Quantum focusing conjecture

Abstract

We propose a universal inequality that unifies the Bousso bound with the classical focusing theorem. Given a surface $\ensuremath{\sigma}$ that need not lie on a horizon, we define a finite generalized entropy ${S}_{\mathrm{gen}}$ as the area of $\ensuremath{\sigma}$ in Planck units, plus the von Neumann entropy of its exterior. Given a null congruence $N$ orthogonal to $\ensuremath{\sigma}$, the rate of change of ${S}_{\mathrm{gen}}$ 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.