Bell correlations at finite temperature
Matteo Fadel
1
and Jordi Tura
2
1
Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
2
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
November 19, 2018
We show that spin systems with infinite-
range interactions can violate at thermal
equilibrium a multipartite Bell inequality,
up to a finite critical temperature T
c
. Our
framework can be applied to a wide class
of spin systems and Bell inequalities, to
study whether nonlocality occurs naturally
in quantum many-body systems close to
the ground state. Moreover, we also show
that the low-energy spectrum of the Bell
operator associated to such systems can be
well approximated by the one of a quan-
tum harmonic oscillator, and that spin-
squeezed states are optimal in displaying
Bell correlations for such Bell inequali-
ties.
1 Introduction
One of the key features of quantum physics, which
challenges our every-day intuition, is the exis-
tence of nonlocal correlations [1]: Local measure-
ments on a quantum system may lead to correla-
tions whose nature departs entirely from the local
hidden variable model (LHVM) paradigm, mean-
ing that such correlations cannot be simulated lo-
cally by deterministic strategies, even if assisted
by shared randomness [2].
Since Bell showed that quantum mechanics
can produce correlations that do not admit a
LHVM description [1], nonlocal correlations have
had a fundamental interest per sé. Much work
has been devoted to understand and character-
ize them (see [3] for a review). Moreover, Bell
correlations constitute a resource that enables
novel device-independent (DI) quantum informa-
tion processing (QIP) tasks (such as DI quantum
key distribution [4, 5], DI randomness amplifi-
cation/expansion [6, 7] or DI self-testing [8–10]),
Matteo Fadel: matteo.fadel@unibas.ch
Jordi Tura: jordi.tura@mpq.mpg.de
thus providing an extra layer of security in QIP
protocols, allowing to reduce their assumptions
to a minimum level.
Bell correlations are strongly intertwined with
entanglement, perhaps the most significant quan-
tum phenomenon [11]. However, it is well known
that entanglement and Bell correlations are in-
equivalent both in the bipartite [12] and in the
multipartite [13–15] cases: Bell correlations are
strictly stronger as they can only arise from en-
tangled states, while the converse is not true in
general (see e.g. [16]).
In the many-body regime, entanglement has
proven to be key to the understanding of a
plethora of physical phenomena [17]. Intensive
studies of entanglement have elucidated its im-
portant role in modern physics’ central topics
such as quantum phase transitions [18], super-
conductivity at high temperature [19], the holo-
graphic principle [20] or quantum chemistry reac-
tions [21, 22]. Furtermore, entanglement lies at
the heart of the tensor network ansatz [23], and
could even be relevant to the understanding of
quantum machine learning models [24].
However, the role played by nonlocal correla-
tions in many-body physics is not so clear yet.
Their complex characterization (see [25]), which
is known to be NP-hard even in the bipartite case
[26], has posed a barrier to the study of multi-
partite scenarios. For this reason, the implica-
tions of nonlocal correlations in systems consist-
ing of a very large number of parties has been
mostly lacking, despite remarkable results have
been obtained throughout the years [27–37]. In-
terestingly, recent theoretical advances developed
in the context of few-body Bell inequalities [38–
46] have allowed for simple ways to detect these
correlations in many-body systems. On the one
hand, it has been shown that many-body observ-
ables such as the energy of the system could sig-
nal the presence of Bell correlations in the system
[42]. On the other hand, routinely measured ex-
Accepted in Quantum 2018-11-16, click title to verify 1
arXiv:1805.00449v2 [quant-ph] 16 Nov 2018
perimental quantities such as total spin compo-
nents and higher moments thereof [47, 48], have
been shown to be sufficient to formulate Bell cor-
relation witnesses that can detect nonlocality in
many-body systems [38–41]. With these tools,
such correlations have been observed experimen-
tally in a Bose-Einstein Condensate of 480 atoms
[49] and in a thermal ensemble of 5 × 10
5
atoms
[50].
