arxiv: 2604.17756 · v1 · submitted 2026-04-20 · ✦ hep-ph · hep-ex· nucl-th· quant-ph

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Polarization, Maximal Concurrence, and Pure States in High-Energy Collisions

Yu-Xuan Liu , Luo-Ting He , Bo-Wen Xiao

Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:11 UTC · model grok-4.3

classification ✦ hep-ph hep-exnucl-thquant-ph

keywords spin polarizationquantum entanglementconcurrencetwo-qubit systemshigh-energy collisionsZ boson decayparity violationpure states

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The pith

Increasing local spin polarization imposes an upper bound on concurrence in two-qubit systems.

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

The paper derives a mathematical upper bound on concurrence for any two-qubit system at a given value of local polarization, showing that stronger polarization reduces the highest achievable entanglement. This bound is saturated by pure states under certain conditions. The relation is then applied to quark-antiquark pairs produced in electron-positron collisions via a Z boson, where parity violation creates the polarization. If the bound is correct, collider experiments will record lower entanglement when the spins are more aligned, and the maximum occurs only for pure states in selected kinematic windows. The framework applies independently of the production process.

Core claim

We establish a quantitative relation between local spin polarization and quantum entanglement in two-qubit systems by deriving an upper bound on the concurrence at fixed local polarization, showing that increasing polarization constrains the maximum achievable entanglement. We further demonstrate that this bound is saturated by pure states in certain cases. As a concrete physical application, we consider the parity-violating process e⁺e⁻ → Z⁰ → q q-bar, which generates final-state spin polarization. We show that the maximal concurrence is attained in specific kinematic regions and is significantly reduced relative to the unpolarized case. These results establish a general, processindependent

What carries the argument

The upper bound on concurrence expressed as a function of the local polarization vector magnitude in a two-qubit system.

If this is right

In the e⁺e⁻ to Z to q q-bar process the maximal concurrence occurs only in particular kinematic regions.
The maximal concurrence drops substantially compared with the unpolarized case once polarization is present.
Pure states saturate the derived bound in specific cases.
The polarization-concurrence relation holds for any two-qubit system and is independent of the underlying production mechanism.

Where Pith is reading between the lines

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

The same bound may constrain entanglement in other polarized particle-production channels such as heavy-ion collisions.
Collider experiments that measure both polarization and a suitable entanglement witness could directly test the bound.
The approach could be extended to mixed states or to systems with more than two qubits if the local-polarization description remains valid.

Load-bearing premise

The final-state quark and antiquark spins can be modeled as a two-qubit system in which the local polarization vector directly determines the upper limit on concurrence.

What would settle it

A measured concurrence in e⁺e⁻ → Z → q q-bar events that exceeds the bound calculated from the simultaneously measured local polarization vector would contradict the claimed relation.

Figures

Figures reproduced from arXiv: 2604.17756 by Bo-Wen Xiao, Luo-Ting He, Yu-Xuan Liu.

**Figure 1.** Figure 1: Numerical verification of the boundary of concurrence for general density matrices. The green points are obtained from 2 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: (a) Feynman diagram of electron–positron scattering via [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: Concurrence of the final-state qq¯ pair in e − e + annihilation near the Z 0 pole as a function of the quark speed u and scattering angle variable z = cos θ. The left and right panels correspond to up-type and down-type quarks, respectively. The maximal values in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

We establish a quantitative relation between local spin polarization and quantum entanglement in two-qubit systems by deriving an upper bound on the concurrence at fixed local polarization, showing that increasing polarization constrains the maximum achievable entanglement. We further demonstrate that this bound is saturated by pure states in certain cases. As a concrete physical application, we consider the parity-violating process $e^+e^- \to Z^0 \to q\bar{q}$, which generates final-state spin polarization. We show that the maximal concurrence is attained in specific kinematic regions and is significantly reduced relative to the unpolarized case. These results establish a general, process-independent framework connecting local polarization, maximal entanglement, and the role of pure states.

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 derives a polarization-dependent upper bound on concurrence for two qubits and applies it to Z decays, but the two-qubit approximation for real quark pairs may not hold after tracing over color.

read the letter

This paper derives an upper bound on the concurrence of a two-qubit system in terms of its local polarization and shows that pure states saturate the bound in some cases. It then applies the bound to the polarized quark pairs produced in e+e- collisions through a Z boson. The new element is the explicit polarization-dependent bound and the demonstration that it is achieved by pure states. The calculation for the Z decay uses the standard model amplitude to get the polarization and then finds that the maximum concurrence drops in regions where polarization is high. The paper handles the two-qubit math in a straightforward way. It connects a measurable quantity like polarization to an entanglement measure without introducing extra parameters. The main concern is whether the final-state quarks can be treated as a simple two-qubit system. The quarks have color charge, and the density matrix for spins is obtained only after tracing over color and summing over all other degrees of freedom. If those operations mix the state in ways not captured by the local polarization alone, the bound may not apply directly. The paper needs to verify that the reduced spin density matrix remains consistent with the two-qubit assumptions used in the bound. This work is aimed at people interested in quantum information ideas in high-energy physics. A reader who follows spin correlations in collider processes could use the bound as a tool to interpret data. It is less useful for someone expecting direct experimental predictions. The derivation looks solid on its own terms. I recommend sending it to peer review so that referees can check the tightness of the bound and whether the physical application survives the tracing over color and kinematics.

Referee Report

1 major / 1 minor

Summary. The manuscript derives an upper bound on the concurrence of two-qubit systems in terms of their local polarization vectors, showing that increasing polarization reduces the maximum achievable entanglement, and demonstrates that this bound is saturated by certain pure states. It applies the framework to the parity-violating process e⁺e⁻ → Z⁰ → q q̄, arguing that the final-state quark spins form an effective two-qubit system whose local polarizations constrain the concurrence, with the maximum value attained only in specific kinematic regions and significantly lower than in the unpolarized case.

Significance. If the bound derivation is non-tautological and the two-qubit modeling holds after all traces, the result supplies a general, process-independent relation between measurable local polarization and entanglement in high-energy collisions. This could provide a quantitative tool for assessing quantum correlations in polarized production processes and highlights the role of pure states in saturating the bound.

major comments (1)

In the section applying the bound to e⁺e⁻ → Z⁰ → q q̄: the assumption that the spin degrees of freedom of the produced quarks form a closed two-qubit system whose reduced Bloch vectors enter the concurrence bound directly is not justified. Tracing over color, integrating over unobserved kinematics, and summing over other final-state particles can introduce mixing; without an explicit demonstration that the effective spin density matrix remains consistent with the abstract two-qubit derivation, the physical claim that polarization constrains concurrence in this process does not follow.

minor comments (1)

The explicit functional form of the derived upper bound on concurrence should be stated clearly in the general two-qubit section, together with the definitions of the local polarization vectors, to allow readers to verify the saturation claim for pure states.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the single major comment below and will revise the manuscript to provide the requested explicit demonstration.

read point-by-point responses

Referee: In the section applying the bound to e⁺e⁻ → Z⁰ → q q̄: the assumption that the spin degrees of freedom of the produced quarks form a closed two-qubit system whose reduced Bloch vectors enter the concurrence bound directly is not justified. Tracing over color, integrating over unobserved kinematics, and summing over other final-state particles can introduce mixing; without an explicit demonstration that the effective spin density matrix remains consistent with the abstract two-qubit derivation, the physical claim that polarization constrains concurrence in this process does not follow.

Authors: We agree that the current manuscript would benefit from a more explicit derivation to confirm that the reduced spin density matrix for the q q̄ pair remains a valid two-qubit state after all traces. In e⁺e⁻ → Z⁰ → q q̄ the Z decay produces a color-singlet quark pair whose spin correlations are fully determined by the electroweak matrix elements; color and kinematic degrees of freedom factorize and do not introduce additional spin mixing once the reduced density matrix is formed. The local polarization vectors are obtained precisely by tracing over the partner spin and integrating over the unobserved production angle. In the revision we will add a dedicated appendix that (i) writes the full 4×4 spin density matrix from the Z decay amplitude, (ii) performs the explicit partial traces over color and kinematics, and (iii) shows that the resulting operator is of the standard two-qubit Bloch form (I⊗I + r·σ⊗I + I⊗s·σ + c_{ij} σ_i⊗σ_j)/4 with the concurrence bound applying directly to the extracted r and s. This will rigorously justify the physical claim without altering the reported results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; mathematical bound derived from standard two-qubit definitions

full rationale

The paper's central derivation establishes an upper bound on concurrence for fixed local polarization in two-qubit systems and shows saturation by pure states. This is presented as a direct mathematical result from the definitions of concurrence (via the standard Wootters formula) and Bloch-vector polarization, without any fitting to data or reduction to self-citation. The e+e- to Z to qqbar application applies the bound to a modeled two-qubit spin subspace but does not alter the independence of the QI derivation itself. No load-bearing self-citations, ansatze smuggled via prior work, or predictions that collapse to fitted inputs are evident from the abstract and claimed structure. The modeling assumption is an applicability question rather than a circularity in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete; the central claim rests on standard definitions from quantum information theory and the applicability of two-qubit concurrence to quark spins.

axioms (1)

domain assumption Spin states of quark-antiquark pairs produced in the process can be treated as two-qubit systems with a local polarization vector.
Invoked when mapping the physical process to the concurrence bound.

pith-pipeline@v0.9.0 · 5424 in / 1458 out tokens · 39781 ms · 2026-05-10T05:11:42.217226+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

76 extracted references · 76 canonical work pages · 1 internal anchor

[1]

Ashman, et al., A Measurement of the Spin Asymmetry and Determination of the Structure Function g(1) in Deep Inelastic Muon-Proton Scattering, Phys

J. Ashmanet al.[European Muon], Phys. Lett. B206(1988), 364 doi:10.1016/0370-2693(88)91523-7

work page doi:10.1016/0370-2693(88)91523-7 1988
[2]

Airapetianet al.[HERMES], Phys

A. Airapetianet al.[HERMES], Phys. Rev. D75(2007), 012007 doi:10.1103/PhysRevD.75.012007 [arXiv:hep- ex/0609039 [hep-ex]]

work page doi:10.1103/physrevd.75.012007 2007
[3]

Schaelet al.[ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group and SLD Heavy Flavour Group], Phys

S. Schaelet al.[ALEPH, DELPHI, L3, OPAL, SLD, LEP Electroweak Working Group, SLD Electroweak Group and SLD Heavy Flavour Group], Phys. Rept.427(2006), 257-454 doi:10.1016/j.physrep.2005.12.006 [arXiv:hep-ex/0509008 [hep-ex]]

work page doi:10.1016/j.physrep.2005.12.006 2006
[4]

Bunce, R

G. Bunce, R. Handler, R. March, P. Martin, L. Pondrom, M. Sheaff, K. J. Heller, O. Overseth, P. Skubic and T. Devlin,et al.Phys. Rev. Lett.36(1976), 1113-1116 doi:10.1103/PhysRevLett.36.1113

work page doi:10.1103/physrevlett.36.1113 1976
[5]

Z. T. Liang and X. N. Wang, Phys. Rev. Lett.94(2005), 102301 [erratum: Phys. Rev. Lett.96(2006), 039901] doi:10.1103/PhysRevLett.94.102301 [arXiv:nucl-th/0410079 [nucl-th]]. 10

work page doi:10.1103/physrevlett.94.102301 2005
[6]

Global Λ hyperon polarization in nuclear col- lisions: evidence for the most vortical fluid

L. Adamczyket al.[STAR], Nature548(2017), 62-65 doi:10.1038/nature23004 [arXiv:1701.06657 [nucl-ex]]

work page doi:10.1038/nature23004 2017
[7]

