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Quark and gluon entanglement in the proton based on a light-front Hamiltonian

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Adding a gluon to the proton's valence state roughly doubles quark spin entanglement.

desk verdict A transparent numerical study whose central claim about gluon-enhanced entanglement is undermined by an uncontrolled comparison between wave functions from two different Hamiltonians and parameter sets. read the letter →

arxiv 2412.11860 v1 pith:CLBA4ZCT submitted 2024-12-16 hep-ph quant-ph

Chen Qian , Siqi Xu , Yang-Guang Yang , Xingbo Zhao This is my paper · ORCID
classification hep-phquant-ph
keywords entanglemententropylight-frontHamiltonianprotonstructuregluonBasisQuantizationBell-CHinequalitypartonhelicityFocksector
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that including one dynamical gluon in the proton's light-front wave function substantially increases quantum entanglement among its partons. Using wave functions from a Hamiltonian diagonalization in two Fock sectors, |qqq> and |qqq>+|qqqg>, it computes spin and longitudinal-momentum entanglement entropies for each parton relative to the rest. The reported numbers show the u-quark spin entanglement rising from 0.219 to 0.414, and the gluon carrying the largest spin entropy, 0.766. If correct, this provides a concrete, calculable sense in which gluons mediate quantum information exchange inside the proton, and it suggests entanglement could be probed experimentally through parton helicity distributions.

What carries the argument

The central objects are the light-front wave functions generated by Basis Light-front Quantization, a nonperturbative method that solves relativistic bound states by diagonalizing the light-front Hamiltonian in a basis. The paper uses two truncations: one with only the valence sector |qqq> and one that adds the single-gluon sector |qqqg>. Entanglement is quantified by the von Neumann entropy of reduced density matrices obtained by partially tracing over all other degrees of freedom, separately for spin and for longitudinal momentum. The gluon is treated as a spin-1 qutrit with an extra 'no gluon' state, while quarks are spin-1/2 qubits, which matters for encoding and for why no Bell-CH inequality is applied to the full system.

What would settle it

Repeat the computation in a single controlled Hamiltonian where the |qqq> and |qqq>+|qqqg> spaces are solved with identical parameters and truncation; if the quark spin entanglement entropy does not rise when the gluon sector is added, the paper's central claim would be refuted. A less direct test would be to measure parton helicity distributions and check whether the predicted x-dependent entanglement pattern is visible.

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Extended reading notes

Core claim

The authors find that spin entanglement entropy of a valence quark with the remainder of the proton increases when the Fock space is expanded from |qqq> to |qqq>+|qqqg>. In the gluon-included wave function, the reported entropies are S_spin = {0.414, 0.483, 0.414, 0.766} for u, d, u, and gluon, versus {0.219, 0.172, 0.219} in the valence-only state. The longitudinal momentum entropies remain large for quarks (about 3.35-3.47) and drop to 2.722 for the gluon. They also report a marginal Bell-CH inequality violation (maximum 0.0905553) in the pure three-quark spin state, which disappears when the gluon is traced out, and they observe that the spin entanglement of a quark at fixed longitudinal momentum fraction xf is roughly symmetric with that of the gluon at 1-xf, a symmetry that holds exactly in a dressed quark test state.

Load-bearing premise

The comparison between the two wave functions assumes they differ only by the inclusion of the gluon sector, but they actually come from different Hamiltonians, different truncations (Nmax=10 vs 9), and different fitted parameters (e.g., confinement strength 0.34 vs 0.54 GeV), so the reported increase in entanglement could partly be an artifact of these model differences.

Editorial extensions

If this is right

  • Adding a dynamical gluon sector raises the spin entanglement of valence quarks with the rest of the proton, e.g., u quark from 0.219 to 0.414.
  • The gluon itself has the largest spin entanglement (0.766), suggesting it is the dominant carrier of quantum information among quarks.
  • In the pure |qqq> spin state the Bell-CH inequality is violated only marginally (maximum 0.0905553), indicating weak nonlocal correlations among the three valence quarks; tracing out the gluon removes the violation.
  • The spin entanglement of a quark at momentum fraction xf is approximately symmetric with that of the gluon at 1-xf, hinting at an exchange-like role for the gluon.
  • The computed entanglement entropies are in principle accessible through measurements of parton helicity distributions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the comparison uses different model parameters, the quantitative increase in entropy may not survive a controlled calculation; the paper's qualitative claim that gluons carry extra entanglement is more robust than the specific numbers.
  • The approximate symmetry S_spin(xf) ≈ S_spin(1-xf) for quark versus gluon suggests a form of momentum-helicity complementarity that, if confirmed, would tie entanglement entropy directly to the shape of helicity distributions.
  • The gluon spin being a qutrit with a 'no gluon' state means standard qubit Bell inequalities do not apply; constructing a qutrit Bell test for the |qqqg> sector could give a sharper nonlocality probe.
  • The method can be extended to higher Fock sectors (sea quarks, multiple gluons) to see how entanglement grows with resolution scale, which the paper lists as a next step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper uses light-front wave functions for the proton generated by Basis Light-front Quantization (BLFQ) in the |qqq> and |qqq>+|qqqg> Fock spaces to compute von Neumann entanglement entropies for the spin and longitudinal momentum of individual partons. It also evaluates a three-body Bell-CH inequality for the valence-quark spin state and studies spin entanglement at fixed longitudinal momentum fractions, comparing the proton with a dressed quark. The central claim is that the dynamical gluon significantly enhances entanglement among the proton's partons, as evidenced by larger spin entanglement entropies in the two-sector wave function than in the valence-only wave function.

Significance. If the central comparison were controlled, the paper would provide a useful nonperturbative, Hamiltonian-based estimate of parton-level entanglement in the proton, complementing existing DIS-based entanglement studies. The manuscript is transparent about the model parameters and uses wave functions from prior BLFQ publications, and the entanglement calculations are clearly described. The paper also attempts to connect the computed quantities to helicity distributions, although this connection remains qualitative. The main limitation is that the headline comparison is not controlled: the two wave functions come from different Hamiltonians, truncations, and parameter sets, so the attribution of the entropy increase to the dynamical gluon is not currently established. There is no circularity issue, because the model parameters were fitted to the proton mass and form factors, not to entanglement entropies.

