Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quantum rotatability


Author: Stephen Curran
Journal: Trans. Amer. Math. Soc. 362 (2010), 4831-4851
MSC (2000): Primary 46L54; Secondary 46L65, 60G09
Published electronically: April 27, 2010
MathSciNet review: 2645052
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Recently, Köstler and Speicher showed that de Finetti's theorem on exchangeable sequences has a free analogue if one replaces exchangeability by the stronger condition of invariance of the joint distribution under quantum permutations. In this paper we study sequences of noncommutative random variables whose joint distribution is invariant under quantum orthogonal transformations. We prove a free analogue of Freedman's characterization of conditionally independent Gaussian families; namely, the joint distribution of an infinite sequence of self-adjoint random variables is invariant under quantum orthogonal transformations if and only if the variables form an operator-valued free centered semicircular family with common variance. Similarly, we show that the joint distribution of an infinite sequence of random variables is invariant under quantum unitary transformations if and only if the variables form an operator-valued free centered circular family with common variance.

We provide an example to show that, as in the classical case, these results fail for finite sequences. We then give an approximation for how far the distribution of a finite quantum orthogonally invariant sequence is from the distribution of an operator-valued free centered semicircular family with common variance.


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


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L54, 46L65, 60G09

Retrieve articles in all journals with MSC (2000): 46L54, 46L65, 60G09


Additional Information

Stephen Curran
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Email: curransr@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-10-05119-6
PII: S 0002-9947(10)05119-6
Keywords: Free probability, quantum rotatability, quantum invariance, semicircle law
Received by editor(s): February 11, 2009
Published electronically: April 27, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.