Adamet al.[STAR], Phys

J. Adamet al.[STAR], Phys. Rev. C98(2018), 014910 doi:10.1103/PhysRevC.98.014910 [arXiv:1805.04400 [nucl-ex]]

work page doi:10.1103/physrevc.98.014910 2018
[8]

Uwer, Phys

P. Uwer, Phys. Lett. B609(2005), 271-276 doi:10.1016/j.physletb.2005.01.005 [arXiv:hep-ph/0412097 [hep- ph]]

work page doi:10.1016/j.physletb.2005.01.005 2005
[9]

Hentschinski, D

M. Hentschinski, D. E. Kharzeev, K. Kutak and Z. Tu, Phys. Rev. Lett.131(2023) no.24, 241901 doi:10.1103/PhysRevLett.131.241901 [arXiv:2305.03069 [hep-ph]]

work page doi:10.1103/physrevlett.131.241901 2023
[10]

Afik and J

Y . Afik and J. R. M. de Nova, Eur. Phys. J. Plus136(2021) no.9, 907 doi:10.1140/epjp/s13360-021-01902-1 [arXiv:2003.02280 [quant-ph]]

work page doi:10.1140/epjp/s13360-021-01902-1 2021
[11]

Afik and J

Y . Afik and J. R. M. de Nova, Phys. Rev. Lett.130(2023) no.22, 221801 doi:10.1103/PhysRevLett.130.221801 [arXiv:2209.03969 [quant-ph]]

work page doi:10.1103/physrevlett.130.221801 2023
[12]

Afik and J

Y . Afik and J. R. M. de Nova, Quantum6(2022), 820 doi:10.22331/q-2022-09-29-820 [arXiv:2203.05582 [quant-ph]]

work page doi:10.22331/q-2022-09-29-820 2022
[13]

Cheng, T

K. Cheng, T. Han and M. Low, Phys. Rev. D111(2025) no.3, 033004 doi:10.1103/PhysRevD.111.033004 [arXiv:2407.01672 [hep-ph]]

work page doi:10.1103/physrevd.111.033004 2025
[14]

S. Wu, C. Qian, Y . G. Yang and Q. Wang, Chin. Phys. Lett.41(2024) no.11, 110301 doi:10.1088/0256- 307X/41/11/110301 [arXiv:2402.16574 [hep-ph]]

work page doi:10.1088/0256- 2024
[15]

A. J. Barr, M. Fabbrichesi, R. Floreanini, E. Gabrielli and L. Marzola, Prog. Part. Nucl. Phys.139(2024), 104134 doi:10.1016/j.ppnp.2024.104134 [arXiv:2402.07972 [hep-ph]]

work page doi:10.1016/j.ppnp.2024.104134 2024
[16]

Ehat¨ aht, M

K. Ehatäht, M. Fabbrichesi, L. Marzola and C. Veelken, Phys. Rev. D109(2024) no.3, 032005 doi:10.1103/PhysRevD.109.032005 [arXiv:2311.17555 [hep-ph]]

work page doi:10.1103/physrevd.109.032005 2024
[17]

Y . Guo, X. Liu, F. Yuan and H. X. Zhu, Research2025(2025), 0552 doi:10.34133/research.0552 [arXiv:2406.05880 [hep-ph]]

work page doi:10.34133/research.0552 2025
[18]

R. A. Morales, Eur. Phys. J. Plus138(2023) no.12, 1157 doi:10.1140/epjp/s13360-023-04784-7 [arXiv:2306.17247 [hep-ph]]

work page doi:10.1140/epjp/s13360-023-04784-7 2023
[19]

S. J. Parke and Y . Shadmi, Phys. Lett. B387(1996), 199-206 doi:10.1016/0370-2693(96)00998-7 [arXiv:hep- ph/9606419 [hep-ph]]

work page doi:10.1016/0370-2693(96)00998-7 1996
[20]

M. M. Altakach, P. Lamba, F. Maltoni, K. Mawatari and K. Sakurai, Phys. Rev. D107(2023) no.9, 093002 doi:10.1103/PhysRevD.107.093002 [arXiv:2211.10513 [hep-ph]]

work page doi:10.1103/physrevd.107.093002 2023
[21]

Aoude, G

R. Aoude, G. Elor, G. N. Remmen and O. Sumensari, [arXiv:2402.16956 [hep-th]]

work page arXiv
[22]

Bernal, Phys

A. Bernal, Phys. Rev. D109(2024) no.11, 116007 doi:10.1103/PhysRevD.109.116007 [arXiv:2310.10838 [hep- ph]]

work page doi:10.1103/physrevd.109.116007 2024
[23]

Brandenburg, Z

A. Brandenburg, Z. G. Si and P. Uwer, Phys. Lett. B539(2002), 235-241 doi:10.1016/S0370-2693(02)02098-1 [arXiv:hep-ph/0205023 [hep-ph]]

work page doi:10.1016/s0370-2693(02)02098-1 2002
[24]

Hentschinski, D

M. Hentschinski, D. E. Kharzeev, K. Kutak and Z. Tu, Rept. Prog. Phys.87(2024) no.12, 120501 doi:10.1088/1361-6633/ad910b [arXiv:2408.01259 [hep-ph]]

work page doi:10.1088/1361-6633/ad910b 2024
[25]

D. F. V . James, P. G. Kwiat, W. J. Munro and A. G. White, Phys. Rev. A64(2001) no.5, 052312 doi:10.1103/PhysRevA.64.052312 [arXiv:quant-ph/0103121 [quant-ph]]. 11

work page doi:10.1103/physreva.64.052312 2001
[26]

Kowalska and E

K. Kowalska and E. M. Sessolo, JHEP07(2024), 156 doi:10.1007/JHEP07(2024)156 [arXiv:2404.13743 [hep- ph]]

work page doi:10.1007/jhep07(2024)156 2024
[27]

X. L. Li, X. Liu, F. Yuan and H. X. Zhu, Phys. Rev. D108(2023) no.9, L091502 doi:10.1103/PhysRevD.108.L091502 [arXiv:2308.10942 [hep-ph]]

work page doi:10.1103/physrevd.108.l091502 2023
[28]

Maltoni, C

F. Maltoni, C. Severi, S. Tentori and E. Vryonidou, JHEP09(2024), 001 doi:10.1007/JHEP09(2024)001 [arXiv:2404.08049 [hep-ph]]

work page doi:10.1007/jhep09(2024)001 2024
[29]

Severi and E

C. Severi and E. Vryonidou, JHEP01(2023), 148 doi:10.1007/JHEP01(2023)148 [arXiv:2210.09330 [hep-ph]]

work page doi:10.1007/jhep01(2023)148 2023
[30]

Subba and R

A. Subba and R. Rahaman, [arXiv:2404.03292 [hep-ph]]

work page arXiv
[31]

Ge, C.-F

C. Severi, C. D. Boschi, F. Maltoni and M. Sioli, Eur. Phys. J. C82(2022) no.4, 285 doi:10.1140/epjc/s10052- 022-10245-9 [arXiv:2110.10112 [hep-ph]]

work page doi:10.1140/epjc/s10052- 2022
[32]

S. Wu, C. Qian, Q. Wang and X. R. Zhou, Phys. Rev. D110(2024) no.5, 054012 doi:10.1103/PhysRevD.110.054012 [arXiv:2406.16298 [hep-ph]]

work page doi:10.1103/physrevd.110.054012 2024
[33]

S. Li, W. Shen and J. M. Yang, Eur. Phys. J. C84(2024) no.11, 1195 doi:10.1140/epjc/s10052-024-13584-x [arXiv:2401.01162 [hep-th]]

work page doi:10.1140/epjc/s10052-024-13584-x 2024
[34]

Mahlon and S

G. Mahlon and S. J. Parke, Phys. Rev. D53(1996), 4886-4896 doi:10.1103/PhysRevD.53.4886 [arXiv:hep- ph/9512264 [hep-ph]]

work page doi:10.1103/physrevd.53.4886 1996
[35]

