Exponential Ergodicity of Classical and Quantum Markov Birth and Death Semigroups

Abstract

With a quantum Markov semigroup (T t )t≥0 on B(h) with a faithful normal invariant state ρ we associate the semigroup (T t )t≥0 on Hilbert-Schmidt operators on h (the L 2(ρ) space) defined by T t (ρ s/2 xρ (1−s)/2) = ρ s/2 T t (x)ρ (1−s)/2. This allows us to study the spectrum of the infinitesimal generator of (T t )t≥0 and deduce informations on the speed of convergence to equilibrium of the given semigroup. We apply this idea to show that some quantum Markov semigroups related to birthand-death processes converge to equilibrium exponentially rapidly in L 2(ρ). Moreover, through unitary transformations, we extend these results to other semigroups as, for instance, the quantum Ornstein-Uhlenbeck semigroup.