  Using non-linear evolution equations of QCD, we compute the von Neumann
entropy of the system of partons resolved by deep inelastic scattering at a
given Bjorken $x$ and momentum transfer $q^2 = - Q^2$. We interpret the result
as the entropy of entanglement between the spatial region probed by deep
inelastic scattering and the rest of the proton. At small $x$ the relation
between the entanglement entropy $S(x)$ and the parton distribution $xG(x)$
becomes very simple: $S(x) = \ln[ xG(x) ]$. In this small $x$, large rapidity
$Y$ regime, all partonic micro-states have equal probabilities -- the proton is
composed by an exponentially large number $\exp(\Delta Y)$ of micro-states that
occur with equal and exponentially small probabilities $\exp(-\Delta Y)$, where
$\Delta$ is defined by $xG(x) \sim 1/x^\Delta$. For this equipartitioned state,
the entanglement entropy is maximal -- so at small $x$, deep inelastic
scattering probes a {\it maximally entangled state}. We propose the
entanglement entropy as an observable that can be studied in deep inelastic
scattering. This will require event-by-event measurements of hadronic final
states, and would allow to study the transformation of entanglement entropy
into the Boltzmann one. We estimate that the proton is represented by the
maximally entangled state at $x \leq 10^{-3}$; this kinematic region will be
amenable to studies at the Electron Ion Collider.
