Probability Theory and Related Fields

, Volume 89, Issue 1, pp 117–129

Quantum random walk on the dual of SU (n)

  • Philippe Biane
Article

Summary

We study a quantum random walk onA(SU(n)), the von Neumann algebra of SU(n), obtained by tensoring the basic representation of SU(n). Two classical Markov chains are derived from this quantum random walk, by restriction to commutative subalgebras ofA(SU(n)), and the main result of the paper states that these two Markov chains are related by means of Doob'sh-processes.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Accardi, L., Frigerio, A., Lewis, J.T.: Quantum stochastic process. RIMS Kokyuroku18, 97–133 (1982)Google Scholar
  2. 2.
    Babillot, M.: Théorie du renouvellement pour des chaines semi-markoviennes transientes. Ann. Inst. Henri Poincare24, 507–569 (1988)Google Scholar
  3. 3.
    Biane, Ph.: Marches de Bernoulli quantiques. Séminaire de Probabilités XXIV. (Lect. Notes Math., vol. 1426, pp. 329–344). Berlin Heidelberg New York: Springer 1990Google Scholar
  4. 4.
    Biane, Ph.: Some properties of quantum Bernoulli random walks. Quantum Probabilities Proceedings, Trento 1989 (to appear)Google Scholar
  5. 5.
    Brocker, T., tom Dieck, T.: Representations of compact Lie groups. (Graduate texts in Mathematics vol. 98). Berlin Heidelberg New York: Springer 1985Google Scholar
  6. 6.
    Dixmier, J.: LesC *-algèbres et leurs représentations. Paris: Gauthier-Villars 1964Google Scholar
  7. 7.
    Feller, W.: An introduction to probability theory and its applications, vol. 1 and 2, 2nd edn. New York: Wiley 1970Google Scholar
  8. 8.
    Parthasarathy, K.R.: A generalized Biane's process. Séminaire de probabilités XXIV. (Lect. Notes Math. vol. 1426, pp. 345–348). Berlin Heidelberg New York: Springer 1990Google Scholar
  9. 9.
    Revuz, D.: Markov chains, 2nd edn. Amsterdam: North-Holland 1982Google Scholar
  10. 10.
    Waldenfels, W. von: The Markov process of total spin. Séminaire de Probabilités XXIV. (Lect. Notes Math., vol. 1426, pp. 357–361). Berlin Heidelberg New York: Springer 1990Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Philippe Biane
    • 1
  1. 1.Laborative de probabilités, Tour 56-66, 3e étageUniversité Paris 6Paris Cedex 05France

Personalised recommendations