# When the Machine Chimes the Bell:
Entanglement and Bell Inequalities with Boosted $t\bar{t}$

Zhongtian Dong<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of Kansas, Lawrence, KS 66045, USA  Dorival Gonçalves<EMAIL_ADDRESS>Department of Physics, Oklahoma State University, Stillwater, OK, 74078, USA
Kyoungchul Kong<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of Kansas, Lawrence, KS 66045, USA  Alberto Navarro
<EMAIL_ADDRESS>Department of Physics, Oklahoma State
University, Stillwater, OK, 74078, USA

###### Abstract

The Large Hadron Collider provides a unique opportunity to study quantum
entanglement and violation of Bell inequalities at the highest energy
available today. In this paper, we will investigate these quantum correlations
with top quark pair production, which represents a system of two-qubits. The
spacelike separation requirement for the two causally disconnected top quarks
requires they fly relativistically away from each other, which motivates the
use of the boosted top-tagging with the semi-leptonic top pair channel.
Although measuring the spin polarization of the hadronic top quark is known to
be challenging, our study indicates that it is feasible to reconstruct the
spin density matrix of the two-qubit system using an optimal hadronic
polarimeter. This is achieved with the aid of jet substructure techniques and
NN-inspired reconstruction methods, which improve the mapping between subjets
and quarks. We find that entanglement can already be observed at more than
$5\sigma$ level with existing data, and violation of Bell inequalities may be
probed above 4$\sigma$ level at the HL-LHC with 3 ab-1 of data.

##  I Introduction

Quantum theory provides a robust framework for describing physical phenomena
at the microscopic level, and its implications are ubiquitous in our daily
life. However, various aspects of quantum theory are highly nontrivial. One
such phenomenon is entanglement, which sets quantum physics apart from
classical physics Einstein  _et al._ (1935); Schrödinger (1935). Even more
perplexing is the violation of Bell inequalities Bell (1964), a consequence of
entanglement that unequivocally identifies this departure from classical
theory. Since the birth of quantum theory, entanglement and violation of Bell
inequalities have been extensively tested in a myriad of experiments, ruling
out the classical hidden variable explanation of quantum phenomena. Typically,
such tests are performed using entangled low-energy photons Freedman and
Clauser (1972); Aspect  _et al._ (1982); Vaziri  _et al._ (2002), ions Rowe
_et al._ (2001), superconducting systems Ansmann  _et al._ (2009), and
nitrogen vacancy centers Pfaff  _et al._ (2012), among others. In recent
decades, a series of experiments have closed all potential loopholes in these
tests Hensen  _et al._ (2015); Giustina  _et al._ (2015); Shalm  _et al._
(2015).

Although these quantum correlations have been probed in the low energy regime,
they have yet to be observed in the very high energy scales. Some proposals
suggested testing Bell inequalities in $e^{+}e^{-}$ collisions Tornqvist
(1981), decays of charmonium Baranov (2009, 2008); Chen  _et al._ (2013), and
positronium Acin  _et al._ (2001). The Large Hadron Collider (LHC) provides an
excellent environment for studying quantum entanglement and Bell inequalities
at the highest energy available today. Only very recently, a few proposals
have been made to test entanglement and Bell inequalities for the final state
with $t\bar{t}$ Afik and de Nova (2021); Fabbrichesi  _et al._ (2021); Severi
_et al._ (2022); Aguilar-Saavedra and Casas (2022); Afik and de Nova (2022,
2023); Severi and Vryonidou (2023); Aoude  _et al._ (2022), a pair of weak
bosons Fabbrichesi  _et al._ (2023a); Barr (2022); Barr  _et al._ (2023);
Aguilar-Saavedra (2023); Aguilar-Saavedra  _et al._ (2023), and
$\tau^{+}\tau^{-}$ Altakach  _et al._ (2023); Fabbrichesi  _et al._ (2023b).

In this paper, we explore the possibility of observing entanglement and
violation of Bell inequalities in top quark pair production, which represents
a system of two-qubits. Previous studies have mainly focused on the dilepton
channel Afik and de Nova (2021); Fabbrichesi  _et al._ (2021); Severi  _et
al._ (2022); Aguilar-Saavedra and Casas (2022), as the top quark spin is
correlated with the direction of charged leptons. They show that entanglement
may be observed for both the $t\bar{t}$ threshold and the boosted region,
while violation of Bell inequalities may be probed only in the boosted region.
However, evaluation of the probability of space-like separated decays shows
that most of the top pairs near threshold decay within each other’s light-
cone. For example, the probability for space-like separation is about 30% at
$m_{t\bar{t}}\sim 400$ GeV, while more than 90% of top quark pairs decay when
they are space–like separated for $m_{t\bar{t}}>800$ GeV Severi  _et al._
(2022). Thus, the threshold region would be difficult to provide a clear
observation of the entanglement of the two top quarks, which motivates us to
study the boosted region. Furthermore, the dilepton channel suffers from lower
statistics and the reconstruction of two top quarks is challenging. Hence, we
will consider the semi-leptonic channel, which will increase the signal
statistics by a factor of six and allow an easier reconstruction. We will
address some technical challenges associated with this analysis, which can be
overcome using an optimal hadronic polarimeter in combination of jet
substructure and machine learning techniques.

Whereas the observation of entanglement and Bell inequalities is interesting
in its own right, it can further shed light on top quark physics. Top quark
physics is one of the most relevant topics within the standard model (SM) and
in the search for new physics. Ultimately, studying entanglement and violation
of Bell inequalities may offer a new perspective on these searches.

This paper is organized as follows. We provide a brief review of entanglement
and Bell inequalities in Section II. We then study the possibility of
observing entanglement and violation of Bell inequalities for the top quark
production in the semi-leptonic channel. We present the parton-level truth in
Section III and present the complete analysis accounting for parton-shower,
hadronization, and detector effects in Section IV. Finally, Section VI is
reserved for summary and outlook.

##  II Theoretical Framework

###  II.1 Quantum Entanglement and Bell Inequalities

A quantum state of two subsystems $A$ and $B$ is separable when its density
matrix $\rho$ can be expressed as a convex sum

$\displaystyle\rho=\sum_{i}p_{i}\rho_{A}^{i}\otimes\rho_{B}^{i}\,,$ (1)

where $\rho^{i}_{A}$ and $\rho^{i}_{B}$ are quantum states of the subsystems
$A$ and $B$, and $\Sigma_{i}p_{i}=1$ with $p_{i}\geq 0$. If the state is not
separable, it is named _entangled_. The Peres-Horodecki criterion provides a
necessary and sufficient condition for entanglement in two-qubit systems Peres
(1996); Horodecki (1997). It exploits the fact that, for a separable density
matrix $\rho$, the partial transpose for a second subsystem

$\displaystyle\rho^{T2}=\sum_{i}p_{i}\rho_{A}^{i}\otimes(\rho_{B}^{i})^{T}\,,$
(2)

results in a non-negative operator. Therefore, if $\rho^{T2}$ displays at
least one negative eigenvalue, the density matrix $\rho$ describes an
entangled system.

An entangled quantum state can result in the violation of Bell-type
inequalities Bell (1964). Remarkably, the violation of such inequalities
demonstrates that there is no hidden variable theory capable of encoding the
generated entanglement. Hence, quantum mechanics cannot be described by
classical laws. The Clauser, Horne, Shimony, and Holt (CHSH) inequality is a
realization of the Bell-type inequalities for bipartite systems Clauser  _et
al._ (1969)

$\displaystyle|\langle A_{1}B_{1}\rangle+\langle A_{2}B_{1}\rangle+\langle
A_{1}B_{2}\rangle-\langle A_{2}B_{2}\rangle|\leq 2\,,$ (3)

where the observer of system $A$ (usually called Alice) can measure two
distinct observables $A_{1}$ or $A_{2}$ and the observer of system $B$
(usually called Bob) can probe $B_{1}$ or $B_{2}$. Each of these observables
has two possible eigenvalues, $\pm 1$. For spin observables, the CHSH
inequality can be violated by certain entangled states, provided that there is
an appropriate choice of spin axes $\hat{a}_{1(2)}$ for $A$ and
$\hat{b}_{1(2)}$ for $B$.

###  II.2 Top Polarization

In this section, we will discuss the top quark polarization and how it can be
used to probe entanglement and a possible violation of the CHSH inequalities
in top quark pair production $t\bar{t}$ at the LHC.

Due to its short lifetime $(\sim 10^{-25}~{}$s), the top quark decays before
hadronization occurs $(\sim 10^{-24}$ s) and spin decorrelation effects take
place $(\sim 10^{-21}$ s) Mahlon and Parke (2010); Zyla  _et al._ (2020). It
implies that the top quark final states correlate with the top quark
polarization axis as

$\displaystyle\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\xi_{k}}=\frac{1}{2}\left(1+\beta_{k}p\cos\xi_{k}\right)\,,$
(4)

where $\Gamma$ is the partial decay width, $\xi_{k}$ is the angle between the
final state particle $k$ and the top quark spin axis in the top quark rest
frame, $p$ is the degree of polarization of the ensemble, and $\beta_{k}$ is
the spin analyzing power for the decay product $k$ Bernreuther and Si (2010).
The spin analyzing power $\beta_{k}$ is $+1$ for the charged lepton $\ell^{+}$
and $\bar{d}$-quark, $-0.3$ for the neutrino $\bar{\nu}$ and $u$-quark, $-0.4$
for the $b$-quark, and $0.4$ for the $W^{+}$ boson. The signs of $\beta_{i}$
coefficients are reversed for anti-top quark decays.

Similarly to the charged lepton $\ell^{\pm}$, the $d$-quark (or down-type
quark) displays maximal spin analyzing power. However, it is a challenging
task to tag a $d$-quark in a collider environment. A possible solution is to
choose the softest of the two light jets from the top decay in the top quark
rest frame. This choice leads to a spin analyzing power
$\beta_{\text{soft}}\simeq 0.5$ Jezabek (1994); Barman  _et al._ (2022). This
spin analyzing power can be uplifted by choosing an optimal hadronic
polarimeter $\beta_{\text{opt}}\simeq 0.64$ Tweedie (2014).

###  II.3 Optimal Hadronic Polarimeter

We begin by considering the top quark decay in the $W$ boson rest frame.
Although the azimuthal angle of the $W$ decay is random, the polar decay angle
($\theta_{W_{\text{hel}}}$) in the $W$ rest frame (often called helicity
angle) is strongly correlated with the polarizations of the $W$. We take the
“$z$-axis” of the decay to be the direction pointing opposite to the $b$-quark
direction, $\hat{z}=-\hat{b}$ in the $W$ boson rest frame.

The cosine of the helicity angle
$c_{W_{\text{hel}}}\equiv\cos\theta_{W_{\text{hel}}}$ is taken to be positive
when the $d$-quark (or down-type quark) is emitted in the forward hemisphere
and the $u$-quark (or up-type quark) in the backward hemisphere. Then, the
distribution of $c_{W_{\text{hel}}}$ for the decay products of $W^{\pm}$ is
given by

$\displaystyle\rho(c_{W_{\text{hel}}})=$ $\displaystyle\frac{3}{8}f_{R}(1\pm
c_{W_{\text{hel}}})^{2}+\frac{3}{4}f_{0}(1-c_{W_{\text{hel}}}^{2})$
$\displaystyle+\frac{3}{8}f_{L}(1\mp c_{W_{\text{hel}}})^{2}.$ (5)

The upper sign corresponds to $W^{+}$ and the lower sign to $W^{-}$. The
polarization fractions can be obtained from the following relations Bern  _et
al._ (2011)

$\displaystyle f_{0}$ $\displaystyle=2-5\langle
c_{W_{\text{hel}}}^{2}\rangle,$ (6) $\displaystyle f_{L}$
$\displaystyle=-\frac{1}{2}\mp\langle
c_{W_{\text{hel}}}\rangle+\frac{5}{2}\langle c_{W_{\text{hel}}}^{2}\rangle,$
(7) $\displaystyle f_{R}$ $\displaystyle=-\frac{1}{2}\pm\langle
c_{W_{\text{hel}}}\rangle+\frac{5}{2}\langle c_{W_{\text{hel}}}^{2}\rangle,$
(8)

with the expectation of an observable $g(c_{W_{\text{hel}}})$ defined as

$\langle
g(c_{W_{\text{hel}}})\rangle\equiv\int_{-1}^{1}g(c_{W_{\text{hel}}})\frac{1}{\sigma}\frac{d\sigma}{dc_{W_{\text{hel}}}}dc_{W_{\text{hel}}}.$
(9)

$f_{0}={m_{t}^{2}}/{(m_{t}^{2}+2m_{W}^{2})}\approx 0.7$,
$f_{L}={(2m_{W}^{2})}/{(m_{t}^{2}+2m_{W}^{2})}\approx 0.3$, and $f_{R}=0$ are,
respectively, the fractions of zero helicity, left-handed helicity, and right-
handed helicity of $W^{+}$ boson in the top quark rest frame. For the $W^{-}$
decay, the quoted values of $f_{R}$ and $f_{L}$ are interchanged, as can be
inferred from Eqs. (7) and (8).

Since the collider detectors do not distinguish the $d$-quark and $u$-quark,
we need to identify $c_{W_{\text{hel}}}\longleftrightarrow-
c_{W_{\text{hel}}}$. However, the quark emitted in the forward direction in
the $W$ rest frame will be harder and more separated from the $b$-quark in the
top rest frame. Similarly, the quark emitted in the backward direction in the
$W$ frame will be softer and more aligned with the $b$-quark in the top rest
frame. Therefore, the hard and soft quarks have a probability of being the
$d$-quark given by

$\displaystyle p(d\rightarrow q_{\text{hard}})$ $\displaystyle=$
$\displaystyle\frac{\rho(\absolutevalue{c_{W_{\text{hel}}}})}{\rho(\absolutevalue{c_{W_{\text{hel}}}})+\rho(-\absolutevalue{c_{W_{\text{hel}}}})}\,,$
(10) $\displaystyle p(d\rightarrow q_{\text{soft}})$ $\displaystyle=$
$\displaystyle\frac{\rho(-\absolutevalue{c_{W_{\text{hel}}}})}{\rho(\absolutevalue{c_{W_{\text{hel}}}})+\rho(-\absolutevalue{c_{W_{\text{hel}}}})}\,.$
(11)

The top spin axis will align with the direction defined by the weighted
average of the hard and soft-quark directions Tweedie (2014), so the optimal
direction is given by

$\displaystyle\vec{q}_{\text{opt}}(\absolutevalue{c_{W_{\text{hel}}}})$
$\displaystyle=$ $\displaystyle p(d\rightarrow
q_{\text{hard}})\,\hat{q}_{\text{hard}}$ (12) $\displaystyle+p(d\rightarrow
q_{\text{soft}})\,\hat{q}_{\text{soft}},$

where $\hat{q}_{\text{hard}}$ and $\hat{q}_{\text{soft}}$ denote the direction
of the hard and soft quarks in the top rest frame, respectively. The spin
analyzer power, $\beta_{\text{opt}}$, as a function of
$\absolutevalue{c_{W_{\text{hel}}}}$ is equal to the length of
$\vec{q}_{\text{opt}}(\absolutevalue{c_{W_{hel}}})$. Its integrated value is
$\beta_{\text{opt}}\simeq 0.64$.

###  II.4 Quantum Tomography

The top quark pair $t\bar{t}$ forms a two-qubit system that can be represented
by the spin density matrix

$\displaystyle\rho=\frac{\mathbb{1}\otimes\mathbb{1}+\sum_{i}\left(B_{i}\sigma_{i}\otimes\mathbb{1}+\bar{B}_{i}\mathbb{1}\otimes\sigma_{i}\right)+\sum_{ij}C_{ij}\sigma_{i}\otimes\sigma_{j}}{4},$
(13)

where $B_{i}$ and $\bar{B}_{i}$ are the spin polarizations and $C_{ij}$ are
the spin correlations for the spin-$1/2$ particles. The spin density matrix
can be simplified when applied to strong $t\bar{t}$ production at the LHC.
Strong top pair production satisfies the P and CP symmetries, leading to
$B_{i}=\bar{B}_{i}=0$ and $C_{ij}=C_{ji}$ Bernreuther  _et al._ (2015).
Electroweak corrections can change these relations; however, they have been
shown to be subleading Frederix  _et al._ (2021). In combination with these
results, the $t\bar{t}$ density matrix can be further simplified in the
helicity basis, where the only non-vanishing parameters are the diagonal
entries $C_{ii}$ and one off-diagonal term $(C_{12}\simeq C_{21})$. The other
off-diagonal terms are generated by P-odd absorptive parts of the mixed QCD-
weak corrections at one-loop, and are negligible Bernreuther  _et al._ (2015).
Henceforth, we shall assume these well-justified relations. In this study, the
top quark spin is measured in the helicity basis $(\hat{k},\hat{r},\hat{n})$
Bernreuther and Si (2010):

  * •

$\hat{k}$ is the direction of the top quark momentum in the top pair rest
frame.

  * •

$\hat{r}=\text{sign}(\cos\theta)(\hat{p}-\cos\theta\hat{k})/\sin\theta$, where
the beam axis $\hat{p}=(0,0,1)$ and $\cos\theta=\hat{k}\cdot\hat{p}$.

  * •

$\hat{n}=\hat{k}\times\hat{r}$.

It was shown that the entanglement and Bell inequalities violation criteria
are written in simpler forms in the helicity basis Aguilar-Saavedra and Casas
(2022). More concretely, for the $t\bar{t}$ system, the Peres-Horodecki
criterion for entanglement results in the relation

$\displaystyle\mathcal{E}\equiv|C_{kk}+C_{rr}|-C_{nn}-1>0.$ (14)

Analogously, the CHSH inequalities can be expressed in terms of spin
correlations

$\displaystyle\mathcal{B}_{1}$
$\displaystyle\equiv|C_{rr}-C_{nn}|-\sqrt{2}>0,$ (15)
$\displaystyle\mathcal{B}_{2}$
$\displaystyle\equiv|C_{kk}+C_{rr}|-\sqrt{2}>0,$ (16)
$\displaystyle\mathcal{B}_{3}$
$\displaystyle\equiv|C_{kk}+C_{nn}|-\sqrt{2}>0,$ (17)
$\displaystyle\mathcal{B}_{4}$
$\displaystyle\equiv|C_{rr}+C_{nn}|-\sqrt{2}>0.$ (18)

We focus on $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ which lead to stronger
bounds among others, since $C_{nn}<0$ and the coefficients $C_{rr}$ and
$C_{kk}$ are mostly positive in the $(m_{t\bar{t}},\cos\theta_{CM})$ plane, as
we explicitly show in the following section. The coefficients $C_{ij}$ can be
measured with the distributions from the top and anti-top quark decay products
$a$ and $b$

$\displaystyle\frac{1}{\sigma}\frac{d^{2}\sigma}{d\cos\theta^{i}_{a}d\cos\bar{\theta}^{j}_{b}}=\frac{1}{4}\left(1+\beta_{a}\beta_{b}C_{ij}\cos\theta^{i}_{a}\cos\bar{\theta}^{j}_{b}\right)\,,$
(19)

where $\theta_{a}^{i}$ is the polar angle for the final state $a$ with respect
to the $i$-th axis in the top quark rest frame and $\bar{\theta}_{b}^{j}$ is
the polar angle for the final state $b$ with respect to the $j$-th axis in the
anti-top rest frame. We can single out the $C_{ij}$ coefficients from Eq. (19)
as

$\displaystyle
C_{ij}=\frac{4}{\beta_{a}\beta_{b}}\frac{N(c_{\theta_{a}^{i}}c_{\bar{\theta}_{b}^{j}}>0)-N(c_{\theta_{a}^{i}}c_{\bar{\theta}_{b}^{j}}<0)}{N(c_{\theta_{a}^{i}}c_{\bar{\theta}_{b}^{j}}>0)+N(c_{\theta_{a}^{i}}c_{\bar{\theta}_{b}^{j}}<0)}\,,$
(20)

where $c_{\theta}\equiv\cos\theta$, $c_{\bar{\theta}}\equiv\cos\bar{\theta}$,
and $N$ denotes the number of events.

