Proof of the quantum null energy condition for free fermionic field theories

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.