Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On dilating quantum dynamical semigroups with classical brownian motion

  • 58 Accesses

  • 5 Citations

Abstract

We show that any quantum dynamical semigroup can be written with the help of the solution of an operator-valued classical stochastic differential equation. Moreover, this equation leads to a canonical construction of a unitary dilation in terms of Wiener spaces. We also briefly discuss an equivalent description on a Fock-like space.

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

References

  1. 1.

    Accardi L., Frigerio A., and Lewis J. T., Publ. RIMS, Kyoto University 18, 97 (1982).

  2. 2.

    Lewis J. T., Phys. Rep. 77, 339 (1981).

  3. 3.

    Quantum Probability and Applications I & II, Lecture Notes in Math., Nos. 1055 & 1136, Springer-Verlag, New York.

  4. 4.

    Barnett C., Streater R. F., and Wilde I., J. Funct. Anal. 48, 172 (1982).

  5. 5.

    Hudson R. L. and Parthasarathy K. R., Commun. Math. Phys. 93, 301 (1984).

  6. 6.

    Applebaum D. B. and Hudson R. L., Commun. Math. Phys. 96, 473 (1984).

  7. 7.

    Hudson, R. L. and Parthasarathy, K. R., ‘Unification of Fermion and Boson Stochastic Calculus’, preprint.

  8. 8.

    Evans, D. E. and Lewis, J. T., ‘Dilations of Irreversible Evolutions in Algebraic Quantum Theory’, Commun. DIAS Ser. A 24 (1977).

  9. 9.

    Kümmerer, B., ‘A Dilation Theory for Completely Positive Operators on W *-Algebras’, Thesis, Tübingen, 1982.

  10. 10.

    Hudson R. L. and Parthasarathy K. R., Acta Appl. Math. 2 353 (1984).

  11. 11.

    Frigerio, A., ‘Covariant Markov Dilations of Quantum Dynamical Semigroups’, preprint.

  12. 12.

    Gorini V., Kossakowski A., and Sudarshan E. C. G., J. Math. Phys. 17, 821 (1976).

  13. 13.

    Lindblad G., Commun. Math. Phys. 48, 119 (1976).

  14. 14.

    Doob J. L., Stochastic Processes, Wiley, New York, 1953.

Download references

Author information

Additional information

Institute of Theoretical Physics and Astrophysics, Gdansk, Poland

Bevoegdverklaard navorser NFWO, Belgium

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alicki, R., Fannes, M. On dilating quantum dynamical semigroups with classical brownian motion. Lett Math Phys 11, 259–262 (1986). https://doi.org/10.1007/BF00400224

Download citation

Keywords

  • Differential Equation
  • Statistical Physic
  • Brownian Motion
  • Group Theory
  • Stochastic Differential Equation