Quantum Stochastic Dynamical Semigroup

Anilesh Mohari, Kalyan B. Sinha

V. P. Belavkin

Jaeseong Heo, Un Cig Ji

We discuss pairs (ϕ, Φ) of maps, where ϕ is a map between C*-algebras and Φ is a ϕ-module map between Hilbert C*-modules, which are generalization of representations of Hilbert C*-modules. A covariant version of Stinespring's theorem for such a pair (ϕ, Φ) is established, and quantum stochastic processes constructed from pairs ({ϕt}, {Φt}) of families of such maps are studied. We prove that the quantum stochastic process J = {Jt} constructed from a ϕ-quantum dynamical semigroup Φ = {Φt} is a j-map for the quantum stochastic process j = {jt} constructed from the given quantum dynamical semigroup ϕ = {ϕt}, and that J is covariant if the ϕ-quantum dynamical semigroup Φ is covariant.

Fatemeh Afsharnejad

Debashish Goswami, Kalyan B. Sinha

Hudson-Parthasarathy (H-P) type quantum stochastic dilation of a class of C 0 semigroups of completely positive maps ( quantum dynamical or Markov semigroups) on a von Neumann or C * algebra, with unbounded generators, is constructed under some assumptions on the semigroup and its generator.The assumption of symmetry with respect to a semifinite trace allows the use of Hilbert space techniques,while that of covariance with respect to an action of a Lie group on the algebra gives a better control on the domain of the generator.A dilation of the dynamical semigroup is obtained, under some further assumptions on the domain of the generator, with the help of a conjugation by a unitary quantum stochastic process satisfying Hudson-Parthasarathy equation in Fock space.

Wawrzyniec Guz