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arXiv:2511.09623

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

Kun-Feng Lyu, Rahul Muraleedharan, Kuver Sinha

hep-phhep-thquant-phSubmitted Nov 12, 2025• Announced Nov 14, 2025• Revised Jun 26, 2026 (v2)
v1 · Nov 12, 2025v2 · Jun 26, 2026
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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.

Journal reference
JHEP 06 (2026) 210
DOI
10.1007/JHEP06(2026)210
Comments
41 pages, 4 figures, comments welcome

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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 develop a constructive, entanglement-first derivation of quantum field theory in which all physical structure arises from a single operator relation, ρ = e −S(H) . The fundamental object is the correlation matrix Cij = ⟨λi |ρ|λj ⟩, from which locality, spectral geometry, renormalization flow, and interaction structure are derived without prior assumptions of spacetime, fields, or gauge symmetry. We show that the conventional kernel formulation is a projection of the full entanglement structure and is therefore insufficient to reconstruct physical dynamics, establishing operator entanglement as the unique fundamental layer. Within this framework, non-Abelian gauge structure emerges from symmetry-induced degeneracies of the entanglement operator, yielding a natural construction of gauge connections and curvature. The Yang–Mills action is derived as the leading gauge-invariant functional consistent with entanglement-induced locality and spectral scaling, rather than postulated. In the strong-coupling regime, exponential decay of correlations implies clustering, a nonzero mass gap, and an area law for Wilson loops, providing a nonperturbative mechanism for confinement. We further construct the full quantum theory via an entanglement-induced measure, verifying all Osterwalder–Schrader axioms and establishing Hilbert space reconstruction, uniqueness of the continuum limit, universality, vacuum uniqueness, and operator-domain control. The resulting framework provides a unified and constructive foundation for quantum field theory in which Yang–Mills dynamics and nonperturbative phenomena arise directly from operator entanglement.
We study entanglement dynamics among helicity degrees of freedom in quantum electrodynamics (QED) scattering processes. For generic initial states, we consider scattering at fixed momentum, corresponding to a generalized measurement described by a positive operator-valued measure, resulting in a post-measurement state. Such processes are modeled in terms of quantum maps, whose spectral structure fully determines the associated entanglement dynamics. For scattering involving fermions only, maximal entanglement present in the initial state is always preserved. Moreover, iterating the corresponding quantum maps on arbitrary initial states, we obtain the fixed points of the maps, which, in the largest number of cases, are asymptotic (pure) maximally entangled states. The structure of the maps also accounts for the entanglement dynamics in processes involving both fermions and photons. The defining properties of these maps originate from discrete symmetries of the QED interaction.
We study the interplay between crossing symmetry and entanglement in 2→22 \to 22→2 scattering within local quantum field theories that possess an SU(N)SU(N)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)SU(N)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)SU(N)SU(N) global symmetry must generate entanglement in at least one scattering channel.
Swapping and singlet projection as generic entanglement witnesses for quantum Heisenberg spin-sss systems
Based on the SU(NNN) representation of group theory, we derive the generalized SU(NNN) invariant swapping operator for the quantum Heisenberg spin-sss systems (N=2s+1N=2s+1N=2s+1). It has been demonstrated that the partial transposition of the swapping operator is equal to the singlet pairing projection in the tensor product space of the fundamental SU(NNN) representation and its conjugate one. For general SU(2) invariant bipartite spin-sss systems, the expectation value of the swapping operator is found to be the leading term in the negativity expression with respect to the corresponding density matrix. Generalized to the many-body SU(2) invariant states, we prove that the expectation values of the swapping operator and the singlet pairing projector can be regarded as two generic entanglement witnesses, which can be detectable in future experiments.
The entangling power of a unitary operator quantifies its ability to generate entanglement from product states and provides a natural probe of quantum many-body dynamics. Entanglement extremization at points of enhanced symmetry has previously been observed in high-energy scattering. In this work we compute the time-averaged entangling power of anisotropic Heisenberg spin chains across two-site models and finite-size systems, as well as the entangling power of the two-magnon SSS-matrix in the thermodynamic limit. For two-site models we establish a monotonic hierarchy: the entangling power decreases as the symmetry group grows, reaching its minimum at the SU(2)SU(2)SU(2) XXX point. Finite-size XXZ chains exhibit sharp dips at the SU(2)SU(2)SU(2) points Δ=±1Δ=\pm 1Δ=±1 and the free-fermion point Δ=0Δ=0Δ=0, with the free-fermion dip decaying much more slowly with system size. In the thermodynamic limit, we decompose the two-magnon SSS-matrix into quantum logic gates -- Identity, SWAP, and σz⊗σzσ_z\otimesσ_zσz​⊗σz​ -- and show that the entangling power vanishes for all scattering energies at the SU(2)SU(2)SU(2) points, where the SSS-matrix reduces to the Identity gate, while the free-fermion point achieves the maximum -- the opposite of the finite-size many-body behavior. The entangling power can serve as an {\em operator} diagnostic for symmetry and selected aspects of integrability in quantum simulations of spin-chain dynamics.
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.