  Progress on the study of synchronisation in quantum systems has been largely
driven by specific examples which resulted in several examples of frequency
entrainment as well as mutual synchronisation. Here we study quantum
synchronisation by utilising Liouville space perturbation theory. We begin by
clarifying the role of centers, symmetries and oscillating coherences in the
context of quantum synchronisation. We then analyse the eigenspectrum of the
Liouville superoperator generating the dynamics of the quantum system and
determine the conditions under which synchronisation arises. We apply our
framework to derive a powerful relationship between energy conservation,
degeneracies and synchronisation in quantum systems. Finally, we demonstrate
our approach by analysing two mutually coupled thermal machines and the close
relationship between synchronisation and thermodynamic quantities.
