  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.
