Averaging of quantum dynamical semigroups

Mustafa Elsheikh, J O Ogun, Yu. N. Orlov +2 more

Wawrzyniec Guz

Vsevolod Zhanovich Sakbaev

We study the dynamics of quantum system with degenerated Hamiltonian. To this end we consider the approximating sequence of regularized Hamiltonians and corresponding sequence of dynamical semigroups acting in the space of quantum states. The limit points set of the sequence of regularized semigroups is obtained as the result of averaging by finitely additive measure on the set of regularizing parameters. We establish that the family of averaging dynamical maps does not possess the semigroup property and the injectivity property. We define the functionals on the space of maps of the time interval into the quantum states space such that the maximum points of this functionals coincide with the trajectories of the family of averaging dynamical maps.

Andrzej Posiewnik

John Edward Gough, Yurii Nikolaevich Orlov, Vsevolod Zhanovich Sakbaev +1 more

We present a Law of Large Numbers principle for uniformly continuous random quantum dynamical semigroups. Random iterates of independent copies of these semigroups are shown to be Chernoff equivalent to the quantum dynamical semigroup by the average generator.

David Taj

We consider the class of quantum mechanical master equations defined on a generic Banach space, arising by projecting weakly perturbed one-parameter groups of isometries. We show that the possible semigroup approximations are far from unique. However, uniqueness can be reestablished through the introduction of a dynamical time averaging map. The generator of the resulting Contraction Semigroup is always well defined, irrespective of the dimensions of the projected subspace, and of the spectral properties of its free dynamics. We show how our approach includes and generalizes the preexisting literature.