Crossing Symmetry and Entanglement

TL;DR

This work develops a quantum-information framework for 2-to-2 scattering with SU(N) global symmetry by treating internal quantum numbers as qudits and the S-matrix as a quantum operation on a two-qudit Hilbert space. It shows that SU(N)-invariant amplitudes are spanned by a minimal gate set $\{\mathbf{S}_{\mathbb{I}},\mathbf{S}_{W},\mathbf{U}\}$, with crossing symmetry inducing recoupling between invariant bases and linking entanglement properties across scattering channels. The main finding is that crossing symmetry forbids complete entanglement suppression in all channels for interacting theories; if one channel is separable, the crossed channel must generate entanglement, leaving only the free theory as a fully separable case. This establishes a deep connection between the analytic structure of the S-matrix, locality, and symmetry from an information-theoretic perspective, and suggests new principles governing the organization of quantum field theories.

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.

Figures (1)

• Figure 1: Scattering of qudits $\ket{i,j}\to\ket{k,l}$ with different tensor product basis projections.