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Communications on Stochastic Analysis

Generic Fock quantum Markov semigroups with instantaneous states

Anis Ben Ghorbal, Franco Fagnola, Skander Hachicha, Habib Ouerdiane

Published Aug 1, 2008• 1 citation
Publisher

Abstract

We construct a generic quantum Markov semigroup with instantaneous states exploiting the invariance of the diagonal algebra and the explicit form of the action of the pre-generator on off-diagonal matrix elements.Our semigroup acts on a unital C * -algebra and is strongly continuous on this algebra (Feller property).We discuss the generic hydrogen type atoms as an example.

DOI
10.31390/cosa.2.2.01

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We construct a generic quantum Markov semigroup with instantaneous states exploiting the invariance of the diagonal algebra and the explicit form of the action of the pre-generator on off-diagonal matrix elements.Our semigroup acts on a unital C * -algebra and is strongly continuous on this algebra (Feller property).We discuss the generic hydrogen type atoms as an example.
We study a class of generic quantum Markov semigroups on the algebra of all bounded operators on a Hilbert space h arising from the stochastic limit of a discrete system with generic Hamiltonian HS, acting on h, interacting with a Gaussian, gauge invariant, reservoir. The selfadjoint operator HS determines a privileged orthonormal basis of h. These semigroups leave invariant diagonal and off-diagonal bounded operators with respect to this basis. The action on diagonal operators describes a classical Markov jump process. We construct generic semigroups from their formal generators by the minimal semigroup method and discuss their conservativity (uniqueness). When the semigroup is irreducible we prove uniqueness of the equilibrium state and show that, starting from an arbitrary initial state, the semigroup converges towards this state. We also prove that the exponential speed of convergence of the quantum Markov semigroup coincides with the exponential speed of convergence of the classical (diagonal) semigroup towards its unique invariant measure. The exponential speed is computed or estimated in some examples.
In this paper, we study some relevant properties of generic quantum Markov semigroups, in particular related to their asymptotic behavior. We can describe the structure of the set of fixed points and of the invariant states in terms of the Hamiltonian’s spectrum and of the communication classes of the classical Markov process associated with the diagonal part of the semigroup. Moreover we study the decoherence-free algebra and we complete the characterization of environmental decoherence for a generic quantum Markov semigroup.
Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to construct a class of quantum Markov semigroups from QBN which enjoy Feller property. As an application of our results, we examine a special quantum Feller semigroup associated with QBN, when it reduced to a certain Abelian subalgebra, the semigroup gives rise to the semigroup generated by Ornstein–Uhlenbeck operator. Finally, we find a sufficient condition for the existence of faithful invariant states that are diagonal for the semigroup.
We introduce the class of generic quantum Markov semigroups. Within this class we study the class corresponding to the Fock case which is further split into four subclasses each of which contains both bounded and unbounded generators, depending on some global characteristics of the intensities of jumps. For the first two of these classes we find an explicit solution which reduces the problem of finding the quantum semigroup to the calculation of two classical semigroups, one of which is diagonal (in a suitable basis) and the other one is triangular (in the same basis). In the bounded case our formula gives the unique solution. In the unbounded case it gives one solution, which we conjecture to be the minimal one.