#  Inherent quantum resources in the stationary spin chains

Marcin Płodzień<EMAIL_ADDRESS>ICFO-Institut de Ciencies Fotoniques,
The Barcelona Institute of Science and Technology, 08860 Castelldefels
(Barcelona), Spain  Jan Chwedeńczuk<EMAIL_ADDRESS>Faculty of
Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland  Maciej
Lewenstein  ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of
Science and Technology, 08860 Castelldefels (Barcelona), Spain  ICREA, Passeig
Lluis Companys 23, 08010 Barcelona, Spain

###### Abstract

The standard way to generate many-body quantum correlations is via a dynamical
protocol: an initial product state is transformed by interactions that
generate non-classical correlations at later times. Here, we show that many-
body Bell correlations are inherently present in the eigenstates of a variety
of spin-1/2 chains. In particular, we show that the eigenstates and thermal
states of the collective Lipkin-Meshkov-Glick model possess many-body Bell
correlations. We demonstrate that the Bell correlations present in the
eigenstates of the Lipkin-Meshkov-Glick model can take on quantized values
that change discontinuously with variations in the total magnetization.
Finally, we show that these many-body Bell correlations persist even in the
presence of both diagonal and off-diagonal disorder.

††preprint:

##  I Introduction

The system is said to be quantum if any of its properties cannot be explained
within the framework of classical physics. In complex systems these quantum
deviations from classicality are often manifested through the many-body
correlations. The original discussions among prominent figures such as
Einstein [epr] and Schrödinger [sch] gave rise to concepts such as
entanglement [horodecki2009entanglement], Einstein-Podolsky-Rosen steering
[uola2020steering] and, many years later, the correlations that are
responsible for the violation of the postulates of local realism by some
quantum systems [bell, brunner2014bell]. Modern quantum technologies, such as
quantum computing, communication, or sensing [Frerot_2023,
srivastava2024introduction] are powered by these correlations and therefore
depend on our control of these resources at all stages: namely the ability to
generate, store and certificate quantum correlations [PhysRevLett.117.210504,
Acin_2018, Eisert2020, Kinos2021, PhysRevLett.125.150503, Laucht_2021,
Zwiller2022, Fraxanet2022, Sotnikov2022, huang2024certifying,
PRXQuantum.2.010201].

One of the main ways to generate strongly correlated quantum states is via
dynamical protocols. Starting from a state in which all components of a
quantum system are uncorrelated, their mutual interactions generate quantum
correlations at later times. For example, the analogue quantum simulators use
the One-Axis Twisting (OAT) protocol [kitagawa1993squeezed,
wineland1994squeezed] to create strong and scalable quantum resources. The
power of the OAT lies in its ability to generate spin-squeezed, many-body
entangled, and many-body Bell-correlated states. The problem of generating
nonlocality for the arbitrary depth $k$ has been addressed in
[PhysRevLett.95.120502, Tura2014, Tura2014a, Tura2015, PhysRevA.100.032307,
Baccarri2019, Aloy_2021, PRXQuantum.2.030329, PhysRevA.105.L060201] using a
set of Bell inequalities based on second-order correlators, violation of which
implies $k$-producibility of nonlocality with $k\leq 6$ for large number of
parties. Recently, Bell inequalities utilizing many-body correlators
[PhysRevLett.65.1838, PhysRevLett.100.200407, PhysRevLett.88.210401,
PhysRevLett.99.210405, Reid2012, PhysRevA.84.032115, PhysRevLett.131.070201]
were used to demonstrate that the OAT can actually generate arbitrary depth of
nonlocality [plodzien2020producing, plodzien2022one, yanes2022one,
PhysRevA.107.013311, PhysRevB.108.104301, plodzien2024generation,
yanes2024exploring]. The OAT protocol has been studied theoretically and
realised experimentally, and its relevance to quantum information science and
high precision metrology has been demonstrated [doi:10.1126/science.aad8665,
PhysRevLett.118.140401, pezze2018quantum, wolfgramm2010squeezed,
muller2023certifying].

In this work we focus on the stationary approach. Instead of analyzing many-
body Bell correlations generated during dynamical protocols, such as the OAT
discussed above, we show that the many-body Bell correlations are inherently
available in the eigenstates, and thermal equilibrium Gibbs states of
spin-$1/2$ chains described by the family of the Lipking-Meshkov-Glick (LMG)
model [LIPKIN1965188, MESHKOV1965199, GLICK1965211, PhysRevLett.49.478,
LermaH_2014, PhysRevB.28.3955, Carrasco_2016, PhysRevA.97.012115]. We provide
extensive analytical results that support the numerical approach and allow the
determination of the critical temperature above which the Bell correlations
vanish.

The LMG model has attracted interest in recent years providing a versatile
platform for proposals to realise quantum thermal machines and quantum heat
engines [PhysRevResearch.6.013310, PhysRevE.109.034112, HE20126594,
PhysRevE.96.022143, HUANG2018604, Lobejko2020thermodynamicsof, Cakmak2020,
PhysRevResearch.2.023145, Chen2019, centamori2023spinchain, Chakour2021,
PhysRevA.106.032410, Solfanelli_2023, PhysRevB.109.024310] The bipartite
entanglement of the LMG model has been studied extensively over the last two
decades [PhysRevA.70.062304, PhysRevA.71.064101, PhysRevLett.101.025701,
SenDe2012, PhysRevA.85.052112, PhysRevB.101.054431, Hengstenberg2023,
BAO2020412297, Du_2011, PhysRevA.109.052618], and the model itself has been
realised on many physical platforms, such as Bose-Einstein condensates
[Chen:09], superconducting circuits [Larson_2010], atoms in optical cavities
[Sauerwein2023, 10.1063/1.5121558, PhysRevLett.100.040403, Grimsmo_2013,
PhysRevA.97.023807], or with quantum circuits [Hobday2023,
hobday2024variance]. The present work contributes to this rapidly developing
field and supports the importance of the LMG systems for modern quantum
technologies. We note that the multipartite entanglement in a ground state of
the Dicke model has recently been analyzed in [lourenco2024genuine].

The paper is organised as follows. In the Sec.II we provide preliminary
overview of the work presented. In Sec.II.1, we give an introduction to the
considered LMG model, and in Sec.II.2, we give an introduction to the many-
body Bell correlator which characterizes the depth of Bell correlations of an
arbitrary state of a spin-$1/2$ chain. Next, in Sec.III we present results for
three limiting cases of the LMG model with all-to-all couplings between the
spin, and we present results for finite-range interactions. We conclude in
Sec.IV.

