Random-walk approximation to vacuum cocycles

摘要

Quantum random walks are constructed on operator spaces with the aid of matrix-space lifting, a type of ampliation intermediate between those provided by spatial and ultraweak tensor products. Using a form of Wiener–Ito decomposition, a Donsker-type theorem is proved, showing that these walks, after suitable scaling, converge in a strong sense to vacuum cocycles: these are vacuum-adapted processes that are Feller cocycles in the sense of Lindsay and Wills. This is employed to give a new proof of the existence of homomorphic quantum-stochastic ilations for completely positive contraction semigroups on von Neumann algebras and separable unital C- algebras. The analogous approximation result is also established within the standard quantum stochastic framework, using the link between the two types of adaptedness.