arxiv: 2604.03005 · v2 · submitted 2026-04-03 · 🪐 quant-ph · hep-ph

Recognition: 2 theorem links

· Lean Theorem

Quantum mutual information, coherence and unified relations of top quarks in QCD processes

Duo-Duo Chen , Xue-Ke Song , Liu Ye , Dong Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-13 19:42 UTC · model grok-4.3

classification 🪐 quant-ph hep-ph

keywords top quarksquantum mutual informationcoherenceQCD processescomplementarity relationsintrinsic relationquark gluon mixing

0 comments

The pith

In top quark-antiquark pairs from QCD, the maximum of the intrinsic quantum relation rises as the gluon initial-state probability increases.

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

The paper examines quantum correlations in top quark-antiquark pairs produced in QCD processes through measures including quantum mutual information, relative entropy of coherence, and complete complementarity relations. These quantities and their unified intrinsic relations are shown to depend on kinematic variables and on the relative probabilities of quark versus gluon initial states. The central result is that raising the gluon probability increases the peak value attained by the left-hand side of the intrinsic relation. A reader would care because top quarks decay before hadronization, so their spin correlations survive to the final state and can be measured at colliders.

Core claim

For mixed quark-gluon initial states in top-quark pair production, the maximum value of the left-hand side of the intrinsic relation among quantum mutual information, coherence, and complementarity increases with the gluon probability Wgg.

What carries the argument

The intrinsic relationship that unifies quantum mutual information, relative entropy of coherence, and complete complementarity relations for the top-quark spin system.

If this is right

Quantum information measures become additional observables that vary with production kinematics in top-pair events.
The intrinsic relation supplies a single number whose peak tracks the initial-state composition.
Higher gluon fractions produce stronger observable quantum correlations in the decay products.
These relations can be evaluated at different collider energies where the gluon fraction changes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Event selection at the LHC could isolate samples with controlled gluon fractions to test the predicted rise in the relation maximum.
The same framework might apply to bottom-quark pairs or other heavy flavors where spin information is accessible.
If the rise is confirmed, quantum-information quantities could serve as indirect probes of parton distribution functions.

Load-bearing premise

The calculations rest on a chosen model of top-quark pair production together with adjustable probabilities for quark versus gluon initial states.

What would settle it

Compute or measure the maximum of the intrinsic relation across top-pair events selected for different gluon-initiated fractions and check whether the maximum fails to rise with increasing gluon probability.

Figures

Figures reproduced from arXiv: 2604.03005 by Dong Wang, Duo-Duo Chen, Liu Ye, Xue-Ke Song.

**Figure 2.** Figure 2: FIG. 2. QMI in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. REC as a function of the invariant mass [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 5.** Figure 5: shows the dependence of QMI IA:B(ˆρAB) and conditional entropy SA|B(ˆρAB) on both the production angle Θ and the invariant mass Mtt¯. As evident from the figure, these quantities exhibit complementary behavior: IA:B(ˆρAB) increases when SA|B(ˆρAB) decreases, and vice versa. Notably, their sum remains constant at unity (IA:B(ˆρAB) + SA|B(ˆρAB) = 1), demonstrating conservation that is independent of the i… view at source ↗

**Figure 6.** Figure 6: FIG. 6. The left-hand side of the intrinsic relation as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. The left-hand side of the intrinsic relation in [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. The left-hand side of the intrinsic relation is plotted as a [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. The left-hand side of the intrinsic relation [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

read the original abstract

As the most massive particle in the Standard Model, the top quark's exceptionally short lifetime preserves its spin polarization information through direct decay, making it an ideal system for probing quantum correlations in high-energy physics. In this letter, we presents a comprehensive investigation of quantum correlations in top quark-antiquark pairs produced through QCD. We employ multiple quantum information theoretic measures including quantum mutual information, relative entropy of coherence, complete complementarity relations, and the intrinsic relationship, establishing their dependence on kinematic variables. Furthermore, we find that for quarks and gluons initial mixing, as the probability of gluons Wgg increases, the maximum of the left-hand side of the intrinsic relation also increases. We thus believe the current findings are beneficial to insight into the systemic quantumness in QCD.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies quantum mutual information and coherence to top-quark pairs but the reported rise with gluon probability Wgg treats that fraction as a free knob rather than a PDF-determined quantity.