##  II Preliminaries

###  II.1 The Lipkin-Meskhov-Glick model

The LMG model describes a family of 1D chains of $L$ spin-$1/2$ particles
depicted by the Hamiltonian

$\displaystyle\hat{H}_{\gamma}=-\frac{1}{L}\sum_{i<j=1}^{L}\left(\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{x}^{(j)}+\gamma\hat{\sigma}_{y}^{(i)}\hat{\sigma}_{y}^{(j)}\right)-h\sum_{i=1}^{L}\hat{\sigma}_{z}^{(i)}.$
(1)

Here $\hat{\sigma}_{n}^{(i)}$ is the $n$th Pauli matrix of the $i$th spin
($n=x,y,z$, $i\in\\{1\ldots L\\}$), while the anisotropy parameter $\gamma$
and the magnetic field coupling constant $h$ differentiate the members of this
family. The pair-wise interactions are of infinite range, which allows to
express the Hamiltonian using the collective-spin operators

$\displaystyle\hat{S}_{n}=\frac{1}{2}\sum_{i=1}^{L}\hat{\sigma}_{n}^{(i)}$ (2)

as follows (up to an additive $L$-dependent constant)

$\displaystyle\hat{H}_{\gamma}=-\frac{2}{L}\left(\hat{S}_{x}^{2}+\gamma\hat{S}_{y}^{2}\right)-2h\hat{S}_{z}.$
(3)

The spectrum of this Hamiltonian is spanned by $L+1$ symmetric Dicke states,
namely the eigenstates of the total spin operator
$\mathbf{\hat{S}}^{2}=\hat{S}_{x}^{2}+\hat{S}_{y}^{2}+\hat{S}_{z}^{2}$ and,
for instance, the $\hat{S}_{z}$, i.e.,

$\displaystyle\mathbf{\hat{S}}^{2}|S,m\rangle$
$\displaystyle=S(S+1)|S,m\rangle$ (4a) $\displaystyle\hat{S}_{z}|S,m\rangle$
$\displaystyle=m|S,m\rangle,$ (4b)

where $S=L/2$ and $m\in\\{-S,\ldots,S\\}$.

Here we focus on the three emblematic cases of the LMG model, when the
anisotropy parameter takes the values $\gamma=-1$, $\gamma=0$ and $\gamma=1$,
giving the Hamiltonians (again, up to an additive constant)

$\displaystyle\hat{H}_{-1}=-\frac{2}{L}\left(\hat{S}_{x}^{2}-\hat{S}_{y}^{2}\right)-2h\hat{S}_{z},$
(5a) $\displaystyle\hat{H}_{0}=-\frac{2}{L}\hat{S}_{x}^{2}-2h\hat{S}_{z},$
(5b) $\displaystyle\hat{H}_{+1}=\frac{2}{L}\hat{S}_{z}^{2}-2h\hat{S}_{z}.$
(5c)

Particularly interesting is the $\gamma=1$, where the spectrum is spanned by
the Dicke states $|S,m\rangle$’s, giving the parabolic dependence of the
eigenenergies on $m$, i.e.,

$\displaystyle E_{m}(h)$ $\displaystyle=\frac{2}{L}m^{2}-2hm.$ (6)

Its minimal energy is reached when $m$ is equal to

$\displaystyle m=m_{0}(h)\equiv I\left[\frac{hL}{2}\right],$ (7)

i.e., the closest integer (for even $L$) or half-integer ($L$ odd) to the
minimum of this function. The ground state is hence

$\displaystyle|\psi^{(h)}_{0}\rangle=|S,m_{0}(h)\rangle$ (8)

For instance, when $h=0$, the ground state is

$\displaystyle|\psi^{(h=0)}_{0}\rangle=|S,0\rangle$ (9)

and the double-degenerated excited states lie symmetrically at $m$ positive
and negative. On the other hand, when $h$ is so large that $m_{0}\geqslant S$
the ground-state is

$\displaystyle|\psi^{(h)}_{0}\rangle=|S,S\rangle$ (10)

and the excited part of the spectrum is spanned by all the $m$’s smaller than
$S$—an observation that will be relevant in the Results section, Sec.III.

This concludes the preliminary section on the LMG model. We now introduce a
correlation function that turns out to be a useful tool that allows us to
certify the presence and depth of many-body Bell correlations in this system.

###  II.2 Bell correlations

When assessing the presence and the strength of Bell correlations in an
$L$-body system, we must refer to a corresponding model that is consistent
with the postulates of local realism. Here, we assume that each of $L$ parties
can measure a pair of observables $\sigma^{(k)}_{x}$ and $\sigma^{(k)}_{y}$,
each yielding binary outcomes, i.e., $\sigma_{x/y}=\pm 1$, with
$k\in\\{1\ldots L\\}$. The correlation function (also referred to as the
correlator) of these results is here defined as

$\displaystyle\mathcal{E}_{L}=\left|\langle\prod_{k=1}^{L}\sigma_{+}^{(k)}\rangle\right|^{2},$
(11)

with $\sigma_{+}^{(k)}=1/2(\sigma_{x}^{(k)}+i\sigma_{y}^{(k)})$. This
correlation is consistent with the local and realistic theory if the average
can be expressed in terms of an integral over the random (“hidden”) variable
$\lambda$ distributed with a probability density $p(\lambda)$ as follows

$\displaystyle\langle\prod_{k=1}^{L}\sigma_{+}^{(k)}\rangle=\int\\!\\!d\lambda\,p(\lambda)\prod_{k=1}^{L}\sigma_{+}^{(k)}(\lambda)$
(12)

We substitute this expression into Eq.(11) and use the Cauchy-Schwarz
inequality for complex integrals together with the fact that
$\left|\sigma_{+}^{(k)}(\lambda)\right|^{2}=1/2$ for all $k$, giving

$\displaystyle\mathcal{E}_{L}$
$\displaystyle=\left|\int\\!\\!d\lambda\,p(\lambda)\prod_{k=1}^{L}\sigma_{+}^{(k)}(\lambda)\right|^{2}$
$\displaystyle\leqslant\int\\!\\!d\lambda\,p(\lambda)\left|\sigma_{+}^{(k)}(\lambda)\right|^{2}=2^{-L}.$
(13)

We thus conclude that

$\displaystyle\mathcal{E}_{L}\leqslant 2^{-L}$ (14)

is the $L$-body Bell inequality, because its violation defies the postulates
of local realism expressed in Eq. (12). It is valid for systems that yield
binary outcomes of local, single-particle observables. This is the case of the
$L$-spin LMG model, hence the quantum-mechanical equivalent of Eq. (11),
labelled with ($q$) and defined as

$\displaystyle\mathcal{E}_{L}^{(q)}=\left|\langle\bigotimes_{k=1}^{L}\hat{\sigma}_{+}^{(k)}\rangle\right|^{2},$
(15)

will be used in the remainder of this work to witness Bell correlations. For
convenience, we introduce

$\displaystyle\mathcal{Q}_{L}^{(q)}=\log_{2}\left(2^{L}\mathcal{E}_{L}^{(q)}\right)$
(16)

do that the Bell inequality from Eq. (14) takes a more “user-friendly” form

$\displaystyle\mathcal{Q}_{L}^{(q)}\leqslant 0.$ (17)

The value of the Bell correlator ${\cal Q}^{(q)}_{L}$ provides information
about the structure of the many-body quantum state and depth of many-body Bell
correlations. Namely, the maximal value of the correlator that is reachable by
quantum mechanical systems is $\mathcal{Q}_{L,{\rm max}}^{(q)}=L-2$, which is
obtained with the Greenberg-Horne-Zeilinger (GHZ) state

$\displaystyle|\psi_{\rm
ghz}\rangle=\frac{1}{\sqrt{2}}\left(|1\rangle^{\otimes L}+|0\rangle^{\otimes
L}\right).$ (18)

Next, all the values of $\mathcal{Q}_{L}^{(q)}$ that lie in the region
$\mathcal{Q}_{L}^{(q)}\in]0,L-2]$ signal the presence of Bell correlations of
different depth. In particular, when

$\displaystyle\mathcal{Q}_{L,{\rm
max}}^{(q)}-(n+1)<\mathcal{Q}_{L}^{(q)}\leqslant\mathcal{Q}_{L,{\rm
max}}^{(q)}-n$ (19)

with $n\in\mathds{N}$, then maximally $n$ spins are not Bell-correlated with
the remaining.

As a last remark of this introductory part, we stress that
$\mathcal{E}_{L}^{(q)}$ (and hence $\mathcal{Q}_{L}^{(q)}$) is

$\displaystyle\mathcal{E}_{L}^{(q)}=\left|\varrho_{\rm ghz}\right|^{2},$ (20)

where $\varrho_{\rm ghz}$ governs the GHZ coherence between all the spins
$|1\rangle^{\otimes L}$ up and all down $|0\rangle^{\otimes L}$. This
observation—that information on Bell correlations is encoded in a single
element of the density matrix—will be relevant in the forthcoming sections of
this work.

For a thorough discussion of Bell correlator presented above, we recommend the
following works: [zukowski2002bell, cavalcanti2007bell, he2011entanglement,
he2010bell, cavalcanti2011unified, spiny.milosz, PhysRevLett.126.210506,
PhysRevLett.129.250402].

####  II.2.1 Bell correlator for symmetric states

The LMG infinite-range model requires a symmetrized Bell correlator, i.e.,
such that uses the collective spin operators as defined in Eq. (2). The
product of $L$ rising operators $\hat{\sigma}_{+}^{(k)}$ that address
individual particles, as present in Eq. (15), should be replaced as follows

$\displaystyle\bigotimes_{k=1}^{L}\hat{\sigma}_{+}^{(k)}\ \ \longrightarrow\ \
\hat{S}_{+}^{L}.$ (21)

Note however, that when expressing the collective rising operator using Eq.
(2) we have

$\displaystyle\hat{S}_{+}^{L}=\left[\frac{1}{2}\sum_{k=1}^{L}\hat{\sigma}_{+}^{(k)}\right]^{L},$
(22)

which gives $L!$ times more ordered products of $L$ distinct operators. This
combinatory factor, stemming from the symmetry of the setup, must be taken
into account. Hence the inequality (14) adapted to the symmetric case (denoted
by the tilde) reads

$\displaystyle\tilde{\mathcal{E}}_{L}^{(q)}=\left|\frac{1}{L!}\langle\hat{S}_{+}^{L}\rangle\right|^{2}\leqslant
2^{-L}$ (23)

or in analogy to Eq. (17)

$\displaystyle\tilde{\mathcal{Q}}_{L}^{(q)}=\log_{2}\left(2^{L}\tilde{\mathcal{E}}_{L}^{(q)}\right)\leqslant
0.$ (24)

For a detailed discussion of the symmetrized many-body Bell correlators, see
[10.21468/SciPostPhysCore.5.2.025].

####  II.2.2 Optimized correlator

According to the previous sections, the correlator $\mathcal{Q}_{L}^{(q)}$
takes a product of $L$ operators rising the spin projection along the
$z$-axis, each in the form of

$\displaystyle\hat{\sigma}_{+}^{(k)}=\frac{1}{2}\left(\hat{\sigma}_{x}^{(k)}+i\hat{\sigma}_{y}^{(k)}\right)$
(25)

or, for the symmetrized case, the $\tilde{\mathcal{Q}}_{L}^{(q)}$ is fed with

$\displaystyle\hat{S}_{+}=\hat{S}_{x}+i\hat{S}_{y}.$ (26)

Naturally, none of the arguments invoked above lead to the formulation of the
Bell inequality (17), does change upon a choice of different orientations of
local axes. We can always prepare the rotation of the rising operator for the
$k$-th spin,

$\hat{\sigma}_{+}^{(k)}\to\hat{R}_{k}^{\dagger}(\vec{\theta}_{k})\hat{\sigma}_{+}^{(k)}\hat{R}_{k}(\vec{\theta}_{k}),$
(27)

where
$\hat{R}_{k}(\vec{\theta}_{k})=e^{-i\vec{\theta}_{k}\cdot\vec{\sigma}_{k}}$,
$\vec{\theta}_{k}=\\{\theta^{(k)}_{x},\theta^{(k)}_{y},\theta^{(k)}_{z}\\}$,
$\vec{\sigma}_{k}=\\{\hat{\sigma}_{x}^{(k)},\hat{\sigma}_{y}^{(k)},\hat{\sigma}_{z}^{(k)}\\}$,
or for the symmetrized case

$\hat{S}_{+}\to\hat{P}^{\dagger}(\vec{\theta})\hat{S}_{+}\hat{P}(\vec{\theta}),$
(28)

$\hat{P}(\vec{\theta})=e^{-i\vec{\theta}\cdot\vec{S}}$,
$\vec{\theta}=\\{\theta_{x},\theta_{y},\theta_{z}\\}$,
$\vec{S}=\\{\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}\\}$. This freedom of choice
allows us to optimize the correlator $\mathcal{Q}_{L}^{(q)}$, i.e., adapt it
to the geometry of the many-body state that is under consideration.

