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Entanglement entropy is a valuable tool for characterizing the correlationstructure of quantum field theories. When applied to gauge theories, subtletiesarise which prevent the factorization of the Hilbert space underlying thenotion of entanglement entropy. Borrowing techniques from extended topologicalfield theories, we introduce a new definition of entanglement entropy for bothAbelian and non-Abelian gauge theories. Being based on the notion ofexcitations, it provides a completely relational way of defining regions.Therefore, it naturally applies to background independent theories, e.g.gravity, by circumventing the difficulty of specifying the position of theentangling surface. We relate our construction to earlier proposals and arguethat it brings these closer to each other. In particular, it yields thenon-Abelian analogue of the "magnetic centre choice", as obtained through anextended-Hilbert-space method, but applied to the recently introduced fusionbasis for 3D lattice gauge theories. We point out that the differentdefinitions of entanglement theory can be related to a choice of (squeezed)vacuum state.

On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity