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Infinite Dimensional Analysis Quantum Probability and Related Topics

BASINS OF ATTRACTION OF INVARIANT STATES OF A QUANTUM MARKOV SEMIGROUP

Franco Fagnola, Emanuela Sasso, Veronica Umanità

Published Nov 1, 2023• 0 citations
Publisher

Abstract

We determine the basins of attraction of invariant states of a quantum Markov semigroup with a faithful normal invariant state and atomic decoherence-free subalgebra.
DOI
10.1142/9789811275999_0004

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Let [Formula: see text] be a quantum Markov semigroup on [Formula: see text] with a faithful normal invariant state ρ. The decoherence-free subalgebra [Formula: see text] of [Formula: see text] is the biggest subalgebra of [Formula: see text] where the completely positive maps [Formula: see text] act as homomorphisms. When [Formula: see text] is the minimal semigroup whose generator is represented in a generalised GKSL form [Formula: see text], with possibly unbounded H, Lℓ, we show that [Formula: see text] coincides with the generalised commutator of [Formula: see text] under some natural regularity conditions. As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and Lℓ. We give examples of quantum Markov semigroups [Formula: see text], with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.
THE DECOHERENCE-FREE SUBALGEBRA OF A QUANTUM MARKOV SEMIGROUP ON B(h)\mathcal{B}({\rm{h}})B(h)