PLB140426

140426

S0370-2693(26)00279-010.1016/j.physletb.2026.140426The Authors

Fig. 1

QMI as a function of invariant mass Mtt¯ and production angle Θ for tt¯ pairs production processes. (a): gg→tt¯; (b): qq¯→tt¯.

Fig. 1 dummy alt text

Fig. 2

QMI in tt¯ production with mixed gluons (gg) and quarks (qq¯) initial state, where the gluons probability Wgg varies: (a): Wgg=0.2; (b): Wgg=0.4; (c): Wgg=0.6; (d): Wgg=0.8.

Fig. 2 dummy alt text

Fig. 3

REC as a function of the invariant mass Mtt¯ and the production angle Θ in a tt¯ pairs. (a): gg→tt¯. (b): qq¯→tt¯.

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Fig. 4

REC Cre(ρ^AB) in tt¯ production for mixed gluons (gg) and quarks (qq¯) initial state with varying gluon probability Wgg: (a): Wgg=0.2; (b): Wgg=0.4; (c): Wgg=0.6; (d): Wgg=0.8.

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Fig. 5

CCR in tt¯ production from mixed initial states of gluons (gg) and quarks (qq¯). Blue circles represent the QMI IA:B(ρ^AB), orange triangles show conditional entropy SA|B(ρ^AB), and green diamonds indicate the CCR. The gluons probability Wgg varies: (a): Wgg=0.2; (b): Wgg=0.4; (c): Wgg=0.6; (d): Wgg=0.8. The invariant mass is fixed to Mtt¯=500 GeV.

Fig. 5 dummy alt text

Fig. 6

The left-hand side of the intrinsic relation as a function of the invariant mass Mtt¯ and the production angle Θ in a tt¯ pairs. (a): gg→tt¯. (b): qq¯→tt¯.

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Fig. 7

The left-hand side of the intrinsic relation in tt¯ production for mixed gluons (gg) and quarks (qq¯) initial state with varying gluons probability Wgg varies: (a): Wgg=0.2; (b): Wgg=0.4; (c): Wgg=0.6; (d): Wgg=0.8.

Fig. 7 dummy alt text

Fig. 8

The left-hand side of the intrinsic relation is plotted as a function of the production angle Θ for fixed invariant masses Mtt¯=400, 500, and 700 GeV (gg→tt¯). Cyan line represents the invariant masses Mtt¯=400, purple dash-dotted line shows Mtt¯=500 and pink dash-dotted line indicates Mtt¯=700.

Fig. 8 dummy alt text

Fig. 9

The left-hand side of the intrinsic relation S(Q^|B)+S(R^|B)+S(ρ^B)+Pvn(ρ^AB)+Cre(ρ^AB) is plotted as a function of the production angle Θ for fixed invariant masses Mtt¯=400, 500, and 700 GeV (gg→tt¯). Cyan line represents the invariant masses Mtt¯=400, purple dash-dotted line shows Mtt¯=500 and pink dash-dotted line indicates Mtt¯=700. (a): The sum of conditional entropies S(Q^|B)+S(R^|B); (b): Predictability measure Pvn(ρ^AB); (c): REC Cre(ρ^AB).

Fig. 9 dummy alt text

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

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

Duo-Duo

Chen

Xue-Ke

Song

Liu

Dong

Wang

⁎

dwang@ahu.edu.cn

School of Physics, Anhui University, Hefei, 230601, PR ChinaSchool of Physics, Anhui University Hefei

230601

PR China

School of Physics, Anhui University, Hefei, 230601, PR China

⁎Corresponding author.

Editor: Shi-Lin Zhu

Abstract

As the most massive particle in the Standard Model, the top quark’s exceptionally short lifetime (τ∼10−25 s) 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 (tt¯) 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.

Keywords

Quantum mutual informationQuantum coherenceUnified relationsTop quarks

Data availability

No data was used for the research described in the article.

Introduction

The top quark plays an important role in particle physics, cosmology and other fields. As the most massive fundamental particle known to exist (mtc2 ≈ 173 GeV), it was first discovered by the D0 and CDF collaborations at the Tevatron in the Fermilab [1–3]. In 2008, the Large Hadron Collider (LHC) [4,5], the highest-energy collider, began operation. The top quark mass, derived from combined Tevatron and LHC Run 1 and Run 2 data, is mtc2 ≈ 173.1 ± 0.6 GeV. The top quark’s large mass results in a correspondingly large decay width, yielding an exceptionally short lifetime τ=1/Γt∼10−25 s. Top quarks are typically produced in top-antitop (tt¯) pairs. The process of hadronization and spin decorrelation cannot affect tt¯ spin correlations measurement because of hadronization (with a time scale ∼10−23) and spin decorrelation (with a time scale ∼10−21). This is what makes the top quark so unique and we can reconstruct its spin from the decay products.The spin correlation of top quark pairs has been investigated by the D0 and CDF collaborations at the Tevatron with proton-antiproton (pp¯) collisions [6,7], and by the ATLAS and CMS collaborations at the LHC with proton-proton (pp) collisions [8–11]. In a particle accelerator, two beams of protons (or antiprotons and protons) are accelerated in opposite directions and collide with each other. These collisions produce large number of particles, including the top quark and anti-top quark pairs. A proton consists of two up quarks (spin-1/2 fermions) and one down quark. Quarks-elementary particles that interact via the strong force-are the fundamental constituents of matter. They are tightly bound together by gluons (massless spin-1 bosons) to form stable protons. An antiprotons is the antimatter counterpart of the proton. It is a composite particle consisting of two anti-up quarks and one anti-down quark. Antiprotons can remain stable in high-vacuum isolation, but upon contact with ordinary matter (e.g., protons), they almost instantaneously annihilate, converting their mass into other particles and energy.Establishing a connection between high-energy physics and quantum information science is extremely important. Afik et al. proposed the detection of entanglement between the spins of top-antitop-quark pairs at the LHC, representing the entanglement observation at the highest energy scale so far in 2021 [12], computed the quantum state of a tt¯ pairs produced from the most elementary quantum chromodynamics (QCD) processes, finding the presence of entanglement and CHSH violation in different regions of phase space in 2022 [13] and also provide the full picture of quantum correlations in top quarks by studying also quantum discord and steering in 2023 [14]. In 2024, Ye et al. employed the entropic uncertainty relations and the quantum Fisher information to explore the formation of quark tt¯ pairs at the LHC through the combination of qq¯ pairs and gg pairs initiated processes [15]. In 2025, Cheng et al. show analytically that the basis that diagonalizes the spin-spin correlations is optimal for maximizing spin correlations, entanglement, and Bell inequality violation [16]. Furthermore, Han et al. employ two complementary approaches for the study of the top anti-top system, namely the decay method and the kinematic method and they highlight subtleties associated with measuring discord for reconstructed quantum states at colliders in 2025 [17].In this work, we present a comprehensive investigation of quantum correlations in top quark-antiquark (tt¯) pairs produced via QCD processes. Rather than being restricted to specific collider settings, we perform a general theoretical scan over the full range of initial-state mixtures (wgg=0--1) to reveal the universal scaling of quantum observables with production mechanisms. We apply several key information theoretic tools—namely QMI, REC, CCR, and an intrinsic relation to quantify and characterize the entanglement structure within the tt¯ system at the QCD. These observables go beyond conventional spin correlation measures by capturing both quantum and classical correlations in a unified framework. Specifically, QMI quantifies the total correlations between the top and antitop quarks. As a standard measure in quantum resource theory, REC quantifies the quantum coherence of a state. The CCR provides a complementary relation that links coherence, predictability, and correlations, revealing how quantum information is distributed among subsystems. The intrinsic relation further connects uncertainty, coherence, and predictability, offering a more complete picture of the quantumness of the system.Unlike traditional observables such as spin correlations or entanglement witnesses, these quantities are continuous values, typically ranging from 0 to 2log2d for QMI and from 0 to log2d for REC in qubit system (d is the dimension of a single particle). A larger value of QMI indicates the stronger total correlations including classical and quantum parts, while a larger REC signifies the greater quantum coherence (i.e., quantumness). Importantly, these observables allow us to distinguish between different types of quantum behavior-such as coherence versus entanglement-and to explore how they vary across kinematic phase space. By measuring them in tt¯ pair production, we gain new insights into the structure of quantum correlations in high-energy collisions. Thereby, the QMI and REC can intrinsically reveal the global correlation and quantumness in the course of particle collisions.The Letter is structured as follows. In Section 2, we review in detail the production of tt¯ pairs through QCD processes. In Section 3, we discuss the quantum mutual information in tt¯ production. In Section 4, we study the relative entropy of coherence in tt¯ production. In Section 5, we investigate the complete complementarity relation among QMI, the conditional entropy, REC, and predictability in tt¯ pairs. In Section 6, we explore the intrinsic relation among entropic uncertainty, REC and predictability in tt¯ pairs. Finally, we end up our Letter with a concise conclusion in Section 7.2

Preliminaries of tt¯ pairs production

A natural Standard Model candidate for an entangled two-qubit system is a particle-antiparticle (denoted generically as PP¯) pairs produced from some initial state Is(1)Is→P+P¯,in this paper, we only discuss the case where P is a fermion of mass m in the Standard Model. The components of the initial state Is annihilate themselves through interactions, resulting in the production of PP¯ pairs. In this paper, we study particle-antiparticle pairs (PP¯), focusing on top-antitop quark pairs (tt¯) produced via QCD processes in high-energy colliders. In the theoretical calculation, we employ leading-order (LO) QCD perturbation theory.The interactions between a light quark and antiquark (qq¯) or a pair of gluons (gg) through QCD result in the production of a tt¯ pair as follows(2)q+q¯→t+t¯g+g→t+t¯,the production of a tt¯ pair is described by the invariant mass Mtt¯ and the direction k^ in the center-of-mass frame (c.m.). In this frame, the top and antitop relativistic momenta are ktμ=(kt0,k), kt¯μ=(kt¯0,-k), satisfying the invariant dispersion relation kt2≡ktμkμt=mt2, and similar for the antitop kt¯2=kt2=mt2.The invariant mass Mtt¯ in the c.m. energy is defined as(3)Mtt¯2≡stt¯≡(kt+kt¯),in the c.m. frame, Mtt¯2=4(kt0)2=4(mt2+k2), the momentum of top quark is related to its velocity β, satisfying |k|=mtβ/1−β2, we obtain(4)β=1−4mt2/Mtt¯2.The production threshold for tt¯ pairs, corresponding to β=0, occurs at the lowest energy, Mtt¯=2mt≈346 GeV.The spin correlations of a tt¯ pair are fully described, at a given partonic center-of-mass energy and scattering angle, by the so-called production spin density matrix ρ˜. In the basis of tensor products of top-quark and anti-top-quark spin states, the density matrix can be parameterized as(5)ρ˜=A˜I4+∑i(B˜i+σi⊗I2+B˜i−I2⊗σi)+∑i,jC˜ijσi⊗σj.The production spin density matrix ρ˜ is characterized by 16 parameters, A˜, B˜i±, C˜ij, with A˜ determining the differential cross section for tt¯ production at fixed energy and top direction.The proper spin density matrix ρ^ of the tt¯ pairs are obtained from normalizing ρ˜(6)ρ^=ρ˜Tr(ρ˜)=ρ˜4A˜,where A˜ represents the differential cross-section for the production of tt¯ at a fixed energy and in a specified top-quark direction.As a result, the spin polarizations Bi± and spin correlations Cij of the tt¯ pairs are normalized by the total cross section coefficient A˜(7)Bi±=B˜i±A˜,Cij=C˜ij±A˜.The spin density matrix ρ^ in the helicity basis is then given by(8)ρ^=14(I4+∑i(Bi+σi⊗I2+Bi−I2⊗σi)+∑i,jCijσi⊗σj).In this framework, the correlation matrix Cij and spin polarizations Bi± are constrained by the LO symmetries. The normalized form of the spin density matrix ρ^, expressed in the helicity basis, is fully characterized by five independent parameters (C˜rr,C˜nn,C˜kk,C˜rk,C˜kr), where ρ^ depends only on β=1−4mt2/Mtt¯2 and Θ. At LO, in the Standard Model, the tt¯ pairs is unpolarized (B˜i±=0), and the correlation matrix satisfies C˜ij=C˜ji. The explicit form of ρ^ is given by(9)ρ^=14A˜(A˜+C˜kkC˜krC˜rkC˜rr−C˜nnC˜krA˜−C˜kkC˜rr+C˜nn−C˜rkC˜rkC˜rr+C˜nnA˜−C˜kk−C˜krC˜rr−C˜nn−C˜rk−C˜krA˜+C˜kk).For qq¯ process, the coefficients of the matrix are(10)A˜qq¯=Fq(2−β2sin2Θ),C˜rrqq¯=Fq(2−β2)sin2Θ,C˜nnqq¯=−Fqβ2sin2Θ,C˜kkqq¯=Fq[2−(2−β2)sin2Θ],C˜rkqq¯=C˜krqq¯=Fq1−β2sin2Θ,Fq=118.For gg process, the coefficients of the matrix are(11)A˜gg=Fg[1+2β2sin2Θ−β4(1+sin4Θ)],C˜rrgg=−Fg[1−β2(2−β2)(1+sin4Θ)],C˜nngg=−Fg[1−2β2+β4(1+sin4Θ)],C˜kkgg=−Fg[1−β2sin22Θ2−β4(1+sin4Θ)],C˜rkgg=C˜krgg=Fg1−β2β2sin2Θsin2Θ,Fg=7+9β2cos2Θ192(1−β2cos2Θ)2.The spin density matrix for the mixed state of qq¯ and gg initial states can be expressed as(12)ρ^(Mtt¯,k^)=∑I=qq¯,ggWI(Mtt¯,k^)ρ^I(Mtt¯,k^).In this work, we treat Wgg as a free parameter ranging from 0 to 1, allowing us to systematically explore how the quantum nature of tt¯ pairs depend on the initial-state composition. This parameter covers the full range of physical scenarios: from low-energy colliders where qq¯ annihilation dominates (e.g., Tevatron, Wgg≈0.1), to high-energy colliders where gg fusion dominates (e.g., LHC at 13 TeV, Wgg≈0.85–0.95).3

