[table]capposition=top capbtabboxtable[][]

# Simultaneous Verification of Genuine Multipartite Nonlocality and Full
Network Nonlocality

Ning-Ning Wang${}^{1,2,3,5\,*}$, Xue Yang4111These authors contribute
equally., Yan-Han Yang4, Chao<EMAIL_ADDRESS>Ming-
Xing<EMAIL_ADDRESS>Bi-Heng Liu1,2,3,5, Yun-Feng
<EMAIL_ADDRESS>Chuan-Feng<EMAIL_ADDRESS>Guang-Can Guo1,2,3,5 1\. CAS Key Laboratory of Quantum Information, University
of Science and Technology of China, Hefei, 230026, P. R. China
2\. Anhui Province Key Laboratory of Quantum Network, University of Science
and Technology of China, Hefei 230026, P. R. China
3\. CAS Center For Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, 230026, P. R. China
4\. School of Information Science and Technology, Southwest Jiaotong
University, Chengdu 610031, P. R. China
5\. Hefei National Laboratory, University of Science and Technology of China,
Hefei, 230088, P. R. China

###### Abstract

Genuine multipartite nonlocality and nonlocality arising in networks composed
of several independent sources have been separately investigated. While some
genuinely entangled states cannot be verified by violating a single Bell-type
inequality, a quantum network consisting of different sources allows for the
certification of the non-classicality of all sources. In this paper, we
propose the first method to verify both types of nonlocality simultaneously in
a single experiment. We consider a quantum network comprising a bipartite
source and a tripartite source. We demonstrate that there are quantum
correlations cannot be simulated if the tripartite source distributes
biseparable systems while the bipartite source distributes even stronger-than-
quantum systems. These correlations can be used to verify both the genuine
multipartite nonlocality of generalized Greenberger-Horne-Zeilinger states and
the full network nonlocality that is stronger than all the existing results.
Experimentally, we observe both types of nonlocality in a high fidelity
photonic quantum network by violating a single network Bell inequality.

Introduction.–Bell’s theorem has revolutionized our understanding of entangled
quantum systems by revealing that they can exhibit correlations that defy the
constraints of any local realistic theory Bell (1964). This groundbreaking
discovery offers a new perspective on the challenge initially posed by the
Einstein-Podolski-Rosen (EPR) argument Einstein  _et al._ (1935). Bell
nonlocality, which transcends EPR’s explanation of steering, sheds light on
the profound quantum effects inherent in general entangled states Gisin
(1991); Gisin and Peres (1992). This phenomenon has been empirically confirmed
in numerous Bell experiments Giustina  _et al._ (2015); Shalm  _et al._
(2015); Hensen  _et al._ (2015), where violations of the Clauser-Horne-
Shimony-Holt (CHSH) inequality Clauser  _et al._ (1969) have been reported.
Beyond its fundamental significance, Bell nonlocality has also proven to be a
crucial resource in device-independent quantum information tasks Acín  _et
al._ (2007); Pironio  _et al._ (2010).

The first extension of bipartite nonlocality is to characterize the global
correlations of multipartite entanglement. The genuine multipartite
nonlocality (GMN) specifically pertains to the correlations observed in
systems with multiple particles, where these correlations cannot be simulated
using classical mixtures of bipartite separable correlations under the no-
signaling principle Svetlichny (1987); Popescu and Rohrlich (1992). This
distinction highlights the fundamentally distinct and complex nature of
nonlocality in multipartite entangled systems. The verification of the GMN
hinges on the utilization of specialized Bell-type inequalities, such as the
Mermin-Ardehali-Belinskii-Klyshko (MABK) and Svetlichny inequalities Mermin
(1990); Ardehali (1992); Belinskii and Klyshko (1993); Svetlichny (1987).
Other methods have been proposed to examine specific entangled states within
this context Chen  _et al._ (2014); Yu  _et al._ (2012). Notably, Scarani and
Gisin Scarani and Gisin (2001) have contributed to this area by demonstrating
that generalized Greenberger-Horne-Zeilinger (GHZ) states Greenberger  _et
al._ (1989) do not violate the MABK inequalities Mermin (1990); Ardehali
(1992); Belinskii and Klyshko (1993). This intriguing result has been further
corroborated for general states Żukowski  _et al._ (2002); Schwemmer  _et al._
(2015); Augusiak  _et al._ (2015). These findings have sparked renewed
interest in the exploration of novel methods to verify GMN Contreras-Tejada
_et al._ (2021a, 2022), presenting an intellectually stimulating and
compelling problem for further investigations within the realm of quantum
nonlocality Gallego  _et al._ (2012); Coiteux-Roy  _et al._ (2021).

