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Banach Center Publications

Quantum random walk revisited

Kalyan B. Sinha

Published Jan 1, 2006• 10 citations
Publisher

Abstract

In the framework of the symmetric Fock space over $L^2 ({\Bbb R}_{+}),$ the details of the approximation of the four fundamental quantum stochastic increments by the four appropriate spin-matrices are studied. Then this result is used to prove the strong

DOI
10.4064/bc73-0-30

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arXiv:math/0702700•Journal of the London Mathematical Society•Feb 17, 2010•8 citations
In the framework of the symmetric Fock space over L2(R+),L^2 ({\Bbb R}_{+}),L2(R+​), the details of the approximation of the four fundamental quantum stochastic increments by the four appropriate spin-matrices are studied. Then this result is used to prove the strong
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 which 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. Key words: quantum random walk; repeated interactions; Donsker’s invariance principle; functional central-limit theorem; quantum stochastic dilation; quantum dynamical semigroup; completely positive contraction semigroup; toy Fock space; discrete approximation; Feller cocycle; vacuum adapted. MSC 2000: 81S25 (primary); 46L07, 46L53, 46N50, 47D06, 60F17 (secondary). 1
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.