  We demonstrate a surprising connection between pure steady state entanglement
and relaxation timescales in an extremely broad class of Markovian open
systems, where two (possibly many-body) systems $A$ and $B$ interact locally
with a common dissipative environment. This setup also encompases a broad class
of adaptive quantum dynamics based on continuous measurement and feedback. As
steady state entanglement increases, there is generically an emergent strong
symmetry that leads to a dynamical slow down. Using this we can prove rigorous
bounds on relaxation times set by steady state entanglement. We also find that
this time must necessarily diverge for maximal entanglement. To test our bound,
we consider the dynamics of a random ensemble of local Lindbladians that
support pure steady states, finding that the bound does an excellent job of
predicting how the dissipative gap varies with the amount of entanglement. Our
work provides general insights into how dynamics and entanglement are connected
in open systems, and has specific relevance to quantum reservoir engineering.