In this work, we study whether Bell-nonlocal
states (i.e., quantum states with the potential
to produce Bell correlations if one were to per-
form a proper loophole-free Bell test on them)
appear naturally in the low-energy spectrum of
physically relevant Hamiltonians. As shown in
[38, 39, 42], partial answers to this question are
given, by noticing that the ground state of some
spin Hamiltonians can violate a Bell inequality.
However, conclusions for more experimentally rel-
evant situations, such as non-zero temperature or
systems with infinite-range interactions are still
missing.
Here, we show that a many-body spin system
at finite temperature can violate a Bell inequal-
ity and we quantify this violation. We also show
that the low-energy part of the spectrum of the
Bell operator corresponding to the optimal mea-
surements of the Bell inequality can be well ap-
proximated by a quantum harmonic oscillator,
enabling a connection to squeezed states. We
observe a critical temperature above which non-
locality cannot be detected with our inequality,
raising the question of whether this phenomenon
can be observed in other systems and with other
inequalities. This could allow to observe Bell-
nonlocal states appearing naturally in solid-state
systems cooled below some transition tempera-
ture.
2 Spin model
We consider a system of N spins, described by
an Ising model with infinite-range interactions.
With s
(i)
= (s
(i)
x
, s
(i)
y
, s
(i)
z
) denoting the vector
of spin operators for the i-th particle, the Hamil-
tonian of the system is
H
0
=
N
X
i=1
B · s
(i)
+ J
N
X
i<j
s
(i)
z
s
(j)
z
, (1)
where B = (B
x
, B
y
, B
z
) is the external magnetic
field and J the strength of the spin-spin coupling.
Since Eq. (1) has a preferred z direction, we can
assume, without loss of generality, that B
y
= 0.
The case of interest for our study corresponds to
B
z
, J > 0 and B
x
< 0, which we shall assume
throughout the rest of the paper. Since J > 0,
the spin-spin interaction is antiferromagnetic.
Using the definition of collective spin operator
S = (S
x
, S
y
, S
z
), given by
S =
N
X
i=1
s
(i)
, (2)
we rewrite Eq. (1) as
H
0
= B
x
S
x
+ B
z
S
z
+
J
2
S
2
z
−
N
4
. (3)
Here, and in the following, we shall not drop the
constant term in the Hamiltonian, as it will be
important later on.
Hamiltonians of this form, where interactions
are of infinite range, are routinely realized in a
number of experiments on many-body systems.
Examples include Bose-Einstein condensates [51–
53], atomic ensembles in optical cavities [54], and
ion crystals [55, 56].
We are interested in the limit N → ∞, for
which the energy of H
0
(cf. Eq. (3)) is minimized
when the collective spin has the maximum length
S = N/2 and pointing direction along x (since
B
x
< 0). To approximate the low-energy spec-
trum of H
0
, we perform the Holstein-Primakoff
transformation, which maps the collective spin to
a bosonic mode [57]
S
x
= (S − a
†
a) , (4a)
S
z
− iS
y
=
p
2S − a
†
a
a , (4b)
S
z
+ iS
y
= a
†
p
2S − a
†
a
, (4c)
with a
†
, a being respectively the bosonic creation
and annihilation operators acting on the Fock
space, and satisfying the canonical commutation
relation [a, a
†
] = 1. Because we are interested
in the low-energy excitations above the ground
state, i.e. the case where S ≈ N/2 and a
†
a ≈ 0,
we further approximate
√
2S − a
†
a ≈
√
2S in
Eqs. (4b) and (4c). Note that a
†
a ≈ 0 implies
we are considering a small number of bosonic ex-
citations. These bosonic modes arise from the
Holstein-Primakoff transformation and can be
seen as “fictitious” bosonic modes. As it can be
seen from (4a), they are excitations of the spin
Accepted in Quantum 2018-11-16, click title to verify 2
system that reduce S
x
. This allows us to approx-
imate Eq. (3) with the expression
H
1
=B
x
S − a
†
a
+
B
z
√
S
√
2
a + a
†
+
+
J
4
S
a + a
†
2
−
N
2
. (5)
This Hamiltonian is diagonalized by perform-
ing the Bogoliubov (squeezing) transformation
[58, 59]
a = cosh(ξ)b + sinh(ξ)b
†
+ w (6a)
a
†
= cosh(ξ)b
†
+ sinh(ξ)b + w
∗
(6b)
with ξ and w defined as
ξ =
1
4
log
B
x
B
x
− JS
, w =
B
z
√
S
(B
x
− JS)
√
2
,
(7)
and b
†
, b being newly defined bosonic creation
and annihilation operators satisfying the canon-
ical commutation relation [b, b
†
] = 1. With
Eqs. (6) we express Eq. (5) in the diagonal form
H
1
=
q
B
x
(B
x
− JS)
b
†
b +
1
2
+
+ B
x
S +
1
2
−
JN
8
+
B
2
z
S
2(B
x
− JS)
.
