Quantum focusing conjecture

We propose a universal inequality that unifies the Bousso bound with the classical focusing theorem. Given a surface σ that need not lie on a horizon, we define a finite generalized entropy Sgen as the area of σ in Planck units, plus the von Neumann entropy of its exterior. Given a null congruence N orthogonal to σ, the rate of change of Sgen per unit area defines a quantum expansion. We conjecture that the quantum expansion cannot increase along N. This extends the notion of universal focusing to cases where quantum matter may violate the null energy condition. Integrating the conjecture yields a precise version of the Strominger-Thompson quantum Bousso bound. Applied to locally parallel light-rays, the conjecture implies a novel inequality, the quantum null energy condition, a lower bound on the stress tensor in terms of the second derivative of the von Neumann entropy. We sketch a proof of the latter relation in quantum field theory.