  We investigate the quantum entanglement in rapidity space of the soft gluon
wave function of a quarkonium, in theories with non-trivial rapidity
evolutions. We found that the rapidity evolution drastically changes the
behavior of the entanglement entropy, at any given order in perturbation
theory. At large $N_c$, the reduced density matrices that "resum" the leading
rapidity-logs can be explicitly constructed, and shown to satisfy
Balitsky-Kovchegov (BK)-like evolution equations. We study their entanglement
entropy in a simplified $1+1$ toy model, and in 3D QCD. The entanglement
entropy in these cases, after re-summation, is shown to saturate the
Kolmogorov-Sinai bound of 1. Remarkably, in 3D QCD the essential growth rate of
the entanglement entropy is found to vanish at large rapidities, a result of
kinematical "quenching" in transverse space. The one-body reduction of the
entangled density matrix obeys a BFKL evolution equation, which can be recast
as an evolution in an emergent AdS space, at large impact-parameter and large
rapidity. This observation allows the extension of the perturbative wee parton
evolution at low-x, to a dual non-perturbative evolution of string bits in
curved AdS$_5$ space, with manifest entanglement entropy in the confining
regime.