(8)
In the limit N → ∞ the last term of Eq. (8) tends
to −B
2
z
/(2J), leaving us with the Hamiltonian H,
defined as
H =
q
B
x
(B
x
− JS)
b
†
b +
1
2
+
+ B
x
S +
1
2
−
4B
2
z
+ J
2
N
8J
. (9)
Note that the Hamiltonian in Eq. (9) describes
a quantum harmonic oscillator of frequency
ω(S) =
q
B
x
(B
x
− JS) , (10)
and its energy spectrum is given by
E(S, n) =ω(S)
n +
1
2
+
+ B
x
S +
1
2
−
4B
2
z
+ J
2
N
8J
, (11)
where n ∈ {0, 1, 2, ...} and S ∈ {N/2, N/2−1, ...}.
Eq. (11) shows us that the ground state of H
0
is
approximated by the ground state of H, which
has n = 0 and S = N/2 (remember B
x
< 0).
It is important to realize that the spectrum in
Eq. (11) has two characteristic spacings: In the
limit N = 2S → ∞, one is E(S, n+1)−E(S, n) =
ω(S) ≈
√
−B
x
JS, and the other is E(S + 1, n) −
E(S, n) ≈ B
x
. This shows that for large S, the
low-energy part of the spectrum is dominated by
excitations of n ≈ 0 and different S. Note that,
for large values of S, E(S, n + 1) −E(S, n) scales
as
√
S, whereas E(S +1, n)−E(S, n) remains ap-
proximately constant. This implies that for suffi-
ciently large values of S, the latter constant be-
comes effectively arbitrarily small, and therefore
many levels of S become accessible.
For a graphical comparison, we plot in Fig. 1
the spectrum of the Hamiltonian Eq. (1), and
of its approximation Eq. (9), for N = 5 × 10
3
,
B = (−1, 0,
√
3), J = 6, and some values of S
(the reason for this specific choice of parameters
will be clear in Sec. 4). We can see that low-
energy excitations of the two models are very sim-
ilar. As expected, energy levels for S ≈ N/2 and
n ≈ 0 are very well described by the Holstein-
Primakoff approximation. Note also that this
match improves as N → ∞.
It should be noted that the state yielding the
ground energy of the family of Hamiltonians con-
sidered is spin-squeezed, as these Hamiltonians
are diagonalized via the spin-squeezing transfor-
mation (cf. Eq. (6)). In Sec. 4 we shall see
that the Hamiltonians we consider can be mapped
into Bell inequalities. For this reason we conclude
that spin-squeezed states are optimal for violating
such inequalities, in the sense that they asymp-
totically (as N goes to infinity) give the maximal
quantum violation [39].
To describe the system at finite-temperature,
it is crucial to keep track of the degeneracy of the
energy levels, originating from the multiplicities
g(S) of the blocks with given S (see e.g. [60, 61]).
This is 1 if S = N/2 and otherwise it is given by
g(S) =
N
N/2 − S
!
−
N
N/2 − S − 1
!
. (12)
The derivation of g(S) follows from representa-
tion theory results such as the Shur-Weyl duality.
We refer the interested reader to [62–64] for their
derivations in quantum information and mathe-
matical contexts.
In the following we will consider two different
Accepted in Quantum 2018-11-16, click title to verify 3
�
�
()
()
-
-
-
-
-
-
/
Figure 1: With N = 5 × 10
3
, sample for some values
of S of the spectrum of the Hamiltonian Eq. (3) com-
pared to the spectrum of the approximated Hamiltonian
Eq. (9), for B = (−1, 0,
√
3) and J = 6. Energy is nor-
malized to N. Note that for a fixed S the energy levels
for the Bell operator are not equally spaced, while they
are for the approximated Hamiltonian (harmonic oscilla-
tor). The thin gray line at −5/4 shows the ground state
energy per particle in the limit N → ∞. The blue line at
−1 is the classical bound associated to the Bell operator
Eq. (27), see Sec. 4 for details.
regimes. The first one is the low temperature
regime, where n ≈ 0 and S ≈ N/2. In this
case we can perform the following approxima-
tion: For N → ∞ and p = N/2 − S N/2,
the leading term in g(S) is given by the factor
N
p
/p!. The second regime of interest corresponds
to S ≈ N/4, and therefore the previous approx-
imation cannot be done. We will discuss how to
treat this case in detail in Sections 3 and 5.
3 Partition function
In this section we introduce the standard tools of
statistical mechanics that are needed to describe
a system at finite-temperature. In particular, ex-
pectation values of interest are computed from
the partition function of the considered statisti-
cal ensemble. As we consider a system in ther-
mal equilibrium at temperature T and with fixed
number of atoms N , we are in the canonical en-
semble. The canonical partition function for the
system described by Eq. (9) is given by
Z =
X
S,n
g(S)e
−βE(S,n)
(13)
with g(S) the degeneracy (independent of n) of
the energy level E(S, n), and β = T
−1
is the in-
verse temperature. Performing the sum over n
leaves us with
Z = e
β
4B
2
z
+J
2
N
8J
X
S
g(S)e
−βB
x
(
S+
1
2
)
2 sinh
βω(S)
2
. (14)
The argument of the summation is a compli-
cated expression, and for this reason there is no
general analytical solution for Z. As we will be
interested in two specific temperature regimes, we
approximate the sum by looking at which terms
are relevant.
In the low temperature regime, T ≈ 0, the sum
over S will be dominated by values of S ≈ N/2.
This can easily be seen e.g. by plotting the terms
of the sum in (14) as a function of S. Then one
sees that only the terms that are close to S ≈
N/2 contribute. This observation allows us to
set ω(S) ' ω(N/2), which becomes independent
of S. Therefore, the denominator of Eq. (14) can
be taken outside the sum. Since in this case the
approximation g(S) ' N
p
/p! holds, we can find
an analytical expression for the summation over
S:
X
S
g(S)e
−βB
x
(
S+
1
2
)
'
X
p
N
p
p!
e
−βB
x
(
N
2
−p+
1
2
)
(15)
= e
−β
B
x
2
(N+1)
e
Ne
βB
x
.
(16)
To summarize, in the low-temperature regime
the canonical partition function can be approxi-
mated by
Z
T ≈0
=
e
β
4B
2
z
+J
2
N
8J
2 sinh
βω(S)
2
e
−β
B
x
2
(N+1)
e
Ne
βB
x
.
(17)
The second temperature regime of our interest
is the one for which T ≈ 1, for a reason that
will be clear in the following. At this tempera-
ture, and for the parameters B and J introduced
later, we observe that the terms in the sum of
Eq. (14) are dominant for S ≈ N/4. In this situ-
ation, we cannot provide an accurate and simple
Accepted in Quantum 2018-11-16, click title to verify 4
approximation to g(S), and therefore we adopt
a different approach. By observing that in the
limit N → ∞ the terms in the summation of
Eq. (14) follow a Gaussian distribution with mean
S = N/4, we approximate them in this way. Fi-
nally, the sum is replaced by an integral which can
be easily computed for a Gaussian. The resulting
(approximated) partition function Z
T ≈1
is an an-
alytical expression too complicated to be written
here (see Appendix), and therefore we will only
present some results computed from it.
The crucial quantity of our interest will be the
total energy of the system at finite temperature,
which is defined as
hHi
T
= −
∂
∂β
log Z. (18)
In the regime T ≈ 0 and N → ∞ we find
hHi
T ≈0
'
4B
x
− J
8
− B
x
e
B
x
β
N+
√
−B
x
JN
2
√
2
.
(19)
For the parameters of our interest, B =
(−1, 0,
√
3) and J = 6, this becomes
hHi
T ≈0
'
1
4
4e
−β
− 5
N + 2
√
3N
. (20)
For these specific parameters, we find in the
regime T ≈ 1 and N → ∞ the simple expression
hHi
T ≈1
'
1
64
9
√
6
√
N(2β + 2 − log(3))−
− 4N(3β + 16 − 3 log(3))
. (21)
4 Bell inequality
In this section we introduce the class of inequal-
ities with which we consider to detect Bell cor-
relations at finite temperature. We note that
our scenario of interest is not the one in which a
loophole-free Bell test is performed, but rather, in
which one certifies the presence of a Bell-nonlocal
state; i.e., a quantum state capable to produce
Bell correlations if a proper loophole-free Bell test
were performed upon it. We also recall the map-
ping between multipartite Bell inequalities and
quantum Hamiltonians. In particular, we show
how the Bell operator associated to the inequali-
ties of our interest is related to the Hamiltonian
in Eq. (1).
We consider the simplest multipartite Bell ex-
periment, in which N spatially separated ob-
servers labelled from 1 to N hold their shares
of a quantum system and have at their disposal
two dichotomic measurements each. The mea-
surements are labelled 0 or 1, and they can yield
outcomes ±1. We denote the k-th measurement
performed on the i-th party M
(i)
k
. We are inter-
ested in Bell inequalities that involve only sym-
metric one- and two-body correlators, as they are
the simplest to be accessed experimentally. These
correlators are defined as
S
k
=
N
X
i=1
M
(i)
k
, (22)
S
kl
=
X
i6=j
M
(i)
k
M
(j)
l
, (23)
and they yield Bell inequalities of the following
form [38, 39]:
I =
X
k
α
k
S
k
+
X
k≤l
α
kl
S
kl
≤ β
C
, (24)
where β
C
∈ is the so-called classical bound and
α
k
, α
kl
are real parameters.
To illustrate our framework, we shall focus on
the multipartite Bell inequality considered in [38],
I = −2S
0
+
1
2
S
00
− S
01
+
1
2
S
11
≥ −2N, (25)
even if our method can be straightforwardly
adapted to any other inequality of the form of
Eq. (24).
Because Ineq. (25) has two settings and two
outcomes, it can be minimized with traceless
real qubit measurements [65]. For this reason,
without loss of generality, it is enough to con-
sider M
(i)
0
= cos(φ)σ
(i)
z
+ sin(φ)σ
(i)
x
and M
(i)
1
=
cos(θ)σ
(i)
z
+ sin(θ)σ
(i)
x
. As shown in [39], only the
angle between θ and φ matters, and therefore it
is also possible to choose φ = π −θ, which allows
us to transform the Bell inequality (25) into the
Bell operator
B
0
= − 2 sin(θ)
N
X
i=1
σ
(i)
x
+ 2 cos(θ)
N
X
i=1
σ
(i)
z
+
+ 2 cos(θ)
2
X
i6=j
σ
(i)
z
σ
(j)
z
−2N , (26)
where A B stands for A − B 0; i.e., A − B
being a positive semidefinite operator. Moreover,
Accepted in Quantum 2018-11-16, click title to verify 5
in the limit of N → ∞ it was found [39] that
the optimal measurement angle is θ = π/6. This
gives, with the definition s
(i)
= σ
(i)
/2, the Bell
operator
B = −
N
X
i=1
s
(i)
x
+
√
3
N
X
i=1
s
(i)
z
+
+ 6
X
i<j
s
(i)
z
s
(j)
z
−N . (27)
Since hBi ≥ −N for all local states, observing
hBi < −N witnesses nonlocality.
At this point, we can motivate our choice of
the parameters B = (−1, 0,
√
3) and J = 6 for
Eq. (1). This particular case is of special interest
as it results in a spin model whose Hamiltonian
is identical to the Bell operator Eq. (27).
Now the idea is simple: since the specific Bell
inequality and spin Hamiltonian we considered
can be mapped one into the other, the energy of
our spin system can be identified to the expec-
tation value of the Bell operator, i.e. hBi
T
=
hH
0
i
T
. For low enough temperatures the en-
ergy of the system will be lower than the classical
bound, implying the presence of Bell correlations.
5 Nonlocality versus temperature
We now investigate the dependence of hBi
T
=
hH
0
i
T
on the temperature T . We plot in Fig. 2
the mean energy (cf. Eq. (18)) per particle as a
function of T , for N = 10
5
. This modest number
of particles allows us to compute numerically the
exact partition function Eq. (14), and from it the
mean energy per particle (orange line in Fig. 2).
Comparing this latter to the classical bound of
the Bell operator Eq. (27) normalized per par-
ticle, −N/N = −1 (blue line in Fig. 2), shows
a violation of the inequality hBi
T
≥ −N up to
a temperature of ≈ 0.9 (see the intersection be-
tween the orange and the blue line in Fig. 2).
In the limit N → ∞, we are interested in find-
ing the critical temperature T
c
above which non-
locality is not detected by the inequality we con-
sidered. Naively, from the low temperature ap-
proximation Eq.(20) we obtain
T
guess
c
=
B
x
log
8+4B
x
−J
8B
x
= (log 4)
−1
≈ 0.72 . (28)
However, we know from Fig. 2 that this estimate
is incorrect, as nonlocality can be detected up
to T ≈ 0.9 > T
guess
c
. Moreover, already from the
fact that T
guess
c
≈ 1, one sees a posteriori that the
low temperature approximation is not sufficiently
accurate in this regime. For this reason, we adopt
the second approximation introduced in Section
3, valid in the regime T ≈ 1.
From the expression of the mean energy given
in Eq. (21), we obtain the exact transition tem-
perature which is
T
c
= (log 3)
−1
≈ 0.91 . (29)
This result is consistent with the validity regime
T ≈ 1 of Eq. (21), and with what we see from
Fig. 2.
For experimentally realistic parameters [53],
B
x
/~ = −1 s
−1
, B
z
/~ =
√
3 s
−1
and J/~ =
6 s
−1
, the resulting transition temperature is T
c
≈
6.95 pK. Let us remark that such temperature is
not the one associated to the motion of the par-
ticles, but to the spin degree of freedom.
We emphasize that the critical temperature
Eq. (29) has been derived from considering the
specific Bell inequality Eq. (25). Therefore, it is
only a lower bound on the temperature that the
considered spin system allows us for displaying
nonlocal correlations. In fact, there might ex-
ist Bell inequalities that are better suited (even
though Inequality (25) has been chosen because
it is experimentally accessible), which allow to
observe Bell correlations in the regime T > T
c
.
Looking for a better, and maybe an upper, bound
on the transition temperature will be addressed in
a future work, for example involving multi-setting
inequalities [41].
Even if the low temperature approximation
Eq. (19) is not suited to describe the regime
around the critical temperature, it enables us to
study the maximum quantum violation of the in-
equality we considered. Note that the ground
state energy of the system, in the limit N → ∞,
tends to
lim
N→∞
E(N/2, 0) = −
5N
4
, (30)
which coincides with the quantum bound of
Ineq. (25), as proven in Ref. [39].
6 Conclusions
We have introduced a framework for the study of
Bell correlations in a wide class of spin models at
Accepted in Quantum 2018-11-16, click title to verify 6
〈�
〉
�
〈〉
�≈
()
〈〉
�≈
()
-
-
-
-
-
-
=β
-
/
Figure 2: With N = 10
5
, mean energy per particle
as a function of the temperature. The orange line is
the exact mean energy per particle computed numeri-
cally (cf. Eq. (1) and Eq. (18)). The green dashed line
corresponds to its low temperature approximation (cf.
Eq. (19)) whereas the red dashed line corresponds to the
approximation that is valid around T ≈ 1 (cf. Eq. (21)).
The shaded orange region is the one for which nonlocal-
ity is detected by considering Eq.(27). The thin vertical
gray line at T
c
= (log 3)
−1
corresponds to the value of
the critical temperature (cf. Eq. (29)). The thin hor-
izontal gray line at −5/4 corresponds to the maximal
quantum violation of Eq.(27) in the limit N → ∞ (cf.
Eq. (30)).
finite temperature. This allowed us to show that
nonlocal correlations appear naturally in many-
body spin systems, when the temperature is be-
low a critical threshold T
c
.
Our results show a clear connection between
two-body permutationally invariant Bell inequal-
ities and bosonic systems, via the Holstein-
Primakoff approximation and the Bogouliubov
transformation. In particular, the low-end part
of the spectrum of the Bell operator correspond-
ing to the measurements leading to the maximal
quantum violation of the Bell inequality, can be
well approximated by a quantum harmonic oscil-
lator, thus extremely simplifying the analysis.
Our work also proves that spin-squeezed states
are asymptotically the optimal ones for the class
of Bell inequalities (25). Therefore, the quan-
tum states that were prepared in previous exper-
iments detecting Bell correlations in many-body
systems belonged to that optimal class (spin-
squeezed BEC [49] or thermal ensemble of atoms
[50]).
Although, for simplicity, we introduced here
our framework with a specific Bell inequality (cf.
Ineq (25)), it can easily be generalized. Of special
interest would be to find other inequalities that
allow to detect Bell correlations within a larger
range of temperatures, and generalizations to in-
equalities with more measurement settings [41]
would constitute a good candidate for this task.
We leave as an open problem also the general-
ization to systems of larger spin. To the best of
our knowledge, currently there is no many-body
Bell inequality suited to e.g. spin-1 systems, and
one would need to resort to methods based on
SDP hierarchies to plausibly obtain them [40].
7 Acknowledgments
We are grateful to Roman Schmied for the
useful discussions and for the help with the
numerical code. M. F. was supported by
the Swiss National Science Foundation through
Grant No 200020_169591. This project has re-
ceived funding from the European Union’s Hori-
zon 2020 research and innovation programme un-
der the Marie-Skłodowska-Curie grant agreement
No 748549.
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8 Appendix
We give here the expression for the approximated
partition function, valid around T ≈ 1 for N →
∞. For B = (−1, 0,
√
3) and J = 6 we find
Z
T ≈1
≈ 2
2N+3
3
β
9
32
p
3
2
√
N−
3N
16
−
3N
4
−
5
6
×
× exp
"
3N
32
−
9
32
r
3
2
√
N +
81
256
!
β
2
+
+
N −
9
16
r
3
2
√
N +
q
3
2
4
√
N
−
1
6
β−
−
109
324N
+
3
32
N log
2
(3) −
5
18
#
. (31)
Accepted in Quantum 2018-11-16, click title to verify 9