Entanglement on curved hypersurfaces: A field-discretizer approach

We propose a covariant scheme for measuring entanglement on general hypersurfaces in relativistic quantum field theory. For that, we introduce an auxiliary relativistic field, "the discretizer,"that by locally interacting with the field along a hypersurface, fully swaps the field's and discretizer's states. It is shown, that the discretizer can be used to effectively cut off the field's infinities, in a covariant fashion, and without having to introduce a spatial lattice. This, in turn, provides us an efficient way to evaluate entanglement between arbitrary regions on any hypersurface. As examples, we study the entanglement between complementary and separated regions in 1+1 dimensions, for flat hypersurfaces in Minkowski space, for curved hypersurfaces in Milne space, and for regions on hypersurfaces approaching null-surfaces. Our results show that the entanglement between regions on arbitrary hypersurfaces in 1+1 dimensions depends only on the spacetime end points of the regions, and not on the shape of the interior. Our results corroborate and extend previous results for flat hypersurfaces.