Figure 1: Parton level distributions for the coefficients $C_{kk}$ (left),
$C_{rr}$ (center), and $C_{nn}$ (right) in the helicity frame as a function of
($m_{t\bar{t}},\cos\theta_{\text{CM}}$). The coefficients are obtained with
the top and anti-top quark decay products {$\ell^{\pm},\vec{q}_{\text{opt}}$}.
The optimal hadronic direction $\vec{q}_{\text{opt}}$ is defined in Eq. (12).

Figure 2: Entanglement probe $\mathcal{E}$ (left) and CHSH violation
indicators $\mathcal{B}_{1}$ (middle), and $\mathcal{B}_{2}$ (right) in the
$(m_{t\bar{t}},\cos\theta_{\text{CM}})$ plane at parton level, using the
charged lepton $\ell^{\pm}$ and the optimal hadronic direction
$\vec{q}_{\text{opt}}$ defined in Eq. (12).

###  II.5 Spin Correlations, Entanglement, and Bell Inequalities

It is worth highlighting the distinction among the concepts of spin
correlations, entanglement, and Bell inequalities. Spin correlations in top
quark physics have been tested since the Tevatron era Mahlon and Parke (1996,
2010); Aaltonen  _et al._ (2011); Abazov  _et al._ (2011) and continue to be
an important subject of study at the LHC Aaboud  _et al._ (2020); Sirunyan
_et al._ (2019). They provide relevant probes for precision physics and beyond
the Standard Model physics searches Buckley and Goncalves (2016a, b);
Goncalves and Lopez-Val (2016); Gonçalves  _et al._ (2018, 2022); Barman  _et
al._ (2022). The existence of spin correlations is ultimately a reflection of
non-vanishing coefficients $C_{ij}$ in the spin density matrix, Eq. (13).

In contrast, the requirements for entanglement and violation of Bell
inequalities further constrain the spin density matrix. In addition to
requiring nonzero contributions $C_{ij}$, these conditions impose particular
relations among the coefficients $C_{ij}$, as presented in Eq. (14), and Eqs.
(15-18). Regarding the comparison between the conditions for entanglement and
violation of Bell inequalities, it is important to stress that entanglement is
a necessary condition for the violation of Bell inequalities. However, the
reverse is not true. The hierarchy of these conditions on the spin density
matrix can be summarized as follows:

$\displaystyle\text{Spin
correl.}\subseteq\text{Entanglement}\subseteq\text{Bell inequalities
violation}.$ (21)

##  III Parton Level Distributions

In this section, we analyze distributions for angular coefficients,
entanglement indicator, and CHSH violation probe at parton level. The results
obtained here will motivate our boosted top quark analysis presented in Sec.
IV.

We start the parton level study generating a $pp\to t\bar{t}\to\ell^{\pm}\nu
2b2j$ sample at the $\sqrt{s}=14$ TeV LHC, where $\ell^{\pm}=e^{\pm}$ or
$\mu^{\pm}$, using MadGraph5_aMC@NLO Alwall  _et al._ (2014) at leading order
within the Standard Model with NNPDF2.3QED for parton distribution function
Ball  _et al._ (2013). The factorization and renormalization scales are set to
$\mu_{F}=\mu_{R}=\left(\sqrt{m_{t}^{2}+p_{Tt}^{2}}+\sqrt{m_{t}^{2}+p_{T\bar{t}}^{2}}\right)/2$.
We normalize the $t\bar{t}$ production cross section to $\sigma=985.7$ pb, as
calculated with the Top++2.0 program to next-to-next-to-leading order in
perturbative QCD, including soft-gluon resummation to next-to-next-to-leading-
log order Czakon and Mitov (2014); Beneke  _et al._ (2012). No event
selections have been applied during the Monte Carlo (MC) generation. Here, we
present the kinematic regime with maximum sensitivity to these probes and show
that the hadronic polarimeter defined in Sec. II.3 works as an efficient proxy
for the $d$-quark.

Figure 3: Entanglement ($\mathcal{E}$) and Bell inequalities measure
($\mathcal{B}_{1},\mathcal{B}_{2}$) (at parton-level) as a function of
$m_{tt}$ and spacelike separation probability. The latter refers to the
fraction of $t$ and $\bar{t}$ decays that are spacelike-separated. We impose
$|\cos\theta_{\text{CM}}|\leq 0.6$ for the left panel and
$|\cos\theta_{\text{CM}}|\leq 0.2$ for the middle and right panels.

In Fig. 1, we present the coefficients $C_{kk}$ (left), $C_{rr}$ (middle), and
$C_{nn}$ (right) in the helicity frame as a function of
$(m_{t\bar{t}},\cos\theta_{\text{CM}})$, where $\theta_{\text{CM}}$ is the top
quark scattering angle in the $t\bar{t}$ center-of-mass frame. The
coefficients are derived with the top and anti-top decay products $\ell^{\pm}$
and $\vec{q}_{\text{opt}}$, where $\vec{q}_{\text{opt}}$ is the optimal
hadronic direction defined in Eq. (12). The coefficients are obtained with Eq.
(20), assuming perfect reconstruction of the neutrino momentum. The
coefficients for the spin density matrix display large dependencies with the
kinematic regime. In the boosted region (large $m_{t\bar{t}}$) for a small
$\cos\theta_{CM}$, $C_{kk}$ and $C_{rr}$ approach $1$ (represented by dark
blue), while $C_{nn}$ approaches $-1$ (represented by dark orange), which
maximizes $\mathcal{E}$, $\mathcal{B}_{1}$, and $\mathcal{B}_{2}$, see Eqs.
14, 15, 16, 17 and 18.

The numerical variation for the coefficients results in sizable kinematic
correlations in the entanglement indicator $\mathcal{E}$ and CHSH probes
$\mathcal{B}_{1,2}$, as illustrated in Fig. 2. The blue shades represent the
region of $(m_{t\bar{t}},\cos\theta_{\text{CM}})$ space where two top quarks
are entangled ($\mathcal{E}>0$) and violate the CHSH inequalities
($\mathcal{B}_{1,2}>0$). Remarkably, there are two kinematic regions that
display large entanglement $\mathcal{E}>0$, namely the $t\bar{t}$ production
threshold and high transverse momentum ($m_{tt}\gg 2m_{t}$ and
$\theta_{\text{CM}}\sim\pi/2$). In contrast, CHSH inequalities are only
violated in the high transverse momentum configuration.

The study of the boosted top quark regime for the semi-leptonic top pair
process $pp\to t\bar{t}\to\ell\nu 2b2j$ is motivated by three key reasons.
First, the semi-leptonic top pair has an event rate approximately six times
higher than that of the dileptonic process, making it a more effective probe
for the high-energy regime. Second, in the high-energy configuration, boosted
techniques can be employed to tag the hadronic top and efficiently identify
the optimal hadronic direction, as described in the following section. Third,
the two top quarks are expected to be causally disconnected from each other in
the boosted regime Severi  _et al._ (2022). Fig. 3 shows the causal
disconnection, where the histogram heights represent the respective
$(\mathcal{E},\mathcal{B}_{1},\mathcal{B}_{2})$ values as a function of the
$t\bar{t}$ invariant mass $m_{t\bar{t}}$. The colors are mapped by the
spacelike probability, which shows the fraction of $t$ and $\bar{t}$ decays
that are spacelike separated, as a function of $m_{t\bar{t}}$. See Appendix C
for more details. At the top quark pair threshold, a large fraction of the
produced $t\bar{t}$ pairs are timelike separated; however, in the high-energy
regime, spacelike separated top quark pairs are prevalent. For example, for
$m_{t\bar{t}}>1$ TeV, approximately 95% of the events display spacelike
separated top quark pairs.

##  IV Analysis

Figure 4: Differential angular distributions with respect to
$\cos\theta^{i}_{\ell}\cos\bar{\theta}^{i}_{q_{\text{opt}}}$ for $i=k$ (left),
$i=r$ (middle) and $i=n$ (right). We display the parton level correlations
between the charged lepton and down-type quark (black dashed), and charge
lepton with optimal direction (black solid). The detector level (red dashed)
and unfolded (red solid) results for the correlations between charged lepton
and optimal hadronic direction are also presented. We apply the selections
$m_{t\bar{t}}\geq 800$ GeV and $|\cos\theta_{\text{CM}}|\leq 0.6$.

In this section, we derive the sensitivity for the entanglement probe
$\mathcal{E}$ and CHSH violation indicators $\mathcal{B}_{1,2}$ at the HL-LHC.
We focus on the boosted regime, using the semi-leptonic top pair final state.
Parton-level events are further processed, accounting for parton shower and
hadronization effects using PYTHIA8 Sjostrand  _et al._ (2008). We simulate
the detector effects for all jets according to ATLAS parameterization The
(2013), and the missing transverse momentum $\not{\vec{p}}_{T}$ by Gaussian
smearing with resolution $\sigma=-0.07+2.92/|\not{\vec{p}}_{T}|$ de Favereau
_et al._ (2014). We begin our analysis by requiring one isolated charged
lepton $\ell^{\pm}=\\{e^{\pm},\mu^{\pm}\\}$ with $p_{T\ell}>10$ GeV and
$|\eta_{\ell}|<3$. We then impose the missing transverse momentum cut,
$\not{E}_{T}=|\not{\vec{p}}_{T}|>30$ GeV and the minimum invariant mass
$m_{t\bar{t}}>600$ GeV. The goal here is to find four momenta of the hadronic
and leptonic top quarks and reconstruct the spin correlation matrix $C_{ij}$,
whose combinations would determine the entanglement probe and the Bell
inequalities measurements.

For the hadronic part of the event, instead of proceeding with the traditional
resolved semileptonic $t\bar{t}$ analysis, we take advantage of the high top
tagging efficiency in the boosted regime. We start by reconstructing jets with
the Cambridge-Aachen algorithm with $R=1.5$ Cacciari  _et al._ (2012),
demanding at least one fat jet with $p_{TJ}>150$ GeV and $|\eta_{J}|<3$. We
require that one of the fat jets to be tagged with the HEPTopTagger Plehn  _et
al._ (2010). To guarantee a robust reconstruction, the tagged top jet should
satisfy the top mass constraint 130 GeV$<m_{\text{top-jet}}<215$ GeV. The high
transverse momentum results in a large top tagging efficiency and a small fake
rate for this algorithm. Furthermore, we observe that the boosted top quark
regime dovetails nicely with both entanglement and CHSH probes, respectively
$\mathcal{E}$ and $\mathcal{B}_{1,2}$, which are enhanced at high energy
scales and display spacelike separated top pairs, as shown in Figs. 2 and 3.

After tagging the hadronic top, we require one of its subjets to be $b$-tagged
with efficiency $\epsilon_{b}=70\%$ and rejection factor of 300 (7) for light
jets (charm-quark jets) Aad  _et al._ (2023a). We tailored the HEPTopTagger’s
$b$-subjet identification process by incorporating machine learning
algorithms. Although the standard HEPTopTagger approach mostly depends on the
mass relationships among the three subjets $(m_{12},m_{13},m_{23})$ Plehn  _et
al._ (2010), our custom version for the $b$-jet identification employs
additional machine learning techniques. Once three subjets within the top jet
are identified using the top tagging algorithm, we run the Lorentz Boost
Network (LBN) Erdmann  _et al._ (2019) to determine which one of the three
subjects is the $b$-jet candidate. The network takes the 4-momenta of the
three subjets as inputs ($p_{T}$-ordered), and outputs a categorical label (1
of 3). We used three hidden layers, with the LBN-specific hyperparameter “M”
(the number of hypothetical particles or the number of rest frames) set to
$12$. The network forms $12$ linear combinations of input particles and boosts
them to different frames, then the features of the boosted particles are fed
into a simple DNN. The set of features of the boosted particles is selected to
be (energy, transverse momentum, $\eta,~{}\phi$, and mass) of each particle,
as well as the cosine of the angle between each pair of particles. The network
is trained on subjets identified by the standard HEPTopTagger algorithm. The
efficiency and purity of the LBN with $b$-tagging criteria (checking the
$B$-hadrons inside the subjet) is 75% and 96%, respectively, while they are
68% and 94% when using the default HEPTopTagger. Hence, the main advantage of
using HEPTopTagger+LBN would be the higher efficiency when compared to the
standard top tagging algorithm.

After identifying the $b$-tagged subjet candidate, one can select a proxy for
the down-type jet candidate from the remaining two subjets. One common
strategy is to choose the softer of the two light subjets in the rest frame of
the top jet. This initial choice provides a spin analyzing power of about
$\beta_{\text{soft}}\sim 0.5$ Jezabek (1994); Barman  _et al._ (2022).
However, the spin analyzing power can be improved by selecting an optimal
hadronic polarimeter, which results in $\beta_{\text{opt}}\approx 0.64$
Tweedie (2014). We follow the latter approach. See Sec. II.3 for further
details.

Since we have only one heavy hadronic particle in our signal event, namely the
hadronic top quark, we study the remaining event reconstruction using a
smaller jet radius. This suppresses the underlying event contamination. Thus,
we remove all the hadronic activity associated with the successfully top-
tagged jet and re-cluster the remaining hadronic activity by applying the
anti-$k_{T}$ jet algorithm with radius $R=0.4$. Jets are defined with
$p_{Tj}>30$ GeV and $|\eta_{j}|<3$. We demand a $b$-tagged jet in the leptonic
top decay, assuming a $b$-tagging efficiency of 85% and 1% mistag rate for
light jets. This large $b$-tagging performance is in agreement with the HL-LHC
projections performed by ATLAS CER (2017), which account for the new central
tracking systems that will be in operation at the HL-LHC. The dominant
backgrounds are $W+{\rm jets}$ and single top production $tW$ followed by
other subleading contributions from $t\bar{t}V$, $t\bar{t}h$, $Z+{\rm jets}$,
and diboson production. With a relatively high invariant mass cut and the
signal selection criteria, ATLAS collaboration finds that these backgrounds
are as small as $\sim 4\%$ of $t\bar{t}$ production Aad  _et al._ (2022). With
hadronic top tagging, the background will be even further suppressed in our
analysis framework, resulting in approximately 2% of the signal rate. This
strategy, in particular, effectively mitigates the dominant background
contribution from $W+{\rm jets}$. Refer to Appendix A for more comprehensive
information regarding the background estimation.

A simple approach to find the momentum of the leptonic top quark would be via
analytic reconstruction. One can use the on-shell condition of top quark or
$W$ boson to fix the $z$-component of the neutrino momentum. However,
considering the finite width effects and the poor detector resolution for the
missing transverse momentum, it frequently leads to complex solution. To
overcome these problems, one can scan over the three momentum of the neutrino
until the on-shell conditions are met. Alternatively, one can attempt to find
all three components of the missing neutrino via

$\displaystyle\chi^{2}=\dfrac{(m_{t}-m_{b\ell\nu})^{2}}{\sigma_{t}^{2}}+\dfrac{(m_{W}-m_{\ell\nu})^{2}}{\sigma_{W}^{2}}+\dfrac{(\not{\vec{p}}_{T}-\vec{p}_{\nu
T})^{2}}{\sigma_{MET}^{2}},$ (22)

where $\sigma_{t}$, $\sigma_{W}$, and $\sigma_{MET}$ are the mass resolution
of top quark, $W$ boson, and the uncertainty in the missing transverse
momentum measurement, respectively. While the mass fits are very good, as
required by construction, we find that the shape of angular distributions with
the top quark momentum obtained via $\chi^{2}$ can significantly differ from
the expected shape of parton-level distributions.

To overcome this problem, in the present study, we used the LBN with the
following input features: the 4-momenta of the lepton and $b$-jet associated
with leptonic top, as well as the $x$ and $y$ components of
$\not{\vec{p}}_{T}$ as an artificial 4-vector, with the energy and $z$
component set to zero. We employed the same LBN setup as in the hadronic top
case, with the output layer consisting of three dimensions, corresponding to
the three-momentum component of the neutrino.

To train the network, we developed a custom loss function defined as
$L=\text{Mean squared
loss}+\lambda_{1}(m_{b\ell\nu}-m_{t})^{2}+\lambda_{2}(m_{\ell\nu}-m_{W})^{2}$,
where $m_{b\ell\nu}$ and $m_{\ell\nu}$ represent the invariant masses of the
reconstructed top quark and $W$ boson, and $m_{t}$, $m_{W}$ are their true
masses. The hyperparameters $\lambda_{1}$ and $\lambda_{2}$ were tuned to 0.8
and 0.4, respectively. We find that the LBN method outperforms the $\chi^{2}$
approach in terms of correct angular distributions.

Once we find the momenta of the hadronic and leptonic top quarks, we compute
the double angular distributions
${d^{2}\sigma}/{d\cos\theta^{i}_{\ell}d\cos\bar{\theta}^{j}_{\rm
q_{\text{opt}}}}$ in the helicity basis as in Eq. (19). Fig. 4 shows the
differential angular distributions with respect to
$\cos\theta^{i}_{\ell}\cos\bar{\theta}^{i}_{q_{\text{opt}}}$ for $i=k$ (left),
$i=n$ (middle), and $i=r$ (right). Parton-level distributions with the charged
lepton and down-type quark (dashed) and optimal hadronic direction (solid) are
shown in black, while detector-level distributions with the optimal direction
(dashed) and unfolded distributions (solid) are shown in red, respectively. We
observe that the profiles of parton-level distributions are very well retained
at each stage of the analysis.

As shown in Fig. 4, the overall angular distributions do not change
significantly in these different stages of the simulation. However, the
correlation coefficients are very sensitive to small effects in the angular
distributions (due to cuts, detector effects, imprecise estimation of the
optimal hadronic direction, etc.), affecting the entanglement and Bell
inequalities estimations. To reach a more robust quantitative conclusion, we
perform unfolding with the TSVDUnfold package Höcker and Kartvelishvili
(1996). It uses Singular Value Decomposition (SVD) of the response matrix.
Details of the unfolding algorithm and adopted parameters are described in
Appendix B.

Figure 5: Entanglement indicator as a function of the luminosity. The yellow
and blue areas represent the regions for $\mathcal{E}\pm 2\sigma$ and
$\mathcal{E}\pm 5\sigma$, respectively.

Figure 6: Bell indicators $\mathcal{B}_{1}$ (left panel) and $\mathcal{B}_{2}$
(right panel) as a function of the luminosity. The yellow and blue areas
represent the regions for $\mathcal{B}_{i}\pm 2\sigma$ and $\mathcal{B}_{i}\pm
5\sigma$ ($i=1,2$), respectively.

##  V Results

To maximize the entanglement probe and CHSH violation indicator, we focus on
the highly relativistic regime of ($m_{t\bar{t}}$, $\cos\theta_{\text{CM}}$)
instead of the entire phase space. In the case of entanglement, we choose the
region defined by $m_{t\bar{t}}\geq 800$ GeV and $|\cos\theta_{\text{CM}}|\leq
0.6$, which leads to the cross section of $382.8$ fb. We use approximately
$7.8$ M events for training (corresponding to $20$ ab-1 luminosity). The
unfolded result for the entanglement probe $\mathcal{E}$ is shown in Fig. 5 as
a function of the luminosity. We have randomly selected the events from the
test data set corresponding to the adopted luminosity. The yellow and blue
regions represent the regions for $\mathcal{E}\pm 2\sigma$ and $\mathcal{E}\pm
5\sigma$, respectively, where $\sigma$ is the error estimated using TSVDUnfold
Höcker and Kartvelishvili (1996). Remarkably, Fig. 5 indicates that ATLAS and
CMS collaborations have already accumulated sufficient data to observe
entanglement between two top quarks, using the semi-leptonic top pair final
state in the boosted regime.

For the Bell inequalities, we impose a more stringent phase space restriction,
$m_{t\bar{t}}\geq 1.3$ TeV and $|\cos\theta_{\text{CM}}|\leq 0.2$, reducing
the cross section down to $8.25$ fb.111Refer to Appendix D for a discussion on
loopholes associated with the measurement of Bell inequalities at LHC. We
randomly select the test data corresponding to $3$ ab-1 and use the remaining
data ($288$ K events, corresponding to $35$ ab-1 luminosity) for training.
Fig. 6 displays the unfolded CHSH violation indicators, $\mathcal{B}_{1}$
(left) and $\mathcal{B}_{2}$ (right), as functions of luminosity. Fig. 6
indicates that the 5$\sigma$ observation of the CHSH violation is very
promising with the full HL-LHC luminosity, especially when results from CMS
and ATLAS can be combined.

##  VI Summary

Indicator |  Parton-level |  Unfolded |  Significance $(\mathcal{L}=3$ ab-1)
---|---|---|---
$\mathcal{B}_{1}$ | $0.267\pm 0.023$ | $0.274\pm 0.057$ | $4.8$
$\mathcal{B}_{2}$ | $0.204\pm 0.023$ | $0.272\pm 0.058$ | $4.7$
Table 1: Parton-level and unfolded values of $\mathcal{B}_{1}$ and
$\mathcal{B}_{2}$ at the HL-LHC with $\mathcal{L}=3$ ab-1. The parton-level
uncertainty only accounts for the Monte Carlo error and is derived to obtain a
robust benchmark.

We performed a comprehensive analysis to investigate the feasibility of
detecting quantum entanglement and the violation of Bell inequalities in top
quark pair production at the LHC, with a focus on the semi-leptonic top pair
final state. This study has significant implications, as it offers an
excellent opportunity to test these quantum correlations at high-energy
scales.

To ensure a high probability of spacelike separation between the two top
quarks, a large invariant mass of the $t\bar{t}$ system is required. Boosted
top tagging is very well motivated for such circumstances. By using the semi-
leptonic top pair final state, we take advantage of its six times higher event
rate compared to the dileptonic top pair final state, explored in previous
studies Afik and de Nova (2021); Fabbrichesi  _et al._ (2021); Severi  _et
al._ (2022); Aguilar-Saavedra and Casas (2022).

We showed that the final state with lepton and optimal hadronic direction, in
the semi-leptonic channel, can effectively probe the top quark pair spin
density matrix, prompting access to the entanglement probe and the CHSH
violation indicator. For more realistic simulation, we have included parton-
shower, hadronization and detector effects. We have shown that HEPTopTagger
and Lorentz Boost Network provide excellent reconstruction of the hadronic and
leptonic top quarks.

Our final results indicate that the detection of entanglement for the semi-
leptonic top pairs is straightforward, in agreement with existing studies in
the literature that use dileptonic tops.222Following the submission of our
work to arXiv, ATLAS reported the detection of quantum entanglement with
dileptonic tops, focusing on the threshold regime Aad  _et al._ (2023b). The
experimental study of the semi-leptonic boosted tops, initially proposed in
the present article, is still pending as of the submission of this paper.
ATLAS and CMS collaborations should be able to observe entanglement in semi-
leptonic tops with their current data sets. However, probing the violation of
Bell inequalities proves more challenging, given its more restrictive nature
compared to entanglement. Nevertheless, we establish the feasibility of such
an observation at a significance level exceeding 4$\sigma$ at the HL-LHC
(refer to Table 1). These results can be statistically boosted even further by
a combination of ATLAS and CMS datasets and accounting as well for the
dileptonic top quark pair final state.

## Acknowledgements

The authors thank Alexander Khanov, Mayukh Lahiri, Myeonghun Park, Joonwoo
Bae, and John Ralston for useful discussions. DG and AN thank the U.S.
Department of Energy for the financial support, under grant number DE-SC
0016013. ZD is supported in part by the US Department of Energy under Grant
no. DE-SC0024407. KK is supported in part by the US DOE under Award No DE-
SC0024407. This investigation was supported in part by the University of
Kansas General Research Fund allocation 2151077. Some computing for this
project was performed at the High Performance Computing Center at Oklahoma
State University, supported in part through the National Science Foundation
grant OAC-1531128.

##  Appendix A Backgrounds

Table 2 displays the cross sections for both signal and background after the
complete reconstruction of the hadronic and leptonic top quarks, as described
in Sec. IV. The cross sections are obtained in two stages: after additional
cuts used for entanglement study ($m_{t\bar{t}}>800$ GeV and $|\cos\theta_{\rm
CM}|<0.6$) in the first row, and Bell inequalities ($m_{t\bar{t}}>1.3$ TeV and
$|\cos\theta_{\rm CM}|<0.2$) in the second row. In both cases, backgrounds
display subleading effects, resulting to approximately $2\%$ of the signal
rate. Notably, the imposition of the hadronic top tagging and the requirement
of two $b$-tags play crucial roles in suppressing the $W+$jets and $tW$
backgrounds.

Cuts |  Signal |  $t\bar{t}V$ |  $tW$ |  $W$+jets
---|---|---|---|---
Entanglement | 382.8 | 0.74 | 7.27 | 1.03
Bell inequalities | 8.25 | 0.02 | 0.14 | 0.02
Table 2: Cross sections for signal and backgrounds in $fb$ at the HL-LHC. $V$
includes $Z$, $W^{\pm}$ and $h$. Entanglement and Bell inequalities refer to
the kinematic selections imposed on the respective analyses.

##  Appendix B Unfolding Algorithm

We start the unfolding procedure with the MC truth distribution $x^{init}$ and
the corresponding MC measured distribution $b^{init}$. The response matrix
$A_{ij}$ is the probability that an event generated in the true bin $j$ will
be found in the measured bin $i$. The response matrix is then used to solve
the system $Ax=b$, where $b$ is the measured distribution (the distribution we
want to unfold). Before using SVD of the response matrix, a suitable rescaling
is performed and a new distribution $w_{i}=x_{i}/x_{i}^{init}$ is introduced
so that $A$ can be filled with the number of events rather than probabilities.
An additional rescaling may be needed if the covariance matrix of $b$ is not
diagonal, which is not the case in our analysis.

Then we perform the ridge regression (also known as Tikhonov regularization)
by minimizing

$\displaystyle(\tilde{A}w-\tilde{b})^{T}(\tilde{A}w-\tilde{b})-\tau(Cw)^{T}(Cw),$
(23)

where $C$ is a constant matrix that minimizes the curvature of $w$. Using the
SVD of $\tilde{A}C^{-1}=USV^{T}$ and introducing $d=U^{T}\tilde{b}$ and
$z=V^{T}C^{-1}w$, the solution becomes $z_{i}=d_{i}/s_{i}$, where $s_{i}$ is
the $i$-th singular value of $\tilde{A}C^{-1}$. The regularization parameter
$\tau$ is related to the singular values of $\tilde{A}C^{-1}$. We find the
best choice of $\tau$ from the $\log d_{i}$ distribution, which is the
coefficient of the decomposition of $\tilde{b}$ in a basis defined by the
$i$-th column of $U$. Usually, only the first $m$ terms are statistically
significant. The best choice of this parameter is then $\tau=s_{m}^{2}$,
_i.e._ , the square of the $m$-th singular value Höcker and Kartvelishvili
(1996).

To compute the $C_{ij}$ coefficients defined in Eq. (20) we define the
asymmetries as

$\displaystyle
A=\frac{\sum_{\text{bins}>0}\,N_{i}-\sum_{\text{bins}<0}\,N_{i}}{\sum_{\text{all
bins}}\,N_{i}}\,,$ (24)

where $N_{i}$ is the content of bin $i$ and $\text{bins}>0$ ($\text{bins}<0$)
denotes the number of bins with
$\cos\theta^{i}_{\ell}\cos\bar{\theta}^{i}_{q_{\text{opt}}}>0$
($\cos\theta^{i}_{\ell}\cos\bar{\theta}^{i}_{q_{\text{opt}}}<0$). Using error
propagation we compute the error in the asymmetry as

$\displaystyle\sigma_{A}=\sqrt{\sum_{i,j}g_{i}g_{j}V_{ij}}\,,$ (25)

where $V$ is the covariance matrix of the unfolded
$\cos\theta^{i}_{\ell}\cos\bar{\theta}^{i}_{q_{\text{opt}}}$ distribution and
$g_{i}$ is a factor defined below

$\displaystyle
g_{i}=\begin{dcases}\frac{-2\sum_{\text{bins}>0}\,N_{i}}{(\sum_{\text{all
bins}}\,N_{i})^{2}}\,,&i\in\mbox{bins\quad}\\!\\!\\!\\!\\!<0\\\
\frac{2\sum_{\text{bins}<0}\,N_{i}}{(\sum_{\text{all
bins}}\,N_{i})^{2}}\,,&i\in\mbox{bins\quad}\\!\\!\\!\\!\\!>0\\\
\end{dcases}\,.$ (26)

Note that Eq. (25) accounts for all the bin-to-bin correlations that the
unfolding might have introduced, as it uses the full covariance matrix of the
unfolded distribution.

In Fig. 4, we illustrate the robustness of the unfolding algorithm, applying
it to the distributions
$\cos\theta^{k}_{\ell}\cos\bar{\theta}^{k}_{q_{\text{opt}}}$ and
$\cos\theta^{r}_{\ell}\cos\bar{\theta}^{r}_{q_{\text{opt}}}$ with 22 bins and
$m=6$, as well as $\cos\theta^{n}_{\ell}\cos\bar{\theta}^{n}_{q_{\text{opt}}}$
with 22 bins and $m=7$ in the signal region for entanglement. However, for the
final results, we choose 2 bins and $m=2$ to unfold all distributions, as we
are interested only in the regions with
$\cos\theta^{i}_{\ell}\cos\bar{\theta}^{i}_{q_{\text{opt}}}>0$ and
$\cos\theta^{i}_{\ell}\cos\bar{\theta}^{i}_{q_{\text{opt}}}<0$. Our analysis
accounts for both the statistical error stemming from the measured
distribution and the MC error in the response matrix, calculated with 100
pseudo-experiments.

##  Appendix C Spacelike Separation

To ensure that the top quarks are spacelike separated when the spin
information is passed on to the decay products, they have to be relatively
apart from one another so that no information can be passed in between when
they decay Severi  _et al._ (2022). In the center of mass frame of $t\bar{t}$,
the distance between their decay location is given by $(t_{1}+t_{2})v$, where
$t_{1}$ and $t_{2}$ are the decay times of the top and anti-top quarks, $v$ is
the magnitude of their velocity. The maximum distance that information can
travel between their decay times is given by $|t_{1}-t_{2}|c$, where $c$ is
the speed of the light. Thus, spacelike separation requires the following
inequality:

$\dfrac{|t_{1}-t_{2}|}{t_{1}+t_{2}}<\frac{v}{c}=\sqrt{1-\frac{4m_{t}^{2}}{m_{t\bar{t}}^{2}}}.$
(27)

Counting the fraction of events where this inequality holds tells us how often
the tops are spacelike separated.

##  Appendix D Loopholes

Measurement of a loophole-free Bell violation is a more intricate task
compared to probing entanglement. The LHC, not originally designed for testing
Bell inequalities, can only explore weak violations. It is noteworthy that
preparing loophole-free setups took decades, with the first measurement
occurring around 2015 Hensen  _et al._ (2015); Giustina  _et al._ (2015);
Shalm  _et al._ (2015), recognized by the 2022 Physics Nobel Prize. At the
LHC, a significant challenge is the inability to apply external intervention
for freely choosing the orientation of the measurement axis for the top pair,
which poses a challenge to the free-will loophole. Additionally, achieving
causal independence of the decays is only possible at a statistical level
Severi  _et al._ (2022). Lastly, the LHC analyses focus on a subset of events,
introducing the detection loophole.

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