Dynamical characterization of quantum coherence

Quantum coherence, rooted in the superposition principle of quantum mechanics, is a crucial quantum resource. Various measures, operational interpretations, and generalizations of quantum coherence have been proposed. In recent years, its role in quantum dynamics and technologies has attracted much attention. We introduce the concept of average quantum distance at a given time, and show that quantum coherence can be interpreted as the average evolution speed for arbitrary time-independent Hamiltonians. We extend the considerations to open quantum dynamics, where we find that quantum coherence can be used to bound the average evolution speed. Secondly, beyond this average setting, we show how quantum coherence and the eigenvalues of the Hamiltonian together determine the instantaneous evolution speed in the general case, with the trade-off being the energy uncertainty. Finally, we use the strategy to analyze the charging and discharging in quantum batteries, demonstrating the potential of our method in quantum technologies. We believe that the results clarify the role quantum coherence in quantum dynamics, paving an alternate avenue, complementary to the existing ones, to understand the capacity of quantum resources in dynamical processes.