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Entropy

Revisiting Entanglement within the Bohmian Approach to Quantum Mechanics

Claudia Zander, Angel Ricardo Plastino

Published Jun 18, 2018• 17 citations
Publisher

Abstract

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.

DOI
10.3390/e20060473

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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.
Since its inception Bohmian mechanics has been generally regarded as a hidden-variable theory aimed at providing an objective description of quantum phenomena. To date, this rather narrow conception of Bohm’s proposal has caused it more rejection than acceptance. Now, after 65 years of Bohmian mechanics, should still be such an interpretational aspect the prevailing appraisal? Why not favoring a more pragmatic view, as a legitimate picture of quantum mechanics, on equal footing in all respects with any other more conventional quantum picture? These questions are used here to introduce a discussion on an alternative way to deal with Bohmian mechanics at present, enhancing its aspect as an efficient and useful picture or formulation to tackle, explore, describe and explain quantum phenomena where phase and correlation ( entanglement ) are key elements. This discussion is presented through two complementary blocks. The first block is aimed at briefly revisiting the historical context that gave rise to the appearance of Bohmian mechanics, and how this approach or analogous ones have been used in different physical contexts. This discussion is used to emphasize a more pragmatic view to the detriment of the more conventional hidden-variable (ontological) approach that has been a leitmotif within the quantum foundations. The second block focuses on some particular formal aspects of Bohmian mechanics supporting the view presented here, with special emphasis on the physical meaning of the local phase field and the associated velocity field encoded within the wave function. As an illustration, a simple model of Young’s two-slit experiment is considered. The simplicity of this model allows to understand in an easy manner how the information conveyed by the Bohmian formulation relates to other more conventional concepts in quantum mechanics. This sort of pedagogical application is also aimed at showing the potential interest to introduce Bohmian mechanics in undergraduate quantum mechanics courses as a working tool rather than merely an alternative interpretation.
This chapter considers a more realistic variation of the measuring procedure, taking into account the fact that the measured particles become entangled with the measuring apparatus at some finite time. The results are also generalized to quantum field theory, where quantum states are represented by wave functions depending on an infinite number of space-time coordinates. Thus, to make Bohmian mechanics covariant, one needs first to reformulate the standard quantum mechanics (QM) in a covariant way such that time is treated on an equal footing with space. Another physically interesting situation is when the entanglement with the environment takes the form, where are coherent states. In general, in the Bohmian interpretation all the other quantities may have a distribution totally different from those predicted by purely probabilistic QM. The space probability density should be generalized to the space-time probability density.
We propose a new approach to the problem of defining the degree of entanglement between two particles in a pure state with Hilbert spaces of arbitrary finite dimensions. The central idea is that entanglement gives rise to correlations between the particles that do not occur in separable states. We individuate the contributions of these correlations to the joint and the conditional probabilities of local measurements outcomes. We use these probabilities to define the measure of entanglement. Our measure turns out to be proportional to the so-called 2-entropy and therefore satisfies the properties required for any measure of entanglement. We conclude with an outlook on the problem of extending our approach to the case of multipartite systems and mixed states.
We consider the concept of velocity fields, taken from Bohmian mechanics, to investigate the dynamical effects of entanglement in bipartite realizations of Young's two-slit experiment. In particular, by comparing the behavior exhibited by factorizable two-slit states (cat-type state analogs in the position representation) with the dynamics exhibited by a continuous-variable Bell-type maximally entangled state, we find that, while the velocity fields associated with each particle in the separable scenario are well-defined and act separately on each subspace, in the entangled case there is a strong deformation in the total space that prevents this behavior. Consequently, the trajectories for each subsystem are not constrained any longer to remain confined within the corresponding subspace; rather, they exhibit seemingly wandering behavior across the total space. In this way, within the subspace associated with each particle (that is, when we trace over the other subsystem), not only interference features are washed out, but also the so-called Bohmian non-crossing rule (i.e., particle trajectories are allowed to get across the same point at the same time).
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.