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

TL;DR

This paper introduces a geometric framework for analyzing non-Hermitian quantum dynamics, focusing on speed limits and optimal control of finite-rank density operators, with applications to shortcuts to adiabaticity.

Contribution

It develops a novel geometric approach to describe non-Hermitian evolution, identifying key directions and deriving a tight speed limit expression.

Findings

Identified orthogonal coherent and incoherent directions in non-Hermitian dynamics

Derived a saturable speed limit expression for non-Hermitian Hamiltonians

Illustrated results with a dissipative qubit example

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.