  There is a simple bound on how fast the entanglement entropy of a subregion
of a many-body quantum system can saturate in a quench: $t_\text{sat}\geq
R/v_B$, where $t_\text{sat}$ is the saturation time, $R$ the radius of the
largest inscribed sphere, and $v_B$ the butterfly velocity characterizing
operator growth. By combining analytic and numerical approaches, we show that
in systems with a holographic dual, the saturation time is equal to this lower
bound for a variety of differently shaped entangling surfaces, implying that
the dual black holes saturate the entanglement entropy as fast as possible.
This finding adds to the growing list of tasks that black holes are the fastest
at. We furthermore analyze the complete time evolution of entanglement entropy
for large regions with a variety of shapes, yielding more detailed information
about the process of thermalization in these systems.
