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

The extended de Finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem. In contrast to the classical result of Ryll-Nardzewski, exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of commuting squares and K?mmerer's generalized Bernoulli shifts. Our main result is applicable to classical probability, quantum probability, in particular free probability, braid group representations and Jones subfactors.Peer reviewe

We introduce 'braidability' as a new symmetry for (infinite) sequences of noncommutative random variables related to representations of the braid group B∞. It provides an extension of exchangeability which is tied to the symmetric group S∞. Our key result is that braidability implies spreadability and thus conditional independence, according to the noncommutative extended de Finetti theorem [Kös08]. This endows the braid groups Bn with a new intrinsic (quantum) probabilistic interpretation. We underline this interpretation by a braided extension of the Hewitt-Savage Zero-One Law.Furthermore we use the concept of product representations of endomorphisms [Goh04] with respect to certain Galois type towers of fixed point algebras to show that braidability produces triangular towers of commuting squares and noncommutative Bernoulli shifts. As a specific case we study the left regular representation of B∞ and the irreducible subfactor with infinite Jones index in the non-hyperfinite II 1 -factor L(B∞) related to it. Our investigations reveal a new presentation of the braid group B∞, the 'square root of free generator presentation' F 1/2 ∞ . These new generators give rise to braidability while the squares of them yield a free family. Hence our results provide another facet of the strong connection between subfactors and free probability theory [GJS07]; and we speculate about braidability as an extension of (amalgamated) freeness on the combinatorial level.

Abstract. We introduce symmetric states and quantum symmetric states on universal unital free product C * -algebras of the form A = * ∞ 1 A for an arbitrary unital C * -algebra A, as a generalization of the notions of exchangeable and quantum exchangeable random variables. We prove existence of conditional expectations onto tail algebras in various settings and we define a natural C * -subalgebra of the tail algebra, called the tail C * -algebra. Extending and building on the proof of the noncommutative de Finetti theorem of Köstler and Speicher, we prove a de Finetti type theorem that characterizes quantum symmetric states in terms of amalgamated free products over the tail C * -algebra, and we provide a convenient description of the set of all quantum symmetric states on A in terms of C * -algebras generated by homomorphic images of A and the tail C * -algebra. This description allows a characterization of the extreme quantum symmetric states. Similar results are proved for the subset of tracial quantum symmetric states, though in terms of von Neumann algebras and normal conditional expectations. The central quantum symmetric states are those for which the tail algebra is in the center of the von Neumann algebra, and we show that the central quantum symmetric states form a Choquet simplex whose extreme points are the free product states, while the tracial central quantum symmetric states form a Choquet simplex whose extreme points are the free product traces.

Abstract. We derive the form of the Belavkin-Kushner-Stratonovich equation describing the filtering of a continuous observed quantum system via non-demolition measurements when the statistics of the input field used for the indirect measurement are in a general coherent state.