  The Quantum Null Energy Condition (QNEC) relates energy to the second
variation of entropy in relativistic quantum field theory. We use the QNEC
inequality to bound entanglement entropy in quenches. At early times the
entanglement entropy grows quadratically in time, and the QNEC provides an
upper bound on the prefactor. We demonstrate that the bound is tight, by
showing that it is saturated in certain quench protocols: boundary state
quenches in conformal field theories in any dimensions. In higher than two
dimensions we compute entanglement entropy using AdS/CFT. Our results are the
first purely field theoretic applications of the QNEC.
