  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.
