroups and quantum flows. Proyecciones 18(3), 144 pp. (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Grothendieck, Un résultat sur le dual d’une C ∗-algèbre. J. Math. Pures Appl. (9) 36(2), 97–108 (1957)

    Google Scholar 

  16. E. Hille, R.S. Phillips, Functional Analysis and Semi-Groups. AMS Colloquium Publications, vol. 31, third printing of the revised 1957 edition (American Mathematical Society, Rhode Island, 1974)

    Google Scholar 

  17. R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras II. Advanced Theory. Graduate Studies in Mathematics, vol. 16. (American Mathematical Society, Providence, 1997)

    Google Scholar 

  18. K. Kraus, General state changes in quantum theory. Ann. Phys. 64(2), 311–335 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Kümmerer, Markov dilations on W ∗-algebras. J. Funct. Anal. 63(2), 139–177 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  20. T.M. Liggett, Continuous Time Markov Processes (American Mathematical Society, Providence, 2010)

    Book  MATH  Google Scholar 

  21. G. Lindblad, On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48(2), 119–130 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  22. G.J. Murphy, C ∗ -Algebras and Operator Theory (Academic, New York, 1990)

    MATH  Google Scholar 

  23. W.L. Paschke, Inner product modules over B ∗-algebras. Trans. Am. Math. Soc. 182, 443–468 (1973)

    MathSciNet  MATH  Google Scholar 

  24. V. Paulsen, Completely Bounded Maps and Operator Algebras. Cambridge Studies in Advanced Mathematics, vol. 78 (Cambridge University Press, Cambridge, 2002)

    Google Scholar 

  25. M. Reed, B. Simon, Methods of Modern Mathematical Physics I. Functional Analysis, revised and enlarged edition (Academic, New York, 1980)

    Google Scholar 

  26. L.C.G. Rogers, D. Williams, Diffusions, Markov Processes and Martingales I. Foundations, 2nd edn. (Cambridge University Press, Cambridge, 2000)

    Google Scholar 

  27. J.-L. Sauvageot, Markov quantum semigroups admit covariant Markov C ∗-dilations. Commun. Math. Phys. 106(1), 91–103 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  28. I. Siemon, A.S. Holevo, R.F. Werner, Unbounded generators of dynamical semigroups. Open Syst. Inf. Dyn. 24(4), 1740015, 24pp. (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. W.F. Stinespring, Positive functions on C ∗-algebras. Proc. Am. Math. Soc. 6(2), 211–216 (1955)

    MathSciNet  MATH  Google Scholar 

  30. M. Takesaki, Theory of Operator Algebras I. Encyclopaedia of Mathematical Sciences, vol. 124 (Springer, Berlin, 2002)

    Google Scholar 

  31. G.F. Vincent-Smith, Dilation of a dissipative quantum dynamical system to a quantum Markov process. Proc. Lond. Math. Soc. 49(1), 58–72 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Lancaster University, Lancaster, UK

    Alexander C. R. Belton

Authors
  1. Alexander C. R. Belton
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander C. R. Belton .

Editor information

Editors and Affiliations

  1. Mathematisches Institut, Universität Göttingen, Göttingen, Germany

    Dorothea Bahns

  2. Department 3 – Mathematik, Universität Bremen, Bremen, Germany

    Anke Pohl

  3. Mathematisches Institut, Universität Göttingen, Göttingen, Germany

    Ingo Witt

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Belton, A.C.R. (2019). Introduction to Classical and Quantum Markov Semigroups. In: Bahns, D., Pohl, A., Witt, I. (eds) Open Quantum Systems . Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-13046-6_1

Download citation

Publish with us

Policies and ethics

Search

Navigation