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Generalized Entanglement, Charges and Intertwiners

The entanglement theory in quantum systems with internal symmetries is rich due to the spontaneous creation of entangled pairs of charge/anti-charge particles at the entangling surface. We call these pair creation operators the bi-local intertwiners because of the role they play in the representation theory of the symmetry group. We define a generalized measure of entanglement entropy as a measure of information erased under restriction to a subspace of observables. We argue that the correct entanglement measure in the presence of charges is the sum of two terms; one measuring the entanglement of charge-neutral operators, and the other measuring the contribution of the bi-local intertwiners. Our expression is unambiguously defined in lattice models as well in quantum field theory (QFT). We use the Tomita-Takesaki modular theory to highlight the differences between QFT and lattice models, and discuss an extension of the algebra of QFT that leads to a factorization of the charged modes.