Z. Dong, D. Gonçalves, K. Kong and A. Navarro, Phys. Rev. D109(2024) no.11, 115023 doi:10.1103/PhysRevD.109.115023 [arXiv:2305.07075 [hep-ph]]

work page doi:10.1103/physrevd.109.115023 2024
[36]

J. A. Aguilar-Saavedra and J. A. Casas, Eur. Phys. J. C82(2022) no.8, 666 doi:10.1140/epjc/s10052-022-10630- 4 [arXiv:2205.00542 [hep-ph]]

work page doi:10.1140/epjc/s10052-022-10630- 2022
[37]

Fabbrichesi, R

M. Fabbrichesi, R. Floreanini, E. Gabrielli and L. Marzola, Eur. Phys. J. C83(2023) no.9, 823 doi:10.1140/epjc/s10052-023-11935-8 [arXiv:2302.00683 [hep-ph]]

work page doi:10.1140/epjc/s10052-023-11935-8 2023
[38]

Ghosh and R

D. Ghosh and R. Sharma, JHEP08(2023), 146 doi:10.1007/JHEP08(2023)146 [arXiv:2303.03375 [hep-th]]

work page doi:10.1007/jhep08(2023)146 2023
[39]

C. D. White and M. J. White, Phys. Rev. D110(2024) no.11, 116016 doi:10.1103/PhysRevD.110.116016 [arXiv:2406.07321 [hep-ph]]

work page doi:10.1103/physrevd.110.116016 2024
[40]

T. Han, M. Low and T. A. Wu, JHEP07(2024), 192 doi:10.1007/JHEP07(2024)192 [arXiv:2310.17696 [hep- ph]]

work page doi:10.1007/jhep07(2024)192 2024
[41]

Y . Du, X. G. He, C. W. Liu and J. P. Ma, Eur. Phys. J. C85(2025) no.11, 1255 doi:10.1140/epjc/s10052-025- 15003-1 [arXiv:2409.15418 [hep-ph]]

work page doi:10.1140/epjc/s10052-025- 2025
[42]

R. A. Morales, Eur. Phys. J. C84(2024) no.6, 581 doi:10.1140/epjc/s10052-024-12921-4 [arXiv:2403.18023 [hep-ph]]

work page doi:10.1140/epjc/s10052-024-12921-4 2024
[43]

L. Yang, Y . K. Song and S. Y . Wei, Phys. Rev. D111(2025) no.5, 054035 doi:10.1103/PhysRevD.111.054035 [arXiv:2410.20917 [hep-ph]]

work page doi:10.1103/physrevd.111.054035 2025
[44]

Fabbrichesi, R

M. Fabbrichesi, R. Floreanini and G. Panizzo, Phys. Rev. Lett.127(2021) no.16, 16 doi:10.1103/PhysRevLett.127.161801 [arXiv:2102.11883 [hep-ph]]

work page doi:10.1103/physrevlett.127.161801 2021
[45]

Aoude, E

R. Aoude, E. Madge, F. Maltoni and L. Mantani, Phys. Rev. D106(2022) no.5, 055007 doi:10.1103/PhysRevD.106.055007 [arXiv:2203.05619 [hep-ph]]. 12

work page doi:10.1103/physrevd.106.055007 2022
[46]

Y . Afik, F. Fabbri, M. Low, L. Marzola, J. A. Aguilar-Saavedra, M. M. Altakach, N. A. Asbah, Y . Bai, H. Banks and A. J. Barr,et al.Eur. Phys. J. Plus140(2025) no.9, 855 doi:10.1140/epjp/s13360-025-06752- 9 [arXiv:2504.00086 [hep-ph]]

work page doi:10.1140/epjp/s13360-025-06752- 2025
[47]

Fabbrichesi, R

M. Fabbrichesi, R. Floreanini and L. Marzola, JHEP11(2025), 005 doi:10.1007/JHEP11(2025)005 [arXiv:2505.02902 [hep-ph]]

work page doi:10.1007/jhep11(2025)005 2025
[48]

