⚛️ high-energy theory

Entanglement Entropy in Non Relativistic Collisions

This paper numerically demonstrates that despite significant variations in wave functions and density matrices across different non-relativistic collision scenarios involving local potentials, the temporal build-up trends and values of entanglement entropy remain surprisingly consistent.

Original author: Bertrand Giraud (IPhT Saclay), Robi Peschanski (IPhT Saclay)

Published 2026-11-05

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CC0 1.0

Original author: Bertrand Giraud (IPhT Saclay), Robi Peschanski (IPhT Saclay)

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet corners of the quantum world, particles do not always behave as solitary travelers. When two objects collide, they can become inextricably linked, a phenomenon where the state of one instantly reflects the state of the other, regardless of the distance separating them. This connection, known as entanglement, is a fundamental feature of nature that challenges our everyday intuition. Physicists have long understood how this link forms when particles move at the speed of light, governed by the laws of relativity. However, a lingering question remained: how does this connection build up in the slower, more familiar world of everyday matter, where particles move at ordinary speeds? Specifically, scientists wanted to know how the "entanglement entropy"—a measure of how deeply two particles are intertwined—grows over time as they crash into one another and then drift apart.

To answer this, researchers at the Institute of Theoretical Physics in France set out to simulate the collision of two simple, non-relativistic particles. They imagined two tiny, invisible spheres moving toward each other in a one-dimensional space, like beads on a wire. These particles were initially separate, each carrying its own distinct identity, with no prior connection between them. The scientists then introduced a force between them, either pushing them apart like two magnets with the same pole facing each other, or pulling them together like opposite poles. They watched, through detailed computer calculations, how the wave-like nature of these particles evolved from the moment they began to approach, through the moment of impact, and into the moment they separated again.

The researchers explored several different scenarios to see if the outcome depended on the specific conditions of the crash. In one case, they set up a barrier that the particles had to either bounce off or tunnel through, a quantum trick where a particle passes through a wall it seemingly shouldn't be able to cross. They tested this at low speeds, where tunneling is common, and at higher speeds, where the particles simply fly over the barrier. They also examined a situation where the particles were attracted to each other, falling into a potential "pocket" before scattering. In every single scenario, the team tracked the changing relationship between the two particles, calculating a specific value that quantifies their entanglement.

What they found was surprisingly consistent. Regardless of whether the particles were repelling or attracting, or whether they were moving slowly or quickly, the process of entanglement followed a remarkably similar path. At the very beginning, when the particles were far apart, the connection was zero. As they approached and interacted, the entanglement grew steadily. It did not jump suddenly or fluctuate wildly; instead, it rose smoothly and then leveled off, settling into a stable, permanent value once the particles had moved far enough apart to stop influencing each other. This pattern looked like a gentle S-curve, rising from nothing to a fixed limit.

The study revealed that the final amount of entanglement reached was of a similar magnitude across all the different physical situations tested. Whether the particles bounced off a wall, tunneled through it, or were drawn together by attraction, the resulting "link" between them ended up being of the same order of strength. The researchers observed that as time passed, the particles moved apart, and the force between them effectively switched off, leaving the system with a constant, unchanging level of entanglement. This suggests that the act of scattering itself is what imprints this connection, and once the collision is over, the degree of entanglement is fixed.

This work provides a clear picture of how quantum connections form in the slow-moving world, filling a gap left by earlier theories that only looked at high-speed collisions. The simulations show that entanglement is not a chaotic or unpredictable result of a crash, but a process that builds up in a predictable, orderly fashion. While the specific details of the particles' paths varied wildly depending on the forces involved, the story of their connection remained the same: a steady rise to a stable limit. The authors note that while their computer models strongly suggest this behavior holds true for all such collisions, a general mathematical proof for this universal pattern is still needed. For now, the evidence points to a quiet, consistent rule governing how the quantum world ties itself together, even in the simplest of collisions.

Technical Summary: Entanglement Entropy in Non-Relativistic Collisions

Problem Statement
Recent relativistic S-matrix theory has established a framework for measuring entanglement in both elastic and inelastic collisions of elementary particles. However, the S-matrix formalism is inherently limited to asymptotic times, leaving open the question of how entanglement evolves at finite times. This paper addresses that gap by investigating the build-up of entanglement entropy between two colliding, non-relativistic particles driven by the time-dependent Schrödinger equation. The study focuses on systems governed by local, finite-range potentials, starting from an initial state defined as a product of two independent wave packets.

Methodology
The authors employ a numerical approach to solve the time-dependent Schrödinger equation for a two-body system in $d$ dimensions (primarily utilizing 1D toy models for computational tractability).

Formalism: The system consists of two spinless particles of equal mass ( $m=1$ ) interacting via a local, Galilean-invariant scalar potential $V(r)$ . The Hamiltonian is separated into center-of-mass (cm) and relative motion components. The cm degree of freedom is treated as a passive, factorizable Gaussian wave packet, while the entanglement dynamics are driven entirely by the relative motion wave function $\psi_{rel}(\vec{r}; t)$ .
Entanglement Measure: Entanglement is quantified using the quantity $I = S_1 + S_2 - S_{12}$ , where $S_i$ are the von Neumann entropies of the reduced density matrices ( $\rho_1, \rho_2$ ) and $S_{12}$ is the entropy of the total pure state density matrix ( $\rho_{12}$ ). For a pure total state, $S_{12}=0$ , making $I$ a direct measure of the deviation of the total state from a product state.
Initial Conditions: The simulations begin with a product state $\psi(1, 2; 0) = \phi_1 \phi_2$ , ensuring zero initial entanglement ( $I=0$ ). The particles are initialized as Gaussian wave packets separated by a distance $2\sigma$ and boosted toward each other with momentum $\pm k$ .
Numerical Implementation: The time evolution of the relative wave function is computed numerically. Reduced density matrices are obtained via partial traces, and entropies are calculated at discrete time steps.

Key Contributions and Results
The study numerically computes the time evolution of entanglement entropy across four distinct physical scenarios involving a Gaussian potential $V(r) = \lambda e^{-\mu r^2}$ :

Repulsive Barrier (Tunneling): A low-energy collision ( $k=0.55$ ) where the particle energy is below the barrier height, resulting in significant tunneling and reflection.
Repulsive Barrier (High Energy): A collision where the relative energy is twice the barrier height ( $k=\sqrt{2}$ ).
Repulsive Barrier (Resonant Energy): A collision where the relative energy equals the barrier height ( $k=1$ ).
Attractive Potential: A collision involving an attractive pocket ( $V = -e^{-x^2}$ ) with energy equal to the magnitude of the previous barrier height ( $k=1$ ).

Findings:

Universal Build-up Trend: Despite significant differences in the physical wave functions and density matrices across the four scenarios (ranging from tunneling effects to simple transmission), the time evolution of the entanglement entropy $I(t)$ exhibits a strikingly similar behavior. In all cases, $I(t)$ increases monotonically and appears to saturate.
Asymptotic Stability: The paper observes that as time progresses and the particles separate (the relative distance $\langle x \rangle$ grows linearly with time), the interaction potential effectively switches off. Consequently, the entanglement entropy reaches a stable, asymptotic value.
Magnitude Consistency: The calculated values for the asymptotic entanglement entropy are of the same order of magnitude across different energy regimes and potential types (repulsive vs. attractive). For instance, in the repulsive cases, $I$ stabilizes around values between 1.1 and 2.2, while the attractive case stabilizes around 1.2.
Comparison to Relativistic Results: The authors note that the magnitudes of entanglement found in this non-relativistic approach are comparable to those found in the relativistic S-matrix approach of Ref. [1].

Significance and Claims
The primary significance of this work is providing a concrete answer to the question of finite-time entanglement evolution, which was previously inaccessible via S-matrix theory alone.

Resolution of Open Question: The study demonstrates that entanglement entropy grows from zero to a finite asymptotic value during the collision process, following a sigmoid-like pattern.
Universality: The authors claim that the specific details of the potential (tunneling vs. reflection, attraction vs. repulsion) do not drastically alter the trend or the order of magnitude of the entanglement build-up.
Modest Conjectures: While the numerical evidence strongly suggests that this sigmoidal saturation behavior is universal for such collisions, the authors explicitly state that a general theorem proving this behavior or providing rigorous bounds for $I$ remains to be established. They confirm the consistency with relativistic results but acknowledge the lack of a general theoretical bound in the non-relativistic regime.

In conclusion, the paper establishes that for non-relativistic collisions with finite-range potentials, entanglement is generated dynamically during the interaction and stabilizes once the particles separate, with the final entropy value being robust against variations in collision energy and potential type.

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