Unification of independence in quantum probability

Abstract

For a family of unital free *-algebras with a family of states on them, we construct a sequence of noncommutative probability spaces, which are tensor product algebras with tensor product states and which approximate the free product of states in the sense of convergence of mixed moments to those of free random variables. The constructed sequence of noncommutative probability spaces is called the hierarchy of freeness with the corresponding states called m-free products of states. The first-order approximation corresponding to m=1 gives the Boolean product of states. Thus the product states for the main types of noncommutative independence can be obtained from tensor products of states. Therefore, this approach can be viewed as a unification of independence. We also show how to associate a cocommutative *-bialgebra with the m-free product of states. This approach is developed in the paper for the more general case of the conditionally free product of states and can be easily extended to families of sequences of states.