  We investigate the proposal by Kharzeev and Levin of a maximally entangled
proton wave function in Deep Inelastic Scattering at low $x$ and the proposed
relation between parton number and final state hadron multiplicity. Contrary to
the original formulation we determine partonic entropy from the sum of gluon
and quark distribution functions at low $x$, which we obtain from an
unintegrated gluon distribution subject to next-to-leading order
Balitsky-Fadin-Kuraev-Lipatov evolution. We find for this framework very good
agreement with H1 data. We furthermore provide a comparison based on NNPDF
parton distribution functions at both next-to-next-to-leading order and
next-to-next-to-leading with small $x$ resummation, where the latter provides
an acceptable description of data.
