On the generators of quantum dynamical semigroups

Abstract

The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB(ℋ) is derived. This is a quantum analogue of the Lévy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.

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References

  1. 1.

    George, C., Prigogine, I., Rosenfeld, L.: Kgl. Danske Videnskab. Selskab Mat.-Fys. Medd38, no 12 (1972)

  2. 2.

    Grecos, A. P., Prigogine, I.: Physica59, 77–96 (1972)

    Google Scholar 

  3. 3.

    Williams, D. N.: Commun. math. Phys.21, 314–333 (1971)

    Google Scholar 

  4. 4.

    Davies, E. B.: Commun. math. Phys.33, 171–186 (1973)

    Google Scholar 

  5. 5.

    Davies, E. B.: Commun. math. Phys.39, 81–110 (1974)

    Google Scholar 

  6. 6.

    Pulè, J. V.: Commun. math. Phys.38, 241–256 (1974)

    Google Scholar 

  7. 7.

    Mehra, J., Sudarshan, E. C. G.: Nuovo Cimento11 B, 215–256 (1972)

    Google Scholar 

  8. 8.

    Kossakowski, A.: Rep. Math. Phys.3, 247–274 (1972)

    Google Scholar 

  9. 9.

    Ingarden, R. S., Kossakowski, A.: Ann. Phys.89, 451–485 (1975)

    Google Scholar 

  10. 10.

    Phillips, R. S.: Pacific J. Math.5, 269–283 (1955)

    Google Scholar 

  11. 11.

    Hille, E., Phillips, R. S.: Functional analysis and semigroups. Providence: Amer. Math. Soc. 1957

    Google Scholar 

  12. 12.

    Dixmier, J.: Les algebres d'operateurs dans l'espace Hilbertien. Paris: Gauthier-Villars 1969

    Google Scholar 

  13. 13.

    Stinespring, W. F.: Proc. Amer. Math. Soc.6, 211–216 (1955)

    Google Scholar 

  14. 14.

    Feller, W.: An introduction to probability theory and its applications, vol. 2. New York: Wiley 1971

    Google Scholar 

  15. 15.

    Arveson, W.: Acta. Math.123, 141–224 (1969)

    Google Scholar 

  16. 16.

    Størmer, E.: Springer lecture notes in physics29, 85–106 (1974)

    Google Scholar 

  17. 17.

    Choi, M. D.: Canadian J. Math.24, 520–529 (1972)

    Google Scholar 

  18. 18.

    Kraus, K.: Ann. Phys.64, 311–335 (1970)

    Google Scholar 

  19. 19.

    Sz.-Nagy, B., Foias, C.: Harmonic analysis of operators on Hilbert space. Amsterdam: North-Holland 1970

    Google Scholar 

  20. 20.

    Russo, B., Dye, H. A.: Duke Math. J.33, 413–416 (1966)

    Google Scholar 

  21. 21.

    Kadison, R. V.: Bull. London Math. Soc.7, 41–44 (1975)

    Google Scholar 

  22. 22.

    Lumer, G., Phillips, R. S.: Pacific J. Math.11, 679–698 (1961)

    Google Scholar 

  23. 23.

    Powers, R. T.: Ann. Math.86, 138–171 (1967)

    Google Scholar 

  24. 24.

    Sakai, S.:C*-algebras andW*-algebras. Berlin, Heidelberg; New York: Springer 1971

    Google Scholar 

  25. 25.

    Topping, D. M.: Lectures on von Neumann algebras. London: Van Nostrand 1971

    Google Scholar 

  26. 26.

    Davies, E. B.: Z. Wahrscheinlichkeitstheorie verw. Geb.23, 261–273 (1972)

    Google Scholar 

  27. 27.

    Davies, E. B.: Commun. math. Phys.15, 277–304 (1969);19, 83–105 (1970);22, 51–70 (1971)

    Google Scholar 

  28. 28.

    Kossakowski, A.: Bull. Acad. Polon. Sci. Ser. math. astr. et phys.21, 649–653 (1973)

    Google Scholar 

  29. 29.

    Kossakowski, A.: Bull. Acad. Polon. Sci. Ser. math. astr. et phys.20, 1021–1025 (1972)

    Google Scholar 

  30. 30.

    Haken, H.: Handb. Phys.25, 2c. Berlin, Heidelberg, New York: Springer 1970

    Google Scholar 

  31. 31.

    Zwanzig, R.: Physica30, 1109–1123 (1964)

    Google Scholar 

  32. 32.

    Emch, G. C., Sewell, G. L.: J. Math. Phys.9, 846–958 (1968)

    Google Scholar 

  33. 33.

    Haake, F.: Springer Tracts Mod. Phys.66, 98–168 (1973)

    Google Scholar 

  34. 34.

    Gorini, V., Kossakowski, A., Sudarshan, E. C. G.: Preprint CPT 244, U. of Texas, Austin

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Communicated by H. Araki

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Lindblad, G. On the generators of quantum dynamical semigroups. Commun.Math. Phys. 48, 119–130 (1976). https://doi.org/10.1007/BF01608499

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Keywords

  • Neural Network
  • Statistical Physic
  • Complex System
  • Nonlinear Dynamics
  • Explicit Form