A Noncommutative de Finetti Theorem: Invariance under Quantum Permutations is Equivalent to Freeness with Amalgamation

We showthat 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 (xi )i?N, we prove that invariance of the joint distribution of the xi's under quantum permutations is equivalent to the fact that the xi 's are identically distributed and free with respect to the conditional expectation onto the tail algebra of the xi 's.