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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.