We study the relationships between ergodicity and environment induced decoherence for Quantum Markov Semigroups on a von Neumann algebra. We show that these properties are equivalent when the set of fixed points is an algebra containing the maximal subalgebra on which the semigroup is authomorphic.
We study environmental decoherence for a quantum Markov semigroup T acting on an arbitrary von Neumann algebra M. In particular, we analyze the relationships between the decomposition of the algebra induced by decoherence and a sort of “isometric-sweeping decomposition” for the space of states. Moreover, when the semigroup has a faithful normal invariant state, we embed the algebra M in its completion M̂ with respect to the scalar product induced by the faithful state, and we compare the decomposition induced by decoherence with some other kind of isometric-sweeping decompositions of M and M̂.
Abstract We demonstrate a method for finding the decoherence-free subalgebra N(T) N ( T ) of a Gaussian quantum Markov semigroup on the von Neumann algebra B(Γ(Cd)) B ( Γ ( C d ) ) of all bounded operator on the Fock space Γ(Cd) Γ ( C d ) on Cd C d . We show that N(T) N ( T ) is a type I von Neumann algebra L∞(Rdc;C)⊗B(Γ(Cdf)) L ∞ ( R d c ; C ) ⊗ ¯ B ( Γ ( C d f ) ) determined, up to unitary equivalence, by two natural numbers dc,df≤d d c , d f ≤ d . This result is illustrated by some applications and 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.