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 extends the quantum null energy condition (QNEC), and includes the possibility of expanding or contracting geodesics. Treating the INEC as a field theory inequality, we use the properties of relative entropy and modular Hamiltonian associated with null deformation of the sphere to recast it as an integrated inequality. At leading order in a perturbation of the vacuum, this inequality becomes a weighted positivity condition on the smeared null energy. As the underlying single-segment light-cone operator is unbounded below for d > 2, we pass to the Lorentzian cylinder ℝ × S^{d − 1}, where the operator is well-defined. We verify that the INEC holds explicitly in the thermal state, in both the perturbative and high-temperature regimes. Furthermore, using the QNEC and INEC as a basis, we briefly speculate about a possible modified Quantum Focusing Conjecture.