A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation

TL;DR

This paper establishes a noncommutative version of the de Finetti theorem, showing that invariance under quantum permutations implies the variables are identically distributed and free with respect to a conditional expectation.

Contribution

It introduces a noncommutative de Finetti theorem linking quantum permutation invariance to freeness with amalgamation, extending classical symmetry concepts.

Findings

Quantum permutation invariance implies freeness with amalgamation.

Identically distributed noncommutative variables are characterized by quantum symmetry.

The result generalizes classical de Finetti to a quantum setting.

Abstract

We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen `exchangeability' (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables, we prove that invariance of their joint distribution under quantum permutations is equivalent to the fact that the random variables are identically distributed and free with respect to the conditional expectation onto their tail algebra.