Crossing symmetry and entanglement beyond massive $2\to2$ $SU(N)$ scattering: gapless theories, higher-point amplitudes, unequal internal spaces, and further quantum resources

McGinnis (2025) recast $2\to2$ scattering amplitudes of local quantum field theories with an $SU(N)$ global symmetry as quantum operations on the Hilbert space of internal quantum numbers, showing that all $SU(N)$-invariant scattering operators are spanned by a minimal set of three unitary gates (the identity, a Wigner swap, and a charge-conjugation-type exchange), that crossing symmetry fixes the recoupling relations between scattering channels, and that any interacting theory with an $SU(N)$ global symmetry must generate entanglement in at least one scattering channel. The rigorous crossing input is the Bros–Epstein–Glaser theorem, which assumes a mass gap, and the framework is confined to four-point amplitudes, equal-dimensional internal spaces, and entanglement as the only analyzed resource. The open problem is the extension program the source itself states: (i) justifying the crossing-based channel relations without a mass gap; (ii) constructing the analogous minimal gate algebras and crossing-imposed recoupling constraints for higher-point ($2\to m$) processes; (iii) characterizing the invariant operator algebras for unequal internal spaces $H_N\otimes H_M$ with $N\neq M$ and for subgroup embeddings such as $\mathbf{5}=\mathbf{2}\oplus\mathbf{3}$ under $SU(2)\times SU(3)\subset SU(5)$; and (iv) determining how other quantum-information resources (measures of maximal entanglement, quantum magic) transform under crossing and recoupling.

Background

Crossing symmetry is the statement, used throughout relativistic scattering theory, that the amplitude for a process such as $a+b\to c+d$ is the boundary value of a single analytic function whose continuation also yields the amplitudes for 'crossed' processes such as $a+\bar{c}\to\bar{b}+d$: a particle moved to the other side of the reaction is reinterpreted as its antiparticle. The only fully rigorous justification of this property in general quantum field theory is the theorem of Bros, Epstein and Glaser, which proves crossing for two-particle amplitudes in theories obeying the Wightman axioms (the standard axiomatic framework for relativistic quantum fields) with a mass gap — a nonzero lower bound on the spectrum of the mass operator, which underwrites the needed analyticity domains; Bros extended the crossing domains to $2\to3$ processes, still under mass-gap assumptions. No comparable theorem exists for massless or gapless theories, where infrared and collinear singularities complicate the analytic structure: modern work either conjectures generalized multiparticle crossing relations and proves them at tree level (Caron-Huot, Giroux, Hannesdottir and Mizera), or reformulates crossing algebraically as an operator equation in the language of modular theory and tensor categories (Correa da Silva, Giorgetti and Lechner).

In parallel, a quantum-information perspective on scattering has developed. 'Entanglement suppression' is the observation that S-matrices which generate minimal entanglement between the scattered subsystems exhibit enhanced, emergent symmetries (Low and Mehen). McGinnis pushed this to a gate-level correspondence: for $2\to2$ scattering of particles carrying an $SU(N)$ global quantum number, each external particle is viewed as a qudit — a $d$-level quantum system, here $d=N$ — so a fixed-helicity (fixed spin projection along the direction of motion) amplitude is an operator on the internal space $H_N\otimes H_N$ (particles in the fundamental representation $\mathbf{N}$, antiparticles in the conjugate $\bar{\mathbf{N}}$). Representation theory then implies that every $SU(N)$-invariant scattering operator is a linear combination of three unitary gates: the identity $\mathbf{S}_{\mathbb{I}}$; the Wigner swap $\mathbf{S}_{W}=\sum_{i,j}|i,j\rangle\langle\bar{j},\bar{i}|=\tfrac{1}{N}\left(\mathbb{I}\otimes\mathbb{I}+2N\sum_{a}T^{a}\otimes T^{a}\right)$, with $T^{a}$ the generators of $SU(N)$ in the fundamental representation; and a charge-conjugation-type exchange operator $\mathbf{U}$ interchanging two distinguished internal states. Each of $\mathbf{S}_{W}$ and $\mathbf{U}$ realizes a $\mathbb{Z}_{2}$ action (the cyclic group of order two): for particle-particle scattering $\mathbf{S}_{W}$ manifests as spin-statistics, while for particle-antiparticle scattering $\mathbf{U}$ manifests as charge-conjugation parity. Crossing symmetry is what fixes the recoupling relations between the two natural operator bases,

$$\begin{pmatrix}\mathbf{I}\\ \mathbf{S}_{W}\end{pmatrix}=\begin{pmatrix}N/2 & N/2\\ 1 & 0\end{pmatrix}\begin{pmatrix}\mathbf{I}\\ \mathbf{U}\end{pmatrix},$$

and it forces entanglement generated from separable states in one scattering channel to be intertwined with the crossed channels. The central theorem is that any interacting quantum field theory realizing an $SU(N)$ global symmetry must generate entanglement in at least one scattering channel; only free theories escape. 'Quantum magic' (nonstabilizerness) is a further quantum resource, complementary to entanglement, measuring the non-Clifford content of a state or operation — its distance from the Clifford group, the elementary gate set (Hadamard, phase, and controlled-NOT gates) that admits efficient classical simulation — and governing classical-simulability barriers; it is the object of an active literature on scattering, so far computed in spin and helicity spaces.

The framework's rigorous crossing input is the mass-gapped Bros–Epstein–Glaser theorem, and its constructions are confined to four-point amplitudes, to both qudits transforming under the same $SU(N)$ in equal-dimensional spaces, and to entanglement as the only analyzed resource. The source paper states precisely these as its principal limitations and defers their resolution to future work, while its companion paper supplies the announced quantum-circuit realization of the gate algebra via linear combinations of unitaries and block encoding.

Problem Statement

Establish the crossing-entanglement correspondence of McGinnis (arXiv:2511.10559) beyond its original setting — massive $2\to2$ scattering, both particles' internal quantum numbers in equal-dimensional fundamental spaces, and entanglement as the only analyzed resource — by resolving the four directions the source itself identifies as its open limitations, four independent research questions each resolvable on its own:

(1) Gapless regime. Determine whether the crossing-based channel relations — the recoupling constraints among the invariant gates and the conclusion that an interacting $SU(N)$-symmetric theory must generate entanglement in at least one scattering channel — survive without a mass gap, i.e. in massless or partially massless theories such as QED or Yang-Mills: either prove the required crossing (analytic continuation) property beyond the mass-gapped Bros–Epstein–Glaser setting, or establish the channel relations by an alternative route; and identify where anomalous thresholds (amplitude singularities produced by on-shell intermediate states in crossed channels) or infrared subtleties obstruct or modify the conclusions.

(2) Higher-point processes. Construct the analogue of the minimal three-gate algebra and of the crossing-imposed recoupling relations for $2\to m$ and general $m$-point processes (rigorous crossing domains are known up to $2\to3$): determine which operators span the space of $SU(N)$-invariant higher-point amplitudes, whether a finite minimal generating set exists, and which constraints crossing imposes among the channels of the different multiplicity orderings.

(3) Unequal internal spaces and subgroup embeddings. Characterize the invariant operator algebra, the minimal gate spanning sets, and the crossing/recoupling constraints for scattering between internal Hilbert spaces of different dimensions, $H_N\otimes H_M$ with $N\neq M$, and for embeddings of subgroup structures into the internal space, such as the branching decomposition $\mathbf{5}=\mathbf{2}\oplus\mathbf{3}$ (with $2+3=5$) for the embedding $SU(2)\times SU(3)\subset SU(5)$.

(4) Resources beyond entanglement. Determine how quantum-information resources other than entanglement — measures of maximal entanglement (the entanglement entropy and Rényi entropies of the outgoing internal-space state) and quantum magic — transform under the crossing and recoupling maps of the framework, and whether crossing enforces necessity-type statements for these resources analogous to the entanglement-necessity theorem.

These four independent directions jointly bear on one objective: deciding whether the correspondence between crossing symmetry and the quantum-information structure of the S-matrix (minimal gate algebra, recoupling constraints, mandatory entanglement) is a general property of quantum field theory or an artifact of the massive four-point, equal-representation, entanglement-only setting in which it was established. Each of the four may be settled independently of the others, and a partial treatment of a direction should state which of its constituent questions it settles and which remain open. The source's fifth announced direction — an explicit quantum-circuit and algorithmic realization of the exchange gate $\mathbf{U}$ — has been discharged by the author's companion paper (arXiv:2511.10550) via linear combinations of unitaries and block encoding (standard quantum-algorithm techniques for realizing a general operator on a quantum computer), and is therefore excluded from the open core.

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