In practice the correlator is optimized numerically and its maximal value is
found. For the LMG model as considered here, the permutational symmetry of the
Dicke states implies that the optimization procedure is collective. However,
in general, when such spin-exchange symmetry is not present, the axes must be
determined individually for each particle.

In the following section, we present the optimized correlator
$\tilde{\mathcal{Q}}_{L}^{(q)}$ for the LMG eigenstates. Before we discuss the
results, let us stress again that the many-body Bell correlations are encoded
in the GHZ coherence term of the density matrix, see Eq. (20), and that the
optimization of the correlator (Heisenberg picture) is equivalent to rotating
the density matrix itself (Schrödinger picture).

##  III Results

With an appropriate tool for the study of spin-$1/2$ chains at hand, we can
start to analyze the many-body Bell correlations in the eigenstates of LMG
models.

###  III.1 Pure states

For a given set of parameters $\\{L,\gamma,h\\}$ we obtain the set of
eigenvalues and eigenvectors of the Hamiltonian, Eq.(1),
$\hat{H}_{\gamma}|\psi^{(h)}_{v}\rangle=E^{(h)}_{v}|\psi^{(h)}_{v}\rangle$,
$v=0\dots L$, and for each eigenvector $|\psi^{(h)}_{v}\rangle$ we find the
optimized value of the correlator $\tilde{\mathcal{E}}_{L}^{(q)}$, Eq.(23),
denoted as $\tilde{\mathcal{E}}_{L}^{(q)}(v,h)$, and the corresponding
$\tilde{\mathcal{Q}}_{L}^{(q)}(v,h)$, Eq.(24).

####  III.1.1 Numerical analysis

We start our analysis with the simple case of $L=4$ spins, $S=2$. In Fig.1 we
present optimized Bell correlator $\tilde{\mathcal{Q}}_{L}^{(q)}$ for
eigenstates $|\psi_{v}\rangle$ of the $\hat{H}_{\gamma}$, as a function of the
magnetic field $h$, $\gamma=\\{1,0,-1\\}$. For each $\gamma$ the corresponding
eigenstates reveal many-body Bell correlations,
$\tilde{\mathcal{Q}}_{L}^{(q)}>0$, for particular values of the magnetic field
$h$.

Note that the cases of $\gamma=0$ and $-1$, displayed in Fig.1(b)-(c), are
characterized with continuous and smooth variations of Bell correlators
calculated with an eigenstate $|\psi^{(h)}_{v}\rangle$. However, when
$\gamma=1$, so that $\hat{H}_{\gamma=1}\propto\hat{S}_{z}^{2}$ , the many-body
Bell correlations, as a function of $h$ for a given $|\psi_{v}^{(h)}\rangle$
are non-continuous, see Fig.1(a).

To explain the origin of these jumps, recall that the $\hat{H}_{\gamma=1}$
eigenstates are given by $|\psi^{(h)}_{v}\rangle=|S,m(h)\rangle$, where
$\langle\hat{S}_{z}\rangle=m(h)=I\left[\frac{hL}{2}\right]$ is state
magnetization, Eq.(7). Now, let us focus the ground state
$|\psi^{(h)}_{0}\rangle$. When $h=0$ it takes the form
$|\psi^{(h)}_{0}\rangle=|S,0\rangle$, with magnetization $m=0$. The growing
$h$ does not change the ground state of the system as long as $h<1/4=h_{1}$.
When $h\geq h_{1}$, it suddenly changes to
$|\psi^{(h_{1})}_{0}\rangle=|S,1\rangle$, $m=1$, and the optimized Bell
correlation jumps to $\tilde{\mathcal{Q}}_{L}^{(q)}(0,h_{1})$. Again, the
ground state remains unchanged as long as $h<h_{2}=3/4$ and once this value is
passed we get $|\psi^{(h_{2})}_{0}\rangle=|S,S\rangle$, $m=2$, and Bell
correlations jump to the value $\tilde{\mathcal{Q}}_{L}^{(q)}(0,h_{2})$.

As we can observe, a change in $h$ implies the jump of the magnetization in
the ground state, which determines the value of the Bell correlations
$\tilde{\mathcal{Q}}_{L}^{(q)}(0,h_{2})$. Equivalently, we can say that for
every ground state $|\psi^{(h)}_{0}\rangle$ there exists an $v$-th eigenstate
$|\psi^{(h=0)}_{v}\rangle$ corresponding to $h=0$, i.e. for each $h$ we can
always can find index $v$ such that
$|\psi^{(h)}_{0}\rangle=|\psi^{(h=0)}_{v}\rangle$, thus
$\tilde{\mathcal{Q}}_{L}^{(q)}(0,h)=\tilde{\mathcal{Q}}_{L}^{(q)}(v,0)$. These
simple observations allow for fully analytical results, as we now show.

Figure 1: Value of the optimized Bell correlator,
$\tilde{\mathcal{Q}}_{L}^{(q)}$, Eq.(24) for an eigenstate
$|\psi^{(h)}_{v}\rangle$ of LMG Hamiltonian $\hat{H}_{\gamma}$, as a function
of the magnetic field for $L=4$ spins. The black solid line corresponds to the
ground state ($v=0$), the red dashed line corresponds to the first excited
state ($v=1$), the green dotted line to the $v=2$, the blue dash-dot line to
$v=3$, and the magenta long-dashed line to $v=4$.

####  III.1.2 Analytical expression for $\tilde{\mathcal{E}}_{L}^{(q)}$ for
$\gamma=1$

One of our main numerical results is that for all eigenstates the optimal
orientation of the two axes defining the rising operators turns out to be

$\displaystyle\hat{S}^{(x)}_{+}=\hat{S}_{y}+i\hat{S}_{z}.$ (29)

This operator raises the spin projection along the $x$ axis (as it is
expressed in terms of two collective spin operators of directions orthogonal
to $x$). The optimal direction for the many-body Bell correlator is consistent
with the following observation. The GHZ state obtained with the OAT protocol,
i.e., with the Hamiltonian $\propto\hat{S}_{z}^{2}$ Hamiltonian
[plodzien2020producing, plodzien2022one], is a superposition as in Eq. (18)
with $|0\rangle$ and $|1\rangle$ as eigenstates of $\hat{\sigma}_{x}$
operators. Hence it is the $x$-axis rising operator that is sensitive to this
macroscopic superposition 111 From now on, to distinguish between the
different orientations of the rising operators, we add the proper superscript
[here $(x)$]. .

It is now our goal to express the expectation value of the Bell correlator fed
with operator from Eq. (29) on the Dicke states. These are the eigenstates of
$\hat{S}_{z}$ hence the calculation requires an additional rotation through
the $y$-axis that would align the $x$ and $z$ orientations, i.e.,

$\displaystyle\hat{S}^{(x)}_{+}=e^{-i\frac{\pi}{2}\hat{S}_{y}}\hat{S}^{(z)}_{+}e^{i\frac{\pi}{2}\hat{S}_{y}}.$
(30)

The action of such a rotation operator on the Dicke state is

$\displaystyle
e^{i\frac{\pi}{2}\hat{S}_{y}}|S,m\rangle=\sum_{m^{\prime}=-S}^{S}d_{m;m^{\prime}}^{S}\left(\frac{\pi}{2}\right)|S,m^{\prime}\rangle,$
(31)

where $d_{m;m^{\prime}}^{S}$ represents an element of the Wigner matrix in the
fixed-$S$ subspace 222See [varshalovich1988quantum] for the details on the
rotation matrices in the SU(2) group. As argued below Eq. (20), the Bell
correlator is fully determined by the GHZ-coherence, hence by the modulus
square of the product of the coefficients multiplying the extreme elements
$m^{\prime}=S$ and $m^{\prime}=-S$ in Eq. (31), namely

$\displaystyle\tilde{\mathcal{E}}_{L}^{(q)}=\left|d_{m;S}^{S}\left(\frac{\pi}{2}\right)d_{m;-S}^{S}\left(\frac{\pi}{2}\right)\right|^{2}=\left[\frac{1}{2^{L}}\binom{L}{m+\frac{L}{2}}\right]^{2}.$
(32)

When $L$ is large, the binomial can be approximated with the Gaussian, giving

$\tilde{\mathcal{E}}_{L}^{(q)}\simeq\frac{2}{\pi L}e^{-4\frac{m^{2}}{L}}.$
(33)

For $m=0$. the Bell limit is surpassed for all $L$, as $\frac{2}{\pi
L}>2^{-L}$ for all integer $L$. It is always maximal at $m=0$, hence it is not
necessarily the ground state that is most non-classical. In particular, for
sufficiently large $h$, the ground state given by Eq. (10) is separable—all
the $L$ spins point in the upward direction.

To verify our analytical findings, in Fig. 2, we present
$\tilde{\mathcal{Q}}_{L}^{(q)}=\log_{2}(2^{L}\tilde{\mathcal{E}}_{L}^{(q)})$,
Eq.(24), as a function of magnetization $m=\langle\hat{S}_{z}\rangle$. For
each set of parameters $\\{L,h\\}$ we calculate all the eigenstates
$|\psi_{v}\rangle$ of $\hat{H}_{\gamma=+1}$ and numerically confirm the
optimal orientation of Eq. (29). We plot $\tilde{\mathcal{Q}}_{L}^{(q)}$ as a
function of $m$, with the blue dots presenting the numerical optimization of
the correlator, while the red diamonds corresponding to our analytical
findings, Eq. (33). The inverted parabola $\tilde{\mathcal{Q}}_{L}^{(q)}(m)$
shows that maximal Bell correlations are for the eigenstates for which the
magnetization is zero, one of important results of the presented work. Note
that the normalized Bell correlator $\tilde{\mathcal{Q}}_{L}^{(q)}$ can be
linked with the energy from Eq. (6), calculated at $h=0$, as follows

$\displaystyle\tilde{\mathcal{Q}}_{L}^{(q)}=-\alpha E_{m}+\beta L+\gamma,$
(34)

where $\alpha=2/\ln 2$, $\beta=1-1/\ln 2$ and
$\gamma=\log_{2}\left(\frac{2}{\pi L}\right)-2/\ln 2$. This is a simple
formula but of high predictive power, showings the benefit of keeping a string
of spins at low energies/temperatures. Note that the Bell correlations are
present mostly in the zero- (or close to) magnetization sector; this
observation is in an agreement with results of Ref. [Tura2015], where Authors
have identified classes of Bell inequalities for which zero magnetization
states are optimal. It is important to note that the Dicke states have been
experimentally realized for few qubits using various physical platforms,
including photonic systems [PhysRevLett.109.173604, PhysRevLett.103.020504,
PhysRevA.83.013821, Zhao:15, Wang2016, Mu:20], ultracold atoms
[PhysRevLett.112.155304], and quantum circuits [Chakraborty2014,
10.1007/978-3-030-25027-0_9, 9275336, PhysRevLett.103.020503, 9951196,
Narisada2023].

Figure 2:  The optimized Bell correlator $\tilde{\mathcal{Q}}_{L}^{(q)}$ (blue
dots)for an eigenstate $|\psi^{(h)}_{v}\rangle$ of LMG Hamiltonian
$\hat{H}_{\gamma}$, as a function of the magnetization
$m=\langle\hat{S}_{z}\rangle$ for $L=4,8,12,16,20$ (from inner to outer lines,
respectively). Red squares represent the analytical expression, Eq.(32).
Dashed blue lines are to guide an eye.

###  III.2 Thermal states

The next question we answer is about the impact of thermal fluctuations on the
Bell correlations in the LMG models. To investigate this problem, we calculate
the thermal density matrix

$\displaystyle\hat{\varrho}_{T}=\frac{1}{{\cal
Z}}\sum_{v}e^{-E_{v}/T}|\psi_{v}\rangle\\!\langle\psi_{v}|,$ (35)

for each $\gamma$, where ${\cal Z}=\sum_{v}e^{-E_{v}/T}$ is the statistical
sum and the summation runs over the whole spectrum. Next, we calculate the
expectation value of the many-body Bell correlator,

$\tilde{\mathcal{E}}_{L}^{(q)}(T)=\big{|}\frac{1}{L!}{\rm
Tr}\left[\hat{\varrho}_{T}\hat{P}^{\dagger}(\vec{\theta})\hat{S}_{+}^{(x)L}\hat{P}(\vec{\theta})\right]\big{|}^{2}$
(36)

optimized over $\vec{\theta}$.

In Fig. 3 we show the $\tilde{\mathcal{Q}}_{L}^{(q)}$, Eq. (24), in the $h-T$
plane density plot for $\gamma=1,0,-1$ for given $L$. We observe that for a
given temperature, the Bell correlator plotted as a function of $h$ shows a
complicated and rich structure, knowledge of which could be relevant for
future applications where the performance of a particular task relies on the
strength of Bell correlations.

In the following paragraphs, we investigate the origins of those fine
structures for the $\gamma=1$ and $\gamma=0$, providing the explanation for

Figure 3: Many-body Bell correlator $\tilde{{\cal Q}}^{(q)}_{L}$ for Gibbs
state at temperature $T$ as a function of the magnetic field $h$ for
$\gamma=\\{1,0,-1\\}$ (from left to right, respectively) for $L=4,8,16,40$.
White background denotes negative values. Figure 4: Many-body Bell correlator
$\tilde{{\cal Q}}^{(q)}_{L}$ as a function of temperature $T$ (left column)
for fixed magnetic field $h=0.04$, and as a function of magnetic field $h$ for
fixed temperature $T=0.025$. Rows correspond to system size $L=4,8,16,40$
(from top to bottom, respectively). The black lines correspond to numerical
results; red dashed line is the analytical formula from Eq.(38), while the
blue dashed-dotted line is the formula obtained in the $L\to\infty$ limit,
Eq.(39).

####  III.2.1 Bell correlator for $T\geqslant 0$ and $\gamma=1$ case

For the $\gamma=1$ case, when the spectrum is spanned by the Dicke states, we
have

$\hat{\varrho}_{T}=\frac{1}{{\cal
Z}}\sum_{m=-L/2}^{L/2}e^{-E_{m}/T}|S,m\rangle\langle S,m|,$ (37)

where the energies are given by Eq. (6). Using Eq. (33) we obtain

$\tilde{\mathcal{E}}_{L}^{(q)}(T)\simeq\frac{2}{\pi L}\frac{1}{{\cal
Z}^{2}}\big{|}\sum_{m=-L/2}^{L/2}(-1)^{m}e^{-\frac{2}{L}(m^{2}-hm)}e^{-\frac{2}{LT}m^{2}}\big{|}^{2},$
(38)

In Fig.4 we present the fixed-$T$ and fixed-$h$ cuts of the phase diagram
shown in Fig.3. The black solid lines corresponds to numerical results, while
the red dashed lines to analytical expression, Eq.(38), presenting a perfect
agreement even for small $L$. Some more insight into the behavior of
$\tilde{\mathcal{E}}_{L}^{(q)}(T)$ from Eq. (38) can be retreived from the
large-$L$ limit, where the sum over discrete $m$ is replaced with an integral.
Since the $m$’s range from $-L/2$ to $L/2$, extending the corresponding
integral limits to $\pm\infty$ requires that the width of the $T$-dependent
Gaussian $\sigma=\sqrt{LT}$ remains much smaller than $L$. Hence for the
validity of the continuous approximation to hold, the following must be
satisfied $T\ll 1/L$. In this regime, the Gaussian integral gives a
straightforward outcome

$\displaystyle\tilde{\mathcal{E}}_{L}^{(q)}(T)=\frac{8}{\pi
L(T+1)}e^{-\frac{h^{2}L}{1+T}}e^{-\frac{\pi^{2}LT}{4(1+T)}}.$ (39)

In Fig.4, we plot $\tilde{\mathcal{Q}}_{L}^{(q)}$ for the above equation as a
blue dash-dotted line, which gives an estimate for the envelope for
$\tilde{\mathcal{E}}_{L}^{(q)}(T)$, improving as $L$ grows. The structures
seen in Figs 3 and 4 as fast oscillations of the correlator as a function of
$h$ are lost in the approximation (39). They are a consequence of discrete-$L$
interference effects in Eq. (38). Clearly, as $L$ grows, these structures
become denser and finer, and—as expected—are smoothed-out in the $L\gg 1$
limit.

The formula (39) allows for an estimate of the critical temperature $T_{\rm
crit}$, above which the Bell correlations vanish for a given $h$. We insert
the expression (39) into the inequality Eq. (16) and obtain that

$\displaystyle T_{\rm crit}=\frac{\ln 2-h^{2}}{\frac{\pi^{2}}{4}-\ln 2}$ (40)

which is in very good agreement with numerical results, visible in Fig. 4. We
stress that the critical temperature does not depend on the system size $L$,
as expected from the thermodynamic limit. In particular, when $h=0$ the
critical temperature is maximal and equal to $T_{\rm crit}\simeq 0.39$.
Moreover, the condition $T_{\rm crit}=0$ gives the maximal $h=\sqrt{\ln
2}\simeq 0.83$ above which the Bell correlations vanish for all $T$, see the
blue line in Fig. 4. Note that the $T_{\rm crit}$ obtained from the many-body
Bell correlator considered in this work is about $2$ times smaller than the
critical temperature reported in [Fadel2018bellcorrelations], based on Bell
inequality formed with two-body correlators only.

####  III.2.2 Bell correlator for $T\geqslant 0$ and $\gamma=0$ case

Some insight into the structure of the $T-h$ graph shown in Fig. 3 can be
obtained also for $\gamma=0$. In this case the LMG system can be modelled with
an approximate Schrödinger-like equation, as follows. First, we write the
stationary Schrodinger equation, acting with the $\hat{H}_{\gamma=0}$
Hamiltonian on the state decomposed in the Dicke basis

$\displaystyle|\psi\rangle=\sum_{m=-L/2}^{L/2}\psi_{m}|S,m\rangle$ (41)

which reads

$\displaystyle-\frac{hL}{2}\left[\psi_{m+1}f_{+}(z_{m})+\psi_{m-1}f_{-}(z_{m})\right]+\frac{L}{4h}\psi_{m}z^{2}_{m}=E\psi_{m}.$
(42)

Here, we introduced a normalized variable

$\displaystyle z_{m}=2\frac{m}{L}$ (43)

that changes between -1 and 1. Also, the coefficients $f_{\pm}(z_{m})$ present
in Eq. (42) are defined as follows

$\displaystyle f_{\pm}(z_{m})=\sqrt{\frac{1\pm z_{m}}{2}\left(\frac{1\mp
z_{m}}{2}+\frac{1}{L}\right)}.$ (44)

Note that we performed the $h\rightarrow-h$ transformation, that can be
generated with a local single-spin transformation (rotation of the Hamiltonian
around $\hat{S}_{x}$ through an angle equal to $\pi$), which does not modify
the Bell content but allows to maintain consistent notation with previous
works [ZP2008, hamza2024bell].

For large $L$, the $1/L$ terms in these square-roots can be neglected and
$z_{m}$ becomes quasi-continuous, allowing to approximate the difference
between the state coefficients $\psi_{m\pm 1}$ with the second derivative over
$z$. This gives a Schrödinger-like equation for the wave-function
$\psi_{m}\rightarrow\psi(z)$ of a fictitious unit-mass particle

$\displaystyle\left(-\frac{2}{L}\sqrt{1-z^{2}}\frac{d^{2}}{dz^{2}}+V_{\rm
eff}(z)\right)\psi(z)=\tilde{E}\psi(z),$ (45)

where the effective potential is

$\displaystyle V_{\rm eff}(z)=-\sqrt{1-z^{2}}+z^{2}\kappa/2$ (46)

and $\kappa=\frac{1}{h}$. The details of this derivation can be found in Ref.
[ZP2008].

Crucially, the spectrum of the Eq. (45) depends on the value of $\kappa$. When
it is large, the second term of Eq. (46) dominates giving an effective
potential that is close to harmonic. Interestingly, when $\kappa$ surpasses
the threshold value $\kappa=-1$, the potential develops two minima located at
$\pm z_{0}$, where $z_{0}=(1-1/\kappa^{2})^{1/2}$ and becomes

$\displaystyle V_{\rm eff}(z)\simeq\frac{1}{2}\kappa(1-\kappa^{2})(z\pm
z_{0})^{2}+\mathrm{const.}$ (47)

The ground state tends to localize around these minima and form a macroscopic
superposition

$\displaystyle\psi_{\rm
gr}\simeq\frac{1}{\sqrt{2}}\left[\psi_{0}(z-z_{0})+\psi_{0}(z+z_{0})\right],$
(48)

where $\psi_{0}(z\pm z_{0})$ is the Gaussian ground state of each of the the
harmonic contributions to Eq. (47). When this superposition forms, the Bell
correlator $\tilde{\mathcal{Q}}_{L}^{(q)}$ becomes large, as it is exactly
this type of macroscopic superposition that it is sensitive to, see Eq. (20).
This is why the central column of Fig. 3 shows non-vansing Bell correlator
when $h$ is sufficiently small.

However, an important feature of this plot is yet to explained—namely why the
$\tilde{\mathcal{Q}}_{L}^{(q)}$ seems more vulnerable to thermal fluctuations
when $h$ is very small (i.e., the area of non-zero Bell correlations shrinks
with $h\rightarrow 0$).

To explain this behavior, note that as $h$ drops, so that
$\kappa\rightarrow-\infty$, the ground state (48) becomes (assymptotically)
degenerate with the first excited state

$\displaystyle\psi_{\rm
1^{st}}\simeq\frac{1}{\sqrt{2}}\left[\psi_{0}(z-z_{0})-\psi_{0}(z+z_{0})\right].$
(49)

In this regime, any non-zero value of $T$ will lead to occupation of both
these states. The GHZ coherences of Eqs (48) and (49) have opposite signs,
therefore the Bell correlations quickly degrade in this thermal mixture. This
is a qualitative explanation, for a more quantitative analysis, see
[hamza2024bell]. When $h\simeq 0$, the ground state is the GHZ state, giving
the maximal Bell correlator, but only at $T=0$. This is represented by a very
thin and dark line in the central column of Fig. 3.

###  III.3 Distance-dependent interactions

So far, we considered the LMG model, which is characterized by the constant
strength all-to-all interactions, see Eq. (1). We now take a step further and
allow for the two-body coupling to depend on the distance between the spins,
by considering the following Hamiltonian

$\displaystyle\hat{H}=-\frac{1}{L}\sum_{i<j}J_{ij}\left(\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{x}^{(j)}+\gamma\hat{\sigma}_{y}^{(i)}\hat{\sigma}_{y}^{(j)}\right)$
(50)

with the power-law dependence of the coupling strength, i.e.,
$J_{ij}=|i-j|^{-\alpha}$, with the exponent ranging from $\alpha=0$ (fully
connected all-to-all case) to $\alpha\rightarrow\infty$ (nearest-neighbors
interaction). While the fully connected models are now realized in
experiments, the distance-dependent interactions are still a natural
playground for most cases, as in the state-of-the-art quantum simulator
platforms [Zhang2017, doi:10.1126/science.aax9743, bornet2023scalable,
Eckner2023, franke2023quantumenhanced, PhysRevX.12.011018,
RevModPhys.93.025001, PhysRevLett.131.033604, PhysRevLett.129.063603]. The
above Hamiltonian is of particular importance for potential experiments aiming
to generate non-classical correlations in spin chains. The behavior of the
two-body Bell correlations around the critical value of the transverse-field
Ising model has been studied in [Piga2019].

Note that the Hamiltonian from Eq. (50) is absent of the permutational
invariance. Hence, we must refer to the single-spin addressing Bell correlator
from Eq. (15) or (17) and optimize the orientations of the
$\hat{\sigma}_{+}^{(k)}$ operators independently for each $k$, which
significantly increases the computational load. In Fig. 5 we present the
optimized Bell correlation for the ground state and using finite $\alpha$,
normalized to the value obtained with the all-to-all case (i.e., such that can
be reproduced by setting $\alpha=0$). The blue dashed line shows the constant
$\alpha$-independent normalized Bell correlations for $\gamma=0$, while the
solid red and dot-dashed black line present a decaying Bell correlator for
$\gamma=1$ and $-1$, respectively.

Figure 5: Many-body Bell correlator calculated with the ground state for the
distance-dependent interactions with $L=8$. The dashed blue line is for
$\gamma=0$, the solid red line for $\gamma=1$ and the dot-dashed black line is
for $\gamma=-1$.

A particularly instructive case is for infinitesimally small $\gamma$’s. We
start with the $\gamma=0$ case, giving the Hamiltonian

$\displaystyle\hat{H}=-\frac{1}{L}\sum_{i<j}J_{ij}\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{x}^{(j)}.$
(51)

Due to the presence of the minus sign in front, the minimal energy when the
sum is maximal, i.e., when the spins all’unisono point either “up” or “down”.
The doubly degenerate ground state is either $|0\rangle_{x}^{\otimes L}$ or
$|1\rangle_{x}^{\otimes L}$. Both are fully separable, a consequence of the
classical structure of the Hamiltonian Eq.(51), only involving operators
$\hat{\sigma}_{x}$. However, even infinitesimal $\gamma$ lifts the ground
state degeneracy. As a result, the GHZ states

$\displaystyle|\psi_{\pm}\rangle=\frac{1}{\sqrt{2}}(|0\rangle_{x}^{\otimes
L}\pm|1\rangle_{x}^{\otimes L}),$ (52)

become of minimal energy. Such highly entangled states give the maximal value
of the correlator, see the discussion around Eq. (18) (for detailed discussion
of the quantum Ising ground state see [Parkinson2010, mbeng2020quantum]) Note
that this argument holds for all $\alpha$’s, a consequence of the minus sign
in front of the sum in the Hamiltonian (51).

The other cases, $\gamma=\pm 1$, call for a numerical-only approach and, as
shown in Fig. 5, the dependence of $\alpha$ is strongly pronounced.

###  III.4 Diagonal and off-diagonal disorder

Figure 6: Relative change of the Bell correlations in the ground state, as a
function of disorder amplitude. The red lines with shaded areas correspond to
diagonal disorder, while the blue lines (with shaded area, not visible)
correspond to off-diagonal disorder. The Bell correlations are robust to off-
diagonal disorder, and preserved with diagonal disorder. Top row shows the
impact of the uniform distribution ${\cal P}^{(1)}$ while the bottom row is
for the experimental distribution ${\cal P}^{(2)}$. The number of spins is
$L=8$

In this last section, we enrich the above distance-dependent model, taking
into account non-vanishing $h$, namely

$\displaystyle\hat{H}=-\frac{1}{L}\sum_{i<j}J_{ij}\left(\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{x}^{(j)}+\gamma\hat{\sigma}_{y}^{(i)}\hat{\sigma}_{y}^{(j)}\right)+\sum_{i}h_{i}\hat{\sigma}_{z}^{(i)}$
(53)

and allowing for the presence of fluctuations of both $h_{i}$ (called the
diagonal disorder) and of the spin couplings $J_{ij}$ (the off-diagonal
disorder). Such fluctuations are inevitable in realistic configurations hence
it is of importance to perform a numerical analysis of their influence on the
Bell correlator.

We consider two types of probability distributions that generate the noisy
random variable $\epsilon$ added to either $h$ or $J_{ij}$, i.e.
$h_{i}=\epsilon$ or $J_{ij}=J_{ji}=1+\epsilon$. These are, the uniform
distribution from $-V/2$ to $V/2$, i.e.,

$\displaystyle\mathcal{P}^{(1)}_{a}(\epsilon)=\frac{V}{2}[-1,1]$ (54)

and alternatively, the probability distribution given by

$\displaystyle\mathcal{P}^{(2)}_{a}(\epsilon)=\frac{1}{\pi\sqrt{\epsilon(V-\epsilon)}}.$
(55)

This former type of noise was recently generated and analyzed in the
experimental realization of the random-spins models in an optical cavity
[Sauerwein2023]. The Fig. 6 summarizes the results for the ground state and
for three the cases of $\gamma$ and $\mathcal{P}^{(i)}_{a}(\epsilon)$, showing
a general tendency—Bell correlations are robust to off-diagonal disorder.
Also, we note that Bell correlations are weakly affected by the diagonal
disorder as long as its amplitude is much smaller than $V\ll 1$.

##  IV Outlook and Conclusions

This work addressed the problem of generating the strongest possible and
scalable many-body quantum correlations in spin-$1/2$ chains. We showed that
the creation of quantum states with strong many-body Bell correlations does
not necessarily have to be done via dynamical protocols such as one-axis
twisting.

We showed that many-body Bell correlations are inherently present in the
eigenstates of a broad family of spin-$1/2$ models described by the Lipkin-
spin-$1/2$ Glick-Meskhov model with all-to-all and power-law interacting
spin-$1/2$ models. Next, we showed that inherent many-body Bell correlations
are also present in the thermal states of the LMG model, and we estimated the
critical temperature above which Bell correlations vanish for the one-axis
twisting Hamiltonian. Finally, we showed that the many-body Bell correlations
are robust to disorder the potential. We hope that this work shed some light
on the properties of the many-body Bell correlations beyond the pure state
description [Fadel2018bellcorrelations, PhysRevA.105.L060201].

The remaining question is whether there are possibilities of experimental
certification of the many-body Bell correlations in the quantum systems. The
measurement of the off-diagonal GHZ element of the density matrix in $L=4$
qubit system has been prepared experimentally in element [Sackett2000].
Recently it was shown the many-body Bell correlations can be extracted with
the quantum state tomography via the classical shadows technique
[plodzien2024generation] up to $L=10$ spins. Another approach is a technique
based on multiple quantum coherence [garttner2017measuring,
PhysRevLett.120.040402], which allows to access to individual elements of the
density matrix, including those that encode information about many-body Bell
correlations.

## Acknowledgement

We thank Guillem Müller-Rigat, and Jordi Tura for reading the manuscript and
their useful comments.

ICFO group acknowledges support from: Europea Research Council AdG NOQIA;
MCIN/AEI (PGC2018-0910.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, Plan National STAMEENA PID2022-139099NB, I00, project
funded by MCIN/AEI/10.13039/501100011033 and by the “European Union
NextGenerationEU/PRTR" (PRTR-C17.I1), FPI); QUANTERA MAQS PCI2019-111828-2);
QUANTERA DYNAMITE PCI2022-132919, QuantERA II Programme co-funded by European
Union’s Horizon 2020 program under Grant Agreement No 101017733); Ministry for
Digital Transformation and of Civil Service of the Spanish Government through
the QUANTUM ENIA project call - Quantum Spain project, and by the European
Union through the Recovery, Transformation and Resilience Plan -
NextGenerationEU within the framework of the Digital Spain 2026 Agenda;
Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social
Fund FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2023-3-0024); Funded by the
European Union. Views and opinions expressed are however those of the
author(s) only and do not necessarily reflect those of the European Union,
European Commission, European Climate, Infrastructure and Environment
Executive Agency (CINEA), or any other granting authority. Neither the
European Union nor any granting authority can be held responsible for them
(HORIZON-CL4-2022-QUANTUM-02-SGA PASQuanS2.1, 101113690, EU Horizon 2020 FET-
OPEN OPTOlogic, Grant No 899794), EU Horizon Europe Program (This project has
received funding from the European Union’s Horizon Europe research and
innovation program under grant agreement No 101080086 NeQSTGrant Agreement
101080086 — NeQST); ICFO Internal “QuantumGaudi” project; European Union’s
Horizon 2020 program under the Marie Sklodowska-Curie grant agreement No
847648; “La Caixa” Junior Leaders fellowships, La Caixa” Foundation (ID
100010434): CF/BQ/PR23/11980043. JCh was supported by the National Science
Centre, Poland, within the QuantERA II Programme that has received funding
from the European Union’s Horizon 2020 research and innovation programme under
Grant Agreement No 101017733, Project No. 2021/03/Y/ST2/00195.

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