Dynamics in an emergent quantum-like state space generated by a nonlinear classical network

This work exploits a framework whereby a graph (in the mathematical sense) serves to connect a classical system to a state space that we call `quantum-like’ (QL). The QL states comprise arbitrary superpositions of states in a tensor product basis. The graph plays a special dual role by directing design of the classical system and defining the state space. We study a specific example of a large, dynamical classical system – a system of coupled phase oscillators – that maps, via a graph, to the QL state space. We investigate how mixedness of the state diminishes or increases as the underlying classical system synchronizes or de-synchronizes respectively. This shows the interplay between the nonlinear dynamics of the variables of the classical system and the QL state space. We prove that maps from one time point to another in the state space are linear maps. In the limit of a strongly phase-locked classical network – that is, where couplings between phase oscillators are very large – the state space evolves according to unitary dynamics, whereas in the cases of weaker synchronization, the classical variables act as a hidden environment that promotes decoherence of superpositions. We examine how similar the properties of QL states are to quantum states. We find that during decoherence of the QL states, the off-diagonal density matrix elements decay and that this decay can be observed in any basis we choose for measurement. More surprisingly, we show that a no-cloning theorem (that is, a state of a QL bit cannot be copied) applies not only to the QL states, but also to the underlying classical system.

💡 Research Summary

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The paper presents a novel framework that maps a classical nonlinear network of coupled phase oscillators onto a “quantum‑like” (QL) state space defined by a graph. The authors first construct the QL state space by selecting graphs whose adjacency matrices possess a single, well‑isolated eigenvalue (an “emergent state”). Expander graphs are used because they guarantee such a spectral gap. Each emergent eigenvector of a subgraph serves as a basis vector; two such subgraphs are linked to form a QL bit, and the Cartesian product of multiple QL‑bit graphs yields a tensor‑product Hilbert‑like space for many‑bit systems.

A direct mapping is then established: every vertex of the graph becomes a Kuramoto phase oscillator, and every edge defines a coupling term proportional to the corresponding adjacency‑matrix entry. The oscillator dynamics obey
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