The quantum mutual information in tt¯

In quantum information theory, QMI provides a quantitative measure of correlations between two quantum systems [18–21]. As a generalization of classical mutual information, QMI serves as a fundamental tool for quantifying both quantum and classical correlations, including entanglement [22,23] that exist between the systems. In a two-qubit mixed state, QMI ranges from 0 (uncorrelated) to 2log2d (maximally correlated), with the maximum achieved by states of either maximal classical correlation or maximal entanglement. The QMI quantifies the total correlation between subsystems: a larger value indicates stronger total correlation. Given an arbitrary bipartite quantum state ρ^AB, the QMI is formally defined by [19](13)I(A:B)=S(ρ^A)+S(ρ^B)−S(ρ^AB)=S(ρ^A)−S(A|B).The von Neumann entropy, denoted as S(ρ^), quantifies the information content of a quantum state ρ^ in a d-dimensional Hilbert space. It is formally defined as S(ρ^)=−Tr(ρ^logρ^). For a bipartite quantum system described by the state ρ^AB, the conditional entropy S(A|B) prior to measurement is given by the difference S(ρ^AB)−S(ρ^B), where ρ^B=TrA(ρ^AB) represents the reduced state of subsystem B. The detailed mathematical expression for QMI is provided in Appendix A.

Fig. 1

shows the QMI analysis for tt¯ pairs production at the QCD. Fig. 1(a) displays the results for the gluon fusion channel (gg→tt¯). where I(A: B) exhibits dependence on both production angle Θ and invariant mass Mtt¯. The QMI demonstrates a gradual decrease with increasing Θ, with maximal entanglement observed near the threshold region at Mtt¯≈346 GeV. Fig. 1(b) displays the quark annihilation channel (qq¯→tt¯), where QMI reaches a minimum near threshold and increases with Mtt¯ as Θ increases. The peak QMI occurs at Mtt¯=1000 GeV and Θ=π/2, with the system maintaining both classical and quantum correlations throughout the kinematic range.

