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

Abstract

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.

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• Family of Exact and Inexact Quantum Speed Limits for Completely Positive and Trace-Preserving Dynamics
• Quantum geometric protocols for fast high-fidelity adiabatic state transfer
• Phase space contraction rate for classical mixed states
• A unified fidelity-based framework for quantum speed limits in open quantum systems