Encoding quantum-like information in classical synchronizing dynamics

In previous work, we introduced a formalism that maps classical networks of nonlinear oscillators onto a quantum-like Hilbert space. We demonstrated that specific network transformations correspond to quantum gates, underscoring the potential of classical many-body systems as platforms for quantum-inspired information processing. In this paper, we extend this framework by systematically identifying the classical dynamics best suited for this purpose. Specifically, we address the question: Can the collective steady state of a classical network encode signatures of quantum information? We prove that the answer is affirmative for a special class of synchronizing many-body systems, namely, a complex-field extension of the Kuramoto model of nonlinearly coupled classical oscillators. Through this approach, we investigate how quantum-like entangled states can emerge from classical synchronization dynamics.