  The quantum null energy condition (QNEC) is a quantum generalization of the
null energy condition which gives a lower bound on the null energy in terms of
the second derivative of the von Neumann entropy or entanglement entropy of
some region with respect to a null direction. The QNEC states that $\langle
T_{kk}\rangle_{p}\geq lim_{A\rightarrow 0}\left(\frac{\hbar}{2\pi
A}S_{out}^{\prime\prime}\right)$ where $S_{out}$ is the entanglement entropy
restricted to one side of a codimension-2 surface $\Sigma$ which is deformed in
the null direction about a neighborhood of point $p$ with area $A$. A proof of
QNEC has been given before, which applies to free and super-renormalizable
bosonic field theories, and to any points that lie on a stationary null
surface. Using similar assumptions and methods, we prove the QNEC for fermionic
field theories.
