Exponential ergodicity of quantum Markov semigroups with a spectral gap

This paper explores ergodic properties of quantum Markov semigroups with a faithful normal invariant state whose induced generator has a spectral gap. We provide a proof that demonstrates the exponential convergence of all normal states in a dense subset to some normal invariant state, with the rate of convergence determined by the spectral gap. Moreover, as an illustrative example, we investigate the restriction of quantum Ornstein-Uhlenbeck semigroups to the diagonal subalgebra of the number operator, highlighting their non-uniform exponential convergence and identifying a normal state that does not show exponential convergence with respect to the rate given by the spectral gap.