⚛️ quantum physics

Entangled quantum trajectories in relativistic systems

This paper establishes a consistent framework for analyzing entangled quantum trajectories in relativistic systems by deriving Euler-Lagrange equations under non-entangling constraints, which are then applied to interacting particles in a Klein-Gordon setting to quantify the dynamic and relativistic impacts on quantum correlations.

Original author: Yannick Noel Freitag, Julien Pinske, Jan Sperling

Published 2026-11-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original author: Yannick Noel Freitag, Julien Pinske, Jan Sperling

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to send a secret message between two people who are moving at different speeds, perhaps one on a fast satellite and another on the ground. In the world of quantum physics, the message is carried by particles that are "entangled," a strange connection where the state of one particle instantly influences the other, no matter the distance. This link is the backbone of future technologies like unhackable communication and powerful computers. However, when these particles move at speeds close to light or experience different gravitational pulls, the rules of time and space change for each of them. This creates a puzzle: does the connection between the particles break, or does it look different depending on who is watching? For decades, scientists have struggled to describe this relationship consistently, because standard methods often suggest that the strength of the connection changes simply because an observer is moving, which feels wrong for a fundamental law of nature.

A team of researchers has now proposed a new way to look at this problem, treating the history of each particle as a continuous path through space and time, rather than just a snapshot of where it is at a single moment. By focusing on these entire paths, they developed a mathematical framework that allows them to track whether two particles are truly connected or just moving independently, regardless of how fast they are traveling or which direction they are facing. Their work suggests that while the appearance of the connection might change for a moving observer, the underlying classification of whether the particles are linked remains the same if you look at their full journey through time.

The researchers started by realizing that the usual way of defining entanglement—checking if two particles are linked at a specific instant—fails when relativity is involved. In the old view, if you take a snapshot of two particles that are not linked, and then zoom past them at high speed, the snapshot might suddenly show them as linked. This would mean that the fundamental nature of their relationship depends on the observer's speed, a conclusion that creates confusion in physics. To fix this, the team introduced the idea of "trajectories," which are like the complete movie of a particle's life, including every moment it exists, rather than just a single frame. They defined a "factorizable" trajectory as one where two particles move independently, each with their own separate story, and an "entangled" trajectory as one where their stories are inextricably woven together across time and space.

To test this idea, the scientists created a set of rules, or equations, that describe how these independent paths should behave if they are truly not connected. They then compared these "independent" paths against the actual paths that particles take when they interact with each other in a realistic physical model. In their model, they simulated two particles moving through space and time, interacting through a force that depends on the distance between them in both space and time. They calculated the paths these particles would take if they were allowed to become entangled, and then calculated what the paths would look like if they were forced to remain separate.

The results showed a clear difference. When the particles were allowed to interact naturally, their paths became deeply intertwined, creating a complex web of correlations that spanned across time and space. When the researchers forced the paths to remain separate, the particles followed much simpler, independent routes. By comparing these two scenarios, they could measure exactly how much the particles were entangled. They found that the entanglement grew stronger as the interaction between the particles increased. Crucially, they tested what happened when they changed the speed of the observer, simulating a shift to a different reference frame. In the old way of looking at things, this shift would have made the particles appear to gain or lose their connection. But in their new framework, the classification of the trajectory as entangled or factorizable remained invariant. The connection was a property of the entire journey, not just a single moment, so the fundamental status of the link did not vanish or appear just because the observer changed their speed.

The team also looked at what happens to the "spatial" connection—the link between where the particles are located—when the observer moves. They found that even if the particles started out with no spatial connection in one frame, a moving observer might see them as spatially linked. However, this was just a change in perspective, a side effect of how time and space mix at high speeds. The true, underlying connection was the entanglement of their entire trajectories, which stayed the same. This distinction is vital because it means scientists can now describe quantum systems in space and time without worrying that their conclusions will change just because they are moving.

This work provides a consistent language for describing quantum mechanics in the relativistic world, bridging the gap between the theory of how things move at high speeds and the theory of how particles connect. It confirms that while the appearance of quantum links can shift depending on the observer, the fundamental reality of those links is stable when viewed through the lens of complete spacetime paths. The researchers suggest that this approach could be used to design future experiments, perhaps involving satellites or high-speed particles, to test these ideas in the real world. By measuring the timing and position of particles with extreme precision, it might be possible to detect these trajectory-based connections, offering a new way to verify how quantum mechanics and relativity work together. The findings do not solve every mystery of the universe, but they offer a reliable map for navigating the complex terrain where time, space, and quantum connection meet.

