Quantum space-time Poincar\'e inequality for Lindblad dynamics

We investigate the mixing properties of primitive Markovian Lindblad dynamics (i.e., quantum Markov semigroups), where the detailed balance is disrupted by a coherent drift term. It is known that the sharp $L^2$-exponential convergence rate of Lindblad dynamics is determined by the spectral gap of the generator. We show that incorporating a Hamiltonian component into a detailed balanced Lindbladian can generically enhance its spectral gap, thereby accelerating the mixing. In addition, we analyze the asymptotic behavior of the spectral gap for Lindblad dynamics with a large coherent contribution. However, estimating the spectral gap, particularly for a non-detailed balanced Lindbladian, presents a significant challenge. In the case of hypocoercive Lindblad dynamics, we extend the variational framework originally developed for underdamped Langevin dynamics to derive fully explicit and constructive exponential decay estimates for convergence in the noncommutative $L^2$-norm. This analysis relies on establishing a quantum analog of space-time Poincar\'{e} inequality. Furthermore, we provide several examples with connections to quantum noise and quantum Gibbs samplers as applications of our theoretical results.