Stochastic dilation of minimal quantum dynamical semigroup

Quantum Stochastic Dilation of Symmetric Covariant Completely Positive Semigroups With Unbounded Generator

“…which completes the proof of the fact. Now, the existence, uniqueness and unitarity of U t follows from [18] , noting the facts that Z = Z in this case and the conservativeness of Tt proved earlier suffices for β λ = βλ = 0. ✷…”

“…For the meaning of such q.s.d.e. with unbounded coefficients we refer to [18] and [11] and for the notation a R, a † R we refer to [14].…”

“…We shall now use the results of [18]. Let Z R denote the class of elements L ≡ ((L i j )) i,j≥0 , L i j ∈ B(h) such that for each j there is…”

“…We also set I = {Z ∈ Z − (D) : L i j (I) = 0∀i, j}, Ĩ = {Z ∈ Z − (D) : Z ∈ I}, and furthermore β λ = {X ≥ 0, X ∈ B(h) : L 0 0 (X) = λX} for λ > 0 , and βλ is defined by replacing Z by Z. With these notations, we have from [18] that the q.s.d.e. dU t = i,j V t Z i j dΛ j i (t); V 0 = I has a unique unitary solution provided Z ∈ I Ĩ and β λ = βλ = {0}.…”

“…a semifinite trace and covariant under the action of a Lie group on the algebra. This additional hypothesis enables us to control the domains of the various operator coefficients appearing in the quantum stochastic differential equations so that the Mohari-Sinha conditions ( [18]) can be applied. The covariance is exploited again as in [9] along with the assumption that the crossed-product von Neumann algebra A > ✁G is isomorphic with the von Neumann algebra generated by A and the representation u g of G in the GNS Hilbert space associated with the trace, to obtain the structure maps, and finally the Evans-Hudson (E-H) flow is constructed essentially along the lines of the proofs in [14].…”