Energy density from second shape variations of the von Neumann entropy

We compute the local second variation of the von Neumann entropy of a region in theories with a gravity dual. For null variations our formula says that the diagonal part of the quantum null energy condition (QNEC) is saturated in every state, thus providing an equivalence between energy and entropy. We prove that the formula holds at leading order in 1/N and further argue that it will not be affected at higher orders. We conjecture that the QNEC is saturated in all interacting theories. We also discuss the special case of free theories, and the implications of our formula for the averaged null energy condition, quantum focusing conjecture (QFC), and gravitational equations of motion. We show that the leading-order gravitational equations of motion, Einstein’s equations, are equivalent to the leading-order saturation of the QFC for Planck-width deformations.