  We discuss quantum entanglement between fast and slow degrees of freedom, in
a two dimensional (2D) large $N_c$ gauge theory with Dirac quarks, quantized on
the light front. Using the 't Hooft wave functions, we construct the reduced
density matrix for an interval in the momentum fraction $x$-space, and
calculate its von Neumann entropy in terms of structure functions, that are
measured by DIS on mesons (hadrons in general). We found that the entropy is
bounded by an area law with logarithmic divergences, proportional to the
rapidity of the meson. The evolution of the entanglement entropy with rapidity,
is fixed by the cumulative singlet PDF, and bounded from above by a
Kolmogorov-Sinai entropy of 1. At low-$x$, the entanglement exhibits an
asymptotic expansion, similar to the forward meson-meson scattering amplitude
in the Regge limit. The evolution of the entanglement entropy in parton-$x$ per
unit rapidity, measures the meson singlet PDF. The re-summed entanglement
entropy along the single meson Regge trajectory, is string-like. We suggest
that its extension to multi-meson states, models DIS scattering on a large 2D
$^\prime$nucleus$^\prime$. The result, is a large rate of change of the
entanglement entropy with rapidity, that matches the current
Bekenstein-Bremermann bound for maximum quantum information flow. This
mechanism may be at the origin of the large entropy deposition and rapid
thermalization, reported in current heavy ion colliders, and may extend to
future electron-ion colliders.