Another intriguing extension involves relaxing the assumption of sources,
leading to the exploration of nonlocality in network scenarios. Specifically,
various results have focused on investigating nonlocal correlations arising
from entanglement-swapping experiments in networks Branciard  _et al._ (2010);
Rosset  _et al._ (2016); Chaves (2016); Luo (2018); Renou  _et al._ (2019);
Pozas-Kerstjens  _et al._ (2019); Šupić  _et al._ (2020); Contreras-Tejada
_et al._ (2021b); Tavakoli  _et al._ (2021); Renou and Beigi (2022); Pozas-
Kerstjens  _et al._ (2023); Gatto Lamas and Chitambar (2023); Tavakoli  _et
al._ (2022), where independent parties can become entangled despite a lack of
causal connections. Beyond the single local hidden variable (LHV) assumption
in standard Bell tests, it is more natural to consider models with independent
hidden variables that replicate the network structure. This has led to the
study of bilocal hidden variable models and the associated phenomenon of non-
bilocality. However, this kind of nonlocality may be simulated by mixing the
classical correlations and nonlocal correlations or post-quantum correlations
Pozas-Kerstjens  _et al._ (2022). A further extension is then to rule out all
of these correlations by defining the stronger local model, that is, any
hybrid network consisting of at least one classical source, while all the
others can be any no-signaling sources. This introduces the concept of full
network nonlocality (FNN) Pozas-Kerstjens  _et al._ (2022). Non-bilocality and
FNN have been verified via various experiments Carvacho  _et al._ (2017);
Saunders  _et al._ (2017); Andreoli  _et al._ (2017); Sun  _et al._ (2019);
Poderini  _et al._ (2020); Huang  _et al._ (2022); Gu  _et al._ (2023); Wang
_et al._ (2023). All of these experiments have only considered networks
consisting of all bipartite entangled two-photon states. While the assumption
for replacing the multipartite quantum source in a network with a classical
source is too strong, it intrigues a natural question of how to characterize
the new nonlocality to rule out biseparable sources Svetlichny (1987).

Figure 1: A causal network with four outcomes (gray box) of four observers,
Alice, Bob, Charlie, and Debby, depends on (black arrows) the input states
from one or two sources (circles) and the measurement settings ($x,y,u$).
Charlie performs a joint measurement on his systems yielding outcomes $c$. (a)
Full network nonlocality (FNN), where FNN rules out the correlations from the
hybrid network consisting of an LHV source (gray circle) and a general
nonlocal source (purple circle). (b) Simultaneous strong FNN (FNN#) and
genuine multipartite nonlocality (GMN), where FNN# rules out the correlations
from the hybrid model consisting of a biseparable source (blue circle) and a
general nonlocal source (orange circle), and GMN rules out all biseparable
correlations (deep blue square).

In this paper, we propose a novel method to verify both nonlocalities in a
single experiment simultaneously. We examine a quantum network comprising a
bipartite entangled state and a tripartite genuinely entangled state, as
depicted in Fig. 1. We demonstrate that local measurements for each observer
reveal quantum correlations that cannot be simulated if the tripartite sources
distribute biseparable systems Svetlichny (1987), while bipartite sources
distribute even stronger-than-quantum systems. The new experiment makes two
significant contributions. First, it verifies the GMN of generalized GHZ
states, which cannot be verified using the standard Svetlichny inequality
Svetlichny (1987), in a more accessible way with the assistance of
entanglement swapping. Second, it introduces a new type of nonlocality in
network scenarios. Since our model involves local biseparable and no-signaling
sources, which is stronger than the mixture of bipartite classical and no-
signaling sources Pozas-Kerstjens  _et al._ (2022), our findings reveal
stronger nonlocality than what can be verified by the FNN framework.

Simultaneously Verifying GMN and strong FNN.–Consider a local network model
consisting of a biseparable source and a bipartite no-signaling source, as
shown in Fig. 1. Here, Alice, Bob, and Charlie share one source while Charlie
and Debby share one source. The outcome denoted by $a$ of Alice depends on the
local measurement setting $x$ and its shares. Similarly, the outcome $b$ of
Bob depends on the local measurement setting $y$ and its shares. The outcome
$d$ of Debby depends on the local measurement setting $u$ and its shares.
There is one measurement setting for Charlie with measurement outcome $c$. The
goal here is to verify two kinds of nonlocalities simultaneously within one
network Bell experiment.

One is to verify the joint correlations generated by Alice, Bob, and Charlie
exhibit GMN to rule out those that can be simulated by the biseparable model
Svetlichny (1987). There are also definitions of the GMN in the resource
framework Gallego  _et al._ (2012); Coiteux-Roy  _et al._ (2021), whereas our
discussion of GMN focuses on the partitions of correlations. Specifically, the
tripartite joint correlations (e.g., $P_{bs}(a,b,c|x,y)$, without the
assumption of measurement input for Charlie) generated when the shared source
is biseparable allow the following decomposition Svetlichny (1987):

$\displaystyle P_{bs}(a,b,c|x,y)$ $\displaystyle=$ $\displaystyle
p_{1}p(a|x)p(b,c|y)+p_{2}p(b|y)p(a,c|x)$ (1)
$\displaystyle+p_{3}p(c)p(a,b|x,y),$

where $p_{i}$ represents the probability distribution characterizing the
classical mixture of biseparable correlations among Alice, Bob, and Charlie.
Additionally, $p(a|x)$, $p(b|y)$, and $p(c)$ represent the probability
distribution of the outcomes of Alice, Bob, and Charlie, respectively.
$p(a,b|x,y)$, $p(b,c|y)$ and $p(a,c|x)$ are the joint probability
distributions of the outcomes of Alice and Bob, Bob and Charlie, and Alice and
Charlie, respectively. These joint probability distributions satisfy the
single restriction of the no-signaling principle Popescu and Rohrlich (1992),
i.e., $\sum_{s}p(s,t|u,v)=p(t|v)$ and $\sum_{t}p(s,t|u,v)=p(s|u)$.

The other is to verify the joint correlations of all parties are strong full
network nonlocal (FNN#). The FNN correlations here are defined analogously to
standard nonlocal ones Pozas-Kerstjens  _et al._ (2022), which do not admit
the specific hybrid model shown in Fig. 1(a), where there is at least one
classical source (left). With local measurements, all parties can generate
joint correlations that allow the following decomposition as

$P(a,b,c,d|x,y,u)=\int
d\lambda\mu(\lambda)p(a|x,\lambda)p(b|y,\lambda)p(c,d|u,\lambda),$ (2)

where $\mu(\lambda)$ is the probability distribution characterizing the local
measurable variable $\lambda$ shared by Alice and Bob and Charlie,
$p(a|x,\lambda)$ is Alice’s response function, $p(b|y,\lambda)$ is Bob’s
response function, and $p(c,d|u,\lambda)$ is a joint response function for
Charlie and Debby, which satisfies the no-signaling principle, i.e.,
$\sum_{c}p(c,d|u,\lambda)=p(d|u)$ and $\sum_{d}p(c,d|u,\lambda)=p(c|\lambda)$.
The FNN can be established by ruling out all the cases by interchanging
sources. The strong FNN correlations (FNN#) are defined based on the model
shown in Fig. 1(b), which is analogous to fully network nonlocal correlations
Pozas-Kerstjens  _et al._ (2022) ruling out correlations from networks
consisting of at least one classical source and others being no-signaling
sources Popescu and Rohrlich (1992). To witness FNN#, we apply different
network realizations, i.e., a network consisting of a tripartite biseparable
source Svetlichny (1987) (left) and one bipartite no-signaling source Popescu
and Rohrlich (1992) (right). This allows the decomposition of the joint
network local correlations as

