A tracial quantum central limit theorem

Kuperberg, Greg

doi:10.1090/S0002-9947-03-03449-4

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 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

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. 1, 117–129. MR 1109477, DOI 10.1007/BF01225828
C. D. Cushen and R. L. Hudson, A quantum-mechanical central limit theorem, J. Appl. Probability 8 (1971), 454–469. 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. 2, 129–134. MR 467880, DOI 10.1007/BF00536048
D. Goderis, A. Verbeure, and P. Vets, Noncommutative central limits, Probab. Theory Related Fields 82 (1989), no. 4, 527–544. 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. 1, 259–296. 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), 99–118.
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. 1, 1–20. 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. 279–349. MR 1775641, DOI 10.1007/BFb0106703