  Quantum synchronization is crucial for understanding complex dynamics and
holds potential applications in quantum computing and communication. Therefore,
assessing the thermodynamic resources required for finite-time synchronization
in continuous-variable systems is a critical challenge. In the present work, we
find these resources to be extensive for large systems. We also bound the speed
of quantum and classical synchronization in coupled damped oscillators with
non-Hermitian anti-PT-symmetric interactions, and show that the speed of
synchronization is limited by the interaction strength relative to the damping.
Compared to the classical limit, we find that quantum synchronization is slowed
by the non-commutativity of the Hermitian and anti-Hermitian terms. Our general
results could be tested experimentally and we suggest an implementation in
photonic systems.
