# Quantum information with top quarks in QCD

Yoav Afik<EMAIL_ADDRESS>Experimental Physics Department, CERN, 1211
Geneva, Switzerland  Juan Ramón Muñoz de Nova<EMAIL_ADDRESS>Departamento
de Física de Materiales, Universidad Complutense de Madrid, E-28040 Madrid,
Spain

###### Abstract

Top quarks represent unique high-energy systems since their spin correlations
can be measured, thus allowing to study fundamental aspects of quantum
mechanics with qubits at high-energy colliders. We present here the general
framework of the quantum state of a top-antitop ($t\bar{t}$) quark pair
produced through quantum chromodynamics (QCD) in a high-energy collider. We
argue that, in general, the total quantum state that can be probed in a
collider is given in terms of the production spin density matrix, which
necessarily gives rise to a mixed state. We compute the quantum state of a
$t\bar{t}$ pair produced from the most elementary QCD processes, finding the
presence of entanglement and CHSH violation in different regions of phase
space. We show that any realistic hadronic production of a $t\bar{t}$ pair is
a statistical mixture of these elementary QCD processes. We focus on the
experimentally relevant cases of proton-proton and proton-antiproton
collisions, performed at the LHC and the Tevatron, analyzing the dependence of
the quantum state with the energy of the collisions. We provide experimental
observables for entanglement and CHSH-violation signatures. At the LHC, these
signatures are given by the measurement of a single observable, which in the
case of entanglement represents the violation of a Cauchy-Schwarz inequality.
We extend the validity of the quantum tomography protocol for the $t\bar{t}$
pair proposed in the literature to more general quantum states, and for any
production mechanism. Finally, we argue that a CHSH violation measured in a
collider is only a weak form of violation of Bell’s theorem, necessarily
containing a number of loopholes.

##  1 Introduction

The Standard Model of particle physics is a relativistic quantum field theory,
based on special relativity and quantum mechanics. Therefore, it allows to
study fundamental properties of quantum mechanics in a genuinely relativistic
environment, at the frontier of the known Physics. However, even though the
Standard Model is inherently a quantum theory, observing basic quantum
phenomena in a high-energy collider can become quite challenging due to the
nature of the measurement process.

Some light on this problem can be shed by the field of quantum information,
where most of the foundations of quantum mechanics find a direct application.
There, a characteristic signature of quantumness is provided by the existence
of correlations that cannot be accounted by a classical probability theory,
arising due to the intrinsic wave nature of quantum mechanics. Here emerges
the concept of entanglement [1, 2, 3], perhaps the most genuine feature of
quantum mechanics. Entanglement plays a key role in quantum technologies like
quantum computation, cryptography, metrology and teleportation [4, 5, 6, 7, 8,
9, 10]. In particular, the study of entanglement in high-energy setups is of
fundamental interest since entanglement is expected to be critically affected
by relativistic effects [11, 12, 13, 14, 15, 16]. A number of works have
already addressed the role of entanglement in the context of high-energy
physics [17, 18, 19, 20, 21, 22].

The simplest system which can exhibit entanglement is that formed by two
qubits. In high-energy physics, an interesting realization of a two-qubit
system is a pair of decaying spin-1/2 particles, since their spin correlations
can be measured from the kinematical distribution of their decay products. The
best candidate from the Standard Model to carry out such a measurement is the
top quark, the most massive fundamental particle known to exist
($m_{t}c^{2}\approx 173\leavevmode\nobreak\ \textrm{GeV}$), first discovered
by the D0 and CDF collaborations at the Tevatron in the Fermilab in 1995 [23,
24]. Top quarks are typically produced in top-antitop ($t\bar{t}$) pairs
which, due to their high mass, quickly decay well before any other process can
affect their spin correlations. As a result, the spins of the $t\bar{t}$ pair
are correlated with the kinematical distribution of the decay products, from
where the original top spin quantum state can be reconstructed.

This feature has rendered top spin correlations a rich subject of study within
high-energy physics [25, 26, 27, 28, 29, 30, 31, 32]. Indeed, spin
correlations between $t\bar{t}$ pairs have already been measured by the D0 and
CDF collaborations at the Tevatron with proton-antiproton ($p\bar{p}$)
collisions [33, 34, 35], and by the ATLAS and CMS collaborations at the Large
Hadron Collider (LHC) with proton-proton ($pp$) collisions [36, 37, 38, 39,
40].

However, despite the extensive literature on the topic, only a very recent
work has studied the entanglement between $t\bar{t}$ pairs [41]. Specifically,
it was proposed that the entanglement of a $t\bar{t}$ pair can be detected at
the LHC with high statistical significance using the current data recorded
during Run 2. This would represent the first measurement ever of entanglement
in a quark pair, and the highest-energy observation of entanglement so far.
Entanglement has been also recently postulated as potentially sensitive to
effects of New Physics beyond the Standard Model [42, 43]. An experimental
protocol for the quantum tomography of a $t\bar{t}$ pair was also developed in
Ref. [41], a canonical technique in quantum information but novel in the high-
energy context. In parallel, some recent works have also addressed the
possibility of measuring the violation of Bell inequalities with $t\bar{t}$
pairs [44, 45, 46], and also with $W^{+}W^{-}$ pairs [47], which are massive
spin-1 bosons. Even the measurement of such a basic quantum phenomenon as
quantum interference can become non-trivial in a high-energy collider [48].
Apart from the intrinsic interest of studying foundational aspects of quantum
mechanics at the frontier of the known Physics, all these works are paving the
way to also use high-energy colliders for the study of quantum information
theory. Due to their genuine relativistic behavior, the exotic character of
the symmetries and interactions involved, as well as their fundamental nature,
high-energy colliders are extremely attractive systems for these purposes.

Here, we provide the general formalism of the quantum state of a $t\bar{t}$
pair created through quantum chromodynamics (QCD) within a genuine quantum
information approach. We discuss that, in general, the total quantum state
that can be probed in a scattering experiment in a collider is given in terms
of the so-called production spin density matrix [26], which is necessarily a
mixed state.

For the specific case of a $t\bar{t}$ pair, we analyze in detail its quantum
state for the most elementary QCD production processes: light quark-antiquark
($q\bar{q}$) or gluon-gluon ($gg$) interactions. We extend the entanglement
analysis of Ref. [41] to also include the violation of Bell inequalities,
finding that both quantum phenomena emerge in certain regions of phase space
for both $q\bar{q}$ and $gg$ processes. Interestingly, the entanglement
structure can be qualitatively understood in terms of basic conservation laws,
without the need of knowledge of the particular details of QCD interactions.

We show that any realistic QCD mechanism of $t\bar{t}$ production can be seen
as a statistical mixture of the previous building blocks. In particular, we
focus on the case of $pp$ and $p\bar{p}$ collisions, corresponding to the LHC
and the Tevatron, respectively. We study the dependence of the quantum state
on the energy of the collisions, extending the current phenomenological
literature restricted to Run 2 of LHC [41, 44, 45, 46]. We find that, for
sufficiently high energies, both types of collisions converge to the same
state due to the dominance of $gg$ processes.

In a high-energy collider, spin correlations are measured from the fit of the
differential cross-section describing the angular distribution of the decay
products [29, 49]. We propose realistic experimental observables for the
characterization of the $t\bar{t}$ quantum state by integrating the signal in
certain regions of phase space. At the LHC, it was shown that an entanglement
signature can be obtained from the measurement of one single magnitude,
proportional to the trace of the spin correlation matrix [41]. We prove in
this work that the violation of Bell inequality can be also signaled from the
measurement of just one parameter, the transverse spin correlations. Moreover,
we carry out a study of the energy dependence of both signatures. Regarding
the Tevatron, we find that an entanglement signature is provided by direct
integration over the whole phase space.

Finally, we analyze in detail the conceptual significance of the experimental
implementation of these techniques. We show explicitly that an entanglement
measurement at the LHC represents the violation of a Cauchy-Schwarz
inequality. In contrast, we do not expect a statistically significant
observation of entanglement at the Tevatron. We argue that the quantum
tomography protocol developed in Ref. [41] can be extended to more general
quantum states, and to any $t\bar{t}$ production process. We discuss that, due
to the nature of the detection process, only weak violations of Bell
inequalities can be measured in a high-energy collider, since some loopholes,
like those related to the free-will or to the detection efficiency, cannot be
closed.

Remarkably, while the probability and the spin density matrix of each
$t\bar{t}$ production process are computed by the high-energy theory, once
both are given, we are simply left with a typical problem in quantum
information involving the convex sum of two-qubit quantum states, where the
usual techniques of the field can be applied. This important observation is
translated into the article presentation, fully developed within a quantum
information language, aimed at making it easily understandable by the physics
community outside the high-energy field.

The paper is arranged as follows. Section 2 discusses in detail the general
framework upon we build the results of this work, presenting the basic tools
of two-qubit systems used throughout this work along with a general discussion
about quantum states in colliders and relativistic particle-antiparticle
production. Section 3 studies the $t\bar{t}$ quantum state in elementary QCD
processes. Section 4 extends the previous results to more realistic processes
occurring in actual colliders, analyzing in detail the energy dependence.
Section 5 translates these ideas into relevant experimental observables.
Section 6 provides some technical remarks about the experimental
implementation of the discussed quantum information techniques. Finally,
Section 7 summarizes the main conclusions, and discusses future perspectives.
Technical details are given in the Appendices.

##  2 General formalism

###  2.1 Two-qubit systems

Quantum states are represented in general by a density matrix $\rho$, a
Hermitian non-negative operator with unit trace in a certain Hilbert space
$\mathcal{H}$, $\rm{tr}(\rho)=1$. These conditions imply that the number of
real parameters characterizing $\rho$ for $\dim\mathcal{H}=N$ is $N^{2}-1$.
Expectation values for observables $O$ are computed by taking the product
trace, $\braket{O}=\rm{tr}(O\rho)$.

The most simple example of density matrix is provided by a qubit, that is, a
two-level quantum system. The density matrix of a qubit takes the simple form

$\rho=\frac{I_{2}+\sum_{i}B_{i}\sigma^{i}}{2}$ (1)

with $I_{n}$ the $n\times n$ identity matrix and $\sigma^{i}$, $i=1,2,3$, the
usual Pauli matrices. A physical (i.e., non-negative) density matrix $\rho$ is
described by Bloch vectors $|\mathbf{B}|\leq 1$, with pure states given by
unit vectors saturating the inequality. The $3$ coefficients $B_{i}$
completely determine the quantum state of the system. For a spin-1/2 particle,
they represent the spin polarization, $B_{i}=\braket{\sigma^{i}}$. An
alternative description of $\rho$ can be provided in terms of angular momentum
coherent states $\ket{\mathbf{\hat{n}}}$ through the $P$-representation (see
Appendix A for a summary of its main properties):

$\rho=\int\mathrm{d}\Omega\leavevmode\nobreak\
P(\mathbf{\hat{n}})\ket{\mathbf{\hat{n}}}\bra{\mathbf{\hat{n}}},\leavevmode\nobreak\
\int\mathrm{d}\Omega\leavevmode\nobreak\ P(\mathbf{\hat{n}})=1$ (2)

where the angular momentum coherent states $\ket{\mathbf{\hat{n}}}$ satisfy
$\mathbf{\hat{n}}\cdot\mathbf{\sigma}\ket{\mathbf{\hat{n}}}=\ket{\mathbf{\hat{n}}}$,
and $\Omega$ is the solid angle associated to the unit vector
$\mathbf{\hat{n}}$. The function $P(\mathbf{\hat{n}})$ is the qubit analogue
of the celebrated Glauber-Sudarshan $P$-function in quantum optics [50]. It is
easily seen that $P(\mathbf{n})$ can be always chosen as non-negative for any
density matrix $\rho$ describing a qubit.

More intriguing quantum states arise in bipartite Hilbert spaces
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ composed of subsystems
$A,B$. A quantum state is called separable iff it can be written as a convex
sum of product states

$\rho=\sum_{n}p_{n}\rho^{A}_{n}\otimes\rho^{B}_{n},\leavevmode\nobreak\
\sum_{n}p_{n}=1,$ (3)

with $p_{n}\geq 0$. An entangled state is defined as a non-separable state.

To illustrate this concept, we consider the case of a pair of qubits, whose
density matrix $\rho$ can be decomposed as

$\rho=\frac{I_{4}+\sum_{i}\left(B^{+}_{i}\sigma^{i}\otimes
I_{2}+B^{-}_{i}I_{2}\otimes\sigma^{i}\right)+\sum_{i,j}C_{ij}\sigma^{i}\otimes\sigma^{j}}{4}$
(4)

Now, the quantum state of the system is determined by $15$ parameters
$B^{\pm}_{i},C_{ij}$, which for spin-1/2 particles are their spin
polarizations $B^{+}_{i}=\braket{\sigma^{i}\otimes I_{2}},\leavevmode\nobreak\
B^{-}_{i}=\braket{I_{2}\otimes\sigma^{i}}$, and their spin correlations
$C_{ij}=\braket{\sigma^{i}\otimes\sigma^{j}}$.

An insightful way to understand entanglement for $2$-qubit systems arises by
considering the $P$-representation

$\rho=\int\mathrm{d}\Omega_{A}\mathrm{d}\Omega_{B}\leavevmode\nobreak\
P(\mathbf{n}_{A},\mathbf{n}_{B})\ket{\mathbf{n}_{A}\mathbf{n}_{B}}\bra{\mathbf{n}_{A}\mathbf{n}_{B}},$
(5)

where
$\ket{\mathbf{n}_{A}\mathbf{n}_{B}}=\ket{\mathbf{n}_{A}}\otimes\ket{\mathbf{n}_{B}}$.
From Eq. (3), it is immediately seen that a state is separable iff it admits a
non-negative $P$-representation. This implies that the spin correlations for a
separable state

$C_{ij}=\braket{\sigma^{i}\otimes\sigma^{j}}=\int\mathrm{d}\Omega_{A}\mathrm{d}\Omega_{B}\leavevmode\nobreak\
P(\mathbf{n}_{A},\mathbf{n}_{B})n^{i}_{A}n^{j}_{B}$ (6)

are purely classical, described by a classical probability distribution. The
same applies to the spin polarizations. Thus, in the language of the
$P$-function, entanglement is equivalent to a $P$-function that necessarily
presents negative values, a genuine distinctive signature of non-classicality
and quantum behavior as well known from quantum optics [50].

In order to signal the presence of entanglement, several theoretical criteria
can be used. Perhaps the most well-known one is the Peres-Horodecki criterion
[51, 52], which simply states that if $\rho$ is separable, taking partial
transpose with respect to the second subsystem

$\rho^{\rm{T_{2}}}=\sum_{n}p_{n}\rho^{a}_{n}\otimes\left(\rho^{b}_{n}\right)^{\rm{T}}$
(7)

also yields a non-negative operator. Hence, a not positive semi-definite
$\rho^{\rm{T_{2}}}$ implies that $\rho$ is entangled. In two-qubit systems,
the Peres-Horodecki criterion is also a necessary condition for entanglement.

A quantitative measurement of the entanglement of two qubits is provided by
the concurrence [53]:

$\mathcal{C}[\rho]\equiv\max(0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4})$
(8)

with $\lambda_{i}$ the eigenvalues of the matrix
$\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$ ordered in decreasing magnitude,
$\tilde{\rho}=(\sigma_{2}\otimes\sigma_{2})\leavevmode\nobreak\
\rho^{*}\leavevmode\nobreak\ (\sigma_{2}\otimes\sigma_{2})$, and $\rho^{*}$
the complex conjugate of $\rho$ in the spin basis of $\sigma_{3}$. The
concurrence is related to the entanglement of formation and satisfies
$0\leq\mathcal{C}[\rho]\leq 1$, where a quantum state is entangled iff
$\mathcal{C}[\rho]>0$. Hence, $\mathcal{C}[\rho]=1$ implies that $\rho$ is
maximally entangled.

The computation of these magnitudes requires full knowledge of the quantum
state. This can be achieved by means of quantum tomography, a technique able
to reconstruct a quantum state from the measurement of a selected set of
observables. For the case of a single qubit, characterized by $3$ parameters,
it is enough to measure the Bloch vector $\mathbf{B}$, which is the spin
polarization for a spin-1/2 particle. For the case of two qubits, the quantum
tomography is performed by measuring the $15$ parameters determining
$\mathbf{B}^{\pm},\mathbf{C}$, which for spin-1/2 particles represent the spin
polarizations and spin correlations, respectively. In actual experiments, an
additional measurement is typically required to ensure the proper
normalization of the density matrix [54]. Nevertheless, in order to detect
entanglement, simpler criteria which require the measurement of just a few
parameters can be formulated. Appendix B contains some general entanglement
criteria specifically developed for $t\bar{t}$ quantum states and used
throughout this work.

An even stronger requirement than entanglement is the violation of Bell-type
inequalities [3]. No local hidden-variable model can give rise to it, implying
a violation of local realism. The Clauser-Horner-Shimoni-Holt (CHSH)
inequality [55] provides a particularly useful form of Bell inequality for
$2\times 2$ systems:

$|C(a_{1},b_{1})-C(a_{1},b_{2})+C(a_{2},b_{1})+C(a_{2},b_{2})|\leq 2$ (9)

where $a_{i},b_{i}$, $i=1,2$ are measurement settings in the Alice, Bob
subsystems $A,B$, and $C(a_{i},b_{j})$ are their correlations. In the case of
spin-1/2 particles, they can be interpreted as measurements of spin
polarizations along certain directions defined by unit vectors
$\mathbf{a}_{i},\mathbf{b}_{j}$,
$C(a_{i},b_{j})=\braket{(\mathbf{a}_{i}\cdot\mathbf{\sigma})\otimes(\mathbf{b}_{j}\cdot\mathbf{\sigma})}$.
Thus, Eq. (9) can be rewritten in vector notation as simply

$|\mathbf{a}^{\textrm{T}}_{1}\mathbf{C}\left(\mathbf{b}_{1}-\mathbf{b}_{2}\right)+\mathbf{a}^{\textrm{T}}_{2}\mathbf{C}\left(\mathbf{b}_{1}+\mathbf{b}_{2}\right)|\leq
2$ (10)

with $\mathbf{C}$ the same correlation matrix of Eq. (4). Any separable state
of the form (3) satisfies Eq. (10). Hence, only entangled states can violate a
CHSH inequality.

In practice, the maximization of the l.h.s. of Eq. (10) is computed from

$\displaystyle\mathcal{B}[\rho]$
$\displaystyle\equiv\max_{\mathbf{a}_{i},\mathbf{b}_{i}}|\mathbf{a}^{\textrm{T}}_{1}\mathbf{C}\left(\mathbf{b}_{1}-\mathbf{b}_{2}\right)+\mathbf{a}^{\textrm{T}}_{2}\mathbf{C}\left(\mathbf{b}_{1}+\mathbf{b}_{2}\right)|$
$\displaystyle=2\sqrt{\mu_{1}+\mu_{2}}$ (11)

where $0\leq\mu_{i}\leq 1$ are the eigenvalues of
$\mathbf{C}^{\textrm{T}}\mathbf{C}$, ordered in decreasing magnitude [56].
Therefore, the CHSH inequality can be violated iff $\mu_{1}+\mu_{2}>1$, where
the maximum possible violation of the CHSH inequality is the Cirel’son bound
$\mathcal{B}[\rho]=2\sqrt{2}$ [57].

###  2.2 Quantum states in colliders

We now study how the physics in real colliders can be described in terms of
quantum states. For illustrative purposes, we first address the simple non-
relativistic scattering of an incident spinless particle with mass $m$ from a
single fixed target, characterized by a potential $V$ (the interested reader
is referred to Ref. [58] for the basics of scattering theory). The process is
determined by the scattering matrix $S$, whose elements in momentum
representation are

$\braket{\mathbf{p}^{\prime}}{S}{\mathbf{p}}=\delta(\mathbf{p}-\mathbf{p}^{\prime})-2\pi
i\delta(E_{\mathbf{p}^{\prime}}-E_{\mathbf{p}})\braket{\mathbf{p}^{\prime}}{T}{\mathbf{p}}$
(12)

where the $T$-matrix satisfies the Lippmann-Schwinger equation $T=V+VG_{0}T$,
with $G_{0}$ the usual retarded free Green’s function, and
$E_{\mathbf{p}}=\mathbf{p}^{2}/2m$ is the kinetic energy. In the Dirac
picture, the wave function of the scattered state resulting from an incident
particle with well-defined momentum $\ket{\mathbf{p}}$ is

$\ket{\Psi}=S\ket{\mathbf{p}}$ (13)

The scattering amplitude
$f(\mathbf{p}\rightarrow\mathbf{p}^{\prime})\equiv-(2\pi)^{2}m\hbar\braket{\mathbf{p}^{\prime}}{T}{\mathbf{p}}$
determines the differential cross-section characterizing the scattering to
some momentum $\mathbf{p}^{\prime}$ with the same energy,

$\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}=|f(\mathbf{p}\rightarrow\mathbf{p}^{\prime})|^{2}$
(14)

$\Omega$ being the solid angle associated to $\mathbf{p}^{\prime}$. Therefore,
the differential cross-section is proportional to the probability of the
process, given by the squared on-shell $T$-matrix element connecting the
initial to the final state.

The scattered state $\ket{\Psi}$ can be also rewritten as a density matrix:

$\displaystyle\rho_{S}$ $\displaystyle=\ket{\Psi}\bra{\Psi}$ (15)
$\displaystyle=\iint\mathrm{d}\mathbf{p}^{\prime}\mathrm{d}\mathbf{p}^{\prime\prime}\ket{\mathbf{p}^{\prime}}\bra{\mathbf{p}^{\prime}}S\ket{\mathbf{p}}\bra{\mathbf{p}}S^{\dagger}\ket{\mathbf{p}^{\prime\prime}}\bra{\mathbf{p}^{\prime\prime}}$

In a real collider, only momentum measurements of the scattered particle can
be performed and thus, not all the information is needed to describe the
quantum state in experiments. Moreover, scattering events along the beam
direction are not typically measured. Therefore, the relevant quantum state
that encodes all the information that can be probed in a collider results from
projecting $\rho_{S}$ onto momentum states $\mathbf{p}^{\prime}\neq\mathbf{p}$
with the operators
$\Pi_{\mathbf{p}^{\prime}}=\ket{\mathbf{p}^{\prime}}\bra{\mathbf{p}^{\prime}}$
as

$\displaystyle\rho$
$\displaystyle=\frac{\int\mathrm{d}\mathbf{p}^{\prime}\leavevmode\nobreak\
\Pi_{\mathbf{p}^{\prime}}\ket{\Psi}\bra{\Psi}\Pi_{\mathbf{p}^{\prime}}}{\int\mathrm{d}\mathbf{p}^{\prime}\leavevmode\nobreak\
|\braket{\Psi}{\mathbf{p}^{\prime}}|^{2}\braket{\mathbf{p}^{\prime}}{\mathbf{p}^{\prime}}}$
(16) $\displaystyle=\frac{\int\mathrm{d}\Omega\leavevmode\nobreak\
|\bra{\mathbf{p}^{\prime}}T\ket{\mathbf{p}}|^{2}\ket{\mathbf{p}^{\prime}}\bra{\mathbf{p}^{\prime}}}{\int\mathrm{d}\Omega\leavevmode\nobreak\
|\bra{\mathbf{p}^{\prime}}T\ket{\mathbf{p}}|^{2}\braket{\mathbf{p}^{\prime}}{\mathbf{p}^{\prime}}}$
$\displaystyle=\frac{1}{\sigma}\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}\frac{\ket{\mathbf{p}^{\prime}}\bra{\mathbf{p}^{\prime}}}{\braket{\mathbf{p}^{\prime}}{\mathbf{p}^{\prime}}}$

where the factor $\braket{\mathbf{p}^{\prime}}{\mathbf{p}^{\prime}}$ in the
denominator ensures proper normalization, $\textrm{tr}\leavevmode\nobreak\
\rho=1$. The quantum state described by the density matrix $\rho$ is thus
mixed, resulting from the incoherent sum of momentum states with a probability
proportional to the differential cross-section, where the total cross-section
$\sigma$ here plays an analogue role to the partition function in statistical
mechanics. As a result, $\rho$ is written only in terms of differential cross-
sections, the observables measured in actual colliders, computed in terms of
$T$-matrix elements. In this way, the expectation value of any momentum
observable $O(\mathbf{P})$ can be obtained from $\rho$ as

$\braket{O}=\textrm{tr}[O\rho]=\frac{1}{\sigma}\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}O(\mathbf{p}^{\prime})$ (17)

For a spin-half particle, things go along the same lines. By noticing that the
spin of the scattered particle is not typically detected in a collider, we now
project using
$\Pi_{\mathbf{p}^{\prime}}=\sum_{\alpha}\ket{\mathbf{p}^{\prime}\alpha}\bra{\mathbf{p}^{\prime}\alpha}$
with $\alpha$ labeling spin indices, finding that the quantum state
characterizing the scattering of an incident particle with momentum
$\mathbf{p}$ and spin $\lambda$ is

$\displaystyle\rho^{\lambda}=\frac{1}{Z}\sum_{\alpha\beta}\int\mathrm{d}\Omega\leavevmode\nobreak\
\bra{\mathbf{p}^{\prime}\alpha}T\ket{\mathbf{p}\lambda}\bra{\mathbf{p}\lambda}T^{\dagger}\ket{\mathbf{p}^{\prime}\beta}$
$\displaystyle\times\frac{\ket{\mathbf{p}^{\prime}\alpha}\bra{\mathbf{p}^{\prime}\beta}}{\braket{\mathbf{p}^{\prime}}{\mathbf{p}^{\prime}}}=\frac{1}{Z}\sum_{\alpha\beta}\int\mathrm{d}\Omega\leavevmode\nobreak\
R^{\lambda}_{\alpha\beta}(\mathbf{p}^{\prime})\frac{\ket{\mathbf{p}^{\prime}\alpha}\bra{\mathbf{p}^{\prime}\beta}}{\braket{\mathbf{p}^{\prime}}{\mathbf{p}^{\prime}}}$
(18)

where we have used
$\braket{\mathbf{p}^{\prime}\alpha}{\mathbf{p}^{\prime}\beta}=\braket{\mathbf{p}^{\prime}}{\mathbf{p}^{\prime}}\delta_{\alpha\beta}$.
In the equation above, the partition function $Z$ is defined to ensure
normalization, and the production spin density matrix, in the following simply
denoted as the $R$-matrix [42], is defined as

$R^{\lambda}_{\alpha\beta}(\mathbf{p}^{\prime})\equiv\bra{\mathbf{p}^{\prime}\alpha}T\ket{\mathbf{p}\lambda}\bra{\mathbf{p}\lambda}T^{\dagger}\ket{\mathbf{p}^{\prime}\beta}$
(19)

The $R$-matrix is not properly normalized, since its trace is proportional to
the differential cross-section of the process,

$\displaystyle\textrm{tr}R^{\lambda}(\mathbf{p}^{\prime})$
$\displaystyle=\sum_{\alpha}R^{\lambda}_{\alpha\alpha}(\mathbf{p}^{\prime})$
(20)
$\displaystyle=\sum_{\alpha}|\bra{\mathbf{p}^{\prime}\alpha}T\ket{\mathbf{p}\lambda}|^{2}\propto\frac{\mathrm{d}\sigma^{\lambda}}{\mathrm{d}\Omega}$

The proportionality factor can be chosen arbitrarily; for the present moment,
we take $R^{\lambda}$ directly proportional to the on-shell $T$-matrix
elements. Thus, the partition function becomes once more proportional to the
total cross-section

$Z=\int\mathrm{d}\Omega\leavevmode\nobreak\
\textrm{tr}\,R^{\lambda}(\mathbf{p}^{\prime})\propto\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\mathrm{d}\sigma^{\lambda}}{\mathrm{d}\Omega}=\sigma^{\lambda}$ (21)

As a $2\times 2$ Hermitian matrix, the most general form of the $R$-matrix is
similar to that of Eq. (1),

$R^{\lambda}=\tilde{A}^{\lambda}+\sum_{i}\tilde{B}^{\lambda}_{i}\sigma^{i},$
(22)

but with an extra parameter $\tilde{A}^{\lambda}$ that determines the
probability of the process,
$\textrm{tr}R^{\lambda}(\mathbf{p}^{\prime})=2\tilde{A}^{\lambda}(\mathbf{p}^{\prime})$.

The proper spin density matrix with unit trace describing the quantum state
for a scattering process along a fixed direction is obtained by normalization
of $R$,

$\rho^{\lambda}_{\alpha\beta}(\mathbf{p}^{\prime})=\frac{R^{\lambda}_{\alpha\beta}(\mathbf{p}^{\prime})}{\textrm{tr}R^{\lambda}(\mathbf{p}^{\prime})}=\frac{R^{\lambda}_{\alpha\beta}(\mathbf{p}^{\prime})}{2\tilde{A}^{\lambda}(\mathbf{p}^{\prime})},$
(23)

and whose spin polarization $\mathbf{B}^{\lambda}$ is given by
$B^{\lambda}_{i}=\tilde{B}^{\lambda}_{i}/\tilde{A}^{\lambda}$. In terms of
these quantum substates, the total quantum state of Eq. (2.2) can be written
as simply

$\rho^{\lambda}=\frac{1}{\sigma^{\lambda}}\sum_{\alpha\beta}\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\mathrm{d}\sigma^{\lambda}}{\mathrm{d}\Omega}\rho^{\lambda}_{\alpha\beta}(\mathbf{p}^{\prime})\frac{\ket{\mathbf{p}^{\prime}\alpha}\bra{\mathbf{p}^{\prime}\beta}}{\braket{\mathbf{p}^{\prime}}{\mathbf{p}^{\prime}}}$
(24)

Hence, we can intuitively understand the total spin-momentum quantum state
$\rho^{\lambda}$ as the incoherent average in momentum of the spin quantum
states $\rho^{\lambda}(\mathbf{p}^{\prime})$ describing the scattering along a
fixed direction, weighted with the probability of that scattering process
(proportional to the differential cross-section). The total cross-section
$\sigma^{\lambda}$ plays once more the role of the partition function.

Finally, we note that in a collider the initial state is typically unpolarized
since spin degrees of freedom cannot be controlled. Thus, the real quantum
state describing the experiment after many events results from the averaged
$R$-matrix

$R(\mathbf{p}^{\prime})=\frac{1}{2}\sum_{\lambda}R^{\lambda}(\mathbf{p}^{\prime})$
(25)

In summary, the quantum state describing the scattered particles in a collider
is a mixed state because of two fundamentally different reasons, related to
the control over the degrees of freedom. (i) Regarding orbital variables:
Momentum distributions of the scattered particles are the only measurable
observables in colliders, given in terms of differential cross-sections. Thus,
even if the scattered state is pure, one can only access to the diagonal part
in momentum, which motivates the use of a reduced mixed density matrix. (ii)
Regarding discrete variables: Internal degrees of freedom of the initial state
cannot be typically controlled and one has to average uniformly over them.
This average over all possible initial states after many scattering events
results in an incoherent mixture that is described by a density matrix.

###  2.3 Relativistic particle-antiparticle pair production

We now switch to the actual case of high-energy colliders, whose physics is
described by relativistic quantum field theories within the framework of the
Standard Model. The Standard Model is composed of fundamental spin-1/2
particles (and their corresponding antiparticles) which interact through the
mediation of gauge bosons of spin-1. These interactions can be either strong
(mediated by massless gluons $g$), weak (mediated by the massive
$Z^{0},W^{\pm}$ bosons), or electromagnetic (mediated by the photon $\gamma$),
corresponding to a Yang-Mills theory
$\text{SU(3)}\otimes\text{SU(2)}\otimes\text{U(1)}$, respectively.
Electromagnetic and weak interactions are both unified within the framework of
electroweak theory, while strong interactions are described by quantum
chromodynamics. The remaining particle of the Standard Model, the Higgs boson,
has spin-0, and is the responsible for the occurrence of mass in massive
particles.

A natural Standard Model candidate for an entangled two-qubit system is a
particle-antiparticle (denoted generically as $P\bar{P}$) pair produced from
some initial state $I$:

$I\rightarrow P+\bar{P}$ (26)

In the following, we restrict to the case where $P$ is a Standard Model
fermion of mass $m$, although the formalism can be easily generalized to the
case where $P$ is some massless spin-1 gauge boson like the photon. Among the
Standard Model fermions, we can distinguish between quarks and leptons, where
the latter do not interact through QCD since they do not have color degrees of
freedom. Typically, the components of the initial state $I$ annihilate
themselves through gauge interactions, giving rise to the $P\bar{P}$ pair. In
the process, total energy and momentum are conserved.

The kinematics of a $P\bar{P}$ pair is determined in the center-of-mass (c.m.)
frame by the invariant mass $M$ and the direction of flight $\hat{k}$ of the
particle $P$. In this frame, the particle/antiparticle four-momenta read
$k^{\mu}=(k^{0},\mathbf{k}),\bar{k}^{\mu}=(k^{0},-\mathbf{k})$, with
$\hat{k}=\mathbf{k}/|\mathbf{k}|$, and satisfy the Lorentz-invariant
dispersion relation

$\bar{k}^{2}=k^{2}\equiv
k^{\mu}k_{\mu}=\left(k^{0}\right)^{2}-\mathbf{k}^{2}=m^{2},$ (27)

where in the following we work in natural units $\hbar=c=1$.

The invariant mass $M$ is the c.m. energy of the pair, defined from the usual
invariant Mandelstam variables as

$M^{2}\equiv s\equiv(k+\bar{k})^{2},$ (28)

In the c.m. frame, $M^{2}=4\left(k^{0}\right)^{2}=4(m^{2}+\mathbf{k}^{2})$,
where the momentum is related to the particle velocity $\beta$ by
$|\mathbf{k}|=m\beta/\sqrt{1-\beta^{2}}$, so

$\beta=\sqrt{1-\frac{4m^{2}}{M^{2}}}$ (29)

Hence, threshold production ($\beta=0$) is at the minimum energy possible for
a $P\bar{P}$ pair, $M=2m$, as naturally expected.

With respect to the quantum state of the $P\bar{P}$ pair, mutatis mutandis, we
define again an $R$-matrix in terms of the on-shell relativistic $T$-matrix
elements as

$R^{I\lambda}_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}(M,\hat{k})\equiv\bra{M\hat{k}\alpha\beta}T\ket{I\lambda}\bra{I\lambda}T^{\dagger}\ket{M\hat{k}\alpha^{\prime}\beta^{\prime}}$
(30)

with $\ket{I\lambda}$ labeling the initial state of the system and

$\ket{M\hat{k}\alpha\beta}\equiv\ket{k\alpha}\otimes\ket{\bar{k}\beta}$ (31)

where the first/second subspace correspond to the particle/antiparticle,
respectively. The spins of the particles are computed in their respective rest
frames, where they are well defined. Once more, if internal degrees of freedom
$\lambda$ of the initial state cannot be controlled (like spin or color), one
uses the averaged $R$-matrix

$R^{I}_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}(M,\hat{k})=\frac{1}{N_{\lambda}}\sum_{\lambda}R^{I\lambda}_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}(M,\hat{k})$
(32)

with $N_{\lambda}$ the number of internal degrees of freedom of $I$. In
analogy to Eq. (2.2), the resulting quantum state is

$\displaystyle\rho^{I}$
$\displaystyle=\frac{1}{Z}\sum_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}\int\mathrm{d}\Omega\leavevmode\nobreak\
R^{I}_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}(M,\hat{k})$ (33)
$\displaystyle\times\frac{\ket{M\hat{k}\alpha\beta}\bra{M\hat{k}\alpha^{\prime}\beta^{\prime}}}{\braket{M\hat{k}}{M\hat{k}}}$

with the partition function $Z$ proportional once more to the total cross-
section

$Z=\int\mathrm{d}\Omega\leavevmode\nobreak\ \textrm{tr}\leavevmode\nobreak\
R^{I}(M,\hat{k})\propto\sigma^{I}$ (34)

The $R$-matrix characterizes the quantum state of any $P\bar{P}$ pair produced
in a relativistic process, which is a mixed state by the very same reasons as
in the non-relativistic case. It takes the general form

$\displaystyle R$
$\displaystyle=\tilde{A}I_{4}+\sum_{i}\left(\tilde{B}^{+}_{i}\sigma^{i}\otimes
I_{2}+\tilde{B}^{-}_{i}I_{2}\otimes\sigma^{i}\right)$ (35)
$\displaystyle+\sum_{i,j}\tilde{C}_{ij}\sigma^{i}\otimes\sigma^{j}$

in a similar fashion to Eq. (4), but once more with an extra coefficient
$\tilde{A}$ proportional to the c.m. differential cross-section.