Cheng and B

K. Cheng and B. Yan, Phys. Rev. Lett.135(2025) no.1, 1 doi:10.1103/gmqz-v4cl [arXiv:2501.03321 [hep-ph]]

work page doi:10.1103/gmqz-v4cl 2025
[49]

W. Qi, Z. Guo and B. W. Xiao, Phys. Rev. D113(2026) no.5, 054048 doi:10.1103/6ycn-x3yj [arXiv:2506.12889 [hep-ph]]

work page doi:10.1103/6ycn-x3yj 2026
[50]

Cheng, T

K. Cheng, T. Han and S. Trifinopoulos, [arXiv:2510.23773 [hep-ph]]

work page arXiv
[51]

S. J. Lin, M. J. Liu, D. Y . Shao and S. Y . Wei, JHEP11(2025), 082 doi:10.1007/JHEP11(2025)082 [arXiv:2507.15387 [hep-ph]]

work page doi:10.1007/jhep11(2025)082 2025
[52]

Low, Phys

M. Low, Phys. Rev. D112(2025) no.9, 096008 doi:10.1103/15c3-mg5l [arXiv:2508.10979 [hep-ph]]

work page doi:10.1103/15c3-mg5l 2025
[53]

Probing quantum en- tanglement with Generalized Parton Distributions at the Electron-Ion Collider,

Y . Hatta and J. Schoenleber, [arXiv:2511.04537 [hep-ph]]

work page arXiv
[54]

X. Q. Xi, K. B. Chen, X. B. Tong, S. Y . Wei and J. Wu, Eur. Phys. J. C85(2025) no.12, 1458 doi:10.1140/epjc/s10052-025-15193-8 [arXiv:2510.06103 [hep-ph]]

work page doi:10.1140/epjc/s10052-025-15193-8 2025
[55]

J. Gu, S. J. Lin, D. Y . Shao, L. T. Wang and S. X. Yang, [arXiv:2510.13951 [hep-ph]]

work page arXiv
[56]

Y . C. Guo, T. Han, M. Low and Y . Su, [arXiv:2602.02719 [hep-ph]]

work page arXiv
[57]

Fucilla and Y

M. Fucilla and Y . Hatta, Phys. Rev. D113(2026) no.3, L031504 doi:10.1103/gbk8-z3dd [arXiv:2509.05267 [hep-ph]]

work page doi:10.1103/gbk8-z3dd 2026
[58]

Zhang, W

W. Zhang, W. Qian, Y . Zhou, Y . Li and Q. Wang, [arXiv:2512.21228 [hep-ph]]

work page arXiv
[59]

H. W. Zhang, X. Cao and T. F. Feng, [arXiv:2602.10389 [hep-ph]]

work page arXiv
[60]

Quantum entanglement in electron-nucleus collisions: Role of the linearly polarized gluon distribution

M. Fucilla, Y . Hatta and B. W. Xiao, [arXiv:2604.11697 [hep-ph]]

work page internal anchor Pith review Pith/arXiv arXiv
[61]

Observation of quantum entanglement with top quarks at the ATLAS detector

G. Aadet al.[ATLAS], Nature633(2024) no.8030, 542-547 doi:10.1038/s41586-024-07824-z [arXiv:2311.07288 [hep-ex]]

work page doi:10.1038/s41586-024-07824-z 2024
[62]

Hayrapetyanet al.[CMS], Rept

A. Hayrapetyanet al.[CMS], Rept. Prog. Phys.87(2024) no.11, 117801 doi:10.1088/1361-6633/ad7e4d [arXiv:2406.03976 [hep-ex]]

work page doi:10.1088/1361-6633/ad7e4d 2024
[63]

B. E. Aboonaet al.[STAR], Nature650(2026) no.8100, 65-71 doi:10.1038/s41586-025-09920-0 [arXiv:2506.05499 [hep-ex]]

work page doi:10.1038/s41586-025-09920-0 2026
[64]

