Premium
Random‐walk approximation to vacuum cocycles
Author(s) -
Belton Alexander C. R.
Publication year - 2010
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdp075
Subject(s) - mathematics , pure mathematics , type (biology) , quantum walk , von neumann architecture , quantum , random walk , banach space , separable space , space (punctuation) , operator (biology) , mathematical analysis , quantum mechanics , quantum algorithm , physics , statistics , ecology , linguistics , philosophy , biochemistry , chemistry , repressor , gene , transcription factor , biology
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–Itô 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 dilations 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.