Genuine tripartite entanglement in Bhabha scattering with an entangled spectator particle

Abstract: From the perspective of quantum information science, we investigate tree-level Bhabha scattering between an incident electron $A$ and a positron B, where $B$ is initially entangled with a spectator electron $C$, which does not participate in the scattering interaction.We find that the quantum electrodynamics (QED) scattering between $A$ and $B$ can drive the global $ABC$ system into a genuine tripartite entangled (GTE) state. Using four canonical tripartite entanglement metrics, we systematically characterize and quantify the GTE of the composite system, and demonstrate that the scattering momentum of the $A$-$B$ pair and the initial $B$-$C$ entanglement are the key resources governing GTE generation.We further analyze the monogamy of quantum correlations, which imposes fundamental constraints on the shareability of quantum resources in multipartite systems. Specifically, we systematically study the monogamy relations for the squared entanglement of formation and squared quantum discord in our scattering model, and find that monogamy constraints are markedly relaxed in the non-relativistic regime, enabling enhanced shareability of quantum correlations across the three particles. This work uncovers novel quantum correlation properties of fundamental QED scattering processes, and provides direct theoretical guidance for the development of QED-based quantum information processing protocols.

• The paper demonstrates that tree-level Bhabha scattering with an entangled spectator generates genuine tripartite entanglement, optimizing four key GTE measures at distinct parameter values.
• It employs four quantitative metrics—GGM, three-π, GMC, and concurrence fill—to reveal a robust hierarchical structure in multipartite entanglement distribution.
• The study finds that nonzero initial spectator entanglement and scattering momentum are essential for GTE, offering insights for QED-based quantum information protocols.

Genuine Tripartite Entanglement in Bhabha Scattering with an Entangled Spectator

Introduction and Context

This paper addresses the generation and quantification of genuine tripartite entanglement (GTE) in a fundamental quantum electrodynamics (QED) process: tree-level Bhabha scattering ($e^-e^+ \rightarrow e^-e^+$), in the presence of an initially entangled spectator electron. The study extends previous bipartite treatments by investigating the full tripartite entanglement dynamics of a composite $ABC$ system, where: electron $A$ scatters with positron $B$, which is initially entangled with a remote electron $C$ (the spectator).

Scattering-based entanglement generation has been explored as both a probe for quantum correlations in high-energy regimes and as a building block for quantum information processing tasks in relativistic quantum systems. The inclusion of an entangled spectator allows the study of correlation transfer and multipartite entanglement propagation under fundamental interactions, bridging QED and quantum information resource theories.

Theoretical Model and Methodology

The initial state is constructed as a product between the incoming electron $A$ and an entangled $B$-$C$ pair:

$$|i\rangle = |p_1, a\rangle_A \otimes \left( \cos\eta |p_2, \uparrow\rangle_B \otimes |q, \uparrow\rangle_C + e^{i\beta} \sin\eta |p_2, \downarrow\rangle_B \otimes |q, \downarrow\rangle_C \right),$$

where the parameter $\eta$ controls the initial entanglement weight of $ABC$0-$ABC$1. The effective post-scattering final state is derived by applying the QED $ABC$2-matrix at tree-level, projecting out the non-interacting forward scattering component. The central observables are the GTE metrics constructed from the reduced density matrix of the $ABC$3 system.

For GTE quantification, four canonical multipartite entanglement measures are employed:

• Generalized Geometric Measure (GGM): Quantifies the optimal distance from the set of non-genuine multipartite entangled states.
• Three-$ABC$4 Entanglement: Based on negativity, capturing the residual entanglement not distributed pairwise.
• Genuine Multipartite Concurrence (GMC): Extends concurrence to multipartite settings via minimization over bipartitions.
• Concurrence Fill: Geometric measure derived from the area of the entanglement triangle formed by pairwise concurrences.

Quantum correlation monogamy is analyzed using two distinct metrics:

• Squared Entanglement of Formation (SEF): Examines the distribution of entanglement among subsystems and quantifies non-shareable entanglement.
• Squared Quantum Discord (SQD): Operationally distinct from entanglement, reflects general non-classical correlations subject to monogamy constraints.

Closed-form expressions for certain limits (e.g., concurrence fill in highly relativistic regimes) and full numerical evaluations across parameter space are provided. The independence of all correlation measures from the relative phase $ABC$5 is explicitly established.

