⚛️ quantum physics

Quantum-like states from classical systems

This paper explores how suitably designed classical systems mediated by graphs can generate quantum-like state spaces featuring superpositions and optimized correlation structures, while critically examining the extent to which these systems can exhibit entanglement compared to classical entanglement in optics.

Original authors: Gregory D. Scholes

Published 2026-03-24

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Gregory D. Scholes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: Building a "Quantum" House out of "Classical" Bricks

Imagine you have a very complex, messy room full of ordinary objects: clocks, pendulums, and people talking. This is a classical system. Usually, these things behave in predictable, boring ways. If you push a pendulum, it swings back and forth. If you flip a light switch, it's either on or off.

Now, imagine you want to build a house that behaves like a Quantum System. In the quantum world (the world of atoms and electrons), things are weird:

They can be in two places at once (Superposition).
Two particles can be linked so that changing one instantly affects the other, no matter how far apart they are (Entanglement).

The paper asks a bold question: Can we build a house that acts "quantum" using only ordinary, classical bricks?

The answer, according to this paper, is yes. But we can't just use random bricks. We need a very specific blueprint. That blueprint is a Graph.

1. The Blueprint: The Graph as a Map

Think of a Graph not as a chart, but as a map of a city.

Vertices (Dots): These are the buildings (or people, or oscillators).
Edges (Lines): These are the roads connecting them.

The author says: "If we design the city map (the graph) in a very specific way, the traffic flow (the state of the system) will start behaving like a quantum particle."

The key to this magic is a special type of city map called an Expander Graph.

2. The Magic Ingredient: Expander Graphs (The "Super-Highway" City)

Imagine a small town where everyone is connected to their immediate neighbors. If you cut one road, the town splits in half. This is a "weak" connection.

Now, imagine a Super-Highway City (an Expander Graph).

It has many roads, but not too many.
No matter how you try to cut the city in half, you have to destroy a huge number of roads to do it.
The city is "tightly knit" at every scale.

Why does this matter?
In this tightly knit city, there is one special "vibe" or "rhythm" that the whole city naturally falls into. It's like a choir where everyone suddenly starts singing the exact same note perfectly, even if they started out of tune. The author calls this the Emergent State.

Because the city is so well-connected, this "perfect rhythm" is very stable. It stands out clearly from all the other messy, random noises in the city. This stable rhythm is our Quantum-Like (QL) State.

3. The Quantum Bit (QL Bit): The Two-Choice Switch

In a normal computer, a bit is either 0 or 1. In a quantum computer, a "qubit" can be 0, 1, or a mix of both at the same time.

The paper shows how to build a QL Bit using our city map:

Take two of our "Super-Highway Cities."
Connect them with a few roads.
Because they are so well-connected, the "rhythm" of the two cities mixes together.
Suddenly, the system has two special rhythms: one where the cities are "in sync" (0+1) and one where they are "out of sync" (0-1).

By tweaking the connections (the roads), we can make the system be in a mix of these two rhythms. Bam! We have created a classical system that mimics a quantum superposition.

4. Making it Bigger: The Graph Product (The Lego Tower)

What if we want a complex quantum computer with many qubits? We can't just build a bigger city; the map would get too messy.

The author uses a clever trick called the Cartesian Product.

Imagine you have a map of a small town (City A).
Imagine you have a map of a different small town (City B).
The "Product" is like building a giant 3D structure where you place a copy of City A at every single intersection of City B.

The Magic Result:
When you do this, the "rhythms" of the new giant city are exactly the combinations of the rhythms of the small towns.

If City A has rhythms $A_1$ and $A_2$ .
And City B has rhythms $B_1$ and $B_2$ .
The giant city naturally produces rhythms like $A_1 \times B_1$ , $A_1 \times B_2$ , etc.

This is exactly how quantum computers work: they combine qubits to create a massive space of possibilities. The paper proves that by stacking these graph maps, we can create a classical system that mimics this massive quantum space.

5. The Tricky Part: Entanglement (The Spooky Connection)

The hardest part of quantum mechanics is Entanglement. This is when two particles are linked so that if you measure one, you instantly know the state of the other, even if they are light-years apart.

Can our classical city map do this?

The Problem: In a real quantum experiment, the particles are separate objects. In our graph, everything is one big connected map. You can't really "separate" the parts without breaking the map.
The Solution: The author proposes using "Witness Bits." Imagine adding a small, separate "spy" city connected to the main city. This spy city can "listen" to the main city's rhythm without breaking the main city's structure.
By measuring the spy city, we can infer the state of the main city.

The paper suggests that while this isn't true quantum non-locality (because the whole system is still one connected classical object), it creates a correlation structure that looks and feels exactly like entanglement. It's like two people in a room who can't talk, but they both have a walkie-talkie connected to the same central station. If one speaks, the other hears it instantly, not because of magic, but because of the shared connection.

6. Why Should We Care? (The "So What?")

This isn't just a math puzzle. It has real-world potential:

Robustness: Quantum computers are fragile; a little noise breaks them. These "Quantum-Like" classical systems are made of messy, random networks (like the cities in the paper). They are surprisingly tough. Even if you cut half the roads (edges) in the city, the "perfect rhythm" (the quantum state) stays intact.
New Computers: We might be able to build computers that use these classical networks to solve problems that are currently too hard for normal computers, mimicking the power of quantum computers without the need for freezing temperatures or vacuum chambers.
Understanding Nature: It helps us understand how complex systems (like the brain or social networks) might naturally develop "quantum-like" behaviors just by being highly connected.

Summary Analogy

Think of a Quantum System as a perfect, invisible dance troupe where dancers can be in two places at once and instantly copy each other's moves.

This paper says: "You don't need magic to get that dance. You just need a really, really well-connected dance floor (an Expander Graph)."

If you put enough dancers on a floor where everyone is connected to everyone else in a specific, tight pattern, they will spontaneously start dancing in perfect, synchronized, "quantum-like" patterns. And if you stack these dance floors on top of each other, you get a massive, complex dance that mimics the behavior of the entire universe's quantum particles—all using ordinary, classical dancers.

1. Problem Statement

The central problem addressed is whether classical systems can be constructed to mimic the state spaces and correlation structures of composite quantum systems (specifically those exhibiting superposition and entanglement) without undergoing "quantization." While previous work has explored "quantum-like" (QL) behavior in stochastic or cognitive models, this paper seeks a concrete, structural mapping where a classical network spontaneously generates a state space isomorphic to a tensor product Hilbert space. The authors aim to determine if such classical systems can exhibit non-classical correlations (entanglement) and how these differ from "classical entanglement" observed in optics.

2. Methodology

The paper employs a mathematical framework based on graph theory and spectral graph theory to map classical networks to quantum-like state spaces.

Graph Representation: Classical systems (e.g., coupled oscillators, neural networks) are modeled as graphs $G(n, m)$ where vertices represent entities and edges represent couplings. The state space is defined by the eigenvectors of the graph's adjacency matrix.
QL Bits (Building Blocks): The fundamental unit is the "QL bit," constructed by coupling two expander graphs (specifically $d$ $d$ -regular random graphs).
- Expander graphs are chosen because they possess a spectral gap: a single "emergent state" (the eigenvector corresponding to the largest eigenvalue, $\lambda_0 = d$ ) is well-separated from a sea of "random states" (following a semicircle law).
- By coupling two such expanders ( $G_{a1}$ and $G_{a2}$ ) with a sparse set of connecting edges, the system hybridizes the two emergent states into two new states ( $A_+$ and $A_-$ ), effectively creating a controllable two-level system capable of superposition.
Graph Products: To generate multi-qubit systems, the authors utilize the Cartesian product of graphs ( $G \square H$ $G □ H$ ).
- Proposition 1: The spectrum of a Cartesian product graph is the sum of the spectra of the constituent graphs, and the eigenvectors are the tensor products of the constituent eigenvectors ( $v_A \otimes v_B$ ).
- This operation maps the classical network structure directly to a tensor product basis in the state space, allowing the construction of $N$ -qubit systems from $N$ QL bits.
Optimization: The authors introduce a contraction technique. While the raw Cartesian product of graphs results in an exponentially large vertex set ( $N^q$ ), they demonstrate that the graph can be "contracted" to a much smaller, optimized graph that retains the essential correlation topology and spectral properties, significantly reducing resource requirements.
Dynamics: The paper models the classical system using phase oscillators (Kuramoto model). It shows that as the network synchronizes, the system naturally evolves toward the pure emergent state, increasing the "purity" of the QL state and demonstrating resilience to disorder.

