We introduce the concept of degree of quantumness in quantum synchronization,
a measure of the quantum nature of synchronization in quantum systems.
Following techniques from quantum information, we propose the number of
non-commuting observables that synchronize as a measure of quantumness. This
figure of merit is compatible with already existing synchronization
measurements, and it captures different physical properties. We illustrate it
in a quantum system consisting of two weakly interacting cavity-qubit systems,
which are coupled via the exchange of bosonic excitations between the cavities.
Moreover, we study the synchronization of the expectation values of the Pauli
operators and we propose a feasible superconducting circuit setup. Finally, we
discuss the degree of quantumness in the synchronization between two quantum
van der Pol oscillators
