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Construction of stationary quantum markov processes through quantum stochastic calculus

Part of the Lecture Notes in Mathematics book series (LNM,volume 1136)

Keywords

  • Cocycle Condition
  • Stationary Markov Process
  • Quantum Dynamical Semigroup
  • Unitary Dilation
  • Faithful Normal State

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References

  1. Evans, D.E., and Lewis, J.T.: Dilations of irreversible evolutions in algebraic quantum theory. Commun. Dublin Institute for Advanced Studies, Ser. A, No. 24, 1977.

    Google Scholar 

  2. Kümmerer, B.: A dilation theory for completely positive operators on W*-algebras. Thesis, Tübingen, 1982; __: Markov dilations on W*-algebras. J. Funct. Anal. (to appear).

    Google Scholar 

  3. Kümmerer, B., and Schröder, W.: A Markov dilation of a non-quasifree Bloch evolution. Commun. Math. Phys. 90, 251–262 (1983).

    CrossRef  MATH  Google Scholar 

  4. Kümmerer, B.: Examples of Markov dilations over the 2 × 2 matrices. In: Accardi, L., Frigerio, A., and Gorini, V. (Eds.): Quantum Probability and Applications to the Quantum Theory of Irreversible Processes; Proceedings, Villa Mondragone, 1982. Lecture Notes in Mathematics 1055, pp. 228–244. Berlin Heidelberg New York Tokyo, Springer-Verlag, 1984.

    CrossRef  Google Scholar 

  5. Frigerio, A., and Gorini, V.: Markov dilations and quantum detailed balance. Commun. Math. Phys. 93, 517–532 (1984).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  6. Hudson, R.L., and Parthasarathy, K.R.: Quantum Ito's formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  7. Applebaum, D.B., and Hudson, R.L.: Fermion Ito's formula and stochastic evolutions. Commun. Math. Phys. (to appear).

    Google Scholar 

  8. Hudson, R.L., and Parthasarathy, K.R.: Stochastic dilations of uniformly continuous completely positive semigroups. Acta Math. Applicandae (to appear).

    Google Scholar 

  9. Frigerio, A.: Covariant Markov dilations of quantum dynamical semigroups. Preprint, 1984.

    Google Scholar 

  10. Maassen H.: The construction of continuous dilations by solving quantum stochastic differential equations. Semesterbericht Funktionalanalysis Tübingen, Sommersemester 1984, 183–204 (1984).

    Google Scholar 

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

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  12. Hudson, R.L., and Streater, R.F.: Itô's formula is the chain rule with Wick ordering. Phys. Lett. 86 A, 277–279 (1981).

    CrossRef  ADS  MathSciNet  Google Scholar 

  13. Accardi, L., Frigerio, A., and Lewis, J.T.: Quantum stochastic processes. Publ. RIMS Kyoto Univ. 18, 97–113 (1982).

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Accardi, L.: On the quantum Feynman-Kac formula. Rend. Sem. Mat. Fis. Milano 48, 135–180 (1980).

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. Hudson, R.L., and Lindsay, J.M.: A non-commutative martingale representation theorem for non-Fock quantum Brownian motion. J. Funct. Anal. (to appear).

    Google Scholar 

  16. Kossakowski, A., Frigerio, A., Gorini, V., and Verri, M.: Quantum detailed balance and KMS condition. Commun. Math. Phys. 57, 97–110 (1977).

    CrossRef  ADS  MathSciNet  MATH  Google Scholar 

  17. Haken, H.: Laser Theory. Handbuch der Physik., vol. XXV/2c. Berlin Heidelberg New York, Springer-Verlag, 1970.

    Google Scholar 

  18. Hepp, K., and Lieb, E.H.: Phase transitions in reservoir-driven open systems, with applications to superconductors and lasers. Helv. Phys. Acta 46, 575–603 (1973).

    Google Scholar 

  19. Agarwal, G.S.: Open quantum Markovian systems and the microreversibility. Z. Phys. 258, 409–422 (1973).

    CrossRef  ADS  MathSciNet  Google Scholar 

  20. Davies, E.B.: A model of heat conduction. J. Stat. Phys. 18, 161–170 (1978).

    CrossRef  ADS  MathSciNet  Google Scholar 

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© 1985 Springer-Verlag

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Frigerio, A. (1985). Construction of stationary quantum markov processes through quantum stochastic calculus. In: Accardi, L., von Waldenfels, W. (eds) Quantum Probability and Applications II. Lecture Notes in Mathematics, vol 1136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074473

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  • DOI: https://doi.org/10.1007/BFb0074473

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