Ergodic quantum processes on finite von Neumann algebras

Abstract

Let (M,τ) be a tracial von Neumann algebra with a separable predual and let (Ω,P) be a probability space. A bounded positive random linear operator on L1(M,τ) is a map γ:Ω×L1(M,τ)→L1(M,τ) so that τ(γω(x)a) is measurable for all x∈L1(M,τ) and a∈M, and x↦γω(x) is bounded, positive, and linear almost surely. Given an ergodic T∈Aut(Ω,P), we study quantum processes of the form γTnω∘γTn−1ω∘⋯∘γTmω for m,n∈Z. Using the Hennion metric introduced in [21], we show that under reasonable assumptions such processes collapse to replacement channels exponentially fast almost surely. Of particular interest is the case when γω is the predual of a normal positive linear map on M. As an example application, we study the clustering properties of normal states that are generated by such random linear operators. These results offer an infinite dimensional generalization of the theorems in [21].