Genuine Tripartite Entanglement Selects Gauge-Invariant Theories

Abstract: Recent studies have explored whether physical constants and symmetries can be selected by extremizing bipartite entanglement entropy. We extend this approach to a three-particle setting. Specifically, we consider a $2+1$-particle system in which particles $B$ and $C$ are initially entangled, and then particles $A$ and $B$ scatter without the direct participation of particle $C$ for both gluon-gluon and graviton-graviton scatterings. Using angular expansions around the forward and backward scattering limits, we analyze bipartite entanglement in the corresponding two-particle system of $A$ and $B$ and two measures of genuine tripartite entanglement (GTE), the concurrence fill and the generalized geometric measure (GGM), in the three-particle system. As is already known, the bipartite entanglement provides no universal criterion for selecting the symmetry-preserving theory: depending on the initial helicities, it may either maximize or minimize the entanglement entropy. By contrast, we find that the GTE-based analysis uniquely identifies the gauge-invariant and diffeomorphism-invariant theories as those that suppress the GTE in gluon-gluon and graviton-graviton scatterings, respectively. This result suggests a nontrivial connection between tripartite entanglement and the gauge and diffeomorphism symmetries of fundamental interactions.

• The paper demonstrates that genuine tripartite entanglement, measured by concurrence fill and GGM, is asymptotically more suppressed at the gauge-invariant value k=1 than at any fixed k≠1 in forward and backward scattering.
• It shows that bipartite concurrence lacks a universal extremum at k=1 because its behavior depends on initial helicities, scattering angle, and color-factor configuration, whereas tripartite suppression consistently favors the invariant theory.
• The result applies to a specific four-point-vertex deformation at tree level and fixed k, leaving open whether the selection rule persists for alternative deformations, finite angles, loop corrections, and other analytically controlled entanglement measures.

Overview

This paper investigates whether genuine tripartite entanglement (GTE) can serve as a selection principle for gauge-invariant and diffeomorphism-invariant theories, in cases where bipartite entanglement extrema fail to do so. The author, Junya Yamagishi, considers a $2+1$-particle system in which particles $B$ and $C$ are initially entangled and particles $A$ and $B$ subsequently scatter via tree-level gluon-gluon or graviton-graviton amplitudes, with particle $C$ acting as a spatially separated spectator. The central result is that, in the forward and backward scattering limits, the gauge-invariant value of the deformation parameter ($k=1$) suppresses GTE — quantified by both the concurrence fill and the generalized geometric measure (GGM) — more strongly than any fixed deformation $k\neq 1$, for all color-factor configurations and initial correlations considered. This contrasts sharply with the bipartite case, where whether $k=1$ is a maximum or minimum of the concurrence depends on the initial helicities.

Deformed amplitudes

The analysis follows the deformation prescription of prior work: the free kinetic terms are left untouched and only the four-point vertex coefficient is rescaled. For QCD, an additional term $$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$ with $B$0 equal to the standard four-gluon color tensor and $B$1 changes the four-point Feynman rule from proportional to $B$2 to $B$3. For perturbative gravity in transverse-traceless gauge, $B$4 rescales the four-graviton vertex analogously. In both cases the full amplitude takes the form $B$5, so that $B$6 restores gauge (or diffeomorphism) invariance. The gluon amplitudes are classified by six inequivalent color-factor configurations $B$7 built from $B$8 structure constants; forward and backward expansions are related by $B$9 together with $C$0, so results carry over between the two limits.

The paper concedes explicitly that this is only one particular symmetry-breaking prescription; whether other deformations obey the same selection rule remains unknown.

Failure of bipartite entanglement as a criterion

For ordinary $C$1 scattering, the final-state concurrence $C$2 exhibits no state-independent extremal behavior at $C$3. At $C$4, $C$5 is a maximum or minimum depending on the initial helicities, reproducing earlier findings; at a nonspecial angle such as $C$6, no comparably simple extremal structure appears. The small-$C$7 expansion (with $C$8) makes this precise: for the initial state $C$9, all helicity-changing amplitudes vanish at $A$0, so the concurrence vanishes identically and is minimized at every angle. For $A$1 with $A$2, however,

$A$3

so $A$4 is a local maximum; when $A$5 the concurrence is independent of $A$6 altogether. As a concrete manifestation of gauge invariance in forward scattering, the paper shows that for the color class $A$7, $A$8, the tree-level equal-helicity pair-flip amplitude $A$9 vanishes only at $B$0 — a statement valid at tree level, since loop-induced helicity flips survive even in the gauge-invariant theory.

