⚛️ quantum physics

Existence, structure, and properties of quantum-like states

This paper demonstrates that composite quantum-like systems, such as wave multipole moments and phase oscillator networks, can effectively mimic the separable states of quantum systems, suggesting their potential existence in biological, engineered, and soft matter contexts.

Original authors: Gregory D. Scholes

Published 2026-03-24

📖 6 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: Can Classical Things "Pretend" to be Quantum?

Imagine you are trying to build a robot that can fly like a bird. You could try to build a tiny, fragile, real bird (a Quantum System). But birds are hard to keep alive; they need very specific conditions, and they are fragile.

Alternatively, you could build a machine made of gears, springs, and wires (a Classical System) that looks and moves exactly like a bird, even though it's made of heavy metal. It won't have a soul or biological DNA, but it can mimic the flight perfectly.

This paper is about building that "metal bird."

The author, Gregory Scholes, wants to show that we can build ordinary, everyday physical systems (like networks of vibrating clocks or waves in water) that mimic the behavior of complex quantum computers. Specifically, he wants to prove we can create "Quantum-Like" (QL) states that look just like the "separable" states of quantum systems.

The Toolkit: The "Graph" Blueprint

To do this, the author uses a tool called Graph Theory.

The Metaphor: Think of a graph not as a chart, but as a map of a city.
- Vertices (dots) are the buildings (or people).
- Edges (lines) are the roads connecting them.

In this paper, the "city" is a physical system. The "roads" represent how different parts of the system talk to each other. The author discovered a special way to draw these maps so that the city's traffic patterns (its "spectrum" or eigenvalues) behave exactly like the math of a quantum bit (a qubit).

The Building Block: The "QL Bit"

In quantum computing, the basic unit is a Qubit. It can be in a state of 0, 1, or a magical mix of both at the same time (superposition).

The author shows how to build a QL Bit using a classical city map:

Two Neighborhoods: Imagine two separate, highly connected neighborhoods (called "expander graphs"). Each neighborhood has its own "vibe" or dominant rhythm.
The Bridge: You build a few special bridges connecting these two neighborhoods.
The Magic: When you adjust the "traffic rules" on these bridges (specifically, the phase or timing of the connection), the two neighborhoods start to dance together. They create two new, combined rhythms.
- One rhythm is when they dance in step (like a linear polarization).
- The other is when they dance opposite (like a circular polarization).

By tweaking the bridges, you can make the system dance in any combination of steps. This mimics the infinite possibilities of a quantum bit, but it's done with classical waves or oscillators.

The Super-System: The "Cartesian Product"

Now, what if you want to build a whole quantum computer, not just one bit? You need to combine many bits.

In quantum physics, you combine bits using something called a Tensor Product. It's like stacking layers of reality on top of each other.

The author shows that if you take your "QL Bit" city maps and combine them using a specific mathematical recipe called the Cartesian Product, you get a massive, complex city.

The Result: This giant city has "emergent states" (overall rhythms) that are mathematically identical to the "separable states" of a multi-bit quantum system.
The Analogy: Imagine you have two sets of Lego bricks. If you snap them together in a specific grid pattern, you create a structure that has the exact same stability and shape as a complex quantum molecule, even though it's just plastic bricks.

What Can We Build? (Real-World Examples)

The paper suggests these "QL Cities" aren't just math; they can be real physical things:

Waves: Think of light or sound waves. If you have a wave with a specific polarization (direction of vibration), it acts like a QL bit.
Oscillator Networks: Imagine a room full of metronomes or pendulums. If you connect them with springs (the edges of the graph), they can synchronize in ways that mimic quantum states.
Electronic Circuits: You could build a circuit board where the connections between components are designed like these graphs.
Soft Matter: Even things like slime molds or fluid droplets might naturally form these patterns.

The Catch: The "No-Cloning" and "Entanglement" Limits

The paper is honest about the limits.

