Abstract
Synchronization in quantum systems has been largely driven by specific examples of frequency entrainment as well as mutual synchronization. Here we study quantum synchronization as a Liouville space perturbation theory. We begin by clarifying the role of centers, symmetries, and oscillating coherences in the context of quantum synchronization. We then analyze the eigenspectrum of the Liouville superoperator generating the dynamics of the quantum system and determine the conditions under which synchronization arises. We apply our framework to derive a powerful relationship between energy conservation, degeneracies, and synchronization in quantum systems. Finally, we demonstrate our approach by analyzing two mutually coupled thermal machines and prove that nondegenerate thermal networks cannot be simultaneously energy conserving and synchronous.