Dilations of quantum dynamical semigroups with classical Brownian motion
type: Item
instance of: scholarly article
title: Dilations of quantum dynamical semigroups with classical Brownian motion
zbMATH Open document ID: 0612.60053
DOI: 10.1007/BF01212314
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: It is shown that any one-parameter completely positive and unity preserving semigroup on the algebra of \(n\times n\) matrices may be represented in terms of solutions of the linear stochastic differential equation on \({\mathbb{C}}^ n\) driven by a multi-dimensional Brownian motion. This leads to a construction of the one-parameter unitary group acting on the Hilbert space which is a tensor product of \({\mathbb{C}}^ n\) and the \(L^ 2\)-space over the set of sample paths of the Brownian motion. By the construction this unitary group dilates the quantum dynamical semigroup mentioned above. For the alternative approach to the dilation problem involving non-commutative Itô-calculus see \textit{R. L. Hudson} and \textit{K. R. Parthasarathy}, ibid. 93, 301-323 (1984; Zbl 0546.60058).
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
zbMATH DE Number: 3990530
zbMATH Keywords: one-parameter completely positive and unity preserving semigroup
zbMATH Keywords: one- parameter unitary group acting on the Hilbert space
zbMATH Keywords: non-commutative Itô-calculus
author: Robert Alicki
author: Mark Fannes
MaRDI profile type: Publication
cites work: Markovian master equations
cites work: On the generators of quantum dynamical semigroups
cites work: Properties of quantum Markovian master equations
cites work: Q4141695
cites work: Quantum probability and applications to the quantum theory of irreversible processes. Proceedings of the International Workshop, held at Villa Mondragone, Italy, September 6-11, 1982
cites work: Quantum probability and applications II. Proceedings of the Second Workshop on Quantum Probability and Applications held in Heidelberg, West Germany, October 1-5, 1984
cites work: The Ito-Clifford integral
cites work: Quantum Ito's formula and stochastic evolutions
cites work: Fermion Ito's formula and stochastic evolutions
cites work: On dilating quantum dynamical semigroups with classical Brownian motion
full work available at URL: https://doi.org/10.1007/bf01212314
OpenAlex ID: W2055220787
description: scientific article; zbMATH DE number 3990530