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arXiv:2607.18464

Deep inelastic scattering as a probe of entanglement: the complete QCD dipole cascade

Martin Hentschinski, Krzysztof Kutak, Wieslaw Placzek, Martin Rohrmoser

hep-phquant-phSubmitted Jul 20, 2026• Announced Jul 22, 2026
v1 · Jul 20, 2026v2 · Jul 22, 2026
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Abstract

We study entanglement entropy in Deep Inelastic Scattering (DIS) using the dipole formulation of the high-energy limit of QCD. We argue that a reduced density matrix arises in low $x$ DIS due to a trace over unobserved color degrees of freedom and we obtain entanglement entropy in terms of dipole multiplicities, directly from the von Neumann entropy. %In the present study we explicitly construct a density matrix representing the entangled state of the photon and proton and perform a trace over unobserved degrees of freedom. Dipole multiplicities are obtained from a solution to low $x$ evolution equations, which we solve numerically. Unlike previous studies, we take into account both transverse-size and azimuthal-angle dependence in the dipole evolution kernel. We study both the exact solution of the equation as well as its double leading-logarithmic approximation (DLLA). We find that for the same initial dipole size, the DLLA solution generates a larger entropy. %We interpret this as information loss due to erasing data on daughter dipole sizes. Finally, we calculate the dipole multiplicities and entanglement entropy and compare our results to the Shannon entropy of hadron multiplicities, as measured by the H1 collaboration.

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We study entanglement entropy in Deep Inelastic Scattering (DIS) using the dipole formulation of the high-energy limit of QCD. We argue that a reduced density matrix arises in low xxx DIS due to a trace over unobserved color degrees of freedom and we obtain entanglement entropy in terms of dipole multiplicities, directly from the von Neumann entropy.
%In the present study we explicitly construct a density matrix representing the entangled state of the photon and proton and perform a trace over unobserved degrees of freedom. Dipole multiplicities are obtained from a solution to low xxx evolution equations, which we solve numerically. Unlike previous studies, we take into account both transverse-size and azimuthal-angle dependence in the dipole evolution kernel. We study both the exact solution of the equation as well as its double leading-logarithmic approximation (DLLA). We find that for the same initial dipole size, the DLLA solution generates a larger entropy. %We interpret this as information loss due to erasing data on daughter dipole sizes. Finally, we calculate the dipole multiplicities and entanglement entropy and compare our results to the Shannon entropy of hadron multiplicities, as measured by the H1 collaboration.
Deep inelastic scattering (DIS) samples a part of the wave function of a hadron in the vicinity of the light cone. Lipatov constructed a spin chain which describes the amplitude of DIS in leading logarithmic approximation. Kharzeev and Levin proposed the entanglement entropy as an observable in DIS [Phys. Rev. D 95, 114008 (2017)], and suggested a relation between the entanglement entropy and parton distributions. Here we represent the DIS process as a local quench in Lipatov's spin chain and study the time evolution of the produced entanglement entropy. We show that the resulting entanglement entropy depends on time logarithmically, $\mathcal{S}(t)=1/3\mathrm{ln}(t/\ensuremath{\tau})$ with $\ensuremath{\tau}=1/m$ for $1/m\ensuremath{\le}t\ensuremath{\le}(mx{)}^{\ensuremath{-}1}$, where mmm is the proton mass and xxx is the Bjorken xxx. The central charge ccc of Lipatov's spin chain is determined here to be c=1c=1c=1; using the proposed relation between the entanglement entropy and parton distributions, this corresponds to the gluon structure function growing at small xxx as $xG(x)\ensuremath{\sim}1/{x}^{1/3}$.
Relying on previous results, that link entanglement entropy and parton distribution functions in deep inelastic scattering, and focusing on the small Bjorken scaling region we present here indications that gluon shadowing might indeed be explained as due to a depletion of the entanglement entropy, between observed and unobserved degrees of freedom, per nucleon within a nucleus, with respect to the free nucleon. We apply to gluon shadowing the general Page approach to the calculation of the entanglement entropy in bipartite systems, giving physical motivations of the results.
Recent investigations into High-Energy QCD have identified entanglement entropy as a crucial observable, linking parton distributions to the structure of the quantum vacuum. While momentum-space entanglement has been extensively studied in Deep Inelastic Scattering (DIS), the spatial realization of this entanglement in confined systems remains an open question. In this Letter, we demonstrate that the confining boundary of the MIT Bag Model acts as an ``Entropic Skin,'' generating maximal Spin-Position entanglement. We calculate the local reduced density matrix for the confined quark and show that the linear boundary condition, iγμnμψ=ψi γ^μn_μψ= ψiγμnμ​ψ=ψ, acts as an entangling gate. The surface entropy density reaches a geometric invariant of ≈0.918\approx 0.918≈0.918 bits (92%92\%92% of the qubit limit), independent of the bag radius. We discuss the implications of this result for Chiral Symmetry breaking and propose that this boundary entropy is the precursor to the pion cloud in effective field theories.
We show that the hierarchy of quantum correlations produced in deep-inelastic scattering (DIS) can serve as a novel probe of the proton's nonperturbative structure. Specifically, we show that quantum entanglement, discord, steering, and magic provide nontrivial and complementary sensitivities to the nucleon's parton distribution functions (PDFs), particularly those encoding the transverse-spin polarization of the interacting quark. This connection leads to a unique probe of the proton's parton-level tensor charges with implications for beyond Standard Model (BSM) physics searches. We propose how quantum information measures can be utilized for precision studies of hadron structure at DIS experiments like the upcoming Electron-Ion Collider (EIC).
We construct the entanglement entropy of a general scattering process between one particle species (denoted as AAA) and the rest of the final state. Assuming a pure initial momentum eigenstate, we show that the reduced density matrix in particle AAA's Fock space is block-diagonal in zero-, one- and multi-particle states, and is diagonal in the momentum basis for the one-particle state. As a result, we could write down the Von Neumann entropy of particle AAA, SAS_ASA​, in terms of scattering probabilities and differential cross sections, assuming no more than one particle AAA is produced in the final state. SAS_ASA​ contains the information of both the total scattering probabilities (or cross sections) and the differential distributions --- the latter depends on a phase-space resolution factor as a result of the infinite-dimensional momentum Hilbert space. Our result is general and model-independent since we consider the full final state from time evolution (S∣i⟩S|i\rangleS∣i⟩) rather than a specific outcome, and is applicable as long as the probability for producing multiple particle AAA is subleading. We also obtain the n≥2n\geq2n≥2 Tsallis and Rényi entropies in our framework and find out that, at the leading order, they are proportional to the total scattering probability and carry no differential information.
We investigate quark-antiquark entanglement in heavy quarkonium within a nonperturbative light-front Hamiltonian framework. By tracing over the antiquark degrees of freedom in the hadronic state vector, we construct the reduced density matrix of the quark subsystem and compute the associated von Neumann entropy. For spin-0 quarkonia, we show that this entropy reduces to the Shannon entropy of the unpolarized transverse momentum dependent parton distribution (TMD), up to constant color and spin contributions. For spin-1 quarkonia, we derive the explicit polarization dependence of the entropy and connect it to polarized and tensor-polarized TMDs. Using light-front wave functions obtained via basis light-front quantization (BLFQ), we evaluate the entanglement entropy for charmonium and bottomonium states, revealing a pronounced sensitivity to the polarization of vector mesons. Furthermore, we resolve the infrared parameter by matching the momentum-space entropy to a harmonic-oscillator representation. Ultimately, these results establish entanglement entropy as a novel probe of nonperturbative quarkonium structure, forging a direct link between quantum information measures and partonic observables.