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Infinite Dimensional Analysis Quantum Probability and Related Topics

On the asymptotic behavior of generic quantum Markov semigroups

Raffaella Carbone, Emanuela Sasso, Veronica Umanità

Published Mar 1, 2014• 22 citations
Publisher

Abstract

We study ergodicity and decoherence for generic quantum Markov semigroups: in particular, we highlight the strong relationships among these properties, the convergence in law of the classical Markov process associated with the diagonal part of the semigroup, and the structure of its communication classes.
DOI
10.1142/s0219025714500015

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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.
We study the structure of generic quantum Markov semigroups, arising from the stochastic limit of a discrete system with generic Hamiltonian interacting with a Gaussian gauge invariant reservoir. We show that they can be essentially written as the sum of their irreducible components determined by closed classes of states of the associated classical Markov jump process. Each irreducible component turns out to be recurrent, transient or have an invariant state if and only if its classical (diagonal) restriction is recurrent, transient or has an invariant state, respectively. We classify invariant states and study convergence towards invariant states as time goes to infinity.
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.