Exchangeable stochastic processes and symmetric states in quantum probability

Abstract

We analyze general aspects of exchangeable quantum stochastic processes, as well as some concrete cases relevant for several applications to Quantum Physics and Probability. We establish that there is a one-to-one correspondence between quantum stochastic processes, either preserving or not the identity, and states on free product C∗ -algebras, unital or not unital, respectively, where the exchangeable ones correspond precisely to the symmetric states. We also connect some algebraic properties of exchangeable processes, that is the fact that they satisfy the product state or the block-singleton conditions, to some natural ergodic ones. We then specialize the investigation for the q -deformed Commutation Relations, q∈(−1,1) (the case q=0 corresponding to the reduced group C∗ -algebra C∗r(F∞) of the free group F∞ on infinitely many generators), and the Boolean ones. A generalization of de Finetti theorem to the Fermi CAR algebra (corresponding to the q -deformed Commutation Relations with q=−1 ) is proven, by showing that any state is symmetric if and only if it is conditionally independent and identically distributed with respect to the tail algebra. Moreover, we show that the Boolean stochastic processes provide examples for which the condition to be independent and identically distributed w.r.t. the tail algebra, without mentioning the a-priori existence of a preserving conditional expectation, is in general meaningless in the quantum setting. Finally, we study the ergodic properties of a class of remarkable states on the group C∗ -algebra C∗(F∞) , that is the so-called Haagerup states

Symmetries and ergodic properties in quantum probability

“…Finally one can see, as in Theorem 3.3 in [15], that exchangeable or stationary stochastic processes correspond to symmetric or shift invariant states.…”

“…defines a state whose GNS representation is precisely (π, H, Ω), see Theorem 3.3 in [15]. Conversely, for each state ϕ ∈ S( * J A) with GNS representation (π ϕ , H ϕ , Ω ϕ ), one can define the collection of * -homomorphisms…”

“…We point out the fact that all above considerations can be extended to the non unital case described in Section 3 of [15] either by using an approximate unity which always exists in any C * -algebra, or by adding a unity to A. The reader is referred to Section 2.2.3 of [13] or Section IV.4 of [27].…”

“…The present paper mainly deals with the investigation of the general structure of stochastic processes and their natural symmetries, like stationarity and exchangeability, by using the standard techniques of Operator Algebras, see [14,15,16]. Although some of the pivotal results have been obtained in the above mentioned papers, our aim is to describe new ones and present all matter in a unified approach.…”

“…We review in a self-containing form de Finetti-type results and ergodic properties for stationary and symmetric states in some concrete C * -algebras, plenty of them coming from physical investigations. The cases of q-deformed, −1 < q < 1 [11,15], Bose [16,26], Fermi [14,15], Boolean [9,15,16] and Monotone [16,24] processes are described in detail.…”