Modular Time Evolution and the QNEC

We establish an inequality restricting the evolution of states in quantum field theory with respect to the modular flow of a wedge, ${\mathrm{\Delta }}^{is}$, for large $\mathrm{\mid }s\mathrm{\mid }$. Our bound is related to the quantum null energy condition, QNEC. In one interpretation, it can be seen as providing a ``chaos-bound’’ $\le 2\pi$ on the Lyapunov exponent with respect to Rindler time, $s$. Mathematically, our inequality is a statement about half-sided modular inclusions of von Neumann algebras.