Emergent Quantum-Like Behavior

• Emergent quantum-like behavior is defined as the appearance of quantum phenomena from classical or stochastic foundations under collective or adaptive constraints.
• Models demonstrate recovery of quantum features—like interference, entanglement, and Planck’s relation—from mechanisms such as subquantum zero-point fields, ballistic diffusion, and synchronized oscillator networks.
• Experimental platforms like hydrodynamic analogs, quantum simulators, and neural networks offer practical insights into simulating quantum phenomena without relying on genuine quantum substrates.

Emergent quantum-like behavior refers to the appearance of phenomena mathematically analogous to those of quantum mechanics in systems whose fundamental degrees of freedom, dynamics, or substrates are classical, non-quantum, or described by deterministic rules, noise-driven dynamics, or collective organization. These emergent behaviors feature core quantum structures—linearity, interference, discrete spectra, entanglement/nonlocal correlations, and even a Planck-like action scale—yet arise as effective descriptions at appropriate scales or under specific thermodynamic or dynamical constraints. This concept underpins a variety of approaches toward understanding quantum phenomena as high-level patterns or attractors of underlying non-quantum systems, and has implications for the foundations of physics, complex systems, information theory, and emergent computation.

1. Classical and Statistical Foundations of Emergent Quantum-Like Dynamics

Multiple models demonstrate that canonical quantum structures can be systematically recovered from entirely classical, stochastic, or deterministic substrates when appropriate constraints or collective effects are imposed:

• Subquantum zero-point field models: A classical oscillator ("bouncer") continually exchanges energy with a thermalized zero-point vacuum field (ZPF). In steady state, the average total energy is

$$\langle E_{\rm tot}\rangle = \langle \hbar\omega \rangle + \left\langle \frac{p_u^2}{2m} \right\rangle = \text{const.}$$

with the stochastic exchange between oscillator and ZPF setting $\hbar$ as the quantum of action, and leading to Planck’s relation $E = \hbar\omega$ purely from classical kinetics (Groessing et al., 2012, Groessing, 2013). Diffusive corrections to the classical Hamilton–Jacobi equation then yield the exact Schrödinger equation.

• Ballistic diffusion and emergent probability currents: The thermal field introduces real-valued velocity fields,

$$\mathbf{v} = \frac{1}{m}\nabla S, \qquad \mathbf{u} = -\frac{\hbar}{2m} \frac{\nabla P}{P}$$

in which the ensemble average over subquantum kicks ("path excitation field") recovers quantum mechanical spreading, interference, and probability currents. All observable quantum predictions—free-space Gaussian dispersion, double-slit interference, and Bohmian trajectories—are reproduced from classical stochastic mechanics without invoking true wavefunctions (Groessing et al., 2012, Groessing, 2013).

• Statistical mechanics on neural or adaptive networks: The hydrodynamic Madelung equations derived from canonical ensembles of neural networks become genuinely quantum only in the grand-canonical setting, where exchange of hidden variables induces multivaluedness in the free energy. Imposing quantization of the free energy leads to a linear Schrödinger equation, with the effective “Planck constant” determined by the hidden-variable chemical potential and learning rate (Katsnelson et al., 2020). This theory equates the classical-to-quantum transition with a phase boundary in neural-network thermodynamics.
• Classical cellular automata with enforced ground state occupation: Systems composed of “fast” variables with large energy gaps, when kept in their uniform (ground) state, enforce entanglement and quantum interference among slow variables. The effective slow-sector dynamics is described by a projected Schrödinger equation, and interference phenomena, violation of Bell inequalities, and quantum measurement paradoxes all emerge as deterministic consequences of underlying classical rules (Hooft, 2020).

2. Key Mechanisms Underlying Quantum-Like Emergence

A coherent picture of emergent quantum-like behavior arises from several recurring physical and mathematical mechanisms:

