Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A tracial quantum central limit theorem

Author: Greg Kuperberg
Translated by:
Journal: Trans. Amer. Math. Soc. 357 (2005), 459-471
MSC (2000): Primary 46L53, 81S25; Secondary 60F05
Published electronically: December 15, 2003
MathSciNet review: 2095618
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be considered a generalization of Johansson's theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity.

References [Enhancements On Off] (What's this?)

  • 1. Rabi N. Bhattacharya and R. Ranga Rao, Normal approximation and asymptotic expansions, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York-London-Sydney, 1976. MR 55:9219
  • 2. Philippe Biane, Quantum random walk on the dual of $\operatorname{SU}(n)$, Probab. Theory Related Fields 89 (1991), 117-129. MR 93a:46119
  • 3. C. D. Cushen and R. L. Hudson, A quantum-mechanical central limit theorem, J. Appl. Probability 8 (1971), 454-469. MR 44:6277
  • 4. Narayan C. Giri and Wilhelm von Waldenfels, An algebraic version of the central limit theorem, Z. Wahrscheinlichkeitstheorie 42 (1978), 129-134. MR 57:7731a
  • 5. D. Goderis, A. Verbeure, and P. Vets, Non-commutative central limits, Probab. Theory Related Fields 82 (1989), 527-544. MR 91b:46057
  • 6. James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Heidelberg-Berlin, 1972. MR 48:2197
  • 7. Kurt Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), 259-296, arXiv:math.CO/9906120. MR 2002g:05188
  • 8. Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras, vol. I, Academic Press, 1983. MR 85j:46099
  • 9. -, Fundamentals of the theory of operator algebras, vol. II, Academic Press, 1986. MR 88d:46106
  • 10. Greg Kuperberg, Random words, quantum statistics, central limits, random matrices, Methods Appl. Anal. 9 (2002), 99-118.
  • 11. Michael A. Nielsen and Isaac L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, 2000.
  • 12. Johan Quaegebeur, A noncommutative central limit theorem for CCR-algebras, J. Funct. Anal. 57 (1984), 1-20. MR 85m:46061
  • 13. Jun John Sakurai, Modern quantum mechanics, 2nd ed., Benjamin/Cummings, 1985.
  • 14. Dan Voiculescu, Lectures on free probability theory, Lectures on probability theory and statistics (Saint-Flour, 1998), Springer, Berlin, 2000, pp. 279-349. MR 2001g:46121

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L53, 81S25, 60F05

Retrieve articles in all journals with MSC (2000): 46L53, 81S25, 60F05

Additional Information

Greg Kuperberg
Affiliation: Department of Mathematics, University of California Davis, Davis, California 95616

Received by editor(s): May 22, 2003
Published electronically: December 15, 2003
Additional Notes: The author was supported by NSF grant DMS #0072342
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society