Dilations of quantum dynamical semigroups with classical Brownian motion
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).
one-parameter completely positive and unity preserving semigroup
one- parameter unitary group acting on the Hilbert space
scientific article; zbMATH DE number 3990530