Quantum Exchangeable Sequences of Algebras

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.