Improved Null Energy Condition

Generalized entropy, defined as the sum of the area of a Cauchy-dividing surface in Planck units and the von Neumann entropy of the fields on one side of the surface, captures a deep connection between entanglement and geometry in semiclassical gravity. It is expected to satisfy fundamental thermodynamical conditions, notably Quantum Focusing Conjecture (QFC), which within certain limits reduces to quantum field theory inequalities. When we consider the limit in which the classical expansion is zero, it leads to the quantum null energy condition (QNEC). The QNEC is the quantum version of the null energy condition, and it has been proven in many cases.

Another limit of QFC, obtained through restriction to quantum extremal surfaces, leads to the Improved Quantum Null Energy Condition (INEC). The INEC is a statement relating the local energy density and the shape derivative of von Neumann entropy in Minkowski space-time. It is a stronger condition than the quantum null energy condition and includes the possibility of expanding or contracting geodesics. Using the properties of relative entropy and a modular Hamiltonian associated with null deformation of the sphere, we show that the INEC holds under an additional assumption relating the stress-energy tensor to the relative entropy. We show that the INEC holds at leading order in a
perturbation of the thermal state on a cylinder, and reduce the general case to a monotonicity
condition on the cap entropy.