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arXiv:2511.10559

Crossing Symmetry and Entanglement

Navin McGinnis

hep-thquant-phSubmitted Nov 13, 2025• Announced Nov 14, 2025• 0 citations
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Abstract

We study the interplay between crossing symmetry and entanglement in $2 \to 2$ scattering within local quantum field theories that possess an $SU(N)$ global symmetry. In particular, we recast scattering amplitudes of fixed helicity as quantum operations on the Hilbert space of internal quantum numbers, where the external states play the role of qudits. The entire space of $SU(N)$-invariant scattering operators between qudits is spanned by a minimal set of three quantum gates. Recoupling relations among quantum gates are shown to follow directly from the crossing properties of the underlying amplitudes and reveal that entanglement generated from separable states in one channel is necessarily intertwined with another. Consequently, we argue any interacting quantum field theory that realizes an $SU(N)$ global symmetry must generate entanglement in at least one scattering channel.

Comments
20 pages, 1 figure, + 1 appendix

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Quantum Computational Structure of SU(N)SU(N)SU(N) Scattering
We study scattering of particles which obey an SU(N)SU(N)SU(N) global symmetry through the lens of quantum computation and quantum algorithms. We show that for scattering between particles which transform in the fundamental or anti-fundamental representations, i.e. qudits, all 2-2 scattering amplitudes can be constructed from only three quantum gates. Further, for any NNN, all 2-2 scattering channels are shown to emerge from the span of a Z2\mathbb{Z}_{2}Z2​ algebra, suggesting that scattering in this context is fundamentally governed by the action of ``bit flips'' on the internal quantum numbers. We frame these findings in terms of quantum algorithms constructed from Linear Combinations of Unitaries and block encoding.
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Symmetry is among the most fundamental and powerful concepts in nature, whose existence is usually taken as given, without explanation. We explore whether symmetry can be derived from more fundamental principles from the perspective of quantum information. Starting with a two-qubit system, we show there are only two minimally entangling logic gates: the identity and the SWAP\mathrm{SWAP}SWAP, which interchanges the two states of the qubits. We further demonstrate that, when viewed as an entanglement operator in the spin-space, the SSS-matrix in the two-body scattering of fermions in the sss-wave channel is uniquely determined by unitarity and rotational invariance to be a linear combination of the identity and the SWAP\mathrm{SWAP}SWAP. Realizing a minimally entangling SSS-matrix would give rise to global symmetries, as exemplified in Wigner's spin-flavor symmetry and Schr\"odinger's conformal invariance in low energy quantum chromodynamics. For Nq{N}_{q}Nq​ species of qubit, the identity gate is associated with an [SU(2)]Nq[SU(2){]}^{{N}_{q}}[SU(2)]Nq​ symmetry, which is enlarged to SU(2Nq)SU(2{N}_{q})SU(2Nq​) when there is a species-universal coupling constant.
We study two-body scattering as an SU(N)-equivariant map acting on tensor-product representation spaces and analyze the entanglement generated by the SSS-matrix. This representation-theoretic perspective separates group structure from dynamics: the decomposition of R ⁣⊗ ⁣R′R\!\otimes\!R'R⊗R′ fixes the invariant operator algebra and therefore the qualitative entangling power of the process. For particles in the fundamental representation, EndSU(N)(N ⁣⊗ ⁣N)=Span{I,S}\mathrm{End}_{\mathrm{SU}(N)}(N\!\otimes\!N)=\mathrm{Span}\{\mathbb{I},\mathbb{S}\}EndSU(N)​(N⊗N)=Span{I,S}, so only the identity and swap directions preserve separability, whereas generic combinations generate entanglement. Adjoint-adjoint scattering involves a larger invariant algebra involving ddd-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⋆(2)=34E_\star^{(2)}=\tfrac{3}{4}E⋆(2)​=43​ for SU(2)SU(2)SU(2) and E⋆(3)≃0.91E_\star^{(3)}\simeq0.91E⋆(3)​≃0.91, independent of kinematics. Dimension-six operators preserve this universality, while dimension-eight deformations populate new color sectors and shift E⋆(N)E_\star^{(N)}E⋆(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.
We explore the interplay among entanglement suppression, quantum statistics, and enhanced symmetries in the nonrelativistic S -wave scattering involving the lowest-lying spin-3/2 baryons, which can be considered as four-dimensional qudits. These baryons form a ten-dimensional representation (decuplet) under the SU(3) light-flavor symmetry and, in this limit, are considered indistinguishable under strong interactions. Treating the S matrix in the spin-3/2 baryon-baryon scattering as a quantum logic gate in the spin space, we study the consequence of entanglement suppression and compute the entanglement power of the S matrix. When the entanglement power vanishes, the S matrix is either an Identity or a SWAP gate and spin-flavor symmetries and/or nonrelativistic conformal invariance emerge, as previously observed in spin-1/2 baryons. In the case of scattering identical particles, the entanglement power never vanishes due to constraints from spin statistics, which we interpret as projection-valued measurements onto symmetric or antisymmetric Hilbert space and define the entanglement power accordingly. When the entanglement power is nonvanishing but sits at a global or local minimum, enhanced symmetries still emerge and the S matrix can be interpreted as an Identity or a SWAP gate acting on the restricted Hilbert space allowed by quantum statistics. In general, when scattering identical spin- s particles, we identify an enhanced SU ( 2 s + 1 ) spin symmetry for the Identity gate.