⚛️ quantum physics

Bell Experiments Revisited: A Numerical Approach Based on De Broglie--Bohm Theory

This paper presents a complete, pedagogical, and rigorous numerical model within the de Broglie–Bohm framework that explicitly demonstrates how a deterministic, nonlocal hidden-variable theory can reproduce all quantum-mechanical predictions of EPR–Bell experiments, including the violation of Bell inequalities.

Original authors: Tim Dartois, Signe Seidelin, Aurélien Drezet

Published 2026-03-25

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Tim Dartois, Signe Seidelin, Aurélien Drezet

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 Picture: The "Ghostly" Dance of Particles

Imagine you are watching a magic show. Two magicians, Alice and Bob, are standing on opposite sides of a giant stage, miles apart. They each have a box containing a mysterious, spinning coin.

In the standard view of quantum mechanics (the "Orthodox" view), these coins are like ghosts. Until you look at them, they aren't heads or tails; they are a blurry mix of both. The moment Alice looks at her coin, it instantly "decides" to be Heads. At that exact same instant, Bob's coin, miles away, instantly "decides" to be Tails. They seem to be communicating faster than light, which Einstein famously called "spooky action at a distance."

For decades, scientists argued: Is there a secret rulebook (hidden variables) that decided the outcome before the show started? Or is the universe truly random and connected in a spooky way?

John Bell came up with a test (a mathematical inequality) to settle this. If the coins were just following a secret rulebook (local realism), the results would stay within a certain limit. If they were "spooky" (quantum mechanics), they would break that limit. Experiments proved that nature breaks the limit. The universe is "spooky."

The Paper's Goal: Bringing the Ghosts Back to Earth

This paper asks a fascinating question: Can we explain this "spooky" behavior without giving up on the idea that particles are real, solid things with definite paths?

The authors use a theory called De Broglie–Bohm (dBB) theory (also known as Pilot-Wave theory).

The Analogy: Imagine a surfer (the particle) riding a wave (the quantum wave function).
In standard quantum mechanics, the surfer doesn't exist until you look; the wave is just probability.
In dBB theory, the surfer is always there, riding a specific path. The wave guides them. The wave is "non-local," meaning it stretches across the whole universe instantly. If you change the wave on Alice's side, the surfer on Bob's side feels it instantly, even though the surfer is just following a local path.

The authors built a computer simulation to show exactly how this works. They wanted to prove that a deterministic theory (where everything is pre-planned by the wave) can still produce the "spooky" results that break Bell's rules.

How the Experiment Works (The Simulation)

The authors didn't use real atoms; they built a digital model. Here is how they set it up:

The Source: A machine creates two "entangled" particles (like our magic coins) and sends them flying in opposite directions toward Alice and Bob.
The Twist (The Coils): Instead of physically rotating the detectors (which is hard to model in a simple 1D computer), they used "magnetic coils." Think of these as spin-flippers.
- Alice can flip her coin's spin by angle $\alpha$ .
- Bob can flip his by angle $\beta$ .
- This changes the "wave" guiding the particles without moving the physical detectors.
The Measurement: The particles hit a "Stern-Gerlach" device (a magnetic magnet).
- If the particle is guided "up," it's a "Spin Up" result.
- If "down," it's "Spin Down."
- In the simulation, the authors tracked the exact path (trajectory) of every single particle from start to finish.

The Results: What Did They See?

The simulation showed three distinct scenarios based on the angle between Alice and Bob's settings:

Perfect Opposites ( $\gamma = 0$ ): When the settings are aligned, the particles always go opposite ways. If Alice goes Up, Bob always goes Down. The simulation showed that Bob's particle actually crosses paths with other particles' paths to get to the "Down" side. It's like a traffic jam where cars swap lanes instantly because the "wave" told them to.
Perfect Twins ( $\gamma = \pi$ ): When the settings are flipped 180 degrees, the particles always go the same way. If Alice goes Up, Bob goes Up.
Total Chaos ( $\gamma = \pi/2$ ): When the settings are at 90 degrees, the particles act like strangers. Their results are random and uncorrelated.

The "Aha!" Moment:
The most important part of the paper is showing how this happens.

In the simulation, Bob's particle doesn't know what Alice did until the very last moment.
However, the Wave Function (the guide) knows everything.
When Alice changes her setting, she changes the shape of the global wave. This instantly changes the "traffic rules" for Bob's particle.
The Analogy: Imagine Alice and Bob are two dancers connected by a giant, invisible elastic band (the wave). If Alice suddenly spins, the elastic band stretches and pulls Bob, forcing him to change his dance move instantly. Bob isn't "thinking" about Alice; he is just reacting to the tension in the band. The band connects them instantly across the universe.