$\displaystyle P(a,b,c,d|x,y,u)=P_{bs}(a,b,c|x,y)P_{ns}(c,d|u),$ (3)

with biseparable correlations $P_{bs}(a,b,c|x,y)$ and no-signalling
correlations $P_{ns}(c,d|u)$. So far, there is no result in exploring the non-
classicality of this kind of network.

In what follows, consider a Bell experiment on the network shown in Fig. 1(b).
We show the non-classicality of quantum correlations generated by generalized
entanglement-swapping Żukowski  _et al._ (1993), where we suppose that
$a,b,d,x,y,u\in\\{0,1\\}$ and $c\in\\{0,1,2\\}$. All the joint correlations
$P(a,b,c,d|x,y,u)$ generated from hybrid networks satisfy the following
network Bell inequalities (see the proof in Supplemental Material):

$\langle S_{4}^{c}\rangle/\langle C^{c}\rangle\leq 4,c=0,1,2,$ (4)

where $S_{4}$ is a combination of a tripartite Svetlichny operator with a
positive-operator valued measurement defined by

$\displaystyle S_{4}^{c}$ $\displaystyle=$ $\displaystyle
A_{0}B_{0}C^{c}D_{1}+A_{0}B_{1}C^{c}D_{0}+A_{1}B_{0}C^{c}D_{0}$ (5)
$\displaystyle-A_{1}B_{1}C^{c}D_{1}+A_{1}B_{1}C^{c}D_{0}+A_{1}B_{0}C^{c}D_{1}$
$\displaystyle+A_{0}B_{1}C^{c}D_{1}-A_{0}B_{0}C^{c}D_{0},$

and the mean value of the correlators $\langle A_{x}B_{y}C^{c}D_{z}\rangle$ is
defined by $\langle
A_{x}B_{y}C^{c}D_{z}\rangle=\sum_{a,b,d=0,1}(-1)^{a+b+d}P(a,b,c,d|x,y,z)$ when
three leaf nodes perform Pauli measurement, and $\langle C^{c}\rangle=P(c)$ is
the probability that Charlie obtains the output $c$. The inequalities (4) mean
that the three leaf nodes perform the Svetlichny test in the case where
Charlie obtains the output $c$. Thus, we can examine the average result of the
three Svetlichny tests in terms of the output probability obtained by Charlie:

$\langle S_{4}^{0}\rangle+\langle S_{4}^{1}\rangle+\langle
S_{4}^{2}\rangle\leq 4,$ (6)

The inequalities (4) and (6) are further applied to witness both the GMN of
partial tripartite correlations and FNN# of four-partite joint correlations,
i.e., any violation of them certifies both the GMN of tripartite source and
the FNN# of hybrid networks.