The actual spin density matrix is obtained from the normalization of $R$:

$\rho=\frac{R}{\mathrm{tr}(R)}=\frac{R}{4\tilde{A}}$ (36)

As a result, the spin polarizations $B^{\pm}_{i}$ and spin correlations
$C_{ij}$ of the $P\bar{P}$ pair are

$B^{\pm}_{i}=\frac{\tilde{B}^{\pm}_{i}}{\tilde{A}},\leavevmode\nobreak\
C_{ij}=\frac{\tilde{C}_{ij}}{\tilde{A}}$ (37)

In terms of these quantum states, one can retrieve the relativistic version of
Eq. (24),

$\displaystyle\rho^{I}$
$\displaystyle=\frac{1}{\sigma^{I}}\sum_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\mathrm{d}\sigma^{I}}{\mathrm{d}\Omega\leavevmode\nobreak\
}\rho^{I}_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}(M,\hat{p})$ (38)
$\displaystyle\times\frac{\ket{M\hat{k}\alpha\beta}\bra{M\hat{k}\alpha^{\prime}\beta^{\prime}}}{\braket{M\hat{k}}{M\hat{k}}}$

##  3 Production of $t\bar{t}$ in elementary QCD processes

A particularly interesting example of particle-antiparticle pair is provided
by a top-antitop quark pair, which will be the subject of study throughout the
rest of the work. Specifically, we restrict to the production of $t\bar{t}$
pairs through QCD processes, although the concepts and techniques developed
here can be straightforwardly extended to other types of interaction and/or
particle-antiparticle pairs. For the theoretical computations, we employ
leading-order (LO) QCD perturbation theory, since it provides analytical
results and a clear picture of the underlying physics. Higher-order
corrections are known to be small and do not change the main results [29, 41,
44, 45, 42].

At LO QCD, only two initial states can produce a $t\bar{t}$ pair: a light
quark-antiquark ($q\bar{q}$) or a gluon ($gg$) pair,

$\displaystyle q+\bar{q}$ $\displaystyle\rightarrow t+\bar{t},$ (39)
$\displaystyle g+g$ $\displaystyle\rightarrow t+\bar{t}.$

Representative Feynman diagrams for these processes in the Standard Model are
depicted in Fig. 1.

Figure 1: Representative Feynman diagrams for $t\bar{t}$ production within the
Standard Model. Spring lines represent gluons and straight lines represent
quarks, all either real or virtual.

Each initial state $I=q\bar{q},gg$ creates a $t\bar{t}$ pair in a spin quantum
state described by the production spin density matrix
$R^{I}(M_{t\bar{t}},\hat{k})$, where the spin and color degrees of freedom of
the initial state have already been averaged, $M_{t\bar{t}}$ is the invariant
mass of the $t\bar{t}$ pair, and $\hat{k}$ is the flight direction of the top
quark. For the characterization of the $R$-matrix, an orthonormal basis needs
to be fixed in order to compute the corresponding spin polarizations and
correlations. The most common choice is the helicity basis [31], defined in
the c.m. frame as $\\{\hat{k},\hat{n},\hat{r}\\}$, with
$\hat{r}=(\hat{p}-\cos\Theta\hat{k})/\sin\Theta$,
$\hat{n}=\hat{r}\times\hat{k}$, where $\hat{p}$ is a unitary vector along the
direction of the initial state and $\Theta$ is the production angle of the top
quark with respect to the beam, $\cos\Theta=\hat{k}\cdot\hat{p}$. We note here
that, due to momentum conservation, the c.m. frame of the initial state is the
same as that of the $t\bar{t}$ pair.

The advantage of the helicity basis is that, although the $t\bar{t}$
relativistic spins are only well-defined in their respective rest frames,
those are equivalent to the c.m. frame via a Lorentz transformation along the
top direction, which does not change the orientation of the helicity basis. A
schematic representation of this basis is provided in Fig. 2.

The production spin density matrix $R^{I}(M_{t\bar{t}},\hat{k})$ in the
helicity basis is only a function of $\beta$ (or, equivalently, of
$M_{t\bar{t}}$) and $\cos\Theta$. Specifically, in the Standard Model, the
correlation matrix $\tilde{C}^{I}_{ij}$ is symmetric and
$\tilde{B}_{i}^{I,+}=\tilde{B}_{i}^{I,-}$. Furthermore, at LO, the $t\bar{t}$
pair is unpolarized, $\tilde{B}_{i}^{I,\pm}=0$, and, in QCD production, their
spins along the $n$-axis are uncorrelated with respect to the remaining
directions, $\tilde{C}^{I}_{nr}=\tilde{C}^{I}_{nk}=0$ [39]. Therefore,
$R^{I}(M_{t\bar{t}},\hat{k})$ is characterized at LO QCD by only $5$
parameters in the helicity basis:
$\tilde{A}^{I},\tilde{C}^{I}_{kk},\tilde{C}^{I}_{nn},\tilde{C}^{I}_{rr},\tilde{C}^{I}_{kr}$.
The values of those coefficients can be computed analytically and are well
known [28, 30, 31]. Here, we fix the normalization of
$R^{I}(M_{t\bar{t}},\hat{k})$ such that the c.m. differential cross-section
for $t\bar{t}$ production from an initial state $I$ is [28]

$\frac{\mathrm{d}\sigma^{I}}{\mathrm{d}\Omega}=\frac{\alpha^{2}_{s}\beta}{M^{2}_{t\bar{t}}}\tilde{A}^{I}(M_{t\bar{t}},\hat{k})$
(40)

with $\alpha_{s}\simeq 0.118$ the strong coupling constant characterizing the
strength of QCD interactions.

Figure 2: Orthonormal helicity basis, defined in the c.m. frame. $\hat{k}$ is the direction of the top and $\hat{p}$ is the direction of the initial beam. The vector $\hat{n}$ is perpendicular to the $\\{\hat{k},\hat{p}\\}$ plane and $\hat{r}=\hat{k}\times\hat{n}$ is the vector orthogonal to $\hat{k}$ within the $\\{\hat{k},\hat{p}\\}$ plane. |
---|---
Figure 3: Concurrence of the spin density matrix $\rho^{I}(\beta,\hat{k})$
resulting from an initial state $I=q\bar{q},gg$ as a function of the top
velocity $\beta$ and the production angle $\Theta$ in the $t\bar{t}$ c.m.
frame. All plots are symmetric under the transformation
$\Theta\rightarrow\pi-\Theta$. Left: $q\bar{q}\rightarrow t\bar{t}$. Right:
$gg\rightarrow t\bar{t}$. Solid black lines represent the critical boundaries
between separability and entanglement $\beta^{\textrm{PH}}_{c1,c2}(\Theta)$,
while dashed black lines represent the critical boundaries for the violation
of the CHSH inequality, $\beta^{\textrm{CH}}_{c1,c2}(\Theta)$.

The actual spin density matrices $\rho^{I}(M_{t\bar{t}},\hat{k})$ are computed
from the normalization of $R^{I}(M_{t\bar{t}},\hat{k})$. As a result of the
above considerations, at LO $\rho^{I}(M_{t\bar{t}},\hat{k})$ is unpolarized,
its correlation matrix is symmetric, and it is already diagonal in the
$\hat{n}$ direction. These last two properties imply that the correlation
matrix can be diagonalized by an appropriated rotation in the
$\\{\hat{k},\hat{r}\\}$ plane, with eigenvalues
$\\{C^{I}_{+},C^{I}_{nn},C^{I}_{-}\\}$,

$C^{I}_{\pm}=\frac{C^{I}_{kk}+C^{I}_{rr}}{2}\pm\sqrt{\left(\frac{C^{I}_{kk}-C^{I}_{rr}}{2}\right)^{2}+C^{I2}_{kr}}$
(41)

and orthonormal eigenvectors $\\{\hat{u}_{+},\hat{n},\hat{u}_{-}\\}$. We will
refer to this orthonormal basis as the diagonal basis.

Regarding entanglement, since $C^{I}_{nn}<0$ in all phase space and for both
initial states [see Eqs. (45), (3.2)], we find that the Peres-Horodecki
criterion is equivalent to $\Delta^{I}>0$ (see Appendix B), where

$\displaystyle\Delta^{I}$
$\displaystyle\equiv\frac{-C^{I}_{nn}+|C^{I}_{+}+C^{I}_{-}|-1}{2}$ (42)
$\displaystyle=\frac{-C^{I}_{nn}+|C^{I}_{kk}+C^{I}_{rr}|-1}{2}$

We note that this definition differs from that of Ref. [41] by a factor $2$,
so the concurrence is directly equal to $\Delta^{I}$ for entangled states:

$\mathcal{C}[\rho^{I}]=\max(\Delta^{I},0)$ (43)

Previous approaches in high-energy physics based on the entanglement entropy
[59, 60, 21] are not useful here since they are only valid for pure states. We
recall that the entanglement of the quantum state
$\rho^{I}(M_{t\bar{t}},\hat{k})$ is Lorentz invariant because it has well-
defined momentum [11, 12].

Regarding the violation of the CHSH inequality, from Eq. (2.1) it is
immediately seen in the diagonal basis to be equivalent to

$\displaystyle\mu^{I}$
$\displaystyle\equiv\textrm{tr}\left[\mathbf{C}^{\textrm{T}}\mathbf{C}\right]-\left(C_{\textrm{min}}^{I}\right)^{2}-1>0$
(44) $\displaystyle\left(C_{\textrm{min}}^{I}\right)^{2}$
$\displaystyle\equiv\min\left\\{\left(C_{+}^{I}\right)^{2},\left(C_{nn}^{I}\right)^{2},\left(C_{-}^{I}\right)^{2}\right\\}$

###  3.1 $q\bar{q}$ processes

For $q\bar{q}$ processes, the coefficients of the $R$-matrix in the helicity
basis are

$\displaystyle\tilde{A}^{q\bar{q}}$
$\displaystyle=F_{q}(2-\beta^{2}\sin^{2}\Theta)$ (45)
$\displaystyle\tilde{C}^{q\bar{q}}_{rr}$
$\displaystyle=F_{q}(2-\beta^{2})\sin^{2}\Theta$
$\displaystyle\tilde{C}^{q\bar{q}}_{nn}$
$\displaystyle=-F_{q}\beta^{2}\sin^{2}\Theta$
$\displaystyle\tilde{C}^{q\bar{q}}_{kk}$
$\displaystyle=F_{q}\left[2-(2-\beta^{2})\sin^{2}\Theta\right]$
$\displaystyle\tilde{C}^{q\bar{q}}_{rk}$
$\displaystyle=\tilde{C}^{q\bar{q}}_{kr}=F_{q}\sqrt{1-\beta^{2}}\sin 2\Theta$
$\displaystyle F_{q}$ $\displaystyle=\frac{1}{18}$

The resulting $t\bar{t}$ quantum state $\rho^{q\bar{q}}(\beta,\hat{k})$ is
entangled in the bulk of phase space as 111We note that the result for
$\Delta^{q\bar{q}}$ in Ref. [41] contained a typo, and a factor $2$ was
missing.

$\displaystyle\Delta^{q\bar{q}}=-C_{nn}^{q\bar{q}}=\frac{\beta^{2}\sin^{2}\Theta}{2-\beta^{2}\sin^{2}\Theta}\geq
0$ (46)

The above inequality is saturated only at threshold ($\beta=0$) or for forward
production ($\Theta=0$). In both limits, the $t\bar{t}$ spins are aligned
along the beam axis in a maximally correlated but separable mixed state:

$\displaystyle C_{ij}^{q\bar{q}}(0,\hat{k})$
$\displaystyle=C_{ij}^{q\bar{q}}(\beta,\hat{p})=\hat{p}_{i}\hat{p}_{j}$ (47)
$\displaystyle\rho^{q\bar{q}}(0,\hat{k})$
$\displaystyle=\rho^{q\bar{q}}(\beta,\hat{p})$
$\displaystyle=\frac{\ket{\uparrow_{\hat{p}}\uparrow_{\hat{p}}}\bra{\uparrow_{\hat{p}}\uparrow_{\hat{p}}}+\ket{\downarrow_{\hat{p}}\downarrow_{\hat{p}}}\bra{\downarrow_{\hat{p}}\downarrow_{\hat{p}}}}{2}$

where $\ket{\uparrow_{\hat{p}}},\ket{\downarrow_{\hat{p}}}$ are the spin
eigenstates along the direction $\hat{p}$. This spin alignment at threshold is
a consequence of spin conservation, since the $t\bar{t}$ pair is produced from
the initial $q\bar{q}$ state via gluon exchange (see Fig. 1), which is a
massless spin-1 boson.

Hence, for low production angles or close to threshold, the degree of
entanglement is expected to be small, as can seen in left Fig. 3, where the
concurrence of $\rho^{q\bar{q}}(\beta,\hat{k})$ is represented.

In the opposite limit of high transverse momentum $p_{T}$ (i.e., high energies
and production angles close to $\Theta=\pi/2$), the $t\bar{t}$ pair is in a
spin-triplet pure state,

$\displaystyle C_{ij}^{q\bar{q}}(1,\hat{n}\times\hat{p})$
$\displaystyle=\delta_{ij}-2\hat{n}_{i}\hat{n}_{j}$ (48)
$\displaystyle\rho^{q\bar{q}}(1,\hat{n}\times\hat{p})$
$\displaystyle=\ket{\Psi_{\infty}}\bra{\Psi_{\infty}}$
$\displaystyle\ket{\Psi_{\infty}}$
$\displaystyle=\frac{\ket{\uparrow_{\hat{n}}\downarrow_{\hat{n}}}+\ket{\downarrow_{\hat{n}}\uparrow_{\hat{n}}}}{\sqrt{2}}$

This state is maximally entangled, $\mathcal{C}[\rho]=1$, as seen in upper
right corner of left Fig. 3.

With respect to the CHSH inequality, it is also violated by $q\bar{q}$
production processes in the bulk of phase space. This is seen in the diagonal
basis, where

$C^{q\bar{q}}_{+}=1,\leavevmode\nobreak\
C^{q\bar{q}}_{-}=-C^{q\bar{q}}_{nn}=\Delta^{q\bar{q}}$ (49)

This basis is often called the off-diagonal basis [61] in the literature of
$q\bar{q}$ processes, and is commonly used because the $t\bar{t}$ spins along
the $\hat{u}_{+}$ direction are perfectly correlated. Because of this,

$\mu^{q\bar{q}}(\beta,\Theta)=\left[\Delta^{q\bar{q}}(\beta,\Theta)\right]^{2}\geq
0$ (50)

Therefore, remarkably, entanglement and CHSH violation are equivalent
conditions for $q\bar{q}$ processes.

###  3.2 $gg$ processes

For $gg$ processes, the coefficients of the $R$-matrix in the helicity basis
are

$\displaystyle\tilde{A}^{gg}$
$\displaystyle=F_{g}\left[1+2\beta^{2}\sin^{2}\Theta-\beta^{4}(1+\sin^{4}\Theta)\right]$
$\displaystyle\tilde{C}^{gg}_{rr}$
$\displaystyle=-F_{g}\left[1-\beta^{2}(2-\beta^{2})(1+\sin^{4}\Theta)\right]$
$\displaystyle\tilde{C}^{gg}_{nn}$
$\displaystyle=-F_{g}\left[1-2\beta^{2}+\beta^{4}(1+\sin^{4}\Theta)\right]$
$\displaystyle\tilde{C}^{gg}_{kk}$
$\displaystyle=-F_{g}\left[1-\beta^{2}\frac{\sin^{2}2\Theta}{2}-\beta^{4}(1+\sin^{4}\Theta)\right]$
$\displaystyle\tilde{C}^{gg}_{rk}$
$\displaystyle=\tilde{C}^{gg}_{kr}=F_{g}\sqrt{1-\beta^{2}}\beta^{2}\sin
2\Theta\sin^{2}\Theta$ $\displaystyle F_{g}$
$\displaystyle=\frac{7+9\beta^{2}\cos^{2}\Theta}{192(1-\beta^{2}\cos^{2}\Theta)^{2}}$
(51)

The resulting $t\bar{t}$ quantum state $\rho^{gg}(\beta,\hat{k})$ is entangled
iff $\Delta^{gg}(\beta,\Theta)>0$, where

$\displaystyle\Delta^{gg}$
$\displaystyle=\frac{\beta^{4}(1+\sin^{4}\Theta)-\beta^{2}(1+\sin^{2}\Theta)}{1+2\beta^{2}\sin^{2}\Theta-\beta^{4}(1+\sin^{4}\Theta)}$
$\displaystyle+\frac{|1-\beta^{2}(1+\sin^{2}\Theta)|}{1+2\beta^{2}\sin^{2}\Theta-\beta^{4}(1+\sin^{4}\Theta)}$
(52)

The second line implies that $\rho^{gg}(\beta,\hat{k})$ is not entangled for
$\beta^{2}(1+\sin^{2}\Theta)=1$, so there are finite regions of separability
in phase space, in contrast to the $q\bar{q}$ case. In particular, two regions
of entanglement can be distinguished: one close to threshold, and another one
for high $p_{T}$, delimited by the lower and upper critical boundaries
$\beta^{\textrm{PH}}_{c1}(\Theta),\beta^{\textrm{PH}}_{c2}(\Theta)$:

$\displaystyle\beta^{\textrm{PH}}_{c1}(\Theta)$
$\displaystyle=\sqrt{\frac{1+\sin^{2}\Theta-\sqrt{2}\sin\Theta}{1+\sin^{4}\Theta}}$
$\displaystyle\beta^{\textrm{PH}}_{c2}(\Theta)$
$\displaystyle=\frac{1}{(1+\sin^{4}\Theta)^{\frac{1}{4}}}$ (53)

These features can be observed in right Fig. 3, where the concurrence of
$\rho^{gg}(\beta,\hat{k})$ is represented, with the black solid lines
signaling the critical boundaries $\beta^{\textrm{PH}}_{c1,c2}(\Theta)$
between entanglement and separability.

The strong entanglement signature close to threshold results from the fact
that the spin polarizations of the $gg$ initial state are allowed to align in
different directions. Due to angular momentum conservation, these features
produce at threshold a $t\bar{t}$ pair in a spin singlet

$\displaystyle C_{ij}^{gg}(0,\hat{k})$ $\displaystyle=-\delta_{ij}$ (54)
$\displaystyle\rho^{gg}(0,\hat{k})$
$\displaystyle=\ket{\Psi_{0}}\bra{\Psi_{0}}$ $\displaystyle\ket{\Psi_{0}}$
$\displaystyle=\frac{\ket{\uparrow_{\hat{n}}\downarrow_{\hat{n}}}-\ket{\downarrow_{\hat{n}}\uparrow_{\hat{n}}}}{\sqrt{2}}$

which is maximally entangled.

In the opposite limit of high $p_{T}$, gluon fusion produces the same
maximally entangled triplet state as $q\bar{q}$ processes, Eq. (48). The
reason behind this similarity is that for high energies the orbital angular
momentum contribution dominates over the spin contribution.

With respect to the violation of the CHSH inequality, the eigenvalues of the
correlation matrix are

$C^{gg}_{\pm}=\frac{-1+\beta^{2}(1+\sin^{2}\Theta)\pm\sqrt{\beta^{4}(1-2\sin^{2}\Theta+5\sin^{4}\Theta)-2\beta^{6}(1-\sin^{2}\Theta+3\sin^{4}\Theta+\sin^{6}\Theta)+\beta^{8}\left[1+\sin^{4}\Theta\right]^{2}}}{1+2\beta^{2}\sin^{2}\Theta-\beta^{4}(1+\sin^{4}\Theta)}$

(55)

As a result, an explicit analytical computation of the critical boundaries
$\beta^{\textrm{CH}}_{c1,c2}(\Theta)$ of the regions where the CHSH inequality
is violated becomes much more cumbersome; even in the relatively simple case
of $\Theta=\pi/2$, the critical values $\beta^{\textrm{CH}}_{c1,c2}(\pi/2)$
are obtained from the zeros of fourth order polynomials in $\beta^{2}$,
finding $\beta^{\textrm{CH}}_{c1}(\pi/2)\approx
0.367<\beta^{\textrm{PH}}_{c1}(\pi/2)=2^{-1/4}\approx 0.541$, and
$\beta^{\textrm{CH}}_{c2}(\pi/2)\approx
0.931>\beta^{\textrm{PH}}_{c2}(\pi/2)=\sqrt{1-1/\sqrt{2}}\approx 0.841$. The
numerically computed boundaries $\beta^{\textrm{CH}}_{c1,c2}(\Theta)$ are
represented by black dashed lines in right Fig. 3. As expected, the regions
where CHSH inequality is violated are within entangled regions of phase space,
$\beta^{\textrm{CH}}_{c1}(\Theta)\leq\beta^{\textrm{PH}}_{c1}(\Theta)\leq\beta^{\textrm{PH}}_{c2}(\Theta)\leq\beta^{\textrm{CH}}_{c2}(\Theta)$.

An important conclusion is that the entanglement structure of $t\bar{t}$
production can be mostly understood from basic laws of angular momentum
conservation between the initial and final state, without the need to invoke
technical details of the specific form of QCD interactions.

##  4 Production of $t\bar{t}$ in realistic QCD processes

Because of color confinement, quarks and gluons are not free particles and
cannot be found isolated in nature. Instead, they are forming hadrons, which
are bound states of quarks through QCD interactions. As a result, we can
understand hadrons as a sea of quarks and gluons, which within this context
are indistinctively denoted as partons [62, 63]. Quantitatively, the
composition of a hadron is modeled by its so-called parton distribution
function (PDF), which determines the participation of each parton in a
particular QCD process at a certain energy scale. Due to its non-perturbative
character, PDF distributions are typically computed by fitting experimental
data (see Appendix C for more technical details about PDF). Here, we focus on
$t\bar{t}$ production from two main types of hadron processes which are of
high relevance for experiments: proton-proton collisions, as in the LHC, and
proton-antiproton collisions, as in the Tevatron.

The above considerations modify the results of the previous section as
following. For given energy and top direction in the c.m. frame, the
$R$-matrix describing the $t\bar{t}$ pair resulting from a hadronic process is
computed in terms of each partonic matrix $R^{I}(M_{t\bar{t}},\hat{k})$ as

$R(M_{t\bar{t}},\hat{k},\sqrt{s})=\sum_{I=q\bar{q},gg}L_{I}(M_{t\bar{t}},\sqrt{s})R^{I}(M_{t\bar{t}},\hat{k})$
(56)

The function $L_{I}(M_{t\bar{t}},\sqrt{s})$ is the so-called luminosity
function [28], and is computed in terms of the PDF describing the colliding
hadrons. The luminosity function can be regarded as the probability
distribution of occurrence of each initial state $I$ in the total hadronic
process for a given $M_{t\bar{t}}$, depending also on the hadron c.m. energy
$\sqrt{s}$ [see Eq. (134)]. We note that the c.m. frame of the colliding
hadrons is not the same as the parton c.m. frame. However, at LO the direction
of the parton initial state $I$ in its c.m. frame is to a very good
approximation that of the initial hadron beam, so one can take safely
$\hat{p}$ along the hadron beam in Fig. 2, which is the one whose direction is
controlled in a collider [29].

With the help of the luminosities and the $R$-matrix, the differential cross-
section characterizing $t\bar{t}$ production from a given hadronic process is
computed as

$\displaystyle\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}}$
$\displaystyle=\frac{\alpha^{2}_{s}\beta}{M^{2}_{t\bar{t}}}\tilde{A}(M_{t\bar{t}},\hat{k},\sqrt{s})$
(57)
$\displaystyle=\sum_{I=q\bar{q},gg}L_{I}(M_{t\bar{t}},\sqrt{s})\frac{\mathrm{d}\sigma^{I}}{\mathrm{d}\Omega}(M_{t\bar{t}},\hat{k})$

where the partonic differential cross-sections are those of Eq. (40). Thus,
the differential cross-section per unit solid angle and per unit c.m. energy
is just the sum of the partonic differential cross-sections from an initial
state $I$, multiplied by the probability distribution $L_{I}$ of producing
each initial state $I$ with energy $M_{t\bar{t}}$.

l18ptt12pt(a) |  l18ptt12pt(b) |  l18ptt12pt(c) |  l18ptt12pt(d)
---|---|---|---
l18ptt12pt(e) |  l18ptt12pt(f) |  l18ptt12pt(g) |  l18ptt12pt(h)
l18ptt12pt(i) |  l18ptt12pt(j) |  l18ptt12pt(k) |  l18ptt12pt(l)
Figure 4: Same as Fig. 3 but for the $t\bar{t}$ quantum state
$\rho(M_{t\bar{t}},\hat{k},\sqrt{s})$ resulting from realistic hadronic
processes. Upper row: Toy model where the probabilities are constant in whole
phase space, $w_{I}(M_{t\bar{t}},\hat{k},\sqrt{s})=w_{I}$. (a) $w_{gg}=0.2$.
(b) $w_{gg}=0.4$.(c) $w_{gg}=0.6$.(d) $w_{gg}=0.8$. Middle row: $t\bar{t}$
production from $pp$ collisions. (e) $\sqrt{s}=1\leavevmode\nobreak\
\textrm{TeV}$. (f) $\sqrt{s}=10\leavevmode\nobreak\ \textrm{TeV}$. (g)
$\sqrt{s}=30\leavevmode\nobreak\ \textrm{TeV}$. (h)
$\sqrt{s}=100\leavevmode\nobreak\ \textrm{TeV}$. Lower row: $t\bar{t}$
production from $p\bar{p}$ collisions. (i)-(l) Same values of $\sqrt{s}$ as
(e)-(h), respectively.

Regarding the actual spin quantum state, since
$\rho^{I}=R^{I}/4\tilde{A}^{I}$, then

$\rho(M_{t\bar{t}},\hat{k},\sqrt{s})=\sum_{I=q\bar{q},gg}w_{I}(M_{t\bar{t}},\hat{k},\sqrt{s})\rho^{I}(M_{t\bar{t}},\hat{k})$
(58)

Similar expressions can be written for the spin polarizations and spin
correlations
$\mathbf{B}^{\pm}(M_{t\bar{t}},\hat{k},\sqrt{s}),\mathbf{C}(M_{t\bar{t}},\hat{k},\sqrt{s})$
of $\rho(M_{t\bar{t}},\hat{k},\sqrt{s})$ in terms of their partonic
counterparts, computed from Eqs. (45), (3.2). The weights $w_{I}$ are obtained
from the luminosities as

$w_{I}(M_{t\bar{t}},\hat{k},\sqrt{s})=\frac{L_{I}(M_{t\bar{t}},\sqrt{s})\tilde{A}^{I}(M_{t\bar{t}},\hat{k})}{\sum_{J}L_{J}(M_{t\bar{t}},\sqrt{s})\tilde{A}^{J}(M_{t\bar{t}},\hat{k})}$
(59)

and represent the probability of production of each individual partonic
quantum state $\rho^{I}(M_{t\bar{t}},\hat{k})$, satisfying
$w_{q\bar{q}}+w_{gg}=1$. They can be simply understood as the probability of
occurrence of the initial state $I$, $L_{I}$, multiplied by the probability of
producing a $t\bar{t}$ pair from $I$, proportional to the differential cross-
section, in turn proportional to the coefficient
$\tilde{A}^{I}(M_{t\bar{t}},\hat{k})$.

We stress that the nature of the specific hadron collision only enters in the
calculations through the probabilities $w_{I}(M_{t\bar{t}},\hat{k},\sqrt{s})$,
which are PDF dependent. Hence, the LO QCD $q\bar{q}$ and $gg$ processes are
the building blocks of any realistic hadronic production process, whose role
is reduced to change the amount of mixing between them. From a more general
perspective, we can regard the production spin density matrices
$R^{I}(M_{t\bar{t}},\hat{k})$ and the luminosities
$L_{I}(M_{t\bar{t}},\sqrt{s})$ as inputs from the theory of high-energy
physics, from which we can compute the physical spin density matrices
$\rho^{I}(M_{t\bar{t}},\hat{k})$ and the probabilities
$w_{I}(M_{t\bar{t}},\hat{k},\sqrt{s})$. Once done, we are simply left with a
typical problem in quantum information involving the convex sum of two-qubit
states, where the usual techniques of the field can be applied.

In order to understand better how the mixture between $q\bar{q}$ and $gg$
channels affects the $t\bar{t}$ quantum state
$\rho(M_{t\bar{t}},\hat{k},\sqrt{s})$, we first use a toy model where both
partonic probabilities are constant,
$w_{I}(M_{t\bar{t}},\hat{k},\sqrt{s})=w_{I}$. We gradually introduce $gg$
processes in a pure $q\bar{q}$ reaction in upper row of Fig. 4. For a small
amount of $gg$ processes, the effect of mixing is reduced to shrink the
entangled region towards the high $p_{T}$ region. By further increasing the
amount of $gg$ processes, at some point entangled $t\bar{t}$ states also
emerge at threshold. We can actually compute from Eq. (42) the critical value
$w^{\textrm{PH}}_{c}$ at which entanglement appears, since

$\Delta(\beta=0,\Theta)=\frac{w_{gg}+|3w_{gg}-1|-1}{2}$ (60)

For $w_{gg}>1/3$, $\Delta(0,\Theta)=2w_{gg}-1$, so

$w^{\textrm{PH}}_{c}=\frac{1}{2}$ (61)

Hence, in order to have entanglement close to threshold, we need at least
$50\%$ of the $t\bar{t}$ pairs to be produced through gluon fusion. Regarding
the violation of the CHSH inequality, the corresponding critical value
$w^{\textrm{CH}}_{c}$ is computed by switching to the off-diagonal basis where
both correlation matrices are diagonal ($\rho^{gg}$ at threshold is a spin
singlet, invariant under rotations). This gives

$\mu(\beta=0,\Theta)=2w^{2}_{gg}-1$ (62)

from where we find

$w^{\textrm{CH}}_{c}=\frac{1}{\sqrt{2}}>w^{\textrm{PH}}_{c}$ (63)

Once this simple model is understood, we switch to real hadron processes. We
analyze $t\bar{t}$ production from $pp$ collisions for different c.m. energies
in central row of Fig. 4. By direct comparison with upper row, we observe that
for low c.m. energies $q\bar{q}$ processes dominate. However, the amount of
$gg$ processes increases with the c.m. energy, eventually dominating the
$t\bar{t}$ production mechanism. A similar calculation is shown in the lower
row of Fig. 4 for $p\bar{p}$ collisions where, although $q\bar{q}$
contributions are stronger at low energies, $gg$ dominance is again recovered
for sufficiently high c.m. energies, and both types of collisions converge to
a similar quantum state.

|
---|---
Figure 5: Study of the dependence of $t\bar{t}$ production mechanism with the
hadron c.m. energy $\sqrt{s}$ for $pp$ collisions (solid blue line) and for
$p\bar{p}$ collisions (dashed red line), corresponding to the LHC and
Tevatron, respectively. Black markers indicate the values for c.m. energies of
actual high-energy colliders. Circles: Run 1
($\sqrt{s}=7,8\leavevmode\nobreak\ \textrm{TeV}$), Run 2
($\sqrt{s}=13\leavevmode\nobreak\ \textrm{TeV}$) and Run 3
($\sqrt{s}=13.6\leavevmode\nobreak\ \textrm{TeV}$) of the LHC. Squares:
Possible upgrade of the LHC ($\sqrt{s}=27\leavevmode\nobreak\ \textrm{TeV}$).
Triangles: Tevatron ($\sqrt{s}=1.96\leavevmode\nobreak\ \textrm{TeV}$). Left:
Gluon fraction $f_{gg}$. Right: $gg$ probability at threshold
$w_{gg}(\beta=0)$. Horizontal black lines mark the critical values
$w^{\textrm{PH}}_{gg,c}=1/2$ for entanglement (solid), and
$w^{\textrm{CH}}_{gg,c}=1/\sqrt{2}$ for CHSH violation (dashed).

We can further quantify the amount of mixing of $q\bar{q}$ and $gg$ processes
in the total $t\bar{t}$ production by computing the relative contribution of
each channel to the total cross-section from Eq. (57):

$f_{I}\equiv\frac{\int\limits^{\sqrt{s}}_{2m_{t}}\mathrm{d}M_{t\bar{t}}\int\mathrm{d}\Omega\leavevmode\nobreak\
L_{I}\frac{\mathrm{d}\sigma^{I}}{\mathrm{d}\Omega}}{\int\limits^{\sqrt{s}}_{2m_{t}}\mathrm{d}M_{t\bar{t}}\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}}}$ (64)

where we stress that the maximum invariant mass $M_{t\bar{t}}$ achievable by
the $t\bar{t}$ pair is precisely the c.m. energy of the hadron pair,
$M_{t\bar{t}}=\sqrt{s}$. The fraction of $gg$ processes $f_{gg}$ for both $pp$
and $p\bar{p}$ collisions is represented in left Fig. 5, where its predicted
increase with the c.m. energy indicated by the entanglement analysis is
clearly observed, as well as the stronger contribution of $q\bar{q}$ processes
for $p\bar{p}$ collisions with respect to $pp$ at low energies. At high
energies, both hadronic processes converge to the same gluon fraction
$f_{gg}$. This analysis suggests that entanglement measurements can be also
used to understand the underlying structure of a certain high-energy process
without direct knowledge of it.

A related plot is that of right Fig. 5, in which the threshold values of the
gluon probability $w_{gg}(\beta=0,\sqrt{s})$ are represented (there is no
angular dependence at threshold). We observe that entanglement
[$w_{gg}(0,\sqrt{s})>w^{\textrm{PH}}_{c}=1/2$] can be achieved in $pp$
collisions for small c.m. energies $\sqrt{s}\gtrsim 5\leavevmode\nobreak\
\textrm{TeV}$, below current LHC energies. In particular, the relevant
experimental values $\sqrt{s}=7,8\leavevmode\nobreak\ \textrm{TeV}$ [36, 37,
38], $\sqrt{s}=13\leavevmode\nobreak\ \textrm{TeV}$ [39, 40] and
$\sqrt{s}=13.6\leavevmode\nobreak\ \textrm{TeV}$ [64] are marked by circles,
corresponding to the c.m. energies of Run 1, Run 2 and the ongoing Run 3 of
the LHC, respectively. The expected value for a possible upgrade of the LHC is
$\sqrt{s}=27\leavevmode\nobreak\ \textrm{TeV}$ [65], marked by a square. In
addition, the c.m. energy for $pp$ collisions in the Future Circular Collider
(FCC), $\sqrt{s}=100\leavevmode\nobreak\ \textrm{TeV}$ [66], is represented
explicitly in Fig. 4h. With respect to CHSH violation
[$w_{gg}(0,\sqrt{s})>w^{\textrm{CH}}_{c}=1/\sqrt{2}$], we observe that we need
larger energies $\sqrt{s}\gtrsim 10\leavevmode\nobreak\ \textrm{TeV}$, with
Run 2 and Run 3 only slightly above this limit. For $p\bar{p}$ collisions,
even threshold entanglement cannot be observed since the c.m. energies
required are well above the maximum energy achieved at the Tevatron,
$\sqrt{s}=1.96\leavevmode\nobreak\ \textrm{TeV}$ [33, 34, 35], marked by a
triangle.

From these plots, we conclude that the Tevatron and the LHC provide
experimental realizations to a very good approximation of the two main
paradigms of $t\bar{t}$ production through QCD, $q\bar{q}$ and $gg$ reactions.

##  5 Experimental observables

###  5.1 Experimental motivation

How are all these spin magnitudes translated into actual observables that can
be measured in high-energy colliders? This is where the unique properties of
the top quark enter in place. As explained in the Introduction, the top quark
is the most massive particle of the Standard Model. This property makes that
even its large width $\Gamma_{t}\simeq 1\leavevmode\nobreak\ \textrm{GeV}$ is
still narrow when compared to its mass $m_{t}\simeq 173\leavevmode\nobreak\
\textrm{GeV}$. In turn, this large width results in a very short lifetime
$\tau=1/\Gamma_{t}\sim 10^{-25}\textrm{s}$. This implies that a $t\bar{t}$
pair decays so fast that any other process, such as hadronisation (with a time
scale $\sim 10^{-23}\textrm{s}$) or spin decorrelation (with a time scale
$\sim 10^{-21}\textrm{s}$), cannot affect their spin quantum state, whose
information is directly translated into the properties of the decay products.
This is what makes the top quark so special, as any other quark hadronizes
before decaying, losing its spin information in the process. On the other
hand, if the top quark did not decay, its spin state could not be extracted
because detectors only measure the momentum of the arriving particles.