The implication is direct: neither maximization nor minimization of bipartite entanglement constitutes a universal selection principle, confirming and extending the negative result of previous studies.

Tripartite measures and the fixed-$B$1 asymptotic comparison

In the $B$2-particle setup, particle $B$3 enters with positive helicity and the initial $B$4 state is $B$5 with $B$6; the phase $B$7 drops out of all measures. The spectator operator is $B$8, and the normalized post-selected final state defines $B$9. Two GTE measures are employed:

• Concurrence fill $C$0: the normalized area of the triangle whose sides are the squared one-versus-rest concurrences $C$1, given by Heron's formula. It vanishes on biseparable states (including the initial state) but is not generally an LOCC monotone.
• GGM: $C$2, an independent quantitative measure used to guard against conclusions resting solely on the non-monotone concurrence fill.

Numerically, at $C$3 the concurrence fill shows no special feature at $C$4 (and graviton minima are slightly displaced), while as $C$5 decreases toward the forward limit the minimum approaches $C$6. This motivates a fixed-$C$7 asymptotic comparison: for any comparison point $C$8 fixed in advance, does there exist $C$9 such that $k=1$0 for $k=1$1? Importantly, this does not claim $k=1$2 is the exact global minimizer at finite angle.

Analytic expansion of $k=1$3 confirms the inequality across three classes of leading behavior:

Configuration	Leading term	Mechanism
$k=1$4; graviton	$k=1$5 or $k=1$6	Vanishes at $k=1$7
$k=1$8	$k=1$9	Value at $k\neq 1$0 plus non-negative $k\neq 1$1 increments
$k\neq 1$2	$k\neq 1$3	Vanishes at $k\neq 1$4; higher orders favor $k\neq 1$5

For the exceptional branch $k\neq 1$6 in the $k\neq 1$7 configuration, the first distinguishing term appears at order $k\neq 1$8, and the difference of coefficients is shown to be strictly negative for all $k\neq 1$9 via a polynomial bound, establishing $k=1$0 at sufficiently small angles. The GGM analysis reproduces the same three-class structure (with powers of $k=1$1 shifted down by 4): $k=1$2, $k=1$3 again decomposes into its $k=1$4 value plus positive increments, and $k=1$5 with $k=1$6 otherwise; the $k=1$7 branch is removed by showing the GGM scales as $k=1$8 at $k=1$9 versus $$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$0 at $$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$1. The selection rule thus holds for two independent measures of GTE.

By contrast, maximization fails to select a universal theory: for the $$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$2 and $$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$3 classes the leading coefficients are unbounded as $$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$4, and for the $$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$5 class the maximizing $$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$6 depends explicitly on $$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$7 through a quadratic equation in $$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$8. Suppression, not maximization, of GTE is therefore the operative principle.

Limitations and open questions

Several restrictions qualify the result. The selection rule is asymptotic: it holds in the forward and backward limits under a fixed-$$\lambda_{\mathrm{QCD}}\Lambda^{abcd}A_\mu^a A^{b\mu}A_\nu^c A^{d\nu}$$9 ordering, not as exact global minimization at finite scattering angle. It is established analytically only for the concurrence fill and GGM; numerical checks with three-$B$00 entanglement and the genuine multipartite concurrence also favor $B$01, but these lack analytic proofs and are excluded from the main claims. The deformation itself is a specific choice — deforming only the four-point vertex while preserving the kinetic term and the two transverse helicity polarizations — and the extension to other symmetry-breaking prescriptions is untested. The pair-flip suppression is a tree-level statement, and loop-induced helicity flips evade it. Finally, the initial states form a restricted subclass without single-particle helicity superpositions, and maximization of the GGM was not analyzed.

Conclusion

The paper demonstrates that within a one-parameter family of four-point-vertex deformations, genuine tripartite entanglement generated in $B$02-particle gluon and graviton scattering uniquely identifies the gauge-invariant and diffeomorphism-invariant theories through its suppression in the forward and backward limits — a selection achieved by entanglement alone, whereas the corresponding two-particle result required supplementing entanglement extrema with magic minimization. This suggests a nontrivial connection between multipartite entanglement structure and the gauge and diffeomorphism symmetries of fundamental interactions, while leaving open whether the rule survives alternative deformations, other entanglement measures proven analytically, and finite-angle regimes.