No-Cloning: In quantum physics, you cannot copy a state perfectly (No-Cloning Theorem). The author shows that even in these classical "QL" systems, you cannot perfectly copy a state. The math of the graph prevents it. This is a cool feature because it means these classical systems can mimic quantum security!
Entanglement (The Hard Part): Quantum systems can be "entangled," meaning two particles are linked so deeply that changing one instantly changes the other, no matter the distance.
- The paper argues that while these classical systems can mimic separable states (independent bits), they cannot easily mimic true entanglement.
- Why? To get entanglement, the graph would have to be "disconnected" in a way that breaks the rules of how these classical networks work. It's like trying to make two cities talk to each other without any roads; it's possible in a quantum world, but very hard in a classical one.

Why Does This Matter? (The "So What?")

Cheaper Quantum Tech: We don't need to build fragile, super-cold quantum computers to do some quantum-like tasks. We might be able to build robust, room-temperature "Quantum-Like" circuits using waves, circuits, or even biological networks.
New Computing: These systems have "intrinsic parallelism." They can solve complex problems (like finding the shortest path in a maze) very quickly, similar to how quantum computers work, but using classical physics.
Understanding Biology: Maybe nature uses these "QL" tricks in our brains or in photosynthesis to process information efficiently without needing full-blown quantum mechanics.

Summary in One Sentence

This paper proves that by drawing the right "map" (graph) for a classical system—like a network of vibrating clocks or waves—we can make it dance to the exact same tune as a quantum computer, allowing us to harness quantum-like power without the need for fragile, super-cold quantum hardware.

1. Problem Statement

The paper addresses the fundamental distinction between classical and quantum systems. While quantum systems exhibit unique correlations (entanglement) and superposition states that enable new technologies, classical systems are generally considered distinct. However, there is a growing interest in "quantum-like" (QL) phenomena in fields such as quantum biology, cognitive science, and classical optics.

The core problem this paper tackles is: Can composite classical systems be constructed to closely mimic the separable states of composite quantum systems? Specifically, the author seeks to prove the existence of physical classical systems whose states can be represented in a tensor product basis, emulating the mathematical structure of quantum bits (qubits) without requiring actual quantum mechanics. The paper also investigates the limits of this mimicry, specifically regarding non-separable (entangled) states.

2. Methodology

The author employs a rigorous framework combining Graph Theory, Linear Algebra, and Classical Wave Mechanics to construct and analyze QL systems.

Graph Representation of Classical Systems: The paper utilizes graph theory to model physical systems. A graph $G(n, m)$ represents the physical structure, where vertices correspond to components (e.g., oscillators or wave modes) and edges represent couplings.
QL Bit Construction:
- A single "QL bit" is modeled by coupling two $d$ -regular expander graphs.
- Expander graphs possess a unique "emergent state" (an eigenvalue separated from the rest of the spectrum).
- By coupling two such subgraphs with specific "edge biases" (complex numbers on the unit circle, $e^{i\theta}$ ), the system creates two new emergent states that are linear combinations of the subgraph states.
- These states are shown to map isomorphically to the group $SU(2)$, effectively mimicking a single qubit.
Composite Systems via Cartesian Products:
- To create composite systems (analogous to multi-qubit systems), the author uses the Cartesian product of graphs ( $G_A \square G_B \dots$ ).
- A key mathematical proposition is utilized: The spectrum (eigenvalues and eigenvectors) of a Cartesian product of graphs corresponds to the tensor product of the spectra of the individual graphs.
- This allows the construction of a classical system where the emergent states are separable superpositions in a tensor product basis ( $H_1 \otimes H_2 \otimes \dots$ ).
Physical Realization: The paper identifies physical candidates for these graphs, including:
- Polarization states of classical waves (light, elastic waves).
- Networks of coupled phase oscillators (Kuramoto model).
- Electronic "topological circuits."
- Multipole moments (dipolar, quadrupolar) in soft matter.

3. Key Contributions

A. Proof of Existence for Separable QL States

The paper provides a formal proof (Theorem 1) that classical systems exist which possess a measurable state space mimicking the separable states of quantum systems.

