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. We use this insight to develop its geometrical description. In particular, we identify mutually orthogonal coherent and incoherent directions and give their physical interpretation. This understanding allows us to optimize the success rate for implementing non-Hermitian driving along prescribed trajectories. We show its significance for shortcuts to adiabaticity. We introduce the geometrical interpretation of a speed limit for non-Hermitian Hamiltonians and analyze its tightness. We derive the explicit expression that saturates such a speed limit and illustrate our results on a minimal example of a dissipative qubit.