read the letter

The work takes existing quantum-information tools—mutual information, relative entropy of coherence, and complete complementarity relations—and computes them for top-antitop pairs produced in QCD. It shows the usual dependence on kinematic variables such as invariant mass and scattering angle, then adds the claim that, when initial states mix quarks and gluons, raising the gluon probability Wgg increases the maximum value on the left-hand side of their intrinsic relation. The motivation is reasonable: the top quark’s short lifetime lets spin correlations survive to the decay products, so in principle these measures can be accessed experimentally. That part is straightforward and correctly identified. The calculations appear to rest on a standard model of pair production with adjustable initial-state weights, which is a common setup in this corner of the literature. The new element is simply the numerical scan over Wgg and the observation that the maximum grows with it. No new framework or first-principles derivation is introduced. The soft spot is the treatment of Wgg. In actual collider kinematics that probability is fixed by parton distributions at the relevant scale and energy; it is not a dial. The paper supplies no derivation from PDFs, no comparison with the physical value near 0.85 at 13 TeV, and no check that the monotonic increase survives inside the narrow physical window. The trend therefore reads as an artifact of the parameterization rather than a model-independent feature. The abstract gives no explicit formulas, error estimates, or order of the QCD calculation, so it is impossible to judge numerical stability or whether higher-order corrections would wash out the effect. This is niche work aimed at the handful of people already combining quantum information with top-quark spin studies. A reader outside that intersection will find little to take away. The paper is coherent on its own terms and shows honest engagement with the measures it uses, so it deserves a serious referee to check the derivations and ask for the PDF comparison. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper examines quantum correlations in top quark-antiquark pairs produced via QCD processes. It applies quantum mutual information, relative entropy of coherence, complete complementarity relations, and an intrinsic relation to study their dependence on kinematic variables. The central finding is that, in a model with mixed quark-gluon initial states, increasing the gluon probability Wgg raises the maximum value attained by the left-hand side of the intrinsic relation.

Significance. If the reported dependence on initial-state probabilities survives a physically constrained treatment, the work would offer a novel link between quantum-information measures and the production dynamics of the heaviest Standard Model particle. Such a connection could illuminate how spin correlations and coherence are preserved through the top-quark lifetime in collider environments.

major comments (2)

[Abstract] Abstract: the claim that the maximum of the left-hand side of the intrinsic relation increases with Wgg is obtained by treating the gluon probability Wgg as a free parameter. No derivation from parton distribution functions is supplied, nor is any comparison made to standard values (e.g., Wgg ≈ 0.85 at 13 TeV). Because Wgg is fixed by PDFs and partonic center-of-mass energy in actual kinematics, the monotonic trend may be an artifact of the parameterization rather than a model-independent feature.
[Abstract] Abstract: the manuscript states that the calculations rest on 'a specific model of top-quark pair production in QCD' together with chosen probabilities for quark versus gluon initial states, yet provides neither the explicit form of the model nor the range over which Wgg is varied. Without these details it is impossible to verify whether the reported increase remains inside the physically allowed interval 0 < Wgg < 1.

minor comments (2)

[Abstract] Grammatical error: 'we presents' should read 'we present'.
[Title and Abstract] The title refers to 'unified relations' but the abstract only mentions an 'intrinsic relationship'; the precise relation between these terms should be clarified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address the two major points raised concerning the treatment of Wgg and the description of our model. Revisions have been made to the abstract and main text to improve clarity and provide the requested details.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the maximum of the left-hand side of the intrinsic relation increases with Wgg is obtained by treating the gluon probability Wgg as a free parameter. No derivation from parton distribution functions is supplied, nor is any comparison made to standard values (e.g., Wgg ≈ 0.85 at 13 TeV). Because Wgg is fixed by PDFs and partonic center-of-mass energy in actual kinematics, the monotonic trend may be an artifact of the parameterization rather than a model-independent feature.

Authors: We agree that Wgg is fixed by PDFs in realistic collider kinematics and is not a completely free parameter. Our approach introduces Wgg as a variable weight in the mixed initial-state model precisely to isolate and demonstrate the dependence of the quantum measures on the gluon fraction. We do not claim the trend is model-independent. In the revised version we have added a comparison noting that Wgg ≈ 0.85 at 13 TeV according to standard PDFs, and we explicitly verify that the increase in the maximum value of the intrinsic relation persists when Wgg is restricted to the physical interval [0,1]. The monotonic behavior is therefore a genuine feature of the model within the allowed range rather than an artifact of unphysical parameterization. revision: partial
Referee: [Abstract] Abstract: the manuscript states that the calculations rest on 'a specific model of top-quark pair production in QCD' together with chosen probabilities for quark versus gluon initial states, yet provides neither the explicit form of the model nor the range over which Wgg is varied. Without these details it is impossible to verify whether the reported increase remains inside the physically allowed interval 0 < Wgg < 1.

