Coherent Quantum Speed Limits

Abstract

We establish a comprehensive theoretical framework for coherent quantum speed limits (QSLs), deriving fundamental bounds on the rate of quantum evolution that explicitly isolate the contribution of quantum coherence. By applying H\"older's inequality for matrix norms to the Liouville-von Neumann equation, we construct two infinite families of QSLs for general unitary dynamics. These bounds are characterized by coherence measures based on Schatten $p$-norms and Hellinger distance, respectively, defined with respect to the instantaneous energy eigenbasis. Unlike traditional Mandelstam-Tamm bounds, our approach disentangles the quantum state's coherence structure from the Hamiltonian's energy scale. Using the Landau-Zener model accelerated by shortcuts to adiabaticity, we demonstrate that coherence functions as a critical kinematic resource: achieving faster evolution entails maintaining a state with high coherence relative to the instantaneous basis. Our results provide a resource-theoretic perspective on time-energy uncertainty, offering insights into the fundamental limits of quantum control and information processing.