## A tracial quantum central limit theorem

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- by Greg Kuperberg PDF
- Trans. Amer. Math. Soc.
**357**(2005), 459-471 Request permission

## 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 Johanssonโs theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity.## References

- R. 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**0436272** - Philippe Biane,
*Quantum random walk on the dual of $\textrm {SU}(n)$*, Probab. Theory Related Fields**89**(1991), no.ย 1, 117โ129. MR**1109477**, DOI 10.1007/BF01225828 - C. D. Cushen and R. L. Hudson,
*A quantum-mechanical central limit theorem*, J. Appl. Probability**8**(1971), 454โ469. MR**289082**, DOI 10.2307/3212170 - N. Giri and W. von Waldenfels,
*An algebraic version of the central limit theorem*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**42**(1978), no.ย 2, 129โ134. MR**467880**, DOI 10.1007/BF00536048 - D. Goderis, A. Verbeure, and P. Vets,
*Noncommutative central limits*, Probab. Theory Related Fields**82**(1989), no.ย 4, 527โ544. MR**1002899**, DOI 10.1007/BF00341282 - James E. Humphreys,
*Introduction to Lie algebras and representation theory*, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR**0323842**, DOI 10.1007/978-1-4612-6398-2 - Kurt Johansson,
*Discrete orthogonal polynomial ensembles and the Plancherel measure*, Ann. of Math. (2)**153**(2001), no.ย 1, 259โ296. MR**1826414**, DOI 10.2307/2661375 - Richard V. Kadison and John R. Ringrose,
*Fundamentals of the theory of operator algebras. Vol. I*, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR**719020** - Richard V. Kadison and John R. Ringrose,
*Fundamentals of the theory of operator algebras. Vol. II*, Pure and Applied Mathematics, vol. 100, Academic Press, Inc., Orlando, FL, 1986. Advanced theory. MR**859186**, DOI 10.1016/S0079-8169(08)60611-X - Greg Kuperberg,
*Random words, quantum statistics, central limits, random matrices*, Methods Appl. Anal.**9**(2002), 99โ118. - Michael A. Nielsen and Isaac L. Chuang,
*Quantum computation and quantum information*, Cambridge University Press, Cambridge, 2000. - J. Quaegebeur,
*A noncommutative central limit theorem for CCR-algebras*, J. Funct. Anal.**57**(1984), no.ย 1, 1โ20. MR**744916**, DOI 10.1016/0022-1236(84)90097-1 - Jun John Sakurai,
*Modern quantum mechanics*, 2nd ed., Benjamin/Cummings, 1985. - Dan Voiculescu,
*Lectures on free probability theory*, Lectures on probability theory and statistics (Saint-Flour, 1998) Lecture Notes in Math., vol. 1738, Springer, Berlin, 2000, pp.ย 279โ349. MR**1775641**, DOI 10.1007/BFb0106703

## Additional Information

**Greg Kuperberg**- Affiliation: Department of Mathematics, University of California Davis, Davis, California 95616
- Email: greg@math.ucdavis.edu
- Received by editor(s): May 22, 2003
- Published electronically: December 15, 2003
- Additional Notes: The author was supported by NSF grant DMS #0072342
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**357**(2005), 459-471 - MSC (2000): Primary 46L53, 81S25; Secondary 60F05
- DOI: https://doi.org/10.1090/S0002-9947-03-03449-4
- MathSciNet review: 2095618