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

We investigate the mixing properties of primitive hypocoercive Lindblad dynamics. By extending the variational framework originally developed for underdamped Langevin dynamics, we derive fully explicit and constructive exponential decay estimates for the convergence in the noncommutative L²-norm. Our analysis relies on establishing a quantum analog of space-time Poincar\'{e} inequalities. To complement these hypocoercive estimates, we also analyze the limiting behavior of the spectral gap for Lindblad dynamics with a large coherent contribution, providing sharper convergence rate estimates in this asymptotic regime. A number of concrete examples are provided as applications of our theoretical results.