W. K. Wootters, Phys. Rev. Lett.80(1998), 2245-2248 doi:10.1103/PhysRevLett.80.2245 [arXiv:quant- ph/9709029 [quant-ph]]

work page doi:10.1103/physrevlett.80.2245 1998
[65]

and Wootters, W.K

S. Hill and W. K. Wootters, Phys. Rev. Lett.78(1997), 5022-5025 doi:10.1103/PhysRevLett.78.5022 [arXiv:quant-ph/9703041 [quant-ph]]

work page doi:10.1103/physrevlett.78.5022 1997
[66]

M., Hillery, M., Milburn, G

P. Rungta, V . Bužek, C. M. Caves, M. Hillery and G. J. Milburn, Phys. Rev. A64(2001) no.4, 042315 doi:10.1103/PhysRevA.64.042315 [arXiv:quant-ph/0102040 [quant-ph]]

work page doi:10.1103/physreva.64.042315 2001
[67]

M. Ali, A. R. P. Rau and G. Alber, Phys. Rev. A81(2010) no.4, 042105 doi:10.1103/PhysRevA.81.042105 [arXiv:1002.3429 [quant-ph]]

work page doi:10.1103/physreva.81.042105 2010
[68]

J. Wang, H. Batelaan, J. Podany and A. F. Starace, J. Phys. B39(2006), 4343-4353 doi:10.1088/0953- 4075/39/21/001 [arXiv:quant-ph/0503116 [quant-ph]]. 13

work page doi:10.1088/0953- 2006
[69]

W. J. Munro, D. F. V . James, A. G. White and P. G. Kwiat, Phys. Rev. A64(2001) no.3, 030302 doi:10.1103/PhysRevA.64.030302 [arXiv:quant-ph/0103113 [quant-ph]]

work page doi:10.1103/physreva.64.030302 2001
[70]

G. Baio, D. Chruscinski, P. Horodecki, A. Messina and G. Sarbicki, Phys. Rev. A99(2019), 062312 doi:10.1103/PhysRevA.99.062312 [arXiv:1904.07650 [quant-ph]]

work page doi:10.1103/physreva.99.062312 2019
[71]

C. J. Zhang, Y . X. Gong, Y . S. Zhang and G. C. Guo, Phys. Rev. A78(2008), 042308 doi:10.1103/PhysRevA.78.042308 [arXiv:0806.2598 [quant-ph]]

work page doi:10.1103/physreva.78.042308 2008
[72]

Bernreuther and A

W. Bernreuther and A. Brandenburg, Phys. Rev. D49(1994), 4481-4492 doi:10.1103/PhysRevD.49.4481 [arXiv:hep-ph/9312210 [hep-ph]]

work page doi:10.1103/physrevd.49.4481 1994
[73]

Bernreuther, M

W. Bernreuther, M. Flesch and P. Haberl, Phys. Rev. D58(1998), 114031 doi:10.1103/PhysRevD.58.114031 [arXiv:hep-ph/9709284 [hep-ph]]

work page doi:10.1103/physrevd.58.114031 1998
[74]

Brandenburg, M

A. Brandenburg, M. Flesch and P. Uwer, Phys. Rev. D59(1999), 014001 doi:10.1103/PhysRevD.59.014001 [arXiv:hep-ph/9806306 [hep-ph]]

work page doi:10.1103/physrevd.59.014001 1999
[75]

Baumgart and B

M. Baumgart and B. Tweedie, JHEP03(2013), 117 doi:10.1007/JHEP03(2013)117 [arXiv:1212.4888 [hep-ph]]

work page doi:10.1007/jhep03(2013)117 2013
[76]

Bernreuther, D

W. Bernreuther, D. Heisler and Z. G. Si, JHEP12(2015), 026 doi:10.1007/JHEP12(2015)026 [arXiv:1508.05271 [hep-ph]]. 14

work page doi:10.1007/jhep12(2015)026 2015