Fig. 2

presents a systematic study of quantum correlations in tt¯ production using a simplified model with fixed partonic probabilities. The analysis examines how varying the mixture between gg and qq¯ channels affects the quantum state ρ^(Mtt¯,k^). QMI is shown as a function of production angle Θ and the invariant mass Mtt¯, with subfigures (a)-(d) corresponding to increasing gluon fusion probabilities Wgg=0.2,0.4,0.6, and 0.8 respectively. At low Wgg, maximal I(A: B) occurs in the large mass, large-angle region (upper right corner). Notably, as Wgg increases to 0.8, the QMI behavior converges toward the pure gg→tt¯ case shown in Fig. 1(a).

The relative entropy of coherence in tt¯

Quantum coherence, a fundamental manifestation of the superposition principle in quantum mechanics, has evolved into a key resource for quantum technologies [24–32]. Originating from the ability of quantum states to exist in superpositions, coherence enables diverse quantum-enhanced applications. The quantification of coherence has been rigorously formalized [33,34], with two principal measures emerging: the l1-norm coherence and the REC. For a two-particle mixed state ρ^AB, the REC Cre(ρ^AB) satisfies 0<Cre(ρ^AB)<log2d, where d is the dimension of a single particle. A larger value of Cre(ρ^AB) indicates stronger quantum coherence in the state. The REC Cre(ρ^AB) is defined mathematically as [25](14)Cre(ρ^AB)=S(ρ^diag)−S(ρ^AB),where S(ρ^AB)=−Tr(ρ^ABlogρ^AB) denotes the von Neumann entropy, and ρ^diag is the non-coherence state and obtained by taking diagonal elements of the matrix ρ^AB.The REC for the system is derived from Eqs. (10), (11) and (14). The detailed mathematical expression for REC is provided in Appendix A. In Fig. 3

(a), we present the REC for tt¯ production via gluon fusion (gg→tt¯), showing its dependence on both the production angle Θ and the invariant mass Mtt¯. Notably, when Mtt¯ is near 346 GeV, Cre(ρ^AB) decreases monotonically with increasing Θ. In contrast, Fig. 3(b) displays the REC for tt¯ production through quark-antiquark annihilation (qq¯→tt¯). Here, Cre(ρ^AB) increases monotonically with Θ at fixed Mtt¯.

Fig. 4

displays the REC Cre(ρ^^AB) for tt¯ production through both quark annihilation (qq¯→tt¯) and gluons fusion (gg→tt¯) channels. Subfigures (a)-(d) correspond to gluon fusion probabilities Wgg=0.2,0.4,0.6, and 0.8, respectively, showing the REC dependence on production angle Θ and invariant mass Mtt¯. With increasing Wgg, we observe that suppression of low quality/low angle on REC is weakening. REC shrinks in the right corner range and expands in the left corner range in Fig. 4(a)–(d).

The complete complementarity relations in tt¯

The CCR [35,36] provides a rigorous framework for quantifying quantum correlation in the composite system. We apply this approach to characterize quantum correlations in tt¯ pairs production at the QCD. The CCR are constructed from von Neumann entropy-based measures of predictability and coherence. For bipartite mixed states, the CCR takes the form [35](15)IA:B(ρ^AB)+SA|B(ρ^AB)+Pvn(ρ^A)+Cre(ρ^A)=log2dA.QMI IA:B(ρ^AB) quantifies the correlations between subsystems A and B. The conditional von Neumann entropy SA|B(ρ^AB)=Svn(ρ^AB)−Svn(ρ^B) quantifies the uncertainty in subsystem A conditioned on subsystem B. For a d-dimensional Hilbert space, SA|B(ρ^AB) is bounded as −log2d⩽SA|B(ρ^AB)⩽log2d (d is the dimension of subsystem A or B). A positive conditional entropy (SA|B(ρ^AB)>0) suggests either a classical state or a quantum state with weak correlations, whereas a negative value (SA|B(ρ^AB)<0) guarantees the presence of quantum entanglement. Here, Svn(ρ^) denotes the von Neumann entropy of the state ρ^, and ρ^A,diag=∑i=1dAρ^iiA|i〉〈i| represents the diagonalized reduced state of subsystem A. Additionally, the predictability measure Pvn(ρ^A), defined as Pvn(ρ^A)≡log2dA−Svn(ρ^A,diag), denotes the predictability of the single subsystem A.When considering the state of a single subsystem A, information shared through its correlations with subsystem B is inevitably lost. The reduced density matrix of subsystem A can be expressed as ρ^A=12I2. ρ^A in a two-dimensional Hilbert space.From Eq. (15) and ρ^A=12I2, we find that both the predictability measure Pvn(ρ^A) and the coherence measure Cre(ρ^A) vanish (become zero). Consequently, the CCR for tt¯ production reduce to functions solely dependent on the QMI IA:B(ρ^AB) and the conditional entropy SA|B(ρ^AB).

Fig. 5

shows the dependence of QMI IA:B(ρ^AB) and conditional entropy SA|B(ρ^AB) on both the production angle Θ and the invariant mass Mtt¯. As evident from the figure, these quantities exhibit complementary behavior: IA:B(ρ^AB) increases when SA|B(ρ^AB) decreases, and vice versa. Notably, their sum remains constant at unity (IA:B(ρ^AB)+SA|B(ρ^AB)=1), demonstrating conservation that is independent of the initial-state mixing between gluons (gg) and quarks (qq¯) production channels.

From Fig. 5, we can know that the sum of IA:B(ρ^AB) and SA|B(ρ^AB) remains conserved at unity (IA:B(ρ^AB)+SA|B(ρ^AB)=1), independent of the initial-state mixing between gluons and quarks channels.6

The intrinsic relation in tt¯

In quantum resource theory, CCR provide a powerful framework for characterizing and quantifying quantum correlations and their interplay. As shown in [35], the generalized CCR for bipartite mixed states provides a comprehensive framework to analyzing quantum spin correlations in top quark pairs production. Building upon this foundation, we now investigate these quantum correlations in tt¯ production within the QCD framework through the formalism of von Neumann entropy, examining their fundamental relationships in entropy space [37–41].For a two-qubit system, the measure of predictability Pvn(ρ^AB), REC Cre(ρ^AB), conditional entropy S(A|B) of the state ρ^AB and entropic uncertainty relations can be defined as(16)Pvn(ρ^AB)=log2(dAdB)−S(ρ^d),Cre(ρ^AB)=S(ρ^diag)−S(ρ^AB),S(A|B)=S(ρ^AB)−S(ρ^B),S(Q^|B)+S(R^|B)⩾log21c+S(A|B).From Eq. (16), we derive for a two-qubit mixed state system the intrinsic relation among entropic uncertainty, REC, and predictability, expressed in terms of von Neumann entropy as(17)S(Q^|B)+S(R^|B)+S(ρ^B)+Pvn(ρ^AB)+Cre(ρ^AB)⩾log21c+log2(dAdB),herein, S(Q^|B) denotes the conditional von Neumann entropy of the post-measurement state ρ^Q^B after measuring observable Q^ on subsystem A, where S(ρ^B)=−Tr(ρ^Blog2ρ^B) is the von Neumann entropy of subsystem B. The parameter c=maxi,j|〈q^i|r^j〉|2 quantifies the maximum overlap between eigenstates |q^i〉 of Q^ and |r^j〉 of R^. For a two-qubit system (where A and B are in two-dimensional Hilbert space), Eq. (17) is reduced to: S(Q^|B)+S(R^|B)+S(ρ^B)+Pvn(ρ^AB)+Cre(ρ^AB)⩾3.In Fig. 6

(a), we show the left hand side of the intrinsic relation for tt¯ production via gluon fusion (gg→tt¯), illustrating its dependence on the production angle Θ and the invariant mass Mtt¯. It is observed that for a fixed Mtt¯, the left hand side of the intrinsic relation increases monotonically with Θ. In Fig. 6(b), the left hand side of the intrinsic relation reaches larger values, especially around Θ ≈ π/4.

Fig. 7

plots the left-hand side of the intrinsic relation in terms of von Neumann entropy as a function of production angle Θ and invariant mass Mtt¯. The inequality S(Q^|B)+S(R^|B)+S(ρ^B)+Pvn(ρ^AB)+Cre(ρ^AB)⩾3 holds for all initial-state mixtures Wgg of the gluon (gg) and quark (qq¯) channels. Panels (a)-(d) reveal that increasing the gluon probability Wgg enhances the value of the left-hand side of the intrinsic relation. Notably, Fig. 7(d) shows the maximum value occurs at small invariant mass Mtt¯ and large production angle Θ.

To further illustrate the dependence on the invariant mass, we present the left-hand side of the intrinsic relation as a function of the production angle Θ (gg→tt¯) for fixed Mtt¯ values of =400, 500, and 700 GeV in Fig. 8

. As shown in the figure, the left-hand side of the intrinsic relation increases monotonically with Θ for each fixed mass. At fixed production angle Θ, the value exhibits a clear dependence on the invariant mass. As shown in the figure, increasing Mtt¯ from 500 GeV to 700 GeV does lead to a decrease for the left-hand side of the intrinsic relation.

Fig. 9

presents a detailed analysis of the intrinsic relation in tt¯ production via the gluon fusion channel (gg→tt¯). The left-hand side of the intrinsic relation S(Q^|B)+S(R^|B)+S(ρ^B)+Pvn(ρ^AB)+Cre(ρ^AB) is plotted as a function of the production angle Θ for three fixed invariant masses: Mtt¯=400, 500, and 700 GeV. The three subfigures correspond to the individual contributions: (a) the sum of conditional entropies S(Q^|B)+S(R^|B), (b) the predictability measure Pvn(ρ^AB), and (c) REC Cre(ρ^AB). Fig. 9(b) shows that the predictability Pvn(ρ^AB) exhibits a decrease followed by stabilization with increasing Θ for fixed invariant masses, indicating that the quantum state becomes less predictable at larger production angles Θ. For invariant masses of 500 and 700 GeV, Fig. 9(c) shows that the REC Cre(ρ^AB) remains relatively stable for small Θ before rising as Θ approaches π/4. This demonstrates that quantum coherence is significantly enhanced in the large-angle regime, with the effect being more pronounced at higher energy scales.

Discussion and summary

This work presents the systematic investigation of quantum correlations in top-quark pairs (tt¯) production, establishing a novel connection between quantum information theory and high-energy physics. By analyzing tt¯ production through QMI, REC, CCR, and their intrinsic relations, we have uncovered several key insights: in the gluon-fusion channel (gg→tt¯), QMI peaks at an invariant mass Mtt¯≈346 GeV, demonstrating strong dependence on both the invariant mass Mtt¯ and the production angle Θ. For qq¯→tt¯ process, the larger the invariant mass Mtt¯ and production angle Θ, the stronger the correlation (QMI) between the two subsystems. When the invariant mass Mtt¯ is around 346 GeV, the REC for gg→tt¯ process decreases with the increasing Θ, while the REC for qq¯→tt¯ process maintains at a relatively large value in most region with Θ increases. The analysis shows that the sum of QMI and conditional entropy remains conserved at unity (i.e., IA:B(ρ^AB)+SA|B(ρ^AB)=1) in tt¯ production, indicating that subsystem A undergoes complete decoherence, collapsing into a classical diagonal state-a phenomenon likely driven by environmental interactions that fully erase its quantum coherence. Furthermore, we have rigorously derived the intrinsic relation connecting uncertainty, REC and predictability in terms of von Neumann entropy. Notably, the left-hand side of this inequality grows with Wgg. These findings highlight the profound quantum-informational structure inherent to high-energy particle collisions, offering a unified perspective that bridges quantum field theory and quantum information science. The framework developed here can be extended to other heavy-quark systems, opening new avenues for exploring quantum correlations in beyond-Standard-Model physics and collider experiments.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Science Foundation of China (Grant Nos. 12475009, 12075001, and 62471001), Anhui Provincial Key Research and Development Plan (Grant No. 2022b13020004), Anhui Province Science and Technology Innovation Project (Grant No. 202423r06050004), Anhui Provincial Natural Science Foundation (Grant no. 2508085ZD001), Anhui Provincial Department of Industry and Information Technology (Grant no. JB24044), and Anhui Provincial University Scientific Research Major Project (Grant No. 2024AH040008).

Appendix A

Based on Eqs. (13) and (14), the analytical expressions of QMI and REC for the production of tt¯ pairs are given by(A.1)IA:B(ρ^AB)=2+14A˜ggA˜qq¯log2([WggA˜qq¯(A˜gg+C˜rrgg−C˜nngg+C˜kkgg)+Wqq¯A˜gg(A˜qq¯+C˜rrqq¯−C˜nnqq¯+C˜kkqq¯)]×log[Wgg(A˜gg+C˜rrgg−C˜nngg+C˜kkgg)4A˜gg−Wqq¯(A˜qq¯+C˜rrqq¯−C˜nnqq¯+C˜kkqq¯)4A˜qq¯]+[A˜ggA˜qq¯−WggA˜qq¯(C˜rrgg+C˜nngg+C˜kkgg)−Wqq¯A˜gg(C˜rrqq¯+C˜nnqq¯+C˜kkqq¯)]×log[14−Wgg(C˜rrgg+C˜nngg+C˜kkgg)4A˜gg−Wqq¯(C˜rrqq¯+C˜nnqq¯+C˜kkqq¯)4A˜qq¯]+[Wqq¯A˜gg(A˜qq¯+C˜nnqq¯)+WggA˜qq¯(A˜gg+C˜nngg)−Q]log[Wqq¯A˜gg(A˜qq¯+C˜nnqq¯)+WggA˜qq¯(A˜gg+C˜nngg)−Q4A˜ggA˜qq¯]+[Wqq¯A˜gg(A˜qq¯+C˜nnqq¯)+WggA˜qq¯(A˜gg+C˜nngg)+Q]log[Wqq¯A˜gg(A˜qq¯+C˜nnqq¯)+WggA˜qq¯(A˜gg+C˜nngg)+Q4A˜ggA˜qq¯])and(A.2)Cre(ρ^AB)=14A˜ggA˜qq¯log2(4A˜ggA˜qq¯log4+2(WggA˜qq¯C˜kkgg+Wqq¯A˜ggC˜kkqq¯−A˜ggA˜qq¯)log[1−WggC˜kkggA˜gg−Wqq¯C˜kkqq¯A˜qq¯]−2[WggA˜qq¯(A˜gg+C˜kkgg)+Wqq¯A˜gg(A˜qq¯+C˜kkqq¯)]log[1+WggC˜kkggA˜gg+Wqq¯C˜kkqq¯A˜qq¯]+[WggA˜qq¯(A˜gg+C˜rrgg−C˜nngg+C˜kkgg)+Wqq¯A˜gg(A˜qq¯+C˜rrqq¯−C˜nnqq¯+C˜kkqq¯)]×log[Wgg(A˜gg+C˜rrgg−C˜nngg+C˜kkgg)4A˜gg−Wqq¯(A˜qq¯+C˜rrqq¯−C˜nnqq¯+C˜kkqq¯)4A˜qq¯]+[A˜ggA˜qq¯−WggA˜qq¯(C˜rrgg+C˜nngg+C˜kkgg)−Wqq¯A˜gg(C˜rrqq¯+C˜nnqq¯+C˜kkqq¯)]×log[14−Wgg(C˜rrgg+C˜nngg+C˜kkgg)4A˜gg−Wqq¯(C˜rrqq¯+C˜nnqq¯+C˜kkqq¯)4A˜qq¯]+[Wqq¯A˜gg(A˜qq¯+C˜nnqq¯)+WggA˜qq¯(A˜gg+C˜nngg)−Q]log[Wqq¯A˜gg(A˜qq¯+C˜nnqq¯)+WggA˜qq¯(A˜gg+C˜nngg)−Q4A˜ggA˜qq¯]+[Wqq¯A˜gg(A˜qq¯+C˜nnqq¯)+WggA˜qq¯(A˜gg+C˜nngg)+Q]log[Wqq¯A˜gg(A˜qq¯+C˜nnqq¯)+WggA˜qq¯(A˜gg+C˜nngg)+Q4A˜ggA˜qq¯]),respectively, where f1gg=(Wgg)2(A˜qq¯)2[4(C˜rkgg)2+(C˜rrgg−C˜kkgg)2], f1qq¯=(Wqq¯)2(A˜gg)2[4(C˜rkqq¯)2+(C˜rrqq¯−C˜kkqq¯)2] and Q=f1gg+f1qq¯+2WggWqq¯A˜ggA˜qq¯[4C˜rkggC˜rkqq¯+(C˜rrqq¯−C˜kkqq¯)(C˜rrgg−C˜kkgg)]. For the production of tt¯ pairs via gluons (quarks), Wgg=1(Wqq¯=1).

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