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Quantum Central Limit Theorems

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Symmetries in Science VIII

Abstract

This work is a review of certain quantum limit theorems that may be viewed as non-commutative versions of the classical central limit theorem. Since various kinds of independences can be introduced in quantum probability, there are many quantum versions of this fundamental result in classical probability. We concentrate here only on various algebraic approaches and even within this scope we do not give a complete survey of the vast literature on the subject.

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References

  1. Accardi L., Frigerio A., Lewis J.T., “Quantum stochastic processes”, Publ. RIMS Kyoto Univ. 18 (1982), 97–133.

    Article  MathSciNet  MATH  Google Scholar 

  2. Boiejko M., Speicher R.,“An example of a generalized Brownian Motion”, Commun. Math. Phys. 137, 519–531 (1991).

    Article  ADS  Google Scholar 

  3. Bolejko M., Speicher R.,“An example of a generalized Brownian Motion II”, in Quantum Probability and Related Topics VII“, Proceedings, New Delhi 1990, World Scientific, Singapore 1993, 67–77.

    Google Scholar 

  4. Cushen D.D., Hudson R.L.,“A quantum mechanical central limit theorem”, J. Appl. Probability 8, 454–469 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  5. Drinfel’d V. G.,“Quantum groups”, Proc. Int. Congress of Math., Berkeley, California, 1986, 799–820.

    Google Scholar 

  6. Fannes M., Quaegebeur J., “Central limits of product mappings between CAR-algebras”. Publ. RIMS Kyoto 19, 469–491 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  7. Goderis D., Verbeure A., Vets P, “Non-commutative central limits”, Probab. Th. Rel. Fields 82, 527–544 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  8. Giri N., von Waldenfels W.,“An algebraic version of the central limit theorem”, Z. Wahr. Verw. Gebiete 42, 129–134 (1978).

    Article  MATH  Google Scholar 

  9. Hudson R.L.,“A quantum mechanical central limit theorem for anticommuting observables”, J. Appl. Probability 10, 502–509 (1973).

    Article  MathSciNet  Google Scholar 

  10. Hudson R.L., Parthasarathy K.R.,“Quantum Ito’s formula and stochastic evolution”, Commun.Math.Phys. 93, 301–323 (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Jimbo M.,“A q-difference analogue of U(g) and the Yang-Baxter equation”, Lett. Math. Phys. 10, 63–69 (1985).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Kümmerer B., Markov dialations and non-commutative Poisson processes, preprint.

    Google Scholar 

  13. Lindsay, J.M., Parthasarathy K.R., “Cohomology of power sets with applications in quantum probability”, Commun Math.Phys. 124, 337–364 (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Lenczewski R., Podgorski K.,“A q-analog of the quantum central limit theorem for SU q (2)“, J. Math. Phys. 33, 2768–2778 (1992).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Lenczewski R., “A q-analog of the quantum central limit theorem for SU q (2), q complex“, J. Math. Phys.34, 480–489 (1993).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Lenczewski R.,“On sums of q-independent SUq(2) quantum variables”, Commun. Math. Phys. 154, 127–134 (1993).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Lenczewski R.,“Addition of independent variables in quantum groups”, Rev. Math. Phys., Vol. 6, No.1, 135–147 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  18. Lenczewski R.,“On certain states on free *-algebras and their GNS representation”, preprint (1994).

    Google Scholar 

  19. Quaegebeur J., “A non-commutative central limit theorem for CCR algebras”, J. Funct. Anal. 57, 1–20 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  20. Schürmann M.,“A central limit theorem for coalgebras”, in Probability measures on groups VIII, Proceedings, Oberwolfach 1985, Ed. Heyer H., Lect. Notes in Math. 1210, 153–157, Springer 1986.

    Chapter  Google Scholar 

  21. Schürmann M.,“Quantum q-white noise and a q-central limit theorem”, Commun. Math. Phys. 140, 589–615 (1991).

    Article  ADS  MATH  Google Scholar 

  22. Schürmann M., White Noise on Bialgebras, Lecture Notes in Math. 1544, Springer -Verlag, Berlin, 1993.

    MATH  Google Scholar 

  23. Speicher R.,”A new example of ”Independence” and ”White Noise”, Probab. Th. Rel. Fields 84, 141–159 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  24. Speicher R., ”A non-commutative central limit theorem”, Math. Z. 209, 55–66 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  25. Speicher R., Bolejko M.,“Interpolations between Bosonic and Fermionic relations given by Generalized Brownian Motions”, preprint.

    Google Scholar 

  26. Voiculescu D., “Symmetries of some reduced free product C*-algebras”, in Operator Algebras and their Connections with Topology and Ergodic Theory, 556–588, Lecture Notes in Mathematics 1132, Springer Verlag, Berlin, 1985.

    Google Scholar 

  27. Voiculescu D., “Addition of certain non-commuting random variables”, J. Funct. Anal. 66, 323–346 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  28. von Waldenfels W.,“An algebraic central limit theorem in the anticommuting case”, Z. Wahr. Verw. Gebiete 42, 135–140 (1979).

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Hugo Steinhaus Center for Stochastic Methods Instytut Matematyki, Politechnika Wroclawska, 50-370, Wroclaw, Poland

    Romuald Lenczewski

Authors
  1. Romuald Lenczewski
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Editor information

Editors and Affiliations

  1. Southern Illinois University at Carbondale in Niigata, Niigata, Japan

    Bruno Gruber

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© 1995 Springer Science+Business Media New York

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Lenczewski, R. (1995). Quantum Central Limit Theorems. In: Gruber, B. (eds) Symmetries in Science VIII. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1915-7_22

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  • DOI: https://doi.org/10.1007/978-1-4615-1915-7_22

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-5783-4

  • Online ISBN: 978-1-4615-1915-7

  • eBook Packages: Springer Book Archive

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