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

TL;DR

This work analyzes how nonlinear classical networks of phase oscillators generate a quantum-like (QL) state space from graph structure. By constructing QL states from expander graphs and mapping Cartesian products to large phase-oscillator networks, the authors show that the emergent QL dynamics are linear in the state space and become effectively unitary in the strongly synchronized limit, while weaker synchronization induces decoherence via environmental dephasing. The study combines a Kuramoto-type dynamics, a unitary edge-bias transform, and ensemble averaging to produce a density-matrix description of the emergent state, demonstrating that purity and coherence are controlled by classical synchronization. The findings reveal a concrete link between nonlinear classical dynamics and quantum-like information processing, with no-cloning extending to the underlying classical system and implications for environment-assisted coherence control.

Abstract

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.

Figures (8)

• Figure 1: Picture of a graph with two vertices and the edge connecting them labeled. The adjacency matrix defined by the graph gives a set of states (the 'state space') and spectrum. A map that preserves the structure of the graph in terms of its edge topology defines the structure of a corresponding classical network, for example a phase oscillator network. The map we use in the present work assigns a phase oscillator to each vertex of the graph and and a coupling between oscillators to each edge.
• Figure 2: As a starting point for QL graphs we need a source of states. We use the emergent state $|a_1\rangle$ from an expander graph, $G_{a1}$, as a state. This state is guaranteed to be separated in the spectrum from all the other states (that we ignore), which is why it is called an emergent state. The other states in the spectrum, labeled the random states, have a semi-circular density-of-states like the spectrum of a random matrix. The emergent state is robust because of this separation from other states in the spectrum.
• Figure 3: (a) To produce a system with two emergent states, a QL bit, we ‘hybridize’ two expander graphs by connecting them with additional edges. Now we have two emergent states that are superpositions of the states of the two subgraphs. (b) From this foundation we generate arbitrary states in the tensor-product basis. We do that by combining the QL bit graphs using an operation called the Cartesian graph product. The states of a product graph are tensor products of states of the graphs combined in the product.
• Figure 4: (a) Schematic drawing of the a QL bit graph. (b) Structure of the Cartesian product. Notice how the four subgraphs define the basis as a product of the basis from each QL bit graph.
• Figure 5: (a) Ensemble-averaged Kuramoto order parameter as a function of time predicted for the system of oscillators. The time points corresponding to the plots of spectra are indicated together with the purity calculated at that time point. The time points are indicated as a fraction of the total time of the simulation. (b) Spectra calculated at one representative time point.

Theorems & Definitions (4)

• Proposition 1
• Proposition 2
• Definition 1
• Proposition 3