Towards a Proof of the Improved Quantum Null Energy Condition

The Improved Quantum Null Energy Condition (INEC) was recently derived from the (restricted) quantum focusing conjecture (QFC), and is a statement about the energy-momentum tensor (EMT) of field theories in Minkowski space-time. It is a stronger condition than the quantum null energy condition (QNEC), and includes the possibility of expanding or contracting geodesics. Using the properties of relative entropy and modular Hamiltonian associated with null deformation of the sphere, we show the INEC holds under an additional assumption relating the EMT to the relative entropy. Furthermore, using the QNEC and INEC as a basis, we briefly speculate about a possible modified Quantum Focusing Conjecture.