Numerical Results: Generation and Control of GTE

GTE Dependence on Initial Entanglement and Scattering Parameters

The most salient results can be summarized as follows:

• Generation Condition: Non-zero GTE in the post-scattering $ABC$6 state requires both a non-zero initial $ABC$7-$ABC$8 entanglement ($ABC$9) and non-zero scattering momentum ($A$0). If either is trivial, GTE vanishes identically.
• Parameter Dependence: GTE exhibits a non-monotonic dependence on both the initial $A$1-$A$2 entanglement weight $A$3 and the dimensionless scattering momentum $A$4. The GTE is strictly maximized for intermediate, not extremal, values of these parameters; in particular, GTE is not maximized when the $A$5-$A$6 initial state is maximally entangled ($A$7).
• Strong Numerical Results: The four GTE measures—GGM, three-$A$8, GMC, and concurrence fill—reach their maxima for nearly identical $A$9 parameter values: $B$0, achieving respective maximum values of 0.342 (GGM), 0.648 (three-$B$1), 0.900 (GMC), and 0.902 (concurrence fill).
• Hierarchical Relationship: For any parameter set within the scattering model, the measures satisfy

$B$2

• Relativistic Suppression: In the ultrarelativistic limit ($B$3), both GTE and residual quantum correlations are strongly suppressed due to stringent monogamy constraints. The non-relativistic regime allows for broader GTE shareability.
• Concurrence Fill Superiority: The concurrence fill, by construction, incorporates information from all bipartite subsystems and provides a more integrated estimate of global entanglement, in contrast to extremal measures that rely on maximal or minimal pairwise quantities.

Monogamy Constraints and Correlation Distribution

• Monogamy in SEF and SQD: The squared entanglement of formation and squared quantum discord always satisfy the monogamy relations throughout the scattering process:

$B$4

Quantitatively, these constraints are most relaxed in the non-relativistic regime (maximizing correlation shareability) and most stringent in the relativistic regime (correlation localization).

• Residual Correlation as GTE Indicator: The magnitude of residual entanglement (or discord), defined as the difference between total and pairwise bipartite measures, is strictly positively correlated with GTE. Relaxation of monogamy is necessary for GTE generation.
• Discord vs. Entanglement: For all configurations, residual discord exceeds residual entanglement, reflecting discord's sensitivity to all non-classical correlations. The suppression or enhancement of these residuals is dominated by the same resource parameters: initial $B$5-$B$6 entanglement and the $B$7-$B$8 scattering momentum.

Implications and Outlook

Theoretical Implications

The results provide a rigorous QED-based demonstration that multipartite quantum correlations can be dynamically generated via scattering events involving initially entangled spectators. The explicit resource-dependence and strong parameter control support using such processes as theoretical testbeds for quantum information protocols—entanglement swapping, remote entanglement distribution, and related tasks—in the context of relativistic quantum systems.

The hierarchy and consistency found among multiple entanglement measures highlight the robustness of GTE phenomena in this setting. The strict interplay between monogamy constraints and GTE generation offers a physical basis for understanding optimal entanglement engineering in multipartite setups rooted in fundamental interactions.

Experimental Prospects

Recent collider data (ATLAS, CMS) have demonstrated experimental access to quantum entanglement in high-energy scattering on TeV scales. Bhabha scattering at future $B$9 colliders (CEPC, FCC-ee), with their extremely high event rates and calibration requirements, provides an ideal arena for direct tests of these predictions. The detailed functional dependence of GTE on accessible kinematic parameters can be exploited for benchmarked, parameter-guided quantum information protocols implemented with fundamental particles.

Future Directions

Key directions for extension include:

• Incorporation of higher-order QED corrections (loop/radiative/inelastic contributions) to assess the stability of GTE production against quantum corrections.
• Application to other QED processes (e.g., Møller, Compton, pair production) to probe universality.
• Realization in practical quantum network protocols with atomic, ionic, or solid-state analogs.
• Study of entanglement and monogamy features in open quantum field theory environments and dissipative scenarios.

Conclusion

This work provides a comprehensive framework for the generation and quantification of genuine tripartite entanglement in Bhabha scattering with an entangled spectator, establishing necessary and sufficient resource conditions based on initial entanglement and scattering kinematics. The analysis unifies insights from high-energy particle physics and quantum information theory, with results robust to initial phase and consistent across several entanglement measures. The tight interplay between monogamy constraints and GTE provides a physical explanation for non-monotonic entanglement generation and shareability. Practical realization of QED-controlled entanglement protocols in current and anticipated experimental facilities is feasible, suggesting concrete pathways toward resource-efficient quantum information processing grounded in the fundamental interactions of quantum field theory.