Technical Summary: Entangled and Disentangled Multi-Time Quantum Trajectories in Relativistic Systems

Problem Statement
The paper addresses a fundamental challenge in relativistic quantum information: defining and quantifying entanglement between quantum particles residing in different inertial frames. While quantum entanglement is a critical resource for technologies like ground-to-satellite communication, its description becomes ambiguous when particles experience relativistic effects (e.g., time dilation, Sagnac effects) or when observers are in non-comoving reference frames. Existing literature presents contradictory findings: some studies suggest entanglement decreases with observer velocity, while others claim it is Lorentz-invariant or observer-dependent. The core issue identified is that standard definitions of entanglement rely on factorizable states at a single time. However, under Lorentz transformations, a state that is spatially factorizable in one frame may become spatially entangled in another, creating an ambiguity where the "entanglement" attribute appears to depend on the reference frame.

Methodology
To resolve this ambiguity, the authors propose a framework based on multi-time quantum trajectories rather than single-time states.

Definition of Trajectory Entanglement: Extending Dirac's multi-time approach, a system of $N$ particles is described by a state $|\psi(t_1, \dots, t_N)\rangle$ where each particle has its own time coordinate. A trajectory is defined as the collection of these states over time. The authors define a factorizable trajectory as a tensor product of individual particle trajectories: $\{|\psi_1(t_1)\rangle\} \otimes \dots \otimes \{|\psi_N(t_N)\rangle\}$ . Crucially, this definition allows for temporal correlations (e.g., particle 1's state depending on particle 2's time) to be distinguished from spatial entanglement.
Relativistic Factorizability Euler–Lagrange (ReFEL) Equations: The authors derive a set of equations of motion that enforce the constraint of trajectory factorizability. Starting from a general action $S = \int L dt_1 \dots dt_N$ , they apply variational principles to the constrained subspace of factorizable trajectories. This yields $N$ coupled Euler–Lagrange equations (Eq. 13) for the individual factors $|\psi_j(t_j)\rangle$ , where the effective Lagrangian for each particle includes terms resulting from tracing out the other particles.
Comparison Protocol: The methodology involves solving two sets of equations for a given system:
- The unrestricted Euler–Lagrange equations (Eq. 11), which allow for general entanglement.
- The restricted ReFEL equations (Eq. 13/17), which force the system to remain in a factorizable trajectory.
  The deviation between these solutions serves as a measure of trajectory entanglement.
Application to Klein–Gordon Systems: The framework is applied to a proof-of-concept model of interacting particles obeying a Klein–Gordon-type equation with a potential dependent on spacetime separation ( $V \propto (t_i - t_j)^2 - (x_i - x_j)^2$ ). The authors solve for Gaussian solutions in both the entangled and factorizable cases.
Entanglement Witnessing: To quantify entanglement, the authors employ continuous-variable entanglement witnesses (based on Ref. [42]) adapted for spacetime coordinates. They calculate the expectation value of a witness operator $\hat{L}$ and compare it against a lower bound derived from the optimal factorizable state ( $g_{fac, min}$ ). A negative value ( $\langle\langle \hat{L} \rangle\rangle - g_{fac, min} < 0$ ) certifies entanglement.

Key Results

Invariance of Trajectory Factorizability: The authors demonstrate that while the spatial factorizability of a state at a fixed time is not invariant under Lorentz boosts (a spatially factorizable state can become spatially entangled after a boost), the factorizability of the trajectory is invariant. If a trajectory is factorizable in one frame, it remains factorizable in any other frame related by local spacetime transformations.
Quantification of Entanglement: In the Klein–Gordon model, the unrestricted solution exhibits increasing entanglement (more negative witness values) as the interaction coupling strength $\kappa$ increases. In contrast, the solution constrained by the ReFEL equations yields zero entanglement by construction.
Relativistic Effects on Spatial Entanglement: The study shows that for interacting particles, a trajectory that is initially spatially unentangled (at $t_1=t_2=0$ ) can develop spatial entanglement when observed from boosted frames. However, the factorizable trajectory (constrained by ReFEL) remains free of spatial entanglement regardless of the boost velocity. This confirms that the "appearance" of spatial entanglement in boosted frames is a consequence of underlying trajectory entanglement, which the ReFEL framework successfully isolates and removes.

Significance and Claims
The paper claims to provide a consistent framework for investigating relativistic entanglement that overcomes the frame-dependence issues of standard state-based definitions.

Resolution of Ambiguity: By shifting the focus from factorizable states to factorizable trajectories, the authors offer a definition of entanglement that is invariant under spacetime transformations acting on individual particles. This resolves the contradiction between literature claiming entanglement is observer-dependent and literature claiming it is invariant.
New Dynamical Tool: The derivation of the ReFEL equations provides a method to describe the dynamics of a system where entanglement is artificially "censored" or suppressed, allowing for a direct comparison with fully entangling dynamics.
Experimental Relevance: The authors suggest that their results, particularly the prediction of trajectory entanglement manifesting as spatial entanglement in boosted frames, could be tested using time-resolved balanced homodyne detection, provided sufficient temporal resolution is achieved to measure relativistic effects.

The work is presented as a theoretical foundation for studying dynamical entanglement in relativistic settings, with potential extensions to spinor-valued Dirac particles and second-quantized field theories.

Technical Summary: Entangled and Disentangled Multi-Time Quantum Trajectories in Relativistic Systems

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