Concentration of ergotropy in quantum many-body systems
Ergotropy$\unicode{x2014}$the maximal amount of unitarily extractable work$\unicode{x2014}$quantifies the "charge level" of a quantum battery. We prove that, in large many-body batteries, it exhibits a concentration of measure phenomenon. Namely, the ergotropy almost coincides with its average for almost all states sampled from the Hilbert$\unicode{x2013}$Schmidt measure. Importantly, the average ergotropy is macroscopic, scaling with the number of particles in the system. This occurs despite the von Neumann entropy also concentrating around a macroscopic average, a surprising coexistence since high entropy is commonly associated with effective thermality and thus passivity. We establish these results by proving that ergotropy, as a function of the state, is Lipschitz continuous with respect to the Bures distance and by applying Lévy's measure concentration lemma. Furthermore, we numerically extend our findings to the Bures measure, which represents the situation with the least amount of prior information about the state. Our continuity and concentration bounds open a new perspective on the noise-robustness of quantum many-body batteries and lay the groundwork for further rigorous studies of their charge stability and work extraction with limited prior knowledge.