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arXiv:2607.24068

Quantum entanglement within quarkonium

Wenyu Zhang, Yiyu Zhou, Yang Li, Qun Wang

hep-phquant-phSubmitted Jul 27, 2026• Announced Jul 28, 2026• 0 citations
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Abstract

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.

Comments
14 pages, 6 figures

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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.
We perform a first-principles, non-perturbative investigation of quantum entanglement between partonic constituents in a strongly coupled 3+1-dimensional scalar Yukawa theory, using light-front Hamiltonian methods with controlled Fock-space truncations. By explicitly constructing reduced density matrices for (mock) nucleon, pion, and anti-nucleon subsystems from light-front wave functions, we compute key entanglement witnesses, including von Neumann entropy, mutual information, and linear entropy, in both quenched (no sea pairs) and unquenched frameworks. We find that the entanglement entropy is closely related to the Shannon entropy of the transverse momentum dependent distribution, establishing a link between quantum information and parton structure. In contrast, the unquenched theory reveals genuinely non-classical correlations: the entanglement entropy cannot be reduced to any Shannon entropy of normalized parton distributions, demonstrating that the full hadronic wave function encodes quantum information beyond classical probabilities. Our findings highlight the role of entanglement as a fundamental probe of non-perturbative dynamics in relativistic quantum field theory and lay the groundwork for extending these concepts to QCD and future collider phenomenology.
Quantum entanglement provides a quantitative probe of the internal structure of hadrons and offers a sensitive means to study the quantum correlation in the hadron wave functions. For baryons, the spin state of the three valence quarks forms a tripartite qubit system, whose entanglement structure can be characterized by the four classes of three-qubit states. In this work, we compare the proton spin entanglement obtained from Basis Light-Front Quantization (BLFQ) with that from a quark-diquark model. By analyzing both bipartite and tripartite entanglement, we find that the quark-diquark model yields a substantially more entangled spin state than the BLFQ wave function in the valence Fock sector. This difference mainly originates from the larger W-type and Bell-type entanglement in the quark-diquark model. Within BLFQ, larger stronger coupling constant and smaller quark mass drive the spin correlation among the valence quarks towards an effective quark-diquark configuration with an active ddd quark and a correlated uuuuuu pair.
Within the collinear factorization framework based on Generalized Parton Distributions (GPDs), we calculate the spin density matrix of exclusively produced quark and antiquark pairs uuˉu\bar{u}uuˉ, ddˉd\bar{d}ddˉ, ssˉs\bar{s}ssˉ, ccˉc\bar{c}ccˉ, bbˉb\bar{b}bbˉ in electron-proton scattering. The presence of both real and imaginary parts in the scattering amplitudes leads to a rich pattern of entanglement between the quark and the antiquark. We map out kinematical regions where the pairs exhibit entanglement, Bell nonlocality and non-stabilizerness (`magic'). We also predict that massive quarks and antiquarks are transversely polarized, similar to the well-known transverse hyperon polarization in unpolarized collisions. In strangeness, charm and bottom productions, the polarization can reach 50-80\% in certain kinematic regions in the low-energy runs of the Electron-Ion Collider.
Spin correlations offer a quantum-information perspective on the partonic final states produced in high-energy scattering. We discuss the spin-density matrix of a heavy quark-antiquark pair in two complementary small-xxx processes. In coherent diffractive production, color-singlet exchange enforces an unusually strong relation between entanglement and Bell nonlocality: a longitudinal photon creates a maximally entangled pair, whereas for a transverse photon the pair is generically both entangled and Bell nonlocal, with a model-independent point of maximal entanglement. In inclusive back-to-back production, the density matrix factorizes into a hard spin tensor and the unpolarized and linearly polarized Weizsäcker--Williams gluon distributions. The latter generates an azimuthal modulation and can increase the concurrence when the dijet relative momentum and imbalance are approximately orthogonal. Strikingly, in the saturation model considered, nonlinear effects wash out this modulation for q⊥≲3Qsq_\perp\lesssim 3Q_sq⊥​≲3Qs​, whereas the dilute BFKL limit, in which G2/G0→1G_2/G_0\to1G2​/G0​→1, yields a maximal modulation independent of the imbalance magnitude. These results connect quantum-information observables with the Pomeron and saturation physics.
We study the color correlation between the static quark and antiquark (qq‾q\overline{q}qq​) in the confined phase via reduced density matrices $\ensuremath{\rho}$ defined in color space. We adopt the standard Wilson gauge action and perform quenched calculations with the Coulomb gauge condition for reduced density matrices. The spatial volumes are L3=83{L}^{3}={8}^{3}L3=83, 1631{6}^{3}163, 3233{2}^{3}323, and 4834{8}^{3}483, with the gauge couplings $\ensuremath{\beta}=5.7$, 5.8, and 6.0. Each element of the reduced density matrix in the subspace of quarks' color degrees of freedom of the qq‾q\overline{q}qq​ pair is calculated from staples defined by link variables. As a result, we find that $\ensuremath{\rho}$ is well written by a linear combination of the strongly correlated qq‾q\overline{q}qq​ pair state with the color-singlet component and the uncorrelated qq‾q\overline{q}qq​ pair state with random color configurations. We compute the Renyi entropies SRenyi{S}^{\text{Renyi}}SRenyi from $\ensuremath{\rho}$ to investigate the qq‾q\overline{q}qq​ distance dependence of the color correlation of the qq‾q\overline{q}qq​ pair and find that the color correlation is quenched as the distance increases.
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).