  We show that in any relativistic system, entanglement entropy obeys a speed
limit set by the entanglement in thermal equilibrium. The bound is derived from
inequalities on relative entropy with respect to a thermal reference state.
Thus the thermal state constrains far-from-equilibrium entanglement dynamics
whether or not the system actually equilibrates, in a manner reminiscent of
fluctuation theorems in classical statistical mechanics. A similar
shape-dependent bound constrains the full nonlinear time evolution, supporting
a simple physical picture for entanglement propagation that has previously been
motivated by holographic calculations in conformal field theory. We discuss
general quantum field theories in any spacetime dimension, but also derive some
results of independent interest for thermal relative entropy in 1+1d CFT.