The Bell Inequality Violation

The authors ran the simulation thousands of times, counting the results.

They calculated the "Bell Score" (a number that measures how "spooky" the connection is).
Local theories (secret rulebooks) say the score must be $\le 2$ .
Quantum mechanics says the score can go up to $2\sqrt{2}$ (about 2.82).
The Simulation: The dBB model produced a score of 2.82.

Conclusion: The simulation proved that you can have a theory where particles have definite paths (determinism) and are real (hidden variables), yet still break the Bell limit. The "magic" comes from the fact that the guide wave is non-local. It connects everything instantly.

The "No-Signaling" Trick (Why We Can't Send Secret Messages)

You might ask: "If Alice can change Bob's particle instantly, why can't she send a secret Morse code message to him?"

The paper explains this beautifully using a map of the "Hidden Variables."

Imagine a giant circular map representing all the possible starting positions of the particles.
Depending on the settings, the map is divided into colored zones (e.g., Blue for "Up/Up", Red for "Down/Down").
The Catch: While the shape of the zones changes instantly when Alice flips her switch, the total amount of Blue vs. Red area on Bob's side stays exactly 50/50.
The Analogy: Imagine Alice is a chef changing the recipe for a cake. Bob is the customer.
- Alice changes the recipe (the wave).
- The cake Bob receives changes flavor instantly (the outcome changes).
- BUT, Bob only sees the flavor of his own slice. He doesn't know if the recipe changed because his slice is still 50% chocolate and 50% vanilla, just like before.
- To see the "spooky" connection, Alice and Bob must meet up later and compare their notes (the data). Only then do they realize, "Hey, every time I got Chocolate, you got Vanilla!"

Summary

This paper is a visual proof that the universe could be deterministic (like a clockwork machine) and still look "spooky" to us.

The Mechanism: Particles have real paths, but they are guided by a wave that stretches across the universe instantly.
The Result: This "non-local" guidance allows the particles to coordinate their dance perfectly, breaking Bell's rules.
The Safety: Even though they coordinate instantly, they can't use it to send text messages faster than light, because the local results always look random to the individual observer.

It's a beautiful demonstration that reality can be both "real" (particles exist) and "spooky" (connected instantly), depending on how you look at the invisible wave that guides them.

1. Problem Statement

Despite the widespread acceptance of the de Broglie–Bohm (dBB) theory (also known as Pilot-Wave theory) as a deterministic, non-local hidden-variable interpretation of quantum mechanics, there is a lack of rigorous, self-contained pedagogical models that explicitly demonstrate how dBB reproduces Bell inequality violations.

The Gap: While dBB is known to be non-local, previous literature often lacks a complete numerical treatment of the EPR–Bell experiment involving spin-1/2 particles in three-dimensional space with varying analyzer orientations.
The Challenge: Constructing a model that maintains the deterministic nature of particle trajectories while reproducing the statistical predictions of quantum mechanics (specifically the violation of Bell inequalities) and illustrating the "no-signaling" theorem (impossibility of superluminal communication) within a non-local framework.

2. Methodology

The authors construct a complete numerical model of an EPR–Bell experiment within the dBB framework, utilizing a one-dimensional approximation to simplify the dynamics while preserving the essential physics.

Theoretical Framework:
- The system is governed by the Schrödinger–Pauli equation for spin-1/2 particles.
- Guidance Equation: Particle trajectories are determined by the velocity field $\vec{v} = \vec{j}/\rho$ , where $\vec{j}$ is the probability current and $\rho$ is the probability density.
- Quantum Equilibrium: Initial particle positions are sampled according to the Born rule ( $\rho = |\Psi|^2$ ) to ensure statistical agreement with standard quantum mechanics.
Experimental Setup (Numerical Model):
- Source: Emits two spin-entangled particles in a singlet state (Bell state) moving in opposite directions toward Alice and Bob.
- Spin Manipulation: Instead of physically rotating Stern–Gerlach (SG) analyzers (which would complicate the 1D model), the authors introduce magnetic coils (spin-flippers) on each branch. These coils rotate the spin of the particles by angles $\alpha$ (Alice) and $\beta$ (Bob) before they enter fixed SG analyzers oriented along the $z$ -axis.
- Measurement: The SG apparatus converts spin information into spatial separation. The final spin outcome is determined by the sign of the particle's final position ( $z > 0$ for spin up, $z < 0$ for spin down).
- Time Ordering: The model explicitly enforces a time lag where Alice measures her particle well before Bob, allowing for a clear analysis of non-local influence.
Numerical Implementation:
- Wave Function Evolution: The authors derive analytical forms for the wave function during free evolution, spin rotation (via magnetic coils), and interaction with SG fields.
- Intermediate Regimes: To handle the transition between free evolution and SG interaction, they use a smooth interpolation of probability currents and densities to maintain quantum equilibrium.
- Sampling: Initial conditions $(z_A, z_B)$ are sampled using the inversion method applied to the cumulative distribution function of the initial Gaussian wave packets.
- Simulation: The coupled first-order differential equations for the trajectories of entangled pairs are integrated numerically for thousands of initial conditions.

