Entanglement Sum Rule from Higher-Form Symmetries

Abstract

We prove an entanglement sum rule for $(d-1)$-dimensional quantum lattice models with finite abelian higher-form symmetries, obtained by minimally coupling a sector on $p$-simplices carrying a $p$-form $G$ symmetry to a sector on $(p+1)$-simplices carrying the dual $(d-p-2)$-form $\widehat G$ symmetry (with $\widehat G$ being the Pontryagin dual of $G$). The coupling is introduced by conjugation with a symmetry-preserving operator $\mathcal{U}$ that dresses symmetry-invariant operators with appropriate Wilson operators. Our main result concerns symmetric eigenstates of the coupled model that arise by acting with $\mathcal{U}$ on direct-product symmetric eigenstates of the decoupled model: provided a topological criterion formulated via the Mayer--Vietoris sequence holds for the chosen bipartition, $\mathcal{U}$ factorizes across the cut when acting on the symmetric state, and the entanglement entropy equals the sum of the entropies of the two sectors. This framework explains and generalizes known examples in fermion-$\mathbb{Z}_2$ gauge theory, identifies when topology obstructs the sum rule, and provides a procedure to construct new examples by gauging higher-form symmetries.