3. Key Contributions

Construction of QL State Spaces: The paper provides a rigorous recipe for constructing classical networks that possess an emergent state space with the mathematical structure of a quantum Hilbert space (specifically, a tensor product basis).
Optimization of Graph Products: A novel method for "contracting" the Cartesian product of graphs is developed. This allows the representation of complex quantum-like correlations in a compact classical graph, making the visualization of correlation structures feasible.
Robustness of Expander Graphs: The work demonstrates that QL states are remarkably robust. Even with significant edge deletion (disorder) in the underlying $d$ -regular graphs, the spectral gap and the emergent state remain stable. This suggests such systems could arise naturally in complex biological or physical networks.
Distinction from Classical Entanglement: The paper critically analyzes the difference between QL entanglement and "classical entanglement" in optics (e.g., non-separable light beams). It argues that while optical non-separability relies on the interference of separate beams (a convex sum), QL states arise from a single unified system (the graph), offering a more intrinsic form of correlation.
Measurement and Nonlocality Proposal: The authors propose a mechanism for "reading out" QL states using witness graphs. By coupling external "witness" QL bits to the product graph, one can measure specific sub-components without disconnecting the system, mimicking the concept of local measurement in a non-local space.

4. Key Results

Spectral Properties: Numerical simulations confirm that the Cartesian product of QL bits produces a spectrum where the emergent states are separated from hybrid and random states. The gap remains constant regardless of the number of bits, ensuring the stability of the quantum-like structure.
Entanglement Generation:
- Separable states are generated naturally by the graph product.
- Entangled states (e.g., Bell states) can be generated by exploiting degenerate eigenvalues in the spectrum. When two product states share the same eigenvalue, the system exists in a mixed state that can be interpreted as a statistical mixture of maximally entangled pure states.
- The paper shows that while the ground state of a synchronized classical system is typically separable, accessing excited states or applying specific gate operations (via edge bias control) allows for the generation of non-separable states.
Phase Synchronization: Simulations of the Kuramoto model on QL graphs show that as the classical network synchronizes, the density matrix of the emergent state becomes increasingly pure, effectively "cooling" the system into a quantum-like state.
Nonlocality: The paper concludes that while QL systems mimic the correlation structure of entangled quantum states, they lack true quantum nonlocality because the "bits" are not spatially separable entities in the product graph. However, the proposed "witness" architecture allows for a form of effective nonlocality where measurements on one part of the graph instantaneously correlate with another via the shared correlation space.

5. Significance

Conceptual Bridge: This work bridges the gap between classical network theory and quantum information science. It suggests that "quantumness" (superposition, tensor products, entanglement) may be an emergent property of specific topological structures (expanders) rather than an exclusive feature of quantum mechanics.
Computational Potential: The authors suggest that these classical QL systems could serve as a resource for universal memcomputing. Because the underlying networks possess intrinsic parallelism and can represent superpositions, they might solve NP-complete problems or perform computations in polynomial time, similar to quantum computers but using classical hardware.
Biological Relevance: The robustness of these states to disorder implies that biological systems (e.g., neural networks or protein interactions) could naturally evolve to support QL-like information processing without requiring quantum coherence.
New Perspective on Entanglement: By contrasting QL states with classical optical entanglement, the paper refines the definition of non-classical correlations, highlighting the importance of system unity versus beam interference.

In summary, Scholes demonstrates that by carefully designing the topology of a classical network using expander graphs and Cartesian products, one can create a "quantum-like" state space that mimics the mathematical properties of quantum entanglement, offering new avenues for classical computing and a deeper understanding of the structural origins of quantum correlations.