  A simple argument shows that negative energy cannot be isolated far away from
positive energy in a conformal field theory and strongly constrains its
possible dispersal. This is also required by consistency with the Bekenstein
bound written in terms of the positivity of relative entropy. We prove a new
form of the Bekenstein bound based on the monotonicity of the relative entropy,
involving a "free" entropy enclosed in a region which is highly insensitive to
space-time entanglement, and show that it further improves the negative energy
localization bound.
