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The decoherence-free subalgebra of a quantum Markov semigroup with unbounded generator. (English) Zbl 1209.46041

Summary: Let 𝒯 be a quantum Markov semigroup on (h) with a faithful normal invariant state ρ. The decoherence-free subalgebra 𝒩 ' (𝒯) of 𝒯 is the biggest subalgebra of (h) where the completely positive maps 𝒯 t act as homomorphisms. When 𝒯 is the minimal semigroup whose generator is represented in a generalised GKSL form (x)=-1 2 (L * L x-2L * xL +xL * L )+i[H,x] with possibly unbounded H and L , we show that 𝒩() coincides with the generalised commutator of {e -itH L e -itH L e itH 1, t0} 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 .

We give examples of quantum Markov semigroups (𝔥), with 𝔥 infinite-dimensional, having a nontrivial decoherence-free subalgebra.


MSC:
46L55Noncommutative dynamical systems
46L57Derivations, dissipations and positive semigroups in C * -algebras
47D07Markov semigroups of linear operators and applications to diffusion processes
47N50Applications of operator theory in quantum physics
81S25Quantum stochastic calculus
82C10Quantum dynamics and nonequilibrium statistical mechanics (general)