• Steady-state throughput and thermodynamic balance: Sustained energy throughput between classical subsystems and structured environments (e.g., ZPF or a chemical reservoir) may naturally select attractors corresponding to quantum dynamics, with action quantization emerging from phase-locking phenomena (Groessing et al., 2012, Groessing, 2013, Katsnelson et al., 2020).
• Path excitation fields and anomalous diffusion: Subquantum stochastic processes give rise to effective probability currents and phase fields, with classical diffusive and convective velocities reproducing quantum interference and probability flows, including the precise spatial and temporal development of Gaussian wave-packets.
• Adaptive environment and quantum potential cancellation: In certain complex adaptive systems (e.g., stochastic Lotka-Volterra models), an effective “quantum potential” (de Broglie–Bohm type) arises in the nonlinear, non-unitary evolution of classical action amplitudes. If adaptive feedback or environmental coupling cancels this quantum potential, the remaining linear equation is mathematically identical to the Schrödinger equation, with emergent coherence, discrete spectrum, and phase stability (Hubsch et al., 2023, Minic et al., 2014).
• Synchronization and linear maps in classical oscillator networks: Carefully engineered graphs encoding the coupling structure of large ensembles of classical phase oscillators can realize an emergent state space of tensor-product form (the “quantum-like” or QL state space) in which the time evolution, in the fully synchronized (phase-locked) limit, is exactly unitary. During desynchronization, intrinsic decoherence mechanisms appear, causing decay of off-diagonal coherence akin to quantum decoherence. Linearity of the time-evolution map persists for the full state (Scholes, 13 Jan 2025, Amati et al., 2024).
• Experience-driven Hamiltonians and non-Markovian quantum theory: In experience-centric quantum theory (ECQT), the Hamiltonian is made an explicit functional of the system’s past state trajectory, yielding a non-Markovian, unitary but nonlinear extension of quantum evolution. Dynamical phases including fixed points, irregular oscillations, and hysteretic switching emerge as attractors in closed quantum systems with long memory (Tavanfar et al., 2023).

3. Phenomenology: Signatures and Observables in Emergent Quantum-Like Systems

Emergent quantum-like descriptions reproduce, both quantitatively and structurally, the essential observables and behaviors of quantum mechanics:

Phenomenon	Emergent mechanism	Reference
Planck’s relation	Steady-state throughput in ZPF or neural networks	(Groessing et al., 2012, Katsnelson et al., 2020)
Schrödinger equation	Diffusion-corrected HJ, free-energy multivaluedness	(Groessing, 2013, Hubsch et al., 2023, Katsnelson et al., 2020)
Gaussian wavepacket spreading	Ballistic or chaotic diffusion	(Groessing et al., 2012, Valani et al., 2024)
Double-slit interference	Superposed path-excitation fields	(Groessing et al., 2012, Groessing, 2013, Hooft, 2020)
Bohmian trajectories	Averaged classical currents, real-valued flows	(Groessing et al., 2012, Groessing, 2013)
Nonlocal correlations	Global diffusion fields, “modular momentum”, experience kernels	(Groessing et al., 2012, Groessing, 2013, Tavanfar et al., 2023)
Entanglement	Interacting subgraphs or collective excitations	(Scholes, 13 Jan 2025, Hillberry et al., 2020, Römer, 2015)
Quantum criticality	Emergent AdS₂ throat, low-energy IR CFT	(0907.2694)
Quantum computation-like gates	Graph-based oscillator collectives	(Amati et al., 2024, Scholes, 13 Jan 2025)
Emergent glassiness	Combinatorial kinetic constraints in closed quantum systems	(Yan et al., 2023)

For example:

• Double-slit interference is reconstructed by summing classical probabilities supplemented by phase memory terms that encode the global geometry of the environment. In quantum-like hydrodynamic walker models, coexisting attractors and crisis-induced intermittency yield probability distributions corresponding to quantum eigenstates, and tunneling is realized as chaotic switching events between wells, generating stationary distributions with nodal structures (Valani et al., 2024).
• Entanglement correlations are generated by contextually extended or synchronizing classical networks, with emergent tensor-product state spaces and no-go theorems (no-cloning, state broadcast) holding as in genuine quantum theory (Scholes, 13 Jan 2025, Römer, 2015).

4. Collective and Critical Phenomena: Emergence at Population and System Scale

Quantum-like correlations, phases, and criticality appear in a range of collective and many-body contexts:

• Disorder-free quantum glasses: Kinetically constrained models (e.g., Rydberg arrays on kagome lattices) spontaneously yield glassy behavior—characterized by the Edwards-Anderson order parameter, slow relaxation of local observables, and frustration-driven energy landscapes—even in the absence of quenched disorder. These systems exhibit quantum spin glass properties and dynamical signatures of nonergodicity (Yan et al., 2023).
• Non-interacting systems with reset-induced correlations: In non-interacting spin ensembles subject to stochastic reset protocols, emergent stationary states exhibit long-range, distance-independent quantum and classical correlations. Conditional reset rules induce non-equilibrium phase transitions, order-parameter singularities, and critical scaling, all without explicit two-body interactions. The process is tunable via reset rates and experimentally accessible on quantum simulators (Magoni et al., 2022).
• Complex adaptive and cognitive systems: The formal extension of quantum-like behavior to complex adaptive systems, cognitive processes, and higher-level social or mental phenomena relies on contextual emergence of observables and states (generalized quantum theory). Measurement, contextuality, and complementarity appear generically as coarsened observables, driving interference, superposition, and entanglement effects beyond physics and into information-processing theory (Römer, 2015).

