De Finetti theorems for easy quantum groups

TL;DR

This paper extends de Finetti theorems to sequences invariant under easy quantum groups, unifying classical and free probability results through combinatorial cumulant techniques.

Contribution

It provides new de Finetti type theorems for easy quantum groups, unifying classical and free cases with combinatorial cumulant methods.

Findings

Characterization of infinite quantum invariant sequences

Unified proof of classical de Finetti and Freedman theorems

Recovery of free de Finetti theorem and operator-valued free semicircular families

Abstract

We study sequences of noncommutative random variables which are invariant under "quantum transformations" coming from an orthogonal quantum group satisfying the "easiness" condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups S_n, O_n, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of K\"ostler and Speicher, and the characterization of operator-valued free semicircular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman.