The de Broglie–Bohm formulation assigns deterministic trajectories to quantum particles, with inter-particle coupling encoded in the phase structure of the joint wavefunction. This makes it a natural framework for studying how quantum entanglement shapes kinematic behavior at the individual-trajectory level. We present a controlled numerical investigation of two-particle scattering against a Gaussian barrier, comparing three initial-state configurations: a separable product state (Stage 1), a bosonic symmetrised state (Stage 2A), and a product state subjected to entanglement generated dynamically via a time-ramped quadratic phase coupling (Stage 2B). The three-stage design enforces a strict causal control: Stage 2B shares identical single-particle marginal densities with the product baseline at t = 0, confirmed by DKL = 0 exactly, so any divergence in scattering statistics is attributable solely to the entanglement generated during evolution. The central finding is that both entangled stages produce a scattering independence ratio Isc <1: joint transmission is suppressed relative to the independent-particle prediction, yielding anti-correlated scattering outcomes rather than the bunching one might naively expect. The effect is moderate for bosonic symmetrisation (Isc = 0.75) and extreme for the phase-coupled stage (Isc = 0.09), where joint transmission collapses to fewer than 2% of trajectories. Inter-particle velocity correlations rise from near zero in the product state to substantial values in both entangled stages, with the dynamically generated case exhibiting a characteristic delayed onset that tracks the phase ramp. Entanglement entropy measurements are mutually consistent with this metric ordering across all stages. These results provide controlled numerical evidence, within a low-dimensional model, that Bohmian non-locality leaves quantifiable signatures in scattering outcome distributions. Beyond the Bohmian context, the independence ratio Isc functions as a transport-correlation diagnostic with potential relevance to weak-measurement trajectory-reconstruction experiments and pilot-wave analogue systems.
The de Broglie–Bohm formulation assigns deterministic trajectories to quantum particles, with inter-particle coupling encoded in the phase structure of the joint wavefunction. This makes it a natural framework for studying how quantum entanglement shapes kinematic behavior at the individual-trajectory level. We present a controlled numerical investigation of two-particle scattering against a Gaussian barrier, comparing three initial-state configurations: a separable product state (Stage 1), a bosonic symmetrised state (Stage 2A), and a product state subjected to entanglement generated dynamically via a time-ramped quadratic phase coupling (Stage 2B). The three-stage design enforces a strict causal control: Stage 2B shares identical single-particle marginal densities with the product baseline at t = 0, confirmed by DKL = 0 exactly, so any divergence in scattering statistics is attributable solely to the entanglement generated during evolution. The central finding is that both entangled stages produce a scattering independence ratio Isc <1: joint transmission is suppressed relative to the independent-particle prediction, yielding anti-correlated scattering outcomes rather than the bunching one might naively expect. The effect is moderate for bosonic symmetrisation (Isc = 0.75) and extreme for the phase-coupled stage (Isc = 0.09), where joint transmission collapses to fewer than 2% of trajectories. Inter-particle velocity correlations rise from near zero in the product state to substantial values in both entangled stages, with the dynamically generated case exhibiting a characteristic delayed onset that tracks the phase ramp. Entanglement entropy measurements are mutually consistent with this metric ordering across all stages. These results provide controlled numerical evidence, within a low-dimensional model, that Bohmian non-locality leaves quantifiable signatures in scattering outcome distributions. Beyond the Bohmian context, the independence ratio Isc functions as a transport-correlation diagnostic with potential relevance to weak-measurement trajectory-reconstruction experiments and pilot-wave analogue systems.
Bohmian mechanics, also referred to as the de Broglie-Bohm pilot-wave theory, represents a deterministic and nonlocal interpretation of quantum mechanics. Central to this framework is a description of quantum motion in terms of particle trajectories, whose reconciliation with relativity remains a fundamental challenge. Although relativistic guidance laws have been proposed, their predicted dynamics has remained experimentally untested. Here we use weak measurements in a single-photon interferometer to jointly reconstruct the photon energy and momentum weak values and the associated relativistic velocity field. The reconstructed average trajectories agree with theoretical predictions. Near destructive interference, the flow exhibits local reconstructed velocities exceeding the speed of light and negative values of the effective squared-mass parameter. Our results establish an experimental basis for investigating the physical meaning of quantum trajectories and their role in describing motion in relativistic spacetime.
We revisit the concept of entanglement within the Bohmian approach to quantum mechanics. Inspired by Bohmian dynamics, we introduce two partial measures for the amount of entanglement corresponding to a pure state of a pair of quantum particles. One of these measures is associated with the statistical correlations exhibited by the joint probability density of the two Bohmian particles in configuration space. The other partial measure corresponds to the correlations associated with the phase of the joint wave function, and describes the non-separability of the Bohmian velocity field. The sum of these two components is equal to the total entanglement of the joint quantum state, as measured by the linear entropy of the single-particle reduced density matrix.