major comments (3)
  1. [Abstract and Secs. III A, III B, V (Figs. 1-2)] The central claim that the dynamical gluon 'significantly enhances entanglement' is based on comparing the spin entanglement entropies S_spin of the |qqq> wave function (Fig. 1) with those of the |qqq>+|qqqg> wave function (Fig. 2). These two wave functions are not the same model with and without a gluon: they are generated from different Hamiltonians [Eqs. (3) and (6)], with different Fock-space truncations (Nmax=10 vs Nmax=9) and different fitted parameters (Table I: kappa=0.34 GeV, m_q/k=0.3 GeV, alpha_s=1.1; Sec. II A 2: kappa=0.54, m_u=0.31 GeV, m_d=0.25 GeV, m_g=0.5 GeV, gS_tilde=2.4). The increase in S_spin from 0.219 to 0.414 for the u quark and the gluon entropy 0.766 could therefore be caused by the change of Hamiltonian, basis, or parameters rather than by the dynamical gluon. To support the headline conclusion, the authors need a controlled test, e.g., computing S_spin for the valence-sector projection of the [25] eigenstate alone (or setting the |qqqg> amplitude to zero within the same Hamiltonian and truncation) and showing that it does not already produce S_spin approximately 0.4 for the valence quarks. Without such a test, the Abstract's attribution of the enhancement to the gluon is not established.
  2. [Sec. IV A and IV B, Eqs. (14)-(19)] The x-resolved 'entanglement entropy' is not the entanglement of a parton at fixed x with the rest of the proton in the global state. In Eqs. (14)-(16) (and similarly Eqs. (18)-(19)), the authors select all configurations containing a parton with longitudinal momentum x_f, form a new wave function, and renormalize its norm to 1. The von Neumann entropy of this post-selected, renormalized state is a property of a conditional subensemble, not of the original proton wave function; renormalization discards the probability weight of the selected configurations, and the entropy of a conditional state is not the conditional entropy S(A|B) of quantum information theory. The conclusions of Sec. IV, including the claim that gluons amplify information exchange between quarks at fixed x, therefore need either a clear operational definition of the quantity being computed or a computation from the global state (e.g., by tracing out all other degrees of freedom without post-selection and then conditioning the reduced density matrix on x_f). As written, the physical interpretation in Sec. V is not supported by the quantity defined in Sec. IV.
  3. [Sec. IV B, Fig. 5 and inset] The comparison between the proton and the dressed quark is also uncontrolled. The dressed quark state is computed with a different Fock-space truncation ({Nmax,K}={7,15.5} rather than {9,16.5}) and represents a different physical system. The conclusion that 'most entanglement of the dynamical gluon in the proton arises from exchange interactions among quarks' (Sec. IV B) relies on the assertion that the dressed quark simulates a quark embedded in the proton, but no quantitative test of this assertion is provided. The smaller S_spin in the dressed quark could be due to the smaller Hilbert space or to other differences unrelated to the self-energy/exchange distinction. This conjecture should be either derived from a controlled calculation (e.g., a dressed quark with the proton's truncation, or a proton calculation with the gluon restricted to self-energy-type couplings) or explicitly labeled as speculative.
minor comments (6)
  1. [Sec. II B, Eq. (10)] Equation (10) is missing the minus sign in the von Neumann entropy formula: S = Tr(rho log2 rho) should be S = -Tr(rho log2 rho); the Shannon form on the right is correct.
  2. [Sec. III B] The text states that the maximal spin entanglement entropy is 1, but for the gluon, which is described as a qutrit, the maximum is log2(3) ~ 1.585; this should be stated for a fair comparison with the quarks.
  3. [Sec. IV A] The definition of the large-x region appears to contain a typo: 'xmin >= 5.5/15.5' should read 'xmax >= 5.5/15.5'.
  4. [Sec. IV] The longitudinal momentum fraction conventions are inconsistent: parton momenta are summed to 16.5 (Sec. III A), but x_f and x_g are written with denominators 15.5 (Sec. IV A) and 15 (Sec. IV B); please clarify the normalization of x.
  5. [Abstract and Sec. V] The statement that the entanglement entropies are 'experimentally accessible' through helicity distributions is not substantiated: no formula or reference connects S_spin(x) to a measurable parton-helicity observable.
  6. [Sec. III A] The Bell-CH violation of 0.0905553 is reported without an uncertainty; since Table I gives alpha_s = 1.1 +/- 0.1, the authors should state whether the violation persists within the parameter uncertainty or treat it as a fixed-parameter result.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: entanglement entropies are computed from BLFQ wave functions that were fit to proton mass and form factors, not to the entanglement observable; the uncontrolled |qqq> vs |qqq>+|qqqg> comparison is a validity concern, not a circular reduction.

full rationale

The derivation chain in this paper is: (1) take BLFQ proton wave functions from Refs. [23-25]; (2) form the full density matrices (Eqs. 12-13); (3) partial-trace over all degrees of freedom except spin or longitudinal momentum (Eq. 8); (4) compute von Neumann/Shannon entropy (Eqs. 9-10). The reported S_spin and S_longitudinal values are therefore outputs of a definite algorithm applied to pre-existing wave functions. Nothing in the paper fits a parameter to the entanglement entropy or defines the wave function in terms of it: the |qqq> model parameters are 'determined by fitting the proton mass and the flavor Dirac form factors' (Sec. II.A.1), and the |qqq>+|qqqg> parameters are set 'by fitting the proton mass and electromagnetic form factors' (Sec. II.A.2). Those observables are external to the entanglement calculation, so the cited wave functions are independent support rather than circular inputs. The main weakness is not circularity but an uncontrolled comparison: the two wave functions come from different Hamiltonians (Eq. 3 vs Eq. 6), different truncations (Nmax=10 vs 9), and different fitted parameters (e.g., kappa=0.34 GeV vs 0.54 GeV), so attributing the 0.219->0.414 S_spin increase specifically to the dynamical gluon is not established by the presented evidence. That is a correctness/robustness objection about missing controls, not a demonstration that a prediction reduces by construction to its input. The self-citations to the prior BLFQ papers are load-bearing only as the source of the input wave functions, but those wave functions were not constructed to reproduce entanglement entropies, so the citation chain does not smuggle in the target result. Accordingly, no circular step of the enumerated kinds is present; score 2 reflects only the minor self-citation dependence and the unproven causal attribution, not circularity.