3. Key Contributions

Unified Pedagogical Model: The paper provides a rigorous, self-contained numerical simulation that bridges the gap between abstract dBB theory and concrete experimental outcomes. It explicitly visualizes how deterministic trajectories lead to quantum correlations.
1D Spin-Flipper Approximation: The authors successfully demonstrate that rotating spins via magnetic coils in a 1D model is mathematically equivalent to rotating SG analyzers in 3D, simplifying the computational complexity without losing physical fidelity.
Visualization of Non-Locality: By plotting trajectories for different relative angles ( $\gamma = \beta - \alpha$ ), the paper visually demonstrates how a local change in Alice's setting instantaneously alters the guidance field and trajectory of Bob's particle, even though Bob's local statistics remain unchanged.
Hidden Variable Space Visualization: The authors map the initial conditions (hidden variables) onto a unit disk. By coloring these points based on the final joint outcomes, they provide a geometric representation of how the "domains" of outcomes shift with experimental settings while preserving marginal probabilities.

4. Results

Trajectory Dynamics:
- $\gamma = 0$ (Singlet): Trajectories exhibit perfect anti-correlation. If Alice's particle is deflected up, Bob's is deflected down. The trajectories cross in physical space (a feature of configuration space dynamics), ensuring the anti-correlation.
- $\gamma = \pi$ : Trajectories exhibit perfect correlation.
- $\gamma = \pi/2$ : The system factorizes into two independent one-particle dynamics; trajectories show no correlation or crossing.
Bell Inequality Violation:
- The authors calculated the Bell–CHSH parameter $M(\theta)$ numerically.
- The results closely match the quantum mechanical prediction: $M(\theta) = 3\cos(\theta/2) - \cos(3\theta/2)$ .
- The simulation confirms the violation of the local bound ( $|M| \le 2$ ), reaching the Tsirelson bound ( $2\sqrt{2}$ ), proving that dBB reproduces quantum correlations.
No-Signaling Theorem:
- Despite the explicit non-locality (Bob's trajectory depends on Alice's setting $\alpha$ ), the marginal statistics for Bob ( $P(B=+)$ and $P(B=-)$ ) remain exactly $0.5$ regardless of Alice's choice of $\alpha$ .
- The geometric representation on the unit disk shows that while the boundaries between outcome domains shift non-locally with $\gamma$ , the area (probability) of each local outcome remains constant. This confirms that no superluminal signaling is possible.

5. Significance

Clarification of dBB Non-Locality: The paper offers a concrete illustration of how a deterministic theory can violate Bell inequalities. It dispels the misconception that dBB is a "local" theory by showing exactly how the guidance equation couples distant particles.
Pedagogical Value: It serves as a powerful tool for teaching the foundations of quantum mechanics, demonstrating that "spooky action at a distance" is a feature of the dynamics (guidance field) rather than a violation of causality in terms of information transfer.
Foundation for Future Work: The authors suggest that this framework can be extended to:
- Three-dimensional simulations with arbitrary analyzer orientations.
- Direct integration of the Pauli equation in interaction zones (avoiding interpolation).
- Non-equilibrium scenarios: Investigating cases where $\rho \neq |\Psi|^2$ , which could theoretically allow for signaling, thereby highlighting the critical role of the quantum equilibrium hypothesis.

In conclusion, the paper successfully demonstrates that the de Broglie–Bohm theory is fully capable of reproducing all statistical predictions of standard quantum mechanics, including Bell inequality violations, while maintaining a deterministic and non-local ontology, all without enabling faster-than-light communication.