Abstract Bohmian mechanics supplements the quantum wavefunction with deterministic particle trajectories, offering an alternate, dynamical language for quantum theory. However, the Bohmian wavefunction evolves independently of these trajectories, and is thus unaffected by the observable properties of the system. While this property is widely assumed necessary to ensure agreement with quantum mechanics, much work has recently been dedicated to understanding classical pilot-wave systems, which feature a two-way coupling between particle and wave. These systems—including the “walking droplet” system of Couder and Fort (Couder and Fort (2006) Phys. Rev. Lett. 97:154101) and its various abstractions (Dagan and Bush (2020) CR Mecanique 348:555–571; Durey and Bush (2020) Front. Phys. 8:300; (2021) Chaos 31:033136; Darrow and Bush (2024) Symmetry 16:149)—allow us to investigate the limits of classical systems and offer a touchstone between quantum and classical dynamics. In this work, we present a general result that bridges Bohmian mechanics with this classical pilot-wave theory. Namely, Darrow and Bush ((2024) Symmetry 16:149) recently introduced a Lagrangian pilot-wave framework to study quantum-like behaviours in classical systems; with a particular choice of particle-wave coupling, they recover key dynamics hypothesised in de Broglie’s early double-solution theory (de Broglie (1970) Foundations Phys. 1:5–15). We here show that, with a different choice of coupling, their de Broglie-like system reduces exactly to single-particle Bohmian mechanics in the non-relativistic limit. Our result clarifies that, while multi-particle entanglement is impossible to replicate in general with local, classical theories, no such restriction exists for single-particle quantum mechanics. Moreover, connecting with the previous work of Darrow and Bush, our work demonstrates that de Broglie’s and Bohm’s theories can be connected naturally within a single Lagrangian framework. Finally, we present an application of the present work in developing a single-particle analogue for position measurement in a de Broglie-like setting.
We develop an explicit configuration-space formulation of EPR-Bell experiments within the framework of de Broglie-Bohm theory, in which joint measurement outcomes arise from a deterministic mapping from initial particle configurations to outcome pairs. This construction induces a partition of the hidden-variable configuration space into domains associated with the different measurement outcomes. Using a reduced-dimensional Stern-Gerlach model, we derive the structure of these domains and identify the corresponding separatrices that define their boundaries. We show that Bell correlations emerge from the geometry of these partitions: the domain boundaries depend nonlocally on the measurement settings, while the marginal outcome distributions remain invariant, providing a direct dynamical realization of no-signaling. Analytical results are supported by numerical simulations, which exhibit quantitative agreement with the predicted domain structure as a consequence of the underlying partition of configuration space induced by the measurement dynamics. This approach provides an explicit configuration-space representation of nonlocal correlations in Bohmian mechanics, linking trajectory dynamics, measurement processes, and statistical predictions within a unified framework.
Abstract We investigate the dynamics of quantum correlations in bipartite systems initially prepared in a squeezed state, comparing closed-system unitary evolution under the Schrödinger equation with open-system dynamics governed by the Caldeira-Leggett master equation in the high-temperature, weak-coupling regime, all within the Bohmian mechanics framework. Quantum nonlocality is quantified via the sensitivity of the Bohmian velocity of one particle to the position of the other. Our results show that in both distinct (local) and common bath scenarios, nonlocal correlations initially grow from zero, reach a peak, and then decay. In the case of local baths, the decay is smooth and monotonic; although the peak value increases with temperature, its temporal width (measured via the full width at half maximum) decreases, indicating a shorter duration of nonlocal correlations. For a common bath, the initial growth and decay are followed by revivals and oscillations, whose amplitude and timing vary with temperature. These non-monotonic behaviors arise despite the Markovian nature of the underlying dynamics and reflect the nontrivial role of system-bath correlations. We also analyze how both temperature and the squeezing decay parameter affect the structure of Bohmian trajectories and the evolution of nonlocal correlations. This trajectory-based, velocity-sensitivity measure offers an intuitive and quantitative understanding of entanglement degradation, decoherence, and their characteristic time scales. Our findings emphasize how the structure of the environment critically shapes the observable dynamics of quantum correlations, even in Markovian regimes.
Abstract The recent experimental proposals by Bose et al. and Marletto et al. (BMV) outline a way to test for the quantum nature of gravity by measuring gravitationally induced differential phase accumulation over the superposed paths of two ∼ 10−14 kg masses. These authors outline the expected outcome of these experiments for semi-classical, quantum gravity and collapse models. It is found that both semi-classical and collapse models predict a lack of entanglement in the experimental results. This work predicts the outcome of the BMV experiment in Bohmian trajectory gravity - where classical gravity is assumed to couple to the particle configuration in each Bohmian path, as opposed to semi-classical gravity where gravity couples to the expectation value of the wave function, or of quantized gravity, where the gravitational field is itself in a quantum superposition. In the case of the BMV experiment, Bohmian trajectory gravity predicts that there will be quantum entanglement. This is surprising as the gravitational field is treated classically. A discussion of how Bohmian trajectory gravity can induce quantum entanglement for a non superposed gravitational field is put forward.