Assumptions & free parameters 15 free parameters · 8 assumptions · 0 invented entities

The paper introduces no new particles or forces. The free parameters are all inherited from the BLFQ models of Refs. [23-25], where they were fitted to proton mass and form factors. The central claim depends on these parameters through the input wave functions. The main modeling assumptions are the Fock-space truncation, the effective gluon mass, and the post-selection procedure for x-resolved entanglement.

free parameters (15)
  • mq/k = 0.3 GeV
    Quark kinetic mass in the |qqq> model; fitted to proton mass and form factors in Ref. [23,24].
  • mq/g = 0.2 GeV
    Quark mass in interaction terms in the |qqq> model; fitted.
  • kappa_qqq = 0.34 GeV
    Confinement strength for the |qqq> model; fitted.
  • alpha_s = 1.1
    Fixed strong coupling in the one-gluon exchange potential; fitted.
  • mu = 0.31 GeV
    Up quark mass in the |qqq>+|qqqg> model; fitted.
  • md = 0.25 GeV
    Down quark mass in the |qqq>+|qqqg> model; fitted.
  • mg = 0.5 GeV
    Effective gluon mass in the |qqq>+|qqqg> model; fitted.
  • kappa_qqqg = 0.54 GeV
    Confinement strength in the |qqq>+|qqqg> model; fitted.
  • mf = 1.8 GeV
    Quark mass in the vertex interaction, treated as an independent phenomenological parameter; fitted.
  • gS_tilde = 2.4
    Strong coupling in the |qqq>+|qqqg> model; fitted.
  • b = 0.7 GeV
    Harmonic oscillator basis scale parameter; chosen/fitted.
  • binst = 3 GeV
    UV cutoff for the instantaneous interaction; chosen by hand.
  • Nmax_K_qqq = Nmax=10, K=16.5
    Basis truncation for the |qqq> wave function; chosen by hand.
  • Nmax_K_qqqg = Nmax=9, K=16.5
    Basis truncation for the |qqq>+|qqqg> wave function; chosen by hand.
  • Xmin_threshold = 5.5/15.5
    Threshold for defining small and large x regions in Sec. IV; chosen by hand.
assumptions (8)
  • standard math Von Neumann (Shannon) entropy measures bipartite entanglement for pure states (Eqs. 9-10).
    Used throughout to quantify entanglement between a parton and the rest of the proton.
  • standard math Partial trace over unobserved subsystems yields the reduced density matrix (Eq. 8).
    Fundamental operation for computing entanglement entropies.
  • standard math The Bell-CH inequality bound of 0 for local hidden variable theories (Eq. 11).
    Used to test quantum nonlocality of the three-quark spin state.
  • domain assumption The BLFQ light-front Hamiltonian with Fock-space truncation to |qqq> and |qqq>+|qqqg> sectors approximates the proton state (Sec. II A).
    The entire calculation inherits this approximation from Refs. [23-25].
  • domain assumption The effective gluon mass and the confining potential model the nonperturbative dynamics (Sec. II A2).
    The one-dynamical-gluon wave function depends on this model choice.
  • domain assumption The gluon spin is encoded as a qutrit with an additional 'no gluon' state |Omega> (Sec. III B).
    This encoding is necessary to represent the absence of the gluon in the |qqq> sector.
  • ad hoc to paper Post-selected and renormalized wave functions in Sec. IV represent meaningful conditional states of the proton.
    The x-region analysis selects subsets of Fock-space configurations and renormalizes them; the resulting entanglement entropy is for these conditional states, not the original full state.
  • ad hoc to paper The dressed quark system with {Nmax,K}={7,15.5} simulates a quark embedded in the proton (Sec. IV B).
    Used to infer that gluon self-energy contributes little entanglement, but the truncation differs from the proton case.

