Entanglement, Yang-Mills, and the Scattering Matrix as an SU(N)-equivariant Kernel

TL;DR

The paper develops a representation-theoretic, SU(N)-equivariant framework to study two-body scattering and the entanglement generated by the S-matrix. By decomposing external representations, it shows that the invariant operator algebra—and thus the entangling power—is fixed by group theory, while dynamics select specific coefficients within this space. In Yang-Mills theory, CKD constrains the color kernel to a fixed ray at right angle, yielding universal group-invariant maximal entanglement in color space and a MaxE→MaxE property in helicity space that singles out the YM locus; deformations by higher-dimensional operators reveal EFT content via entanglement shifts. The results demonstrate that entanglement provides a unifying, tomographic lens on algebraic, geometric, and dynamical aspects of scattering, offering a new diagnostic of gauge consistency and effective operators.

Abstract

We study two-body scattering as an SU(N)-equivariant map acting on tensor-product representation spaces and analyze the entanglement generated by the $S$-matrix. This representation-theoretic perspective separates group structure from dynamics: the decomposition of $R\!\otimes\!R'$ fixes the invariant operator algebra and therefore the qualitative entangling power of the process. For particles in the fundamental representation, $\mathrm{End}_{\mathrm{SU}(N)}(N\!\otimes\!N)=\mathrm{Span}\{\mathbb{I},\mathbb{S}\}$, so only the identity and swap directions preserve separability, whereas generic combinations generate entanglement. Adjoint-adjoint scattering involves a larger invariant algebra involving $d$-tensors and is intrinsically entangling. In Yang-Mills theory one can use color-kinematics duality to show that the color kernel lies on a fixed ray of this operator space, yielding a universal maximum of the outgoing entanglement for scattering at right angles, $E_\star^{(2)}=\tfrac{3}{4}$ for $SU(2)$ and $E_\star^{(3)}\simeq0.91$, independent of kinematics. Dimension-six operators preserve this universality, while dimension-eight deformations populate new color sectors and shift $E_\star^{(N)}$, suggesting that entanglement in color space functions as a tomographic probe of effective operators. In helicity space, requiring maximally entangled inputs to scatter into maximally entangled outputs uniquely selects the Yang-Mills quartic coupling and enforces the color Jacobi identity, restating the on-shell Ward constraints as conditions on entanglement preservation. Our results suggest that the information-theoretic viewpoint unifies algebraic, geometric, and dynamical aspects of scattering.

Figures (4)

• Figure 1: Entanglement power as a function of $z$ and $\vartheta$ as in Eq.(\ref{['eq:fund_ent_formula']}) for the case of Fundamental-Fundamental scattering. Here we take $N = 3$ as the benchmark.
• Figure 2: The sampling of points from on the plane of final state entanglement $E(\text{out})$ and the dot product of $u$ and $v$ vectors.
• Figure 3: Entanglement diagnostics of the Yang–Mills locus. All panels use a maximally entangled (MaxE) input and fixed $\theta$ (no phase-space integration). (A) With fixed polarization ($\xi=0$), $E$ singles out $\kappa=1$ for any $(\chi_t,\chi_u)$: this is the quartic test. (B) Varying the on-shell polarization shift $\varepsilon\to\varepsilon+\xi\, p$ at $\kappa=1$ exposes Jacobi via $\Delta E_\xi$; the heatmap is dark only on $\chi_u=-1-\chi_t$. (C) Worst-case over $\xi$ vs. $\kappa$ shows that the robust MaxE$\to$MaxE point occurs iff both conditions hold, i.e. on the YM locus.
• Figure 4: The entanglement as the function of the scattering polar angle $\theta$ for different 3 initial states.

Theorems & Definitions (1)

• proof : Idea