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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.

Department of Physical Chemistry University of the Basque Country UPV/EHU, Apartado 644

LP2N Laboratoire Photonique Numérique et Nanosciences Université Bordeaux-IOGS-CNRS:UMR 5298

Departamento de Física Universidade Federal de São Carlos

Universidade Estadual Paulista (Unesp) Campus Experimental de Itapeva

International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) and Physics Department Shanghai University

IKERBASQUE Basque Foundation for Science, Maria Diaz de Haro 3

Universidade Estadual Paulista (Unesp) Campus Experimental de Itapeva