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Cite this review

Pith. "Pith review of Quark and gluon entanglement in the proton based on a light-front Hamiltonian." pith.science (2026). https://pith.science/paper/CLBA4ZCT

@misc{pith2026241211860,
  author       = {Pith},
  title        = {Pith review of: Quark and gluon entanglement in the proton based on a light-front Hamiltonian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CLBA4ZCT}},
  note         = {Machine review of arXiv:2412.11860}
}
abstract

Given that the wave function of a proton can be derived relativistically and nonperturbatively from a light-front quantized Hamiltonian, investigating the quantum correlation between quarks and gluons offers a novel perspective on the internal structure of partons within a proton. In this work, we address this topic by computing the spin and longitudinal momentum entanglement of each parton inside the proton. The utilized wave functions are generated using Basis Light-front Quantization (BLFQ), incorporating both the valence quarks and one dynamical gluon Fock sectors, $\left|qqq\right\rangle$ and $\left|qqq\right\rangle +\left|qqqg\right\rangle$. Our calculations indicate that the dynamical gluon significantly enhances entanglement among the proton's partons. Additionally, we examine the spin entanglement of quarks and gluons at fixed values of longitudinal momentum fraction, revealing that the presence of a gluon may amplify the informational exchanges between quarks. Finally, these findings suggest the potential for experimental verification of the entanglement between partons by measuring parton helicity distributions in the proton.

Figures

Figures reproduced from arXiv: 2412.11860 by the authors.

Figure 1
Figure 1. The entanglement entropy of spin and longitudinal momentum between each valence quark and the remaining part of the BLFQ proton in the wave function |qqq⟩. The green dots denote the entropy of the spin states Sspin for q1, q2 and q3: {0.219, 0.172, 0.219}. The orange dots denote the entropy of the longitudinal momentum Slongitudinal for q1, q2 and q3: {3.510, 3.493, 3.510}. matrix of |qqq⟩ is ρqqq = |q1q2q3⟩ ⟨q1q2q3… view at source ↗
Figure 2
Figure 2. The entanglement entropy of spin and longitudinal momentum between each valence quark and the remaining part of the BLFQ proton wave function |qqq⟩ + |qqqg⟩. The green dots denote the entropy of the spin states Sspin for q1, q2, q3, and g: {0.414, 0.483, 0.414, 0.766}. The orange dots denote the entropy of the longitudinal momentum Slongitudinal for q1, q2, q3, and g: {3.474, 3.352, 3.474, 2.722}. From [PITH_FULL_I… view at source ↗
Figure 3
Figure 3. The spin entanglement entropy between the parton xmin ≤ 5.5/15.5, xmax ≥ 5.5/15.5 and the remaining partons. The entanglement of a small x region is the dark green dots and line, and that of a large x region is the dark blue dots and line. by renormalizing the new wave function to 1. Then, we exchange xf in the configurations with x1 (no exchange needed when xf = x1), and permute all the other quantum numbers includ… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The entanglement entropy between the parton xf = 0.5/15.5 to 15.5/15.5 and rest partons. Here the dark green dots and line show the spin entanglement entropy in terms of xf . by the partons with largest (smallest) xf in that region. B. The state |qqq⟩ + |qqqg⟩ By inclu…
Figure 5
Figure 5. Figure 5: Comparison of the spin entanglement entropy of quark and gluon with fixed longitudinal momentum xf inside the proton wave function |qqq⟩ + |qqqg⟩. Meanwhile, the inset shows the entanglement entropy of quarks and gluons inside the dressed quark |q⟩ + |qg⟩. The blue dot…

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Forward citations

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Reviewed August 11, 2026 · model on record in the stance chip above.