5. Structural and Theoretical Extensions: Quantum-Like State Spaces and Interactomes

The structure of emergent quantum-like behavior is significantly enriched when viewed through the lens of mathematical and information-theoretic generalizations:

• Tensor-product and operator algebras from classical synchronizing networks: Large networks of classical oscillators (Kuramoto-type or graph-based) admit engineered resource graphs whose collective eigenmodes span a tensor-product state space, supporting “quantum-like bits” (QL-bits) with gates, measurements, and interference structure matching quantum computation. All familiar quantum information measures (purity, negativity, concurrence) are recoverable, and non-Kolmogorovian interference is realized by contextual probability rules (Scholes, 13 Jan 2025, Amati et al., 2024).
• Landscape of generalized quantum theories: Standard quantum mechanics sits as a node within a vast hypergraph of context-based quantum theories (“quantum interactome”), including experience-driven dynamics, symmetry-deformed models, and emergent or simulated quantum systems. Relations between these theories are mediated by simulation, reduction, duality, and context extension, pointing to potential universality in emergent quantum-like behavior in both physical and information-theoretic contexts (Tavanfar et al., 2023).
• Hydrodynamic and field-theoretic analogs: Quantum-to-classical transitions and decoherence phenomena can manifest as turbulence–like transitions (mock quantum turbulence) in the hydrodynamic or stochastic field-theoretic representation of mock quantum systems, with scaling behavior inherited from classical turbulence and stationarity maintained by environmental feedback maintaining phase-coherent flows (Hubsch et al., 2023, Minic et al., 2014).

6. Experimental Realizations and Implications

Emergent quantum-like dynamics has been realized—or is directly testable—on several current experimental platforms:

• Hydrodynamic quantum analogs: Walking droplets on vibrated fluids, particularly in confining double-well potentials, replicate quantized energy levels, chaos-driven tunneling, and quantum-like probability distributions through deterministic but memory-driven dynamics (Valani et al., 2024).
• Quantum simulators and programmable arrays: Rydberg atom arrays, trapped ions, and superconducting qubits can realize both reset-driven collective quantum-like correlations (Magoni et al., 2022) and glassy quantum behavior sans disorder (Yan et al., 2023).
• Designable classical networks: Classical phase oscillator networks and coupler graphs instantiate QL-bits, gates, and even robust interference—suggesting new architectures for classical simulation of quantum information protocols without genuine superposition or entanglement fragility (Amati et al., 2024, Scholes, 13 Jan 2025).

7. Interpretive and Foundational Significance

The emergence of quantum-like behavior in classical or hybrid systems challenges the irreducibility of quantum mechanics and highlights alternative organizing principles:

• Quantum-like phenomena can arise as attractors or invariant measures in nonlinear, dissipative, or complex adaptive systems, without any recourse to underlying quantum measurement or indeterminacy. The invariance of certain structures (probability current, interference phase, entangling correlations) is a consequence of environment-mediated steady-state or contextually partitioned dynamics, not of microphysical postulates.
• The existence of a non-trivial emergent “Planck constant,” gapless excitation spectra, and Born-type probability laws in complex or adaptive contexts suggests a universality or robustness of quantum mathematical structure far beyond its traditional domain.
• Observable extension and contextual emergence argue for a non-hierarchic worldview, with quantum-like non-causal order relations and entanglement-type correlations accepted as fundamental organizing principles in both physical and non-physical domains (Römer, 2015).

In summary, emergent quantum-like behavior appears across a diverse array of models and physical regimes, with rigorous mathematical correspondence to standard quantum structures but rooted in deterministic, stochastic, or adaptive classical substrates. This field provides a framework for understanding how quantum phenomena might arise as collective, thermodynamic, or informational consequences, and opens new pathways for simulation, computation, and theoretical unification (Groessing et al., 2012, Groessing, 2013, Hooft, 2020, Katsnelson et al., 2020, Valani et al., 2024, Magoni et al., 2022, Römer, 2015, Scholes, 13 Jan 2025, Amati et al., 2024, Hubsch et al., 2023, Minic et al., 2014, Tavanfar et al., 2023, 0907.2694, Weiss et al., 2016, Yan et al., 2023).