By tuning the "edge bias" (phase difference) in the coupling edges of the graph, the system can generate a continuous set of states representing all elements of the group $SU(2) \times \dots \times SU(2)$ .
This establishes that classical networks can structurally replicate the Hilbert space of composite quantum systems, provided the states are separable.

B. The No-Cloning Theorem in Classical Graphs

The paper demonstrates that the No-Cloning Theorem (Theorem 2) holds for these QL systems.

Even though the underlying network is classical, the specific structure of the adjacency matrix (which encodes phase relationships) prevents the unitary cloning of a QL state.
The proof shows that no unitary transformation can map a composite state $|a\rangle_A \otimes |b\rangle_B$ to $|a\rangle_A \otimes |a\rangle_B$ without altering the underlying edge structure of the graph, illustrating that the "quantum-like" behavior arises from the topological constraints of the network.

C. Analysis of Non-Separable (Entangled) States

The paper explores the limits of this mimicry regarding entanglement:

Classical Entanglement: It acknowledges that classical systems can exhibit non-separable states (e.g., "classically entangled" light), but these are strictly local correlations and do not violate Bell inequalities.
Graph Disconnection: The analysis suggests that generating maximally entangled states in this graph model requires disconnected graphs.
CNOT Gate Analogy: Applying a CNOT gate operation to the adjacency matrix of a separable QL graph results in a disconnected graph. Since a disconnected graph lacks a natural definition of a product basis (relative phases become random), the paper concludes that true quantum-like entanglement cannot be generated by classical graphs in a systematic way.

D. Optimization and Topology

The paper introduces an optimization technique (Appendix A) to reduce the resource cost of these graphs. By contracting the Cartesian product, the author shows that the essential correlation structure can be preserved in a much smaller graph, revealing that the topology of QL states mirrors the structure of Boolean posets (partially ordered sets) and Hasse diagrams.

4. Key Results

Mimicry of Separable States: Composite QL systems constructed via Cartesian graph products can exactly replicate the mathematical structure of separable quantum states.
Physical Realizability: Suitable physical systems exist, including:
- Waves: Polarization states of light or elastic waves (dipole/quadrupole moments).
- Oscillators: Networks of coupled phase oscillators where edge bias represents phase differences.
- Circuits: Topological electronic circuits with complex edge biases.
Limitation on Entanglement: While classical systems can mimic separable states and even non-separable (but locally correlated) states, they cannot naturally generate maximally entangled states without breaking the connectivity required to define the product basis.
Robustness: The nonlinearity inherent in classical oscillator networks (e.g., Kuramoto models) can produce robust superposition states, offering a potential resource for computation.

5. Significance and Implications

Quantum Biology & Soft Matter: The findings suggest that biological systems (e.g., photosynthetic complexes) or soft matter (emulsions, liquid crystals) could utilize classical network structures to achieve "quantum-like" coherence and efficiency without requiring fragile quantum coherence.
Computing and Sensing: The paper proposes a pathway for "intermediate" technologies between classical and quantum computing. Classical networks mimicking QL states could be used for sensing or solving specific problems (like NP-complete problems via memcomputing) by leveraging superposition resources in a robust, classical framework.
Theoretical Clarification: It clarifies the boundary between quantum and classical physics. It demonstrates that the structure of quantum states (specifically separable ones) is not exclusive to quantum mechanics but can emerge from the topology of classical networks. However, it reinforces that true non-local entanglement remains a uniquely quantum feature.
New Design Paradigm: The use of graph theory as a "blueprint" allows engineers to design classical systems with specific quantum-like properties by manipulating edge biases and network topology, opening new avenues for "topological circuits" and engineered materials.

In conclusion, Scholes demonstrates that classical systems can be engineered to possess a state space isomorphic to the separable states of quantum systems using graph theory and phase-coupled networks. While these systems cannot replicate true quantum entanglement, they offer a robust, physically realizable platform for exploring quantum-like phenomena in biology, materials science, and computing.