  We prove the Quantum Null Energy Condition (QNEC), a lower bound on the
stress tensor in terms of the second variation in a null direction of the
entropy of a region. The QNEC arose previously as a consequence of the Quantum
Focussing Conjecture, a proposal about quantum gravity. The QNEC itself does
not involve gravity, so a proof within quantum field theory is possible. Our
proof is somewhat nontrivial, suggesting that there may be alternative
formulations of quantum field theory that make the QNEC more manifest.
  Our proof applies to free and superrenormalizable bosonic field theories, and
to any points that lie on stationary null surfaces. An example is Minkowski
space, where any point $p$ and null vector $k^a$ define a null plane $N$ (a
Rindler horizon). Given any codimension-2 surface $\Sigma$ that contains $p$
and lies on $N$, one can consider the von Neumann entropy $S_\text{out}$ of the
quantum state restricted to one side of $\Sigma$. A second variation
$S_\text{out}^{\prime\prime}$ can be defined by deforming $\Sigma$ along $N$,
in a small neighborhood of $p$ with area $\cal A$. The QNEC states that
$\langle T_{kk}(p) \rangle \ge \frac{\hbar}{2\pi} \lim_{{\cal A}\to
0}S_\text{out}^{ \prime\prime}/{\cal A}$.