Authors: We apologize for the insufficient detail in the abstract. The model is the standard QCD ttbar production via qqbar annihilation and gg fusion; the total spin density matrix is the convex combination (1−Wgg)ρqq + Wggρgg, where the explicit forms of ρqq and ρgg are given in Section II of the manuscript. We have revised the abstract to state that Wgg is varied continuously over the closed interval [0,1] and added a sentence confirming that the reported increase of the maximum value holds for all Wgg in this physically allowed range. These additions make the calculation fully verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; model-based computation of parameter dependence

full rationale

The paper computes standard quantum-information quantities (mutual information, coherence, complementarity relations) for top-quark pairs produced in a QCD model whose only free inputs are the initial-state probabilities Wgg and Wqq. The reported monotonic increase of the maximum of the intrinsic-relation LHS with Wgg is obtained by direct substitution of these probabilities into the density-matrix expressions and numerical evaluation; it is therefore an explicit model output rather than a self-definition, a fitted parameter renamed as a prediction, or a result forced by self-citation. No uniqueness theorem, ansatz smuggling, or renaming of known results is invoked. The derivation chain remains self-contained once the model and the chosen parameterization are accepted.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the central claims rest on standard QCD production kinematics and chosen initial-state probabilities whose detailed justification is not visible.

free parameters (1)

gluon probability Wgg
Treated as a variable parameter whose increase is reported to raise the maximum of the intrinsic relation.

axioms (1)

domain assumption Top-quark pairs are produced according to standard perturbative QCD matrix elements with spin correlations preserved through decay.
Implicit in any calculation of quantum correlations for top quarks at colliders.

pith-pipeline@v0.9.0 · 5428 in / 1208 out tokens · 37930 ms · 2026-05-13T19:42:07.773821+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts

?

contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

we treat W_gg as a free parameter ranging from 0 to 1... as the probability of gluons W_gg increases, the maximum of the left-hand side of the intrinsic relation also increases

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

[1]

Tanabashiet al.(Particle Data Group), Phys

M. Tanabashiet al.(Particle Data Group), Phys. Rev. D98, 030001 (2018)

work page 2018
[2]

Abachi, B

S. Abachi, B. Abbott, M. Abolins, B. S. Acharya, I. Adam, D. L. Adams, M. Adams, S. Ahn, H. Aihara,et al.(D0 Collab- oration), Phys. Rev. Lett.74, 2632 (1995)

work page 1995
[3]

H. Baer, J. Sender, and X. Tata, Phys. Rev. D50, 4517 (1994)

work page 1994
[4]

Burkhardt, H

O. Burkhardt, H. Burkhardt, and S. Myers, Progress in Particle and Nuclear Physics67, 705 (2012)

work page 2012
[5]

Evans, New J

L. Evans, New J. Phys.9, 335 (2007)

work page 2007
[6]

Aaltonen, B

T. Aaltonen, B. ´Alvarez Gonz ´alez, S. Amerio, D. Amidei, A. Anastassov, A. Annovi, J. Antos, G. Apollinari, J. Appel, A. Apresyan,et al.(CDF Collaboration), Phys. Rev. D83, 031104 (2011)

work page 2011
[7]

V . M. Abazov, B. Abbott, B. S. Acharya, M. Adams, T. Adams, G. D. Alexeev, G. Alkhazov, A. Alton, G. Alverson,et al.(D0 Collaboration), Phys. Rev. Lett.107, 032001 (2011)

work page 2011
[8]

G. Aad, B. Abbott, J. Abdallah, S. Abdel Khalek, O. Abdinov, R. Aben, B. Abi, M. Abolins, O. S. AbouZeid,et al.(ATLAS Collaboration), Phys. Rev. Lett.114, 142001 (2015)

work page 2015
[9]

A. M. Sirunyan, A. Tumasyan, W. Adam, F. Ambrogi, E. Asi- lar, T. Bergauer, J. Brandstetter, M. Dragicevic, J. Er ¨o, A. Es- calante Del Valle,et al.(CMS Collaboration), Phys. Rev. D100, 072002 (2019)

work page 2019
[10]

G. Aad, B. Abbott, J. Abdallah, O. Abdinov, R. Aben, M. Abolins, O. AbouZeid, H. Abramowicz, H. Abreu,et al. (ATLAS Collaboration), Phys. Rev. D93, 012002 (2016)

work page 2016
[11]

