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

Abstract

We present a complete and rigorous model of an EPR--Bell-type experiment within the framework of the de Broglie--Bohm theory. The purpose of this work is to show explicitly how a deterministic hidden-variable theory can reproduce all quantum-mechanical predictions, including the violation of Bell inequalities. Combining analytical arguments with numerical simulations, our approach offers a unified and transparent illustration of the central ingredients of de Broglie--Bohm theory, including particle trajectories, spin dynamics, and quantum entanglement. The model is designed to be pedagogical and self-contained, making it suitable for readers seeking a concrete understanding of how a nonlocal hidden-variable theory can describe the EPR--Bell experiment and illustrate Bell's theorem.

Figures (4)

• Figure 1: Schematic of the EPR--Bell experiment considered in this work. A source $S$ emits two spin-entangled particles prepared in a singlet state and sent toward two distant observers, Alice (left) and Bob (right). Each particle first passes through a local magnetic coil, which rotates its spin by an angle $\alpha$ (Alice) or $\beta$ (Bob) about the horizontal axis. Both Stern--Gerlach analyzers are fixed and oriented along the $z$ direction, so that spin information is converted into a spatial separation along $z$. The protocol is explicitly time-ordered: Alice’s particle crosses its coil and Stern--Gerlach apparatus and is measured well before Bob’s particle reaches his analyzer. This temporal asymmetry allows one to identify unambiguously Alice’s outcome prior to Bob’s measurement, while still exhibiting nonlocal correlations in the Bohmian dynamics. The coils thus play the role of Bell’s variable analyzer orientations, without requiring any physical rotation of the Stern--Gerlach devices, and preserve the essentially one-dimensional character of the model.
• Figure 2: DBB trajectories in the EPR--Bell pilot-wave model for three values of the relative coil angle $\gamma=\beta-\alpha$. In each panel, the left (right) plot shows $z_A(t)$ for Alice ($z_B(t)$ for Bob); trajectories belonging to the same entangled pair share the same color. The correlation pattern changes with $\gamma$: perfect anticorrelation at $\gamma=0$, perfect correlation at $\gamma=\pi$, and an essentially factorized (uncorrelated) behavior at $\gamma=\pi/2$. All trajectories shown in the different panels are generated from the same set of initial conditions, allowing a direct comparison of the dynamics for different values of $\gamma$.
• Figure 3: Bell--CHSH parameter for the EPR--Bell pilot-wave simulation. For each value of $\theta$, the correlation combination $M(\theta)=\langle S(0,\theta,\tfrac{\theta}{2},\tfrac{3\theta}{2})\rangle$ is estimated from Bohmian trajectories by extracting binary outcomes $a_\alpha=\mathrm{sgn}(z_A)$ and $b_\beta=\mathrm{sgn}(z_B)$ at late times. Crosses: numerical estimates obtained from $200$ values of $\theta$ with $2000$ entangled pairs per value (quantum-equilibrium sampling). Solid curve: quantum prediction $M(\theta)=3\cos(\tfrac{\theta}{2})-\cos(\tfrac{3\theta}{2})$. The region where $|M(\theta)|>2$ demonstrates the violation of the local bound, while the maximum is consistent with the Tsirelson limit $2\sqrt{2}$.
• Figure 4: Hidden-variable (unit-disk) representation of outcomes in the EPR--Bell pilot-wave model for four relative coil angles $\gamma=\beta-\alpha$. Each point corresponds to one initial condition, uniformly sampled on the unit disk (quantum equilibrium) and mapped to $(z_A(0),z_B(0))$. Colors label the four joint outcomes $(s_A,s_B)\in\{+,-\}^2$ inferred from the late-time signs $s_A=\mathrm{sgn}(z_A)$ and $s_B=\mathrm{sgn}(z_B)$: blue for $(+,+)$, orange for $(+,-)$, green for $(-,+)$, and red for $(-,-)$. The partitions of the disk evolve with $\gamma$: the left/right split reflects Alice’s earlier measurement, while the curved boundaries encode Bob’s conditional outcome statistics and reproduce the quantum probabilities $P(++ )=P(--)=\tfrac{1}{2}\sin^2(\gamma/2)$ and $P(+-)=P(-+)=\tfrac{1}{2}\cos^2(\gamma/2)$. The solid line shows the theoretical separatrix between the outcome domains, and the numerically sampled regions coincide with it (within numerical resolution), demonstrating perfect agreement between simulation and theory. Although the domain shapes depend nonlocally on $\gamma$, the marginal outcome statistics remain setting-independent and balanced: for any $\gamma$, $P_A(+)=P_A(-)=\tfrac{1}{2}$ and $P_B(+)=P_B(-)=\tfrac{1}{2}$, so Alice and Bob each always measure half $+$ and half $-$ outcomes. This illustrates no signaling at the statistical level.