The Decoherence-Free Subalgebra of a Quantum Markov Semigroup With Unbounded Generator

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, L ℓ , 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 L ℓ . We give examples of quantum Markov semigroups Formula: see text, with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.