Charaf, S

O. Charaf, S. Cooper, C. Henderson, P. Rumerio,et al., Phys. Lett. B758, 321 (2016)

work page 2016
[12]

Afik and J

Y . Afik and J. R. M. de Nova, Eur. Phys. J. Plus136(2021)

work page 2021
[13]

Afik and J

Y . Afik and J. R. M. de Nova, Quantum6, 820 (2022)

work page 2022
[14]

Afik and J

Y . Afik and J. R. M. de Nova, Phys. Rev. Lett.130, 221801 (2023)

work page 2023
[15]

Ye, L.-Y

B.-L. Ye, L.-Y . Xue, Z.-Q. Zhu, D.-D. Shi, and S.-M. Fei, Phys. Rev. D110, 055025 (2024)

work page 2024
[16]

Cheng, T

K. Cheng, T. Han, and M. Low, Phys. Rev. D111, 033004 (2025) [arXiv:2407.01672]

work page arXiv 2025
[17]

T. Han, M. Low, N. McGinnis, and S. Su, J. High Energy Phys. 2025, 081 (2025) [arXiv:2412.21158]

work page arXiv 2025
[18]

Kraskov, H

A. Kraskov, H. St¨ogbauer, and P. Grassberger, Phys. Rev. E69, 066138 (2004)

work page 2004
[19]

Groisman, S

B. Groisman, S. Popescu, and A. Winter, Phys. Rev. A72, 032317 (2005)

work page 2005
[20]

Myeongjin, J

S. Myeongjin, J. Lee, and K. Jeong, Quantum Inf. Process.23, 57 (2024)

work page 2024
[21]

Schumacher and M

B. Schumacher and M. D. Westmoreland, Phys. Rev. A74, 042305 (2006)

work page 2006
[22]

Horodecki, P

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys.81, 865 (2009)

work page 2009
[23]

Ghne and G

O. Ghne and G. Tth, Phys. Rep.474, 1 (2009)

work page 2009
[24]

R. J. Glauber, Phys. Rev.131, 2766 (1963)

work page 1963
[25]

Baumgratz, M

T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)

work page 2014
[26]

W. W. Chow, H. C. Schneider, and M. C. Phillips, Phys. Rev. A68, 053802 (2003)

work page 2003
[27]

Cheng and M

S. Cheng and M. J. W. Hall, Phys. Rev. A92, 042101 (2015)

work page 2015
[28]

Liu, L.-Z

S. Liu, L.-Z. Mu, and H. Fan, Phys. Rev. A91, 042133 (2015)

work page 2015
[29]

Dong, G.-B

D.-D. Dong, G.-B. Wei, X.-K. Song, D. Wang, and L. Ye, Phys. Rev. A106, 042415 (2022)

work page 2022
[30]

Dong, L.-J

D.-D. Dong, L.-J. Li, X.-K. Song, L. Ye, and D. Wang, Phys. Rev. A110, 032420 (2024). 9

work page 2024
[31]

L.-J. Li, X. G. Fan, X.-K. Song, L. Ye, and D. Wang, Phys. Rev. A110, 012418 (2024)

work page 2024
[32]

Streltsov, G

A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod. Phys.89, 041003 (2017)

work page 2017
[33]

Streltsov, U

A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Phys. Rev. Lett.115, 020403 (2015)

work page 2015
[34]

Napoli, T

C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. John- ston, and G. Adesso, Phys. Rev. Lett.116, 150502 (2016)

work page 2016
[35]

Bittencourt, M

V . Bittencourt, M. Blasone, S. De Siena, and C. Matrella, Eur. Phys. J. C84, 301 (2024)

work page 2024
[36]

M. L. W. Basso and J. Maziero, J. Phys. A: Math. Theor.53, 465301 (2020)

work page 2020
[37]

J. M. Renes and J.-C. Boileau, Phys. Rev. Lett.103, 020402 (2009)

work page 2009
[38]

Berta, M

M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Ren- ner, Nat. Phys.6, 659 (2010)

work page 2010
[39]

Wang, B.-F

Z.-A. Wang, B.-F. Xie, F. Ming, Y .-T. Wang, D. Wang, Y . Meng, Z.-H. Liu, K. Xu, J.-S. Tang, L. Ye, C.-F. Li, G.-C. Guo, and S. Kais, Phys. Rev. A110, 062220 (2024)

work page 2024
[40]

L. Wu, L. Ye, and D. Wang, Phys. Rev. A106, 062219 (2022)

work page 2022
[41]

Wang and D

T.-Y . Wang and D. Wang, Phys. Lett. B855, 138876 (2024)

work page 2024