Quantum Exchangeable Sequences of Algebras

TL;DR

This paper generalizes quantum exchangeability to sequences of algebra homomorphisms in noncommutative probability spaces, proving a free de Finetti theorem that characterizes infinite quantum exchangeable sequences as freely independent and identically distributed.

Contribution

It introduces a new notion of quantum exchangeability for sequences of algebra homomorphisms and establishes a free de Finetti theorem in this context, extending classical probabilistic results.

Findings

Infinite quantum exchangeable sequences are freely independent and identically distributed.

A free analogue of the Hewitt Savage zero-one law is established.

Finite quantum exchangeable sequences are characterized as noncommutative sampling without replacement.

Abstract

We extend the notion of quantum exchangeability, introduced by K\"ostler and Speicher in arXiv:0807.0677, to sequences (\rho_1,\rho_2,...c) of homomorphisms from an algebra C into a noncommutative probability space (A,\phi), and prove a free de Finetti theorem: an infinite quantum exchangeable sequence (\rho_1,\rho_2,...c) is freely independent and identically distributed with respect to a conditional expectation. As a corollary we obtain a free analogue of the Hewitt Savage zero-one law. As in the classical case, the theorem fails for finite sequences. We give a characterization of finite quantum exchangeable sequences, which can be viewed as a noncommutative analogue of sampling without replacement. We then give an approximation to how far a finite quantum exchangeable sequence is from being freely independent with amalgamation.