  We propose a universal inequality that unifies the Bousso bound with the
classical focussing theorem. Given a surface $\sigma$ that need not lie on a
horizon, we define a finite generalized entropy $S_\text{gen}$ as the area of
$\sigma$ in Planck units, plus the von Neumann entropy of its exterior. Given a
null congruence $N$ orthogonal to $\sigma$, the rate of change of
$S_\text{gen}$ per unit area defines a quantum expansion. We conjecture that
the quantum expansion cannot increase along $N$. This extends the notion of
universal focussing 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 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 this novel relation in quantum field theory.
