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arXiv:2504.03852 · Physical Review A

Encoding quantumlike information in classical synchronizing dynamics

Graziano Amati, Gregory D. Scholes

Published Aug 28, 2025• 4 citations
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Abstract

In previous work we introduced a formalism that maps classical networks of nonlinear oscillators onto a quantumlike 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 following 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 quantumlike entangled states can emerge from classical synchronization dynamics.

DOI
10.1103/nvdp-sr11

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In previous work we introduced a formalism that maps classical networks of nonlinear oscillators onto a quantumlike 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 following 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 quantumlike entangled states can emerge from classical synchronization dynamics.
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Can complex classical systems be designed to exhibit superpositions of tensor products of basis states, thereby mimicking quantum states? We exhibit a one-to-one map between the product basis of quantum states comprising an arbitrary number of qubits and the eigenstates of a construction comprising classical oscillator networks. Specifically, we prove the existence of this map based on Cartesian products of graphs, where the graphs depict the layout of oscillator networks. We show how quantumlike gates can act on the classical networks to allow quantumlike operations in the state space.
Recent work has exposed the idea that interesting quantum-like (QL) probability laws, including interference effects, can be manifest in classical systems. Here, we propose a model for QL states and QL bits. We suggest a way that huge, complex systems can host robust states that can process information in a QL fashion. Axioms that such states should satisfy are proposed. Specifically, it is shown that building blocks suited for QL states are networks, possibly very complex, that we defined based on k -regular random graphs. These networks can dynamically encode a lot of information that is distilled into the emergent states we can use for QL processing. Although the emergent states are classical, they have properties analogous to quantum states. Concrete examples of how QL functions are possible are given. The possibility of a ‘QL advantage’ for computing-type operations and the potential relevance for new kinds of function in the brain are discussed and left as open questions.
Abstract In recent work we have developed a theory for classical systems that mimic the state space of quantum systems. We call the states ‘quantum-like’ (QL) because they show many properties similar to those of a quantum system. Here we study the QL states of a large classical system comprising a network of coupled phase oscillators. In particular, we seek to understand how the nonlinearity of the classical system manifests in the linear representation of the state space. To that end, we study how a classical network is viewed from the perspective of the time evolution of its emergent state. We investigate how mixedness of the QL states diminish or increase as the underlying classical system synchronizes or de-synchronizes respectively. In the limit of a very strongly phase-locked classical network—that is, where couplings between phase oscillators are very large—the state space evolves according to unitary dynamics. When the system is set up so that the classical phase oscillators synchronize with time, we find that the purity measure of the QL states correspondingly increases. Whereas, in cases where the coupling between oscillators is too small to allow the network to synchronize, the classical variables act like a kind of environment that promotes dephasing of initially-prepared superpositions.
Can synchronization -- the widespread spontaneous emergence of coordinated dynamics in classical nonlinear systems -- also occur in the quantum regime? This question has recently sparked intense research into collective behavior and temporal self-organization in quantum systems. Typical quantum features such as the linearity of time evolution and the presence of quantum noise seem to hinder the appearance of synchrony at the microscopic level. At the same time, quantum coherence and quantum correlations may provide novel mechanisms for enhancing synchronization beyond its classical counterpart. We here survey recent theoretical and experimental advances in quantum synchronization, ranging from the characterization of synchronous oscillations and genuinely nonclassical forms of synchrony to many-body synchronization on quantum networks.
Abstract We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by N quantum oscillators (‘nodes’) connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrödinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary N > 2 identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations.