We investigate the quantum entanglement in rapidity space of the soft gluon wave function of a quarkonium in theories with nontrivial 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 , the reduced density matrices that “resum” the leading rapidity logs can be explicitly constructed, and shown to satisfy Balitsky-Kovchegov-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 resummation, 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 Balitsky-Fadin-Kuraev-Lipatov 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 nonperturbative evolution of string bits in curved space, with manifest entanglement entropy in the confining regime.