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, \(q\in (-1,1)\) (the case \(q=0\) corresponding to the reduced group \(C^{*}\)-algebra \(C^*_r({\mathbb F}_\infty )\) of the free group \({\mathbb F}_\infty \) 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 \(C^*({\mathbb F}_\infty )\), that is the so-called Haagerup states.