Quantitatively, we describe the decay of a top quark to some final state $F$
in terms of a decay spin density matrix, defined in the top rest frame as

$\Gamma^{F}_{\alpha^{\prime}\alpha}\equiv\bra{F}T\ket{t\alpha}\bra{t\alpha^{\prime}}T^{\dagger}\ket{F}$
(65)

where $\ket{t\alpha}$ is the top state with zero momentum and spin $\alpha$,
$\ket{F}$ is the final state after the decay, and $T$ is here the on-shell
$T$-matrix determining the amplitude of the decay process. A similar decay
spin density matrix $\bar{\Gamma}^{\bar{F}}_{\beta^{\prime}\beta}$ can be
defined for the decay of the antitop quark to a final state $\bar{F}$.

The idea is to describe the production and decay of a $t\bar{t}$ pair using
the production and decay spin density matrices. Specifically, we consider a
$t\bar{t}$ pair with fixed c.m. energy and momentum produced from a hadron
collision, whose quantum state is specified by the matrix
$R(M_{t\bar{t}},\hat{k},\sqrt{s})$. In the so-called narrow-width
approximation, valid since $\Gamma_{t}\ll m_{t}$ [26], the differential cross-
section characterizing the decay of a $t\bar{t}$ pair to two final states
$F\bar{F}$ is proportional to [29, 49]

$\displaystyle\mathrm{d}\sigma_{F\bar{F}}$
$\displaystyle\sim\sum_{\alpha\alpha^{\prime},\beta\beta^{\prime}}R_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}(M_{t\bar{t}},\hat{k},\sqrt{s})\Gamma^{F}_{\alpha^{\prime}\alpha}\bar{\Gamma}^{\bar{F}}_{\beta^{\prime}\beta}$
$\displaystyle=\textrm{tr}\left[\left(\Gamma^{F}\otimes\bar{\Gamma}^{\bar{F}}\right)R(M_{t\bar{t}},\hat{k},\sqrt{s})\right]$
(66)

Thus, the information about the quantum state of the $t\bar{t}$ pair, encoded
in $R(M_{t\bar{t}},\hat{k},\sqrt{s})$, is contained in the cross-section
characterizing the decay products of the final states $F\bar{F}$.

We can retrieve this information from final states including a lepton $\ell$,
which can be either an electron or a muon, $\ell=e,\mu$. Specifically, we
consider the electroweak leptonic decay of both the top and the antitop quark:

$\displaystyle t$ $\displaystyle\rightarrow b+W^{+}\rightarrow
b+\ell^{+}+\nu_{\ell},$ (67) $\displaystyle\bar{t}$
$\displaystyle\rightarrow\bar{b}+W^{-}\rightarrow\bar{b}+\ell^{-}+\bar{\nu}_{\ell}$

where $F=b\ell^{+}\nu_{\ell}$, with $\ell^{+}$ the antilepton and $\nu_{\ell}$
its associated neutrino, and $\bar{F}$ its conjugate,
$\bar{F}=\bar{b}\ell^{-}\bar{\nu}_{\ell}$. A diagrammatic representation of
the top/antitop decay is provided in Fig. 6.

The expression for the decay spin density matrices vastly simplifies if we
integrate out all the degrees of freedom of the final states $F\bar{F}$ except
for the lepton directions. In that case, due to the rotational invariance in
the top (antitop) rest frames, the resulting decay spin density matrices
$\Gamma_{\ell}$ $(\bar{\Gamma}_{\ell})$ can only be of the form [31]

$\Gamma_{\ell}\propto\frac{I_{2}+\kappa_{\ell}(\hat{\mathbf{\ell}}_{+}\cdot\mathbf{\sigma})}{2},\leavevmode\nobreak\
\bar{\Gamma}_{\ell}\propto\frac{I_{2}+\bar{\kappa}_{\ell}(\hat{\mathbf{\ell}}_{-}\cdot\mathbf{\sigma})}{2}$
(68)

$\hat{\mathbf{\ell}}_{\pm}$ being the antilepton (lepton) directions in each
one of the parent top (antitop) rest frames, and
$\kappa_{\ell}=-\bar{\kappa}_{\ell}\simeq 1$ the so-called spin analyzing
powers of the leptons.

As a result, the cross-section $\sigma_{\ell\bar{\ell}}$ characterizing the
$\ell^{+}\ell^{-}$ angular distribution reads

$\displaystyle\frac{\mathrm{d}\sigma_{\ell\bar{\ell}}}{\mathrm{d}\Omega_{+}\mathrm{d}\Omega_{-}\mathrm{d}M_{t\bar{t}}\mathrm{d}\Omega_{\hat{k}}}\propto\frac{\alpha^{2}_{s}\beta}{M^{2}_{t\bar{t}}}\textrm{tr}\left[(\Gamma_{\ell}\otimes\bar{\Gamma}_{\ell})R\right]$
$\displaystyle\propto\frac{\alpha^{2}_{s}\beta}{M^{2}_{t\bar{t}}}\left[\tilde{A}+\tilde{\mathbf{B}}^{+}\cdot\hat{\mathbf{\ell}}_{+}-\tilde{\mathbf{B}}^{-}\cdot\hat{\mathbf{\ell}}_{-}-\hat{\mathbf{\ell}}_{+}\cdot\tilde{\mathbf{C}}\cdot\hat{\mathbf{\ell}}_{-}\right]$
(69)

where $\Omega_{\pm}$, $\Omega_{\hat{k}}$ are the solid angles associated to
$\hat{\mathbf{\ell}}_{\pm}$, $\hat{k}$. The differential elements
$\mathrm{d}M_{t\bar{t}}\mathrm{d}\Omega_{\hat{k}}$ arise because we are
considering the decay of a $t\bar{t}$ pair with fixed energy and direction.
The total angular differential cross-section describing the leptons arising
from the $t\bar{t}$ decay is then obtained by integration over all possible
top directions and c.m. energies:

$\frac{\mathrm{d}\sigma_{\ell\bar{\ell}}}{\mathrm{d}\Omega_{+}\mathrm{d}\Omega_{-}}=\int\limits^{\sqrt{s}}_{2m_{t}}\mathrm{d}M_{t\bar{t}}\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\mathrm{d}\sigma_{\ell\bar{\ell}}}{\mathrm{d}\Omega_{+}\mathrm{d}\Omega_{-}\mathrm{d}M_{t\bar{t}}\mathrm{d}\Omega_{\hat{k}}}$
(70)

This implies that the normalized angular differential cross-section
characterizing the dileptonic decay is

$\frac{1}{\sigma_{\ell\bar{\ell}}}\frac{\mathrm{d}\sigma_{\ell\bar{\ell}}}{\mathrm{d}\Omega_{+}\mathrm{d}\Omega_{-}}=\frac{1+\mathbf{B}^{+}\cdot\hat{\mathbf{\ell}}_{+}-\mathbf{B}^{-}\cdot\hat{\mathbf{\ell}}_{-}-\hat{\mathbf{\ell}}_{+}\cdot\mathbf{C}\cdot\hat{\mathbf{\ell}}_{-}}{(4\pi)^{2}}$
(71)

where the vectors $\mathbf{B}^{\pm}$ and the matrix $\mathbf{C}$ are the
integrated values of the top/antitop spin polarizations and the spin
correlation matrix, respectively. Therefore, we can measure all the integrated
spin information of a $t\bar{t}$ pair by fitting the angular distribution of
the leptonic decay products.

Figure 6: Representative Feynman diagrams for a leptonic electroweak decay of
a top (left) and antitop (right) quark.

In particular, from the kinematic reconstruction of each event, the momenta of
the $t\bar{t}$ and $\ell^{+}\ell^{-}$ pairs can be determined. By defining the
polar angle with respect to a certain direction given by the unit vector
$\hat{u}_{i}$,
$\cos\theta^{i}_{\pm}\equiv\hat{\mathbf{\ell}}_{\pm}\cdot\hat{u}_{i}$, and
after integrating over the azimuthal angles, we find that the spin
polarizations can be easily obtained from a linear fit of the angular
distributions for each individual lepton as

$\displaystyle\frac{1}{\sigma_{\ell\bar{\ell}}}\frac{\mathrm{d}\sigma_{\ell\bar{\ell}}}{\mathrm{d}\cos\theta_{\pm}^{i}}$
$\displaystyle=\frac{1}{2}(1\pm B_{i}^{\pm}\cos\theta^{i}_{\pm})$ (72)

The spin correlations for a pair of directions $i,j$ can be measured from the
distribution of the product
$x_{ij}\equiv\cos\theta^{i}_{+}\cos\theta^{j}_{-}$, given by integration of
Eq. (71):

$\frac{1}{\sigma_{\ell\bar{\ell}}}\frac{\mathrm{d}\sigma_{\ell\bar{\ell}}}{\mathrm{d}x_{ij}}=\frac{1}{2}\left[1-C_{ij}x_{ij}\right]\ln\frac{1}{|x_{ij}|}$
(73)

Another measurable magnitude of interest is the trace of the correlation
matrix, which can be directly extracted from

$\frac{1}{\sigma_{\ell\bar{\ell}}}\frac{\mathrm{d}\sigma_{\ell\bar{\ell}}}{\mathrm{d}\cos\varphi}=\frac{1}{2}(1-D\cos\varphi),\leavevmode\nobreak\
D=\frac{\textrm{tr}[\mathbf{C}]}{3}$ (74)

where $\varphi$ is the angle between the lepton directions in each one of
their parent top and antitop rest frames,
$\cos\varphi=\hat{\mathbf{\ell}}_{+}\cdot\hat{\mathbf{\ell}}_{-}$.

The determination of the momentum of the $t\bar{t}$ pair also implies that
their spins can be characterized in any orthonormal basis, e.g. in the
helicity basis considered so far. The measurement of the $t\bar{t}$ spin
polarizations and spin correlations through this technique is well-
established, and has already been performed by the CMS collaboration at the
LHC [39], where $\mathbf{B}^{\pm},\mathbf{C}$ were obtained in the helicity
basis, with no restrictions on $t\bar{t}$ phase space. However, no
entanglement signature was provided by these measurements.

###  5.2 Integrated expectation values

The formalism of the previous subsection focused on integrated values over all
$(M_{t\bar{t}},\hat{k})$ phase space. However, as explained, the $t\bar{t}$
momenta can be experimentally reconstructed, allowing to restrict the
contributions to the integral of Eq. (5.1) to specific regions $\Pi$ of phase
space. In that case, the measured spin polarizations and spin correlations
would correspond to integrated values just within the region $\Pi$. This
allows to further explore the $t\bar{t}$ properties in search of richer
physics. For instance, we can take $\Pi$ within the regions of phase space
where entanglement is present (delimited by the black lines in Figs. 3, 4).

Ideally, one would restrict $\Pi$ to a small bin around a given value
$(M_{t\bar{t}},\hat{k})$, since that would allow to study the individual spin
quantum states $\rho(M_{t\bar{t}},\hat{k},\sqrt{s})$. However, in actual
experiments, the signal needs to be integrated in order to collect a
sufficient number of events for achieving statistically significant
measurements. Therefore, an adaptation of the formalism developed in the
previous sections is required.

We begin by noting that the total quantum state $\rho_{\textrm{T}}(\sqrt{s})$
describing the $t\bar{t}$ pairs produced from a certain hadronic process is
obtained by generalizing Eq. (38) as

$\rho_{\textrm{T}}(\sqrt{s})=\frac{1}{\sigma}\sum_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}\int\limits^{\sqrt{s}}_{2m_{t}}\mathrm{d}M_{t\bar{t}}\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}}\rho_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}(M_{t\bar{t}},\hat{k},\sqrt{s})\frac{\ket{M_{t\bar{t}}\hat{k}\alpha\beta}\bra{M_{t\bar{t}}\hat{k}\alpha^{\prime}\beta^{\prime}}}{\braket{M_{t\bar{t}}\hat{k}}{M_{t\bar{t}}\hat{k}}}$
(75)

If we consider now some observable $O$ diagonal in momentum space, with matrix
elements
$O_{\alpha^{\prime}\beta^{\prime},\alpha\beta}(M_{t\bar{t}},\hat{k})=\bra{M_{t\bar{t}}\hat{k}\alpha^{\prime}\beta^{\prime}}O\ket{M_{t\bar{t}}\hat{k}\alpha\beta}/\braket{M_{t\bar{t}}\hat{k}}{M_{t\bar{t}}\hat{k}}$,
its expectation value $\braket{O}=\textrm{tr}[O\rho_{\textrm{T}}(\sqrt{s})]$
is simply computed as

$\braket{O}=\frac{1}{\sigma}\int\limits^{\sqrt{s}}_{2m_{t}}\mathrm{d}M_{t\bar{t}}\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}}\textrm{tr}\left[O(M_{t\bar{t}},\hat{k})\rho(M_{t\bar{t}},\hat{k},\sqrt{s})\right]$
(76)

This expectation value is readily understood as the average of the expectation
values of $O$ for all possible quantum states
$\rho(M_{t\bar{t}},\hat{k},\sqrt{s})$ of the $t\bar{t}$ pair, with a
probability proportional to the differential cross-section of the process,
normalized by the total cross-section for $t\bar{t}$ production.

The above equation can be simply rewritten in terms of the $R$-matrix as

$\braket{O}=\frac{\int\limits^{\sqrt{s}}_{2m_{t}}\mathrm{d}M_{t\bar{t}}\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\alpha^{2}_{s}\beta}{M^{2}_{t\bar{t}}}\mathrm{tr}\left[O(M_{t\bar{t}},\hat{k})R(M_{t\bar{t}},\hat{k},\sqrt{s})\right]}{\int\limits^{\sqrt{s}}_{2m_{t}}\mathrm{d}M_{t\bar{t}}\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\alpha^{2}_{s}\beta}{M^{2}_{t\bar{t}}}\mathrm{tr}\left[R(M_{t\bar{t}},\hat{k},\sqrt{s})\right]}$
(77)

From these considerations, it is immediate to compute the reduced quantum
state $\rho_{\textrm{T},\Pi}(\sqrt{s})$ that describes the $t\bar{t}$ pair
within a certain region $\Pi$ of phase space by projecting only onto the
relevant states $\ket{M_{t\bar{t}}\hat{k}}$ contained in $\Pi$:

$\rho_{\textrm{T},\Pi}(\sqrt{s})=\frac{1}{\sigma_{\Pi}}\sum_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}\int_{\Pi}\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}}\rho_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}(M_{t\bar{t}},\hat{k},\sqrt{s})\frac{\ket{M_{t\bar{t}}\hat{k}\alpha\beta}\bra{M_{t\bar{t}}\hat{k}\alpha^{\prime}\beta^{\prime}}}{\braket{M_{t\bar{t}}\hat{k}}{M_{t\bar{t}}\hat{k}}}$
(78)

with the partition function $\sigma_{\Pi}$ being the $t\bar{t}$ cross-section
in the region $\Pi$

$\sigma_{\Pi}\equiv\int_{\Pi}\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}}$ (79)

Similar relations can be obtained for the expectation value $\braket{O}_{\Pi}$
of any observable $O$ in the region $\Pi$ by restricting the integrals of Eqs.
(76), (77) to the region $\Pi$.

Furthermore, we can define a genuine two-qubit quantum state by taking the
trace in momentum space

$\displaystyle\rho_{\Pi}(\sqrt{s})$
$\displaystyle\equiv\textrm{tr}_{M_{t\bar{t}}\hat{k}}\left[\rho_{\textrm{T},\Pi}(\sqrt{s})\right]$
$\displaystyle=\sum_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}\rho_{\Pi,\alpha\beta,\alpha^{\prime}\beta^{\prime}}\ket{\alpha\beta}\bra{\alpha^{\prime}\beta^{\prime}}$
(80)

In matrix notation,

$\displaystyle\rho_{\Pi}(\sqrt{s})=\frac{1}{\sigma_{\Pi}}\int_{\Pi}\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}}\rho(M_{t\bar{t}},\hat{k},\sqrt{s})$
$\displaystyle=\frac{\sum_{I}\int_{\Pi}\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}\leavevmode\nobreak\
\frac{\alpha^{2}_{s}\beta}{M^{2}_{t\bar{t}}}L_{I}(M_{t\bar{t}},\sqrt{s})R^{I}(M_{t\bar{t}},\hat{k})}{\sum_{I}\int_{\Pi}\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}\leavevmode\nobreak\
\frac{\alpha^{2}_{s}\beta}{M^{2}_{t\bar{t}}}L_{I}(M_{t\bar{t}},\sqrt{s})4\tilde{A}^{I}(M_{t\bar{t}},\hat{k})}$
(81)

As a two-qubit quantum state, $\rho_{\Pi}$ is determined by the respective
integrated spin polarizations and spin correlations
$\mathbf{B}_{\Pi}^{\pm},\mathbf{C}_{\Pi}$, which can be measured from the
cross-section of the dileptonic decay, Eqs. (72), (73). However, since
$\rho_{\Pi}$ is already the result of an integration in phase space, the
orthonormal basis used to compute $\mathbf{B}_{\Pi}^{\pm},\mathbf{C}_{\Pi}$
cannot depend on $(M_{t\bar{t}},\hat{k})$, which invalidates the use of the
helicity or the diagonal basis. A natural choice is then the beam basis
$\\{\hat{x},\hat{y},\hat{z}\\}$, where $\hat{z}=\hat{p}$ points along the
initial beam, and $\hat{x},\hat{y}$ point transverse directions to the beam,
where all directions are fixed in the c.m. frame [29]. The beam basis is
represented in Fig. 7.

If one inserts the density matrices $\rho(M_{t\bar{t}},\hat{k},\sqrt{s})$
computed in the helicity or the diagonal basis in Eq. (5.2), the resulting
integrated density matrix, which we denote as $\bar{\rho}_{\Pi}(\sqrt{s})$,
does not represent an actual spin quantum state of the $t\bar{t}$ pair, Eq.
(5.2), even though it satisfies the usual properties of a density matrix as it
is a convex sum of physical density matrices (i.e., it is a non-negative
Hermitian matrix with unit trace). The coefficients
$\mathbf{B}_{\Pi}^{\pm},\mathbf{C}_{\Pi}$ characterizing this fictitious
quantum state $\bar{\rho}_{\Pi}(\sqrt{s})$ are precisely the expectation
values in $\Pi$ of the spin polarizations and the spin correlations in the
basis chosen for the integration. Nevertheless, since both the set of
separable and Bell-local states are convex, any signature of entanglement/CHSH
violation in the fictitious quantum state $\bar{\rho}_{\Pi}(\sqrt{s})$ implies
entanglement/CHSH violation in some of the physical substates
$\rho(M_{t\bar{t}},\hat{k},\sqrt{s})$, and so in the total spin-momentum
quantum state $\rho_{\textrm{T},\Pi}$.

The upshot of the discussion of this subsection is that we can aim at
obtaining entanglement signatures by applying the previous criteria to the
corresponding integrated expectation values in selected regions $\Pi$ of phase
space where the $t\bar{t}$ pair is entangled. Moreover, we can analyze
directly if the physical two-qubit quantum state $\rho_{\Pi}$ itself is
entangled. In fact, by measuring the coefficients
$\mathbf{B}_{\Pi}^{\pm},\mathbf{C}_{\Pi}$ we can perform the full quantum
tomography of $\rho_{\Pi}$.

Due to its experimental simplicity and availability of analytical results,
leading to a better understanding of the physics involved, in the following we
take the region $\Pi$ as $\Pi=\Sigma\times S^{2}$. That is, we integrate over
all possible top directions so we only have to specify the cuts in the
invariant mass spectrum that delimit the region $\Sigma$.

Figure 7: Orthonormal beam basis, defined in the c.m. frame. The $\hat{z}$ vector points along the direction of the initial hadron beam, $\hat{z}=\hat{p}$, while $\hat{x},\hat{y}$ point fixed transverse directions. |
---|---
Figure 8: Angular-averaged spin correlations as a function of $\beta$. Main
plot shows $C^{I}_{\perp}$ (solid blue) and $C^{I}_{z}$ (dashed-dotted green).
Inset shows $C^{I}_{rr}$ (solid black), $C^{I}_{nn}$ (dashed orange), and
$C^{I}_{kk}$ (dashed-dotted purple). Left: $I=q\bar{q}$. Dashed red line is
$\Delta_{\Omega}^{q\bar{q}}$. Right: $I=gg$. Dashed red line is
$\mathcal{C}[\rho^{gg}]=\max(\delta_{\Omega}^{gg},0)$. Vertical black lines
represent the critical values $\beta^{\textrm{CH}}_{c}$ (dashed) and
$\beta^{\textrm{PH}}_{c}$ (solid) above which there is no CHSH violation and
entanglement, respectively.

###  5.3 Angular integration

We compute first the distributions resulting from averaging over the angular
coordinates. We start with the $R$-matrix for each partonic initial state $I$:

$\displaystyle R^{I}_{\Omega}(M_{t\bar{t}})$
$\displaystyle=\frac{1}{4\pi}\int\mathrm{d}\Omega\leavevmode\nobreak\
R^{I}(M_{t\bar{t}},\hat{k})$ (82)

The invariance under rotations around the beam axis implies that, in the beam
basis, the correlation matrix describing $R^{I}_{\Omega}(M_{t\bar{t}})$ after
azimuthal integration is diagonal,
$\tilde{C}^{I}_{ij}=\delta_{ij}\tilde{C}^{I}_{j}$, with
$\tilde{C}^{I}_{x}=\tilde{C}^{I}_{y}\equiv\tilde{C}^{I}_{\perp}$. Thus, the
matrices $R^{I}_{\Omega}(M_{t\bar{t}})$ are characterized by just $3$
parameters that can be computed analytically from Eqs. (45), (3.2):
$\tilde{A}^{I}(M_{t\bar{t}}),\tilde{C}^{I}_{\perp}(M_{t\bar{t}})$ and
$\tilde{C}^{I}_{z}(M_{t\bar{t}})$, where $\tilde{A}^{I}(M_{t\bar{t}})$ is the
angular average of $\tilde{A}^{I}(M_{t\bar{t}},\hat{k})$. Actual spin density
matrices $\rho_{\Omega}^{I}(M_{t\bar{t}})$ are obtained from
$R^{I}(M_{t\bar{t}})$ by normalization,
$\rho_{\Omega}^{I}(M_{t\bar{t}})=R_{\Omega}^{I}(M_{t\bar{t}})/4\tilde{A}^{I}(M_{t\bar{t}})$.

With respect to entanglement, due to the symmetry around the beam axis of
$\rho^{I}_{\Omega}$, the Peres-Horodecki criterion is equivalent now to
$\delta_{\Omega}^{I}>0$ [see Eq. (132) and ensuing discussion], with

$\delta_{\Omega}^{I}\equiv\frac{-C^{I}_{z}+|2C^{I}_{\perp}|-1}{2},$ (83)

from where the concurrence reads
$\mathcal{C}[\rho^{I}_{\Omega}]=\max(\delta^{I}_{\Omega},0)$.

Regarding the CHSH violation, $\rho^{I}_{\Omega}$ can achieve it iff
$\mu^{I}_{\Omega}(M_{t\bar{t}})>0$, where now

$\mu^{I}_{\Omega}\equiv\max\left\\{2\left(C^{I}_{\perp}\right)^{2},\left(C^{I}_{\perp}\right)^{2}+\left(C^{I}_{z}\right)^{2}\right\\}-1$
(84)

The angular average of the expectation values in the helicity and the diagonal
basis are computed similarly:

$\displaystyle\tilde{C}^{I}_{ij}(M_{t\bar{t}})\equiv\frac{1}{4\pi}\int\mathrm{d}\Omega\leavevmode\nobreak\
\tilde{C}^{I}_{ij}(M_{t\bar{t}},\hat{k})$ $\displaystyle
C^{I}_{ij}(M_{t\bar{t}})=\frac{\tilde{C}^{I}_{ij}(M_{t\bar{t}})}{\tilde{A}^{I}(M_{t\bar{t}})}$
(85)

We note that the matrix $\tilde{C}^{I}_{ij}(M_{t\bar{t}})$ is also diagonal in
the helicity basis after the angular averaging since
$\tilde{C}^{I}_{kr}(M_{t\bar{t}})$ vanishes at LO due to the odd parity under
inversion of the coefficient $\tilde{C}^{I}_{kr}(M_{t\bar{t}},\hat{k})$.

Moreover, a complementary entanglement signature can be obtained from
$\Delta^{I}_{\Omega}(M_{t\bar{t}})>0$, with

$\Delta^{I}_{\Omega}\equiv\frac{-C_{nn}^{I}(M_{t\bar{t}})+|C_{kk}^{I}(M_{t\bar{t}})+C_{rr}^{I}(M_{t\bar{t}})|-1}{2},$
(86)

since, if all substates $\rho^{I}(M_{t\bar{t}},\hat{k})$ for fixed
$M_{t\bar{t}}$ were separable, $\Delta^{I}(M_{t\bar{t}},\hat{k})\leq 0$ and so
would its angular average $\braket{\Delta^{I}(M_{t\bar{t}})}_{\Omega}$, which
satisfies
$\Delta^{I}_{\Omega}(M_{t\bar{t}})\leq\braket{\Delta^{I}(M_{t\bar{t}})}_{\Omega}$.
Hence, $\Delta^{I}_{\Omega}(M_{t\bar{t}})>0$ necessarily implies that some of
the substates $\rho^{I}(M_{t\bar{t}},\hat{k})$ must be entangled. Another way
to put it is that $\Delta^{I}_{\Omega}(M_{t\bar{t}})>0$ is an entanglement
criterion for a fictitious quantum state
$\bar{\rho}^{I}_{\Omega}(M_{t\bar{t}})$ [see discussion after Eq. (5.2)].
Similarly, one can obtain a criterion for CHSH violation
$\mathcal{B}^{I}_{\Omega}>2$ that results from the angular integration of
$\mathcal{B}[\rho^{I}(M_{t\bar{t}},\hat{k})]$, see Eq. (2.1).

Technical details about the analytical calculation of the angular integrals
are provided in Appendix D.

####  5.3.1 $q\bar{q}$ processes

We obtain for $R^{q\bar{q}}_{\Omega}(M_{t\bar{t}})$ that

$\displaystyle\tilde{A}^{q\bar{q}}(M_{t\bar{t}})$
$\displaystyle=\frac{1}{9}\left[1-\frac{\beta^{2}}{3}\right]$ (87)
$\displaystyle\tilde{C}_{\perp}^{q\bar{q}}(M_{t\bar{t}})$
$\displaystyle=\frac{2}{135}f(\beta)$
$\displaystyle\tilde{C}_{z}^{q\bar{q}}(M_{t\bar{t}})$
$\displaystyle=\frac{1}{9}\left[1-\frac{\beta^{2}}{3}-\frac{4}{15}f(\beta)\right]$
$\displaystyle f(\beta)$
$\displaystyle\equiv\frac{\left(1-\sqrt{1-\beta^{2}}\right)^{2}}{2}$

while for the angular averages of the helicity spin correlations we find

$\displaystyle\tilde{C}^{q\bar{q}}_{rr}(M_{t\bar{t}})$
$\displaystyle=\frac{(2-\beta^{2})}{27}$ (88)
$\displaystyle\tilde{C}^{q\bar{q}}_{nn}(M_{t\bar{t}})$
$\displaystyle=-\frac{\beta^{2}}{27}$
$\displaystyle\tilde{C}^{q\bar{q}}_{kk}(M_{t\bar{t}})$
$\displaystyle=\frac{1+\beta^{2}}{27}$

These results imply that the quantum state $\rho_{\Omega}^{q\bar{q}}$ is
completely separable as

$\delta_{\Omega}^{q\bar{q}}(M_{t\bar{t}})=\frac{-1+\frac{\beta^{2}}{3}+\frac{4}{15}f(\beta)}{1-\frac{\beta^{2}}{3}}<0$
(89)

We note that this only means that the angular-averaged state
$\rho_{\Omega}^{q\bar{q}}$ is separable, but not that entanglement disappears
after the angular average. Indeed,

$\Delta_{\Omega}^{q\bar{q}}(M_{t\bar{t}})=\frac{\beta^{2}}{3-\beta^{2}}\geq 0$
(90)

which means that entanglement is still present in the whole energy range
(except exactly at threshold, $\beta=0$). Since in the diagonal basis
$C_{+}(M_{t\bar{t}},\hat{k})=1$, its angular average also satisfies
$C_{+}(M_{t\bar{t}})=1$, and thus there is CHSH violation for any non-zero
$\beta$, $\mathcal{B}^{q\bar{q}}_{\Omega}(\beta)>2$. Left Fig. 8 displays all
the angular-averaged magnitudes for $I=q\bar{q}$.

####  5.3.2 $gg$ processes

We obtain for $R_{\Omega}^{gg}(M_{t\bar{t}})$ that

$\displaystyle\tilde{A}^{gg}(M_{t\bar{t}})$
$\displaystyle=\frac{1}{192}\left[-59+31\beta^{2}+(66-36\beta^{2}+2\beta^{4})\frac{\textrm{atanh}(\beta)}{\beta}\right]$
(91) $\displaystyle\tilde{C}^{gg}_{\perp}(M_{t\bar{t}})$
$\displaystyle=\frac{1-\beta^{2}}{192}\left[9-16\frac{\textrm{atanh}(\beta)}{\beta}\right]+g(\beta)$
$\displaystyle\tilde{C}^{gg}_{z}(M_{t\bar{t}})$
$\displaystyle=\frac{1}{192}\left[-109+49\beta^{2}+(102-72\beta^{2}+2\beta^{4})\frac{\textrm{atanh}(\beta)}{\beta}\right]-2g(\beta)$
$\displaystyle g(\beta)$
$\displaystyle\equiv\frac{f(\beta)}{96\beta^{4}}\left[49-\frac{149}{3}\beta^{2}+\frac{24}{5}\beta^{4}-(49-66\beta^{2}+17\beta^{4})\frac{\textrm{atanh}(\beta)}{\beta}\right]$

while for the helicity spin correlations we find

$\displaystyle\tilde{C}^{gg}_{rr}(M_{t\bar{t}})$
$\displaystyle=-\frac{1}{192}\left[87-31\beta^{2}+66\frac{\frac{\textrm{atanh}(\beta)}{\beta}-1}{\beta^{2}}-\left(102-38\beta^{2}+2\beta^{4}\right)\frac{\textrm{atanh}(\beta)}{\beta}\right]$
(92) $\displaystyle\tilde{C}^{gg}_{nn}(M_{t\bar{t}})$
$\displaystyle=-\frac{1}{192}\left[41-31\beta^{2}-\left(34-36\beta^{2}+2\beta^{4}\right)\frac{\textrm{atanh}(\beta)}{\beta}\right]$
$\displaystyle\tilde{C}^{gg}_{kk}(M_{t\bar{t}})$
$\displaystyle=-\frac{1}{192}\left[-37+31\beta^{2}-66\frac{\frac{\textrm{atanh}(\beta)}{\beta}-1}{\beta^{2}}+\left(66-34\beta^{2}+2\beta^{4}\right)\frac{\textrm{atanh}(\beta)}{\beta}\right]$

We note that $C^{gg}_{\perp}(M_{t\bar{t}})<0$ for all the energies of interest
(the sign crossover is produced only in the ultrarelativistic limit
$\beta\approx 0.970$). This implies that for practical purposes we can take
$\delta_{\Omega}^{gg}(M_{t\bar{t}})$ as

$\delta_{\Omega}^{gg}=-\frac{1+\textrm{tr}[\mathbf{C}^{gg}]}{2}=\frac{75-31\beta^{2}-\left(68-38\beta^{2}+2\beta^{4}\right)\frac{\textrm{atanh}(\beta)}{\beta}}{-59+31\beta^{2}+(66-36\beta^{2}+2\beta^{4})\frac{\textrm{atanh}(\beta)}{\beta}}$
(93)

From this expression, we compute the critical top velocity
$\beta^{\textrm{PH}}_{c}\approx 0.632$ below which the state
$\rho^{gg}_{\Omega}$ is still entangled, with the associated critical mass
being $M^{\textrm{PH}}_{c}\approx 446\leavevmode\nobreak\ \mathrm{GeV}$.

On the other hand, since in the entangled region
$|C^{gg}_{\perp}|>|C^{gg}_{z}|$, $\rho^{gg}_{\Omega}$ can violate the CHSH
inequality iff

$\mu^{gg}_{\Omega}=2\left(C^{gg}_{\perp}\right)^{2}-1>0$ (94)

This condition is satisfied below the critical value
$\beta^{\textrm{CH}}_{c}\approx 0.378<\beta^{\textrm{PH}}_{c}$, corresponding
to a critical mass $M^{\textrm{CH}}_{c}\approx 374\leavevmode\nobreak\
\mathrm{GeV}$.

We can also reproduce these results from the spin correlations in the helicity
basis. Indeed, since

$C^{gg}_{kk}+C^{gg}_{rr}=-\frac{50-\left(36-4\beta^{2}\right)\frac{\textrm{atanh}(\beta)}{\beta}}{-59+31\beta^{2}+(66-36\beta^{2}+2\beta^{4})\frac{\textrm{atanh}(\beta)}{\beta}}$
(95)

we have that $C^{gg}_{kk}+C^{gg}_{rr}<0$ for $\beta<\beta_{\Delta}\approx
0.864$, and thus, in the energy range where the state is entangled,
$\delta_{\Omega}^{gg}=\Delta_{\Omega}^{gg}=-(1+\textrm{tr}[\mathbf{C}^{gg}])/2$.
In fact, this result can be obtained in any orthonormal basis due to the
rotational invariance of the trace,
$\textrm{tr}[\mathbf{C}^{gg}]=2C^{gg}_{\perp}+C^{gg}_{z}=C^{gg}_{rr}+C^{gg}_{nn}+C^{gg}_{kk}$,
reflecting the symmetry of the spin-singlet state.

However, entanglement is again lost at high energies for $\rho^{gg}_{\Omega}$:
even if we use the full $\Delta_{\Omega}^{gg}$,

$\Delta_{\Omega}^{gg}=\frac{50-31\beta^{2}-\left(50-36\beta^{2}+2\beta^{4}\right)\frac{\textrm{atanh}(\beta)}{\beta}+\left|25-\left(18-2\beta^{2}\right)\frac{\textrm{atanh}(\beta)}{\beta}\right|}{-59+31\beta^{2}+(66-36\beta^{2}+2\beta^{4})\frac{\textrm{atanh}(\beta)}{\beta}}$
(96)

we do not get any entanglement signature, in contrast to $q\bar{q}$ processes.
Right Fig. 8 displays all the angular-averaged magnitudes for $I=gg$.

####  5.3.3 General considerations

We analyze the global results of Fig. 8 for both processes in the light of the
2D plots of Fig. 3 for $\rho^{I}(M_{t\bar{t}},\hat{k})$. The entanglement loss
in $\rho_{\Omega}^{I}(M_{t\bar{t}})$ for both quark and gluon processes arises
due to the statistical average over all possible top directions. However,
close to threshold, gluon fusion produces a $t\bar{t}$ pair in a spin singlet
[see Eq. (54)], invariant under rotations, and thus unaffected by the angular
average, keeping the entanglement.

Nevertheless, an entanglement signature can be recovered for $q\bar{q}$
processes by averaging $\Delta^{q\bar{q}}$, since the helicity basis changes
accordingly its orientation to produce a constructive sum of the spin
correlations. The same technique does not work for the $gg$ channel at high
energies because there is no entanglement close to forward production
$\Theta=0$, which spoils the high-$p_{T}$ entanglement when averaging over all
top directions.

l31ptt22pt(a) |  l37ptt25pt(b)
---|---
l31ptt22pt(c) |  l39ptt25pt(d)
Figure 9: Upper row: Analysis of $t\bar{t}$ production at the LHC for
$\sqrt{s}=13$ TeV. (a) 2D plot of the concurrence as in Figs. 3, 4. (b)
Integrated spin correlations $C_{\perp}$ (solid blue), $C_{z}$ (dashed dotted
green) and $D$ (dashed red) for $\rho(M_{t\bar{t}},\sqrt{s})$. The horizontal
and vertical solid black lines signal the entanglement limit $D=-1/3$ while
dashed ones signal the CHSH violation limit $|C_{\perp}|=1/\sqrt{2}$. Lower
row: Analysis of $t\bar{t}$ production at the Tevatron for $\sqrt{s}=2$ TeV.
(c) Same as (a). (d) Integrated correlations in the spin helicity basis:
$C_{rr}$ (solid black), $C_{nn}$ (dashed orange), and $C_{kk}$ (dashed-dotted
purple). Dashed red and solid blue lines show the integrated value of $\Delta$
and $\mathcal{B}/2$, respectively. Horizontal and vertical dashed black lines
signal the CHSH violation limit $\mathcal{B}/2=1$.

###  5.4 Mass integration

With the help of the angular-averaged $R$-matrix, it is quite simple to
compute expectation values in the region $\Pi$ of phase space by integrating
in the mass range specified by $\Sigma$. Since mass integration involves
luminosity functions (which we stress that do not depend on the $t\bar{t}$
direction), one has necessarily to consider specific hadron processes.

The angular average of the total matrix $R(M_{t\bar{t}},\hat{k},\sqrt{s})$
describing a realistic hadronic $t\bar{t}$ production process for fixed c.m.
energy is

$R_{\Omega}(M_{t\bar{t}},\sqrt{s})=\sum_{I=q\bar{q},gg}L_{I}(M_{t\bar{t}},\sqrt{s})R_{\Omega}^{I}(M_{t\bar{t}})$
(97)

The actual spin density matrix
$\rho_{\Omega}(M_{t\bar{t}},\sqrt{s})=R_{\Omega}(M_{t\bar{t}},\sqrt{s})/4\tilde{A}(M_{t\bar{t}},\sqrt{s})$
is computed from its partonic counterparts $\rho_{\Omega}^{I}(M_{t\bar{t}})$
as in Eq. (58):

$\rho_{\Omega}(M_{t\bar{t}},\sqrt{s})=\sum_{I=q\bar{q},gg}w_{I}(M_{t\bar{t}},\sqrt{s})\rho_{\Omega}^{I}(M_{t\bar{t}})$
(98)

where the probabilities $w_{I}(M_{t\bar{t}},\sqrt{s})$ are now

$w_{I}(M_{t\bar{t}},\sqrt{s})=\frac{L_{I}(M_{t\bar{t}},\sqrt{s})\tilde{A}^{I}(M_{t\bar{t}})}{\sum_{J}L_{J}(M_{t\bar{t}},\sqrt{s})\tilde{A}^{J}(M_{t\bar{t}})}$
(99)

Finally, by Eq. (5.2), the density matrix $\rho_{\Pi}$ reads in terms of these
angular-averaged magnitudes as

$\displaystyle\rho_{\Pi}(\sqrt{s})$
$\displaystyle=\frac{\sum_{I}\int_{\Sigma}\mathrm{d}M_{t\bar{t}}\leavevmode\nobreak\
\frac{\alpha^{2}_{s}\beta}{M^{2}_{t\bar{t}}}L_{I}(M_{t\bar{t}},\sqrt{s})R_{\Omega}^{I}(M_{t\bar{t}})}{\sum_{I}\int_{\Sigma}\mathrm{d}M_{t\bar{t}}\leavevmode\nobreak\
\frac{\alpha^{2}_{s}\beta}{M^{2}_{t\bar{t}}}L_{I}(M_{t\bar{t}},\sqrt{s})4\tilde{A}^{I}(M_{t\bar{t}})}$
$\displaystyle=\frac{1}{\sigma_{\Sigma}}\int_{\Sigma}\mathrm{d}M_{t\bar{t}}\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}M_{t\bar{t}}}\rho_{\Omega}(M_{t\bar{t}},\sqrt{s})$
(100)

with

$\frac{\mathrm{d}\sigma}{\mathrm{d}M_{t\bar{t}}}=\int\mathrm{d}\Omega\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega\mathrm{d}M_{t\bar{t}}},\leavevmode\nobreak\
\sigma_{\Sigma}=\int_{\Sigma}\mathrm{d}M_{t\bar{t}}\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}M_{t\bar{t}}}$ (101)

Similar considerations apply to the mass integration of the angular-averaged
expectation values, such as the spin correlations in the helicity basis.

Because of their illustrative character and their high-experimental relevance,
we focus on two particular hadronic processes: $p\bar{p}$ collisions at
$\sqrt{s}=2\leavevmode\nobreak\ \textrm{TeV}$ (very close to the actual value
$\sqrt{s}=1.96\leavevmode\nobreak\ \textrm{TeV}$ at the Tevatron), and $pp$
collisions at $\sqrt{s}=13\leavevmode\nobreak\ \textrm{TeV}$ (corresponding to
the c.m. energy of Run 2 at the LHC). As shown in Fig. 5, at the Tevatron
$q\bar{q}$ processes dominate while at the LHC $gg$ processes do. The 2D plot
of their concurrences is depicted in left column of Fig. 9. We focus on
integrating the signal in the two relevant regions for entanglement: close to
threshold and at high $p_{T}$.

####  5.4.1 Threshold analysis

Close to threshold, the angular results of the previous subsection predict
that only the $gg$ channel gives rise to entangled $t\bar{t}$ pairs.
Therefore, we restrict to the LHC example and take
$\Sigma=[2m_{t},M_{t\bar{t}}]$, which means that only events in the window
$[2m_{t},M_{t\bar{t}}]$ are selected, so

$\displaystyle\rho_{\Pi}(\sqrt{s})$
$\displaystyle=\rho(M_{t\bar{t}},\sqrt{s})$ (102)
$\displaystyle\equiv\frac{1}{\sigma(M_{t\bar{t}})}\int^{M_{t\bar{t}}}_{2m_{t}}\mathrm{d}M\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}M}\rho_{\Omega}(M,\sqrt{s})$

with $\sigma(M_{t\bar{t}})$ the total integrated cross-section in the same
mass window. Since it is computed in terms of the angular-averaged substates
$\rho_{\Omega}(M_{t\bar{t}},\sqrt{s})$, the quantum state
$\rho(M_{t\bar{t}},\sqrt{s})$ is also characterized by its integrated
transverse and longitudinal spin correlations $C_{\perp},C_{z}$, represented
in Fig. 9b. The necessary and sufficient condition for
$\rho(M_{t\bar{t}},\sqrt{s})$ to be entangled is that the integrated value
$\delta(M_{t\bar{t}},\sqrt{s})$ satisfies $\delta(M_{t\bar{t}},\sqrt{s})>0$.
Remarkably, this implies that a directly measurable entanglement witness $W$,
which satisfies $W<0$ only for entangled states [67], is provided by the
observable $D$ of Eq. (74),

$W\equiv D+1/3$ (103)

The concurrence is also readily computed from $D$ as
$\mathcal{C}[\rho]=\max(-1-3D,0)/2$. The integrated value of $D$ is
represented in Fig. 9b, along with the critical value
$\beta^{\textrm{PH}}_{c}(\sqrt{s})$ (marked by horizontal and vertical solid
black lines) below which $W<0$.

Regarding the CHSH violation, its presence can be also signaled by just one
parameter, $C_{\perp}(M_{t\bar{t}},\sqrt{s})$, as given by the condition
$\mu(M_{t\bar{t}},\sqrt{s})=2\left[C_{\perp}(M_{t\bar{t}},\sqrt{s})\right]^{2}-1>0$.
The critical value $\beta^{\textrm{CH}}_{c}(\sqrt{s})$ below which there is
CHSH violation, $|C_{\perp}(\beta^{\textrm{CH}}_{c},\sqrt{s})|=1/\sqrt{2}$, is
marked in Fig. 9b by horizontal and vertical dashed black lines.

We compute the critical values
$\beta_{c}^{\textrm{CH}}(\sqrt{s})\leq\beta_{c}^{\textrm{PH}}(\sqrt{s})$ as a
function of the c.m. energy and represent them in Fig. 10. We compare these
critical values with those for the angular-averaged substates
$\rho^{gg}_{\Omega}(M_{t\bar{t}})$, $\beta_{c}^{\textrm{CH}}\approx
0.378<\beta_{c}^{\textrm{PH}}\approx 0.632$ (horizontal dashed and solid
lines). We observe that, for sufficiently large c.m. energies, the critical
value $\beta_{c}^{\textrm{PH}}(\sqrt{s})$ exceeds its angular-averaged
counterpart, even though $\rho(M_{t\bar{t}},\sqrt{s})$ contains some mixing
with $q\bar{q}$ processes that reduces the entanglement (see lower part of
right Fig. 3 and Fig. 9a). This exceeding arises from the fact that the total
quantum state $\rho(M_{t\bar{t}},\sqrt{s})$ is a convex sum of the substates
$\rho^{I}_{\Omega}(M_{t\bar{t}})$. Therefore, one needs higher energies to
include a sufficient amount of separable states in order to dilute the
entanglement. Nevertheless, if the integration window $\Sigma$ was entirely
placed in the region of separability of $\rho^{I}_{\Omega}(M_{t\bar{t}})$, no
entanglement would be observed. In contrast, for the CHSH violation,
$\beta_{c}^{\textrm{CH}}(\sqrt{s})$ is always below its angular-averaged
counterpart, at least within the range of energies considered. This is because
of the critical effect introduced by the mixing with $q\bar{q}$ processes at
threshold, see Fig. 5, which cannot be overcome by the opposite effect of mass
integration due to the lower size of the CHSH-violating window. Indeed, for
the considered energy of Run 2, $\sqrt{s}=13\leavevmode\nobreak\
\textrm{TeV}$, the weight of the $gg$ channel at threshold is barely above the
critical value $1/\sqrt{2}$, Fig. 5b.

####  5.4.2 High-$p_{T}$ analysis

For high $p_{T}$ at the Tevatron, entanglement is not present in the two-qubit
quantum state $\rho_{\Pi}$ but is instead signaled by the integrated value of
$\Delta_{\Omega}$. Indeed, Fig. 9c suggests to choose now the mass window
$\Sigma$ as $[M_{t\bar{t}},\sqrt{s}]$:

$\Delta(M_{t\bar{t}},\sqrt{s})=\frac{1}{\bar{\sigma}(M_{t\bar{t}})}\int^{\sqrt{s}}_{M_{t\bar{t}}}\mathrm{d}M\leavevmode\nobreak\
\frac{\mathrm{d}\sigma}{\mathrm{d}M}\Delta_{\Omega}(M,\sqrt{s})$ (104)

with $\bar{\sigma}(M_{t\bar{t}})=\sigma-\sigma(M_{t\bar{t}})$. We define in a
similar fashion the integrated value of the CHSH violation,
$\mathcal{B}(M_{t\bar{t}},\sqrt{s})$.

The values of $\Delta(M_{t\bar{t}},\sqrt{s})$ and
$\mathcal{B}(M_{t\bar{t}},\sqrt{s})/2$, along with the integrated helicity
spin correlations, are represented in Fig. 9d. Interestingly, even though the
$t\bar{t}$ pairs produced close to threshold are not entangled (see lower Fig.
9c), the integration in the mass range makes $\Delta(M_{t\bar{t}},\sqrt{s})>0$
in the whole mass spectrum, similarly to the increase of the critical value of
$\beta_{c}^{\textrm{PH}}(\sqrt{s})$ for $gg$ processes. However, this does not
apply to the CHSH violation, which can only be observed now at high energies
$\beta>\beta^{\textrm{CH}}_{c}\approx 0.652$.

Within the angular-averaged scheme considered here, no entanglement can be
detected for $gg$ processes at high $p_{T}$. We note, however, that
entanglement and CHSH violation can be observed at the LHC for high $p_{T}$ if
one also introduces a cut in the top direction, as suggested by Fig. 9a and
implemented in Refs. [44, 45].

Figure 10: Critical values $\beta_{c}^{\textrm{PH}}(\sqrt{s})$ (solid blue)
and $\beta_{c}^{\textrm{CH}}(\sqrt{s})$ (dashed red) below which entanglement
and CHSH violation can be observed, respectively, close to threshold as a
function of the c.m. energy $\sqrt{s}$ for $pp$ collisions. The horizontal
lines mark the critical values $\beta_{c}^{\textrm{CH}}$ (dashed) and
$\beta_{c}^{\textrm{PH}}$ (solid) for $\rho^{gg}_{\Omega}$. Black markers
indicate values for c.m. energies at the LHC. Circles: Run 1
($\sqrt{s}=7,8\leavevmode\nobreak\ \textrm{TeV}$), Run 2
($\sqrt{s}=13\leavevmode\nobreak\ \textrm{TeV}$) and Run 3
($\sqrt{s}=13.6\leavevmode\nobreak\ \textrm{TeV}$). Squares: Possible upgrade
of the LHC ($\sqrt{s}=27\leavevmode\nobreak\ \textrm{TeV}$).

##  6 Experimental remarks

We complement here the theoretical predictions for observables of the previous
section with some experimental remarks about the actual measurement scheme in
high-energy colliders, and the potential statistical significance of the
discovery. Nevertheless, a dedicated analysis of the expected experimental
sensitivity of each prediction is beyond the scope of the present work.

###  6.1 Entanglement detection

The detection of entanglement is more delicate than it could be naively
expected from Figs. 3, 4 since, even though entanglement is present in a wide
region of phase space, one has to select carefully the phase space regions
where the signal is integrated in order to obtain an entanglement signature.
In fact, the recent measurement of the CMS collaboration at the LHC with the
data of Run 2 [39], with no restrictions in phase space, yielded no
entanglement signature, $D=-0.237\pm 0.011>-1/3$.

Our analysis reveals that there are two main regions where entanglement can be
in principle detected: close to threshold and for high transverse momentum.

####  6.1.1 Threshold

Close to threshold, as shown in Fig. 5, the only experiment with high enough
c.m. energy to give rise to entanglement is the LHC. The idea here is to
measure $D$ from the cross-section of Eq. (74), applying an upper cut in the
mass spectrum for the integrated signal. Although our calculation is
restricted to LO, in general, the criterion $W=D+1/3<0$ still provides a
sufficient condition for the entanglement of $\rho(M_{t\bar{t}},\sqrt{s})$
without any assumption on its specific form [see Eq. (124)].

The entanglement witness $W<0$ can be rewritten as the violation of a Cauchy-
Schwarz inequality, a typical entanglement signature in other fields such as
quantum optics, condensed matter, or analog Hawking radiation [50, 68, 69,
70]. This is proven by using the $P$-representation of Eq. (5). If a state is
separable, $P(\mathbf{n}_{A},\mathbf{n}_{B})>0$, and then

$\displaystyle|\textrm{tr}\leavevmode\nobreak\ \mathbf{C}|$
$\displaystyle=\left|\braket{\mathbf{\sigma}\cdot\mathbf{\sigma}}\right|$
$\displaystyle=\left|\int\mathrm{d}\Omega_{A}\mathrm{d}\Omega_{B}\leavevmode\nobreak\
P(\mathbf{n}_{A},\mathbf{n}_{B})\mathbf{n}_{A}\cdot\mathbf{n}_{B}\right|$
$\displaystyle\leq\int\mathrm{d}\Omega_{A}\mathrm{d}\Omega_{B}\leavevmode\nobreak\
P(\mathbf{n}_{A},\mathbf{n}_{B})\left|\mathbf{n}_{A}\cdot\mathbf{n}_{B}\right|$
$\displaystyle\leq\int\mathrm{d}\Omega_{A}\mathrm{d}\Omega_{B}\leavevmode\nobreak\
P(\mathbf{n}_{A},\mathbf{n}_{B})=1$ (105)

Hence, the directly measurable observable $D$ represents in the range of
values $-1\leq D<-1/3$ a genuine non-classical feature, which can be
qualitatively understood from the fact that the classical average of the
scalar product of two vectors with unit length is never larger than one,
$3|D|=|\textrm{tr}\leavevmode\nobreak\
\mathbf{C}|=|\braket{\mathbf{\sigma}\cdot\mathbf{\sigma}}|\leq 1$. A similar
entanglement criterion based on the trace of the correlation matrix was
derived for Heisenberg spin chains [71].

A dedicated analysis of the entanglement discovery was performed in Refs. [41,
45], finding that high-statistical significance can be expected, potentially
more than $5$ statistical deviations ($5\sigma$), the standard candle for
discovery in high-energy physics.

Figures 5, 10 show that the effect of increasing the c.m. energy saturates for
large energies. Therefore, we do not expect that future hadron colliders with
higher collision energies, such as possible upgrades of the LHC or the FCC,
will strongly increase the entanglement in $t\bar{t}$ production. However, we
do expect that a larger data collection will enhance the sensitivity of the
measurements by reducing the statistical uncertainties. For example, the LHC
is expected to collect about $20-30$ times more events with respect to the
currently recorded data [72], a possible upgrade of the LHC to access higher
energies (HE-LHC) is expected to collect about $100$ times more events [65],
and the FCC is expected to collect about $140-220$ times more events [73].
Another possibility to increase entanglement is to enhance the contribution
from the $gg$ channel by further rejecting events from the $q\bar{q}$ channel,
as recently proposed in Ref. [46].

####  6.1.2 High $p_{T}$

For high $p_{T}$, both partonic processes $q\bar{q}$ and $gg$ give rise to the
same triplet state due to orbital angular momentum dominance. Nevertheless,
because the procedure used here averages over all possible top directions,
only $q\bar{q}$ processes yield an entanglement signature in our scheme. A
detailed study of the statistical significance of an entanglement detection
for high $p_{T}$ at the LHC has been already provided in Ref. [45], predicting
also a potential $5\sigma$ discovery.

Regarding the Tevatron, entanglement is detected from the integrated value of
$\Delta$, which is in general a sufficient condition for the presence of
entanglement [see Eq. (123)]. There are important conceptual differences
between the detection of entanglement close to threshold (involving the
measurement of $D$) and that at high $p_{T}$ (involving the measurement of
$\Delta$). First, close to threshold, entanglement is detected from one single
magnitude, $D$, while $\Delta$ requires the measurement of $3$ magnitudes,
which are the diagonal spin correlations in the helicity or the diagonal basis
(we note that a recent work has proposed to measure $\Delta$ also from a
single parameter [46]). Second, $D+1/3<0$ reveals that the physical two-qubit
quantum state $\rho_{\Pi}$ is entangled, while $\Delta>0$ signals entanglement
in the fictitious quantum state $\bar{\rho}_{\Pi}$, and so in the total spin-
momentum quantum state $\rho_{\textrm{T},\Pi}$.

At the Tevatron, the relatively low c.m. energy and data recorded yielded high
uncertainties on spin-correlation observables [33, 34, 35]. Therefore, by
considering typical uncertainties of the carried measurements, we do not
expect in principle to find there an entanglement signature with high-
statistical significance.

####  6.1.3 Lorentz invariance

Finally, we would like to discuss the possible role of the reference frame
choice on the entanglement detection. While our calculations assume that the
$t\bar{t}$ momenta for each production process are well-defined and therefore
their spin-entanglement is Lorentz invariant, in realistic situations their
wave function is necessarily described by wave packets with finite width, so
their entanglement is no longer independent of the reference frame. However,
within the current experimental scheme, the $t\bar{t}$ spins are by definition
measured in their respective rest frames, since these are precisely the frames
where the lepton directions $\mathbf{\ell}_{\pm}$ are defined in Eq. (68).

###  6.2 Quantum tomography

A protocol for the quantum tomography of the $t\bar{t}$ pair was developed in
Ref. [41], based on the fact that the experimental procedure of Section 5.1
allows to measure the integrated value of the spin correlations and spin
polarizations. Thus, by implementing a cut in the invariant mass spectrum, one
can extract all the parameters characterizing the total quantum state
$\rho(M_{t\bar{t}},\sqrt{s})$ of Eq. (102). As explained, such a
reconstruction requires the use of a fixed orthonormal basis in space, like
the beam basis.

We extend here the proposed quantum tomography protocol to more general
situations. First, we note that the protocol is not restricted to the state
$\rho(M_{t\bar{t}},\sqrt{s})$, but it is rather valid for any arbitrary
integrated quantum state $\rho_{\Pi}$ by measuring all the $15$ parameters
$B^{\pm}_{i},C_{ij}$ in the beam basis. We can further simplify the process if
we integrate over the azimuth around the beam axis, since then rotational
invariance is recovered and thus, at LO, only $2$ parameters are needed for
quantum tomography: the integrated transverse and longitudinal spin
correlations $C_{\perp},C_{z}$. In general, by only assuming symmetry around
the beam axis, the quantum tomography of the $t\bar{t}$ pair requires the
measurement of just $4$ parameters, $B^{\pm}_{z},C_{\perp},C_{z}$, with
$B^{\pm}_{z}$ the spin polarizations along the beam axis. Nevertheless, from
the recent CMS measurement [40], the values of these polarizations are still
expected to be quite small. A summary of the parameters needed for the quantum
tomography of the $t\bar{t}$ pair is presented in Table 1.

Assumption | Coefficients | #Parameters
---|---|---
Symmetry + LO | $C_{\perp},C_{z}$ | 2
Symmetry | $B^{\pm}_{z},C_{\perp},C_{z}$ | 4
None | $B^{\pm}_{i},C_{ij}$ | 15
Table 1: Summary of the parameters needed to be measured in order to perform
the quantum tomography of the $t\bar{t}$ pair for different assumptions on the
form of $\rho_{\Pi}$. “Symmetry” denotes symmetry around the beam axis,
obtained after integration over the azimuth in the beam basis.

Moreover, since the quantum tomography protocol only depends on the decay
properties of the $t\bar{t}$ pair and not on the nature of the production
process, it can be extended to any $t\bar{t}$ production mechanism, such as
positron-electron ($e^{+}e^{-}$) collisions. Furthermore, even though we have
focused on the case of a dileptonic decay of the $t\bar{t}$ pair, Eqs. (68),
(5.1) are valid for any pair of detectable decay products by just replacing
$\mathbf{\ell}_{\pm}$ and $\kappa_{\ell},\bar{\kappa}_{\ell}$ by the
corresponding flight directions in the parent rest frames and spin analyzing
powers. We conclude by noting that the proposed quantum tomography protocol
for the $t\bar{t}$ pair goes beyond the general approach to high-energy
processes presented in Ref. [60].

###  6.3 CHSH violation

Since entanglement is a necessary condition for the violation of CHSH
inequalities, CHSH violations are generally expected to be measured with lower
statistical significance than entanglement [45]. This is clearly seen by
examining Fig. 10, where we see that the upper critical value for a CHSH
violation is well below that for entanglement. Remarkably, in a similar
fashion to entanglement, the CHSH violation close to threshold is also
inferred from the value of just one parameter, $C_{\perp}$. Nevertheless, one
can always measure the full CHSH violation in a more conventional way through
Eq. (10). Due to the rotational invariance in the perpendicular plane to the
beam, the experimental scheme is similar to the usual case of Bell
inequalities with spin-singlet states, and one chooses four vectors
$\mathbf{a}_{1},\mathbf{a}_{2},\mathbf{b}_{1},\mathbf{b}_{2}$ in that plane
satisfying
$\mathbf{a}_{1}\cdot\mathbf{a}_{2}=\mathbf{b}_{1}\cdot\mathbf{b}_{2}=0$ and
$\mathbf{a}_{1}\cdot\mathbf{b}_{1}=\mathbf{a}_{2}\cdot\mathbf{b}_{2}=\mathbf{a}_{2}\cdot\mathbf{b}_{1}=-\mathbf{a}_{1}\cdot\mathbf{b}_{2}=1/\sqrt{2}$.
Since the correlation matrix should be isotropic in this plane, this
experimental scheme is equivalent to measure $|C_{\perp}|>1/\sqrt{2}$. We note
that our choice of spin directions is fixed a priori, so no statistical bias
is introduced [45].

Regarding the statistical significance of the CHSH violation close to
threshold, we expect it to be much lower than that of entanglement. However,
even if a significant detection is not achievable with the current Run 2 data,
it could be performed with the increased c.m. energies (as seen from Fig. 10)
and larger amount of data provided by future runs and/or colliders. At the
Tevatron, since entanglement detection already seems quite challenging, we do
not expect to achieve a statistically significant observation of a CHSH
violation.

As for the case of entanglement, the scheme presented here does not allow to
observe CHSH violation at the LHC at high $p_{T}$. Nevertheless, a thorough
analysis of the potential detection of CHSH violation in this regime and its
statistical significance is presented in Refs. [44, 45].

So far, we have discussed the violation of the CHSH inequality of Eq. (10).
However, a measurement of a genuine violation of Bell theorem implies much
more than just measuring some linear combination of spin correlations above
certain critical value, like in the case of entanglement. In order to fully
rule out a local hidden-variable model, one needs to perform a loophole-free
test that ensures that the experimental setup satisfies all the hypotheses of
Bell theorem. Loophole-free violations of Bell inequalities were measured for
the first time in 2015 [74, 75], and only in 2018 the so-called “free will”
loophole was completely closed [76].

Consequently, a careful analysis of the measurement process in a collider is
in order. Due to its illustrative character, we examine the paradigmatic case
of Run 2 at the LHC. Run 2 consisted of colliding a bunch of protons every
$25$ ns, yielding typically 60-80 collisions per bunch [77]. The products of
these collisions are recorded by different detectors surrounding the $pp$
interaction point that trace their direction and momentum/energy. This process
goes on for several years (Run 1 went from 2009 to 2012, and Run 2 went from
2015 to 2018) to collect sufficient amount of statistics.

As a result, in our specific case, the experiments of Alice and Bob should be
regarded as the detector recording of the $\ell^{+}\ell^{-}$ pair resulting
from a $t\bar{t}$ decay. First, one needs to ensure that the detection events
are causally disconnected. This in principle could be done, since the lepton
pair is typically ultrarelativistic due to the large top mass, and in addition
causally-connected detection events can be rejected after reconstruction of
the lepton momenta.

However, a major problem arises due to the fact that the direction of the spin
measurements cannot be controlled, since we only detect the lepton directions.
In fact, technically, one does not even have access to the $t\bar{t}$ spins,
but rather infers their expectation values since they are correlated with the
lepton directions. We stress that Eq. (68) is only valid after integrating out
all the remaining degrees of freedom of the decay products, and cannot be used
on an event by event basis. In other words: there is not a measurement setting
that yields a $\pm 1$ in each detection event or that can be controlled by
Alice or Bob. Therefore, the free-will loophole cannot be closed and even the
specific measurement setting of the CHSH inequality is not achievable; only
once the spin correlation matrix has been extracted from the fit of the
angular differential cross-section of the decay products, one can aim at
measuring a CHSH violation. Moreover, many of the events are not useful for
the analysis, which gives rise to the so-called detection loophole.

The emergence of all these loopholes is quite natural because high-energy
colliders were not specifically designed for testing Bell inequalities. As a
result, in a high-energy collider, we can only detect a weak violation of the
CHSH inequality, in the sense that some loopholes can never be closed. Related
discussions on the validity of Bell tests in high-energy colliders can be
found in Refs. [44, 45, 47].

##  7 Conclusions and outlook

In this work, we have provided the general framework to study the quantum
state of $t\bar{t}$ pairs produced in QCD processes. We have discussed that,
due to the nature of the measurement scheme, the most general quantum state
that can be probed in a collider is fully determined by the production spin
density matrix. This has to be necessarily a mixed state since a) only
momentum measurements are carried out in a collider and b) one has to average
over the internal degrees of freedom (like spin or color) of the initial
state.

We have analyzed $t\bar{t}$ production for the most elementary QCD partonic
reactions, extending the work of Ref. [41] to also include the analysis of
CHSH violation. For $q\bar{q}$ processes, entanglement and CHSH violation are
equivalent conditions, and are present in whole phase space. For $gg$
processes, entangled $t\bar{t}$ pairs are produced at threshold in a spin-
singlet state, and at high $p_{T}$ in a spin-triplet state, both violating the
CHSH inequality. Remarkably, all these features can be essentially understood
in terms of basic conservation laws of angular momentum, without invoking the
specific details of QCD interactions.

We have shown that, at least at LO, any $t\bar{t}$ production from a real
hadronic process can be written in terms of these basic building blocks
through the luminosity functions, which determine the probability of a certain
partonic reaction at a given c.m. energy in terms of PDF. In particular, we
have focused on $pp$ and $p\bar{p}$ collisions, which are those carried out at
the LHC and the Tevatron, respectively. We have performed a detailed analysis
on how the $t\bar{t}$ quantum state depends on the c.m. energy of the
collisions, finding that at low energies the $q\bar{q}$ channel dominates,
with the $gg$ contribution increasing with the c.m. energy for both types of
collisions. For the LHC, the $gg$ channel already dominates at the energies of
its first run. For the Tevatron, however, due to its relatively low-energy
operating point, $q\bar{q}$ processes dominate. Thus, both colliders represent
a perfect example of each elementary QCD process.

Thinking about potential experimental realizations, we have proposed a number
of realistic observables for the characterization of the $t\bar{t}$ quantum
state that provide signatures of entanglement and CHSH violation.
Interestingly, at the LHC these signatures are obtained by the measurement of
a single magnitude: for entanglement, the trace of the correlation matrix is a
good entanglement witness, as already predicted in Ref. [41], while CHSH
violation can be signaled by measuring the spin correlations in the transverse
plane to the beam.

Finally, we analyze in detail the experimental implementation of these ideas.
We explicitly show that an entanglement measurement at the LHC represents the
violation of a Cauchy-Schwarz inequality. Regarding the Tevatron, we find that
a statistically significant observation of entanglement seems quite
challenging due to the relatively large expected uncertainties.

We also extend the quantum tomography protocol developed in Ref. [41] to more
general quantum states, arguing that it can be applied in general to any
$t\bar{t}$ quantum state. In particular, if one averages over the beam axis,
rotational symmetry is recovered and the quantum tomography can be implemented
from the measurement of just $4$ parameters, related to the transverse and
longitudinal spin correlations, and to the longitudinal spin polarizations.
Furthermore, since the protocol does not depend on the specific production
process, it can be extended to any $t\bar{t}$ production mechanism, like
electroweak production from $e^{+}e^{-}$ collisions.

Regarding the CHSH violation, since it is a stronger requirement than
entanglement, its statistical significance is expected to be lower. Moreover,
we argue that, due to the nature of the detection process, only weak
violations of Bell inequalities can be measured in a high-energy collider,
since some loopholes, like those related to the free-will or to the detection
efficiency, cannot be closed. This is not surprising: after all, high-energy
colliders were not designed to test Bell inequalities.

From a quantum information perspective, top quarks allow to export fundamental
concepts in quantum information, such as entanglement, CHSH violation or
quantum tomography, to the high-energy field. This opens the prospect of using
high-energy colliders to study quantum information problems at the highest-
energy scale available. The genuine relativistic behavior, the exotic
character of the interactions and symmetries involved, and the fundamental
nature of this environment make it especially attractive for such purpose.
Indeed, the detection of entanglement or CHSH violation in $t\bar{t}$ pairs
would represent their highest-energy detections ever, many orders of magnitude
above standard laboratory setups. Another interesting experiment is the
implementation of the quantum tomography of the $t\bar{t}$ pair, which could
be used for instance to measure quantum discord [78, 79].

A very interesting perspective is provided by future colliders, where
$t\bar{t}$ pairs are expected to be produced from $e^{+}e^{-}$ collisions,
such as the FCC-$ee$ [80]. In particular, in future linear $e^{+}e^{-}$
colliders such as the International Linear Collider (ILC) [81] and the the
Compact Linear Collider (CLIC) [82, 83], spin degrees of the initial
$e^{+}e^{-}$ state can be controlled to a large extent, in contrast to the QCD
production discussed in this work. These colliders are then revealed as quite
promising scenarios to study quantum information problems, where the
techniques of this work can be straightforwardly adapted. Alternative
candidates to top quarks for the study of quantum information problems are
gauge bosons arising from Higgs decays and $\tau$ leptons [47, 84, 43].

From the high-energy perspective, the introduction of quantum information
concepts can provide new relevant observables in the field. For example,
entanglement measurements can help to understand the underlying mechanism of a
certain production process. A very intriguing extension of this work is to
explore New Physics beyond the Standard Model by measuring the quantum state
of the $t\bar{t}$ pair, comparing the experimental results with the
predictions of the different theories available. Indeed, some recent works
already address the possibility of using entanglement to trace signatures of
New Physics [42, 43].

We conclude by stressing that the work has been fully developed within a
genuine quantum information framework, since once the production spin density
matrix and the luminosity functions are computed by the theory of high-energy
physics, all the calculations are reduced to study convex combinations of two-
qubit quantum states using standard tools of quantum information theory. Our
work then provides a simple and general approach for non-particle physicists
to the quantum information aspects of high-energy colliders, a fascinating
arena to study the foundations of quantum mechanics.

## Acknowledgements

We thank E. Madge, M. Vos, F. Sols, G. Sierra and J. J. García-Ripoll for
valuable comments. JRMdN acknowledges funding from European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 847635, and also from Grant FIS2017-84368-P from Spain’s MINECO.

##  Appendix A $P$-representation and coherent states for qudits

We review here the main properties of the $P$-representation for qudits
introduced in Ref. [85]. We describe a Hilbert space of dimension $N$ as an
irreducible representation of SU(2) with angular momentum $j$ such that
$N=2j+1$. The qudit basis is relabeled as $\ket{j,m}$, with
$m=-j,-j+1\ldots,j$ the eigenvalues of the $z$-component of the angular
momentum, $J_{z}\ket{j,m}=m\ket{j,m}$. Angular momentum coherent states
$\ket{j,\mathbf{\hat{n}}}$ are defined as those simultaneously eigenstates of
$\mathbf{J}^{2}$ and $\mathbf{n}\cdot\mathbf{J}$, i.e.,
$\mathbf{J}^{2}\ket{j,\mathbf{\hat{n}}}=j(j+1)\ket{j,\mathbf{\hat{n}}}$ and
$\mathbf{n}\cdot\mathbf{J}\ket{j,\mathbf{\hat{n}}}=j\ket{j,\mathbf{\hat{n}}}$,
where we work in the usual orthonormal basis in spherical coordinates
$\\{\mathbf{\hat{n}},\mathbf{\hat{n}}_{\theta},\mathbf{\hat{n}}_{\phi}\\}$:

$\displaystyle\mathbf{\hat{n}}$
$\displaystyle=[\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta]$
$\displaystyle\mathbf{\hat{n}}_{\theta}$
$\displaystyle=[\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta]$ (106)
$\displaystyle\mathbf{\hat{n}}_{\phi}$ $\displaystyle=[-\sin\phi,\cos\phi,0]$

One can easily compute $\ket{j,\mathbf{\hat{n}}}$ from the ladder operators
$J_{\pm,\mathbf{\hat{n}}}=\mathbf{\hat{n}}_{\theta}\cdot\mathbf{J}\pm
i\mathbf{\hat{n}}_{\phi}\cdot\mathbf{J}$ as
$J_{+,\mathbf{\hat{n}}}\ket{j,\mathbf{\hat{n}}}=0$, finding

$\displaystyle\ket{j,\mathbf{\hat{n}}}$
$\displaystyle=\sum^{2j}_{n=0}\sqrt{\binom{2j}{n}}\left[\cos\frac{\theta}{2}\right]^{2j-n}\left[\sin\frac{\theta}{2}\right]^{n}$
$\displaystyle\times e^{-i(j-n)\phi}\ket{j,j-n}$ (107)

As the more usual coherent states, these states form an overcomplete basis of
the Hilbert space,

$\frac{2j+1}{4\pi}\int\mathrm{d}\Omega\ket{j,\mathbf{\hat{n}}}\bra{j,\mathbf{\hat{n}}}=\sum_{m}\ket{j,m}\bra{j,m}=I_{2j+1}$
(108)

The $P$-representation for a quantum state $\rho$ is defined from these
coherent states as

$\rho=\int\mathrm{d}\Omega\leavevmode\nobreak\
P(\mathbf{\hat{n}})\ket{j,\mathbf{\hat{n}}}\bra{j,\mathbf{\hat{n}}},\leavevmode\nobreak\
\int\mathrm{d}\Omega\leavevmode\nobreak\ P(\mathbf{\hat{n}})=1$ (109)

This definition does not completely fix $P(\mathbf{\hat{n}})$. In order to
check it, we consider its expansion in spherical harmonics

$\displaystyle P(\mathbf{\hat{n}})$
$\displaystyle=\sum^{\infty}_{\ell=0}\sum^{\ell}_{m=-\ell}p_{\ell
m}Y^{m}_{\ell}(\mathbf{\hat{n}})$ (110) $\displaystyle
Y^{m}_{\ell}(\mathbf{\hat{n}})$
$\displaystyle=\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}}\sin^{m}\theta
P_{\ell}^{m}(\cos\theta)e^{im\phi}$

where $P_{\ell}^{m}(\cos\theta)$ are the associated Legendre polynomials.
Normalization of $P(\mathbf{\hat{n}})$ implies $p_{00}=1/\sqrt{4\pi}$. Since
spherical harmonics form an orthonormal basis, when inserting Eq. (A) in Eq.