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arXiv:1712.02848 · Annales Henri Poincaré

Strong Convergence of Quantum Random Walks Via Semigroup Decomposition

Alexander C. R. Belton, Michał Gnacik, J. Martin Lindsay

Published Apr 21, 2018• 3 citations
arXivHTMLPDF
DOI
10.1007/s00023-018-0676-4

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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

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Random-walk approximation to vacuum cocycles

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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.

arXiv:math/0702700•Journal of the London Mathematical Society•Feb 17, 2010•8 citations
We give a simple and direct treatment of the strong convergence of quantum random walks to quantum stochastic operator cocycles, via the semigroup decomposition of such cocycles. Our approach also delivers convergence of the pointwise product of quantum random walks to the quantum stochastic Trotter product of the respective limit cocycles, thereby revealing the algebraic structure of the limiting procedure. The repeated quantum interactions model is shown to fit nicely into the convergence scheme described.
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.