A geometrical description of non-Hermitian dynamics: speed limits in finite rank density operators

Non-Hermitian dynamics in quantum systems preserves the rank of the state density operator. Using this insight, we develop a geometric framework to describe its time evolution. In particular, we identify mutually orthogonal coherent and incoherent directions and provide their physical interpretation. This understanding enables us to optimize the success rate of non-Hermitian driving along prescribed trajectories, with direct relevance to shortcuts to adiabaticity. Next, we explore the geometric interpretation of a speed limit for non-Hermitian Hamiltonians and analyze its tightness. We derive the explicit expression that saturates this bound and illustrate our results with a minimal example of a dissipative qubit.