  In an attempt to find a quasi-local measure of quantum entanglement, we
introduce the concept of entanglement density in relativistic quantum theories.
This density is defined in terms of infinitesimal variations of the region
whose entanglement we monitor, and in certain cases can be mapped to the
variations of the generating points of the associated domain of dependence. We
argue that strong sub-additivity constrains the entanglement density to be
positive semi-definite. Examining this density in the holographic context, we
map its positivity to a statement of integrated null energy condition in the
gravity dual. We further speculate that this may be mapped to a statement
analogous to the second law of black hole thermodynamics, for the extremal
surface.
