Deep inelastic scattering as a probe of entanglement

Using nonlinear 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 q2=−Q2. 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 microstates have equal probabilities-the proton is composed by an exponentially large number exp(ΔY) of microstates that occur with equal and exponentially small probabilities exp(−ΔY), where Δ is defined by xG(x)∼1/xΔ. For this equipartitioned state, the entanglement entropy is maximal-so at small x, deep inelastic scattering probes a 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≤10−3; this kinematic region will be amenable to studies at the Electron Ion Collider.