In this work, we give a characterization of the partial order on Lindblad generators using gradient forms. By embedding this order into Vern Paulsen’s Archimedean ordered‐space framework, we prove that it is stable under certain perturbations. In particular, we show that functional inequalities for the associated semigroups—such as Poincar\'e‐ and log–Sobolev–type bounds—are preserved in this ordered setting. Our results offer a unified, order‐theoretic perspective on stability in non-commutative dynamics.
This is joint work with Marius Junge.