Relative entanglement entropy for widely separated regions in curved spacetime

We give an upper bound of the relative entanglement entropy of the ground state of a massive Dirac-Majorana field across two widely separated regions A and B in a static slice of an ultrastatic Lorentzian spacetime. Our bound decays exponentially in dist(A, B) at a rate set by the Compton wavelength and the spatial scalar curvature. The physical interpretation of our result is that, on a manifold with positive spatial scalar curvature, one cannot use the entanglement of the vacuum state to teleport one classical bit from A to B if their distance is of the order of the maximum of the curvature radius and the Compton wavelength or greater.