In quantum scenarios, the network Bell inequalities (4) and (6) can be
violated by quantum correlations. Consider that the tripartite source emits a
generalized GHZ state Greenberger  _et al._ (1989):
$|GHZ(\theta)\rangle=\cos\theta|000\rangle+\sin\theta|111\rangle$ and the
bipartite source emits a generalized EPR state Einstein  _et al._ (1935):
$|EPR(\theta)\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$ with
$\theta\in(0,\frac{\pi}{2})$. Charlie performs the partial Bell state
measurement (PBSM) under the basis
$\\{\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle),\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle),|00\rangle\langle
00|+|11\rangle\langle 11|\\}$ labeled by $c=\\{0,1,2\\}$. When Charlie has an
output of $c=0$, Alice measures the observables $A_{0}=X$ and $A_{1}=Y$, Bob
measures the observables $B_{0}=-\frac{1}{\sqrt{2}}(X+Y)$ and
$B_{1}=\frac{1}{\sqrt{2}}(X-Y)$, and Debby measures the observables $D_{0}=X$
and $D_{1}=-Y$, where $X$ and $Y$ are Pauli operators. The joint conditional
distribution $P(a,b,c=0,d|x,y,u)$ in this scenario gives $\langle
S_{4}^{0}\rangle/\langle C^{0}\rangle=4\sqrt{2}$. When Charlie has an output
of $c=1$, Alice and Bob measure the same observables, while Debby measures the
negative of the previous observables, i.e. $D_{0}=-X$ and $D_{1}=Y$. The joint
conditional distribution $P(a,b,c=1,d|x,y,u)$ in this scenario gives $\langle
S_{4}^{1}\rangle/\langle C^{1}\rangle=4\sqrt{2}$. Both scenarios violate the
inequality (4), thus verifying the GMN of generalized GHZ states and FNN#
simultaneously. For the output of $c=2$, the three leaf nodes become
separable, in which case they all measure the identity operator. The joint
conditional distribution $P(a,b,c=2,d|x,y,u)$ gives $\langle
S_{4}^{2}\rangle/\langle C^{2}\rangle=4$, which equals the classical bound.
Examining the average of these three cases implies

$\langle S_{4}^{0}\rangle+\langle S_{4}^{1}\rangle+\langle
S_{4}^{2}\rangle=4+2(\sqrt{2}-1)\sin^{2}{2\theta}.$ (7)

The quantum correlations generated from the hybrid network can always violate
the inequality (6) regardless of the value of $\theta$. In contrast, the
standard Svetlichny inequality can only certify the GMN for generalized GHZ
states with $\theta>\pi/8$ (see Fig. 3).

As all the observables share the same eigenbases and differ only in the sign
of their eigenvalues, no feedback is required for this experiment test, i.e.,
two measurement inputs for each leaf node are sufficient to examine these
network Bell inequalities. We extend to verify the general GMN of multipartite
GHZ states and FNN# of quantum networks in Supplemental Material.

Figure 2: Sketch of the experimental setup. The network is composed of
bipartite and tripartite sources. The bipartite source is a sandwich-like BBO-
HWP-BBO structure, where the first BBO can be translated along the pump path,
so that the photons collected by single-mode fibers from this crystal can be
tuned, enabling the distribution of generalized EPR states. The tripartite
source distributes generalized GHZ states, which are prepared by Hong-Ou-
Mandel interference of a maximally entangled state with a single-qubit
superposition state. The coherence between pump pulses is erased by randomly
tilted optical windows. Charlie in the central node performs partial Bell
state measurement using pseudo-number-resolving detectors. The other three
nodes perform single-qubit measurements by setting the HWP and QWP to
appropriate angles. BBO beta barium borate, PBS polarizing beam splitter, HWP
half-wave plate, QWP quarter-wave plate, IF interference filter, FC fiber
coupler, QRNG quantum random number generator.

Experiment.–The quantum network is built on a photonic platform, as shown in
Fig. 2. We encode qubits using photonic polarization degrees of freedom, i.e.,
$|0\rangle$/$|1\rangle$ by the horizontally and vertically polarized states
$|H\rangle$/$|V\rangle$. Photons are produced by the spontaneous parametric
downconversion (SPDC) process in beta barium borate (BBO) crystals, which are
pumped by ultraviolet pulses. These pulses, generated by a single laser, are
split to pump two separate sources. To enhance the independence between the
two sources, we insert a randomly tilting optical window controlled by a
quantum random number generator into each pump path, which can apply a
randomly varying phase to each pump beam, thus erasing the coherent
information between the two beams. We measured the mutual information between
any two leaf nodes, and the results were very close to 0 regardless of which
measurement setting was chosen, aligning with the independent source condition
(see Supplemental Material for more details).

Our bipartite and tripartite photon sources can distribute generalized EPR
states and generalized GHZ states, respectively, with $\theta$ tunable between
0 and $\pi/2$ (see Fig. 2 and Supplemental Material for more details). During
the experiment, we set $\theta$ of each state to be as small as possible while
keeping the success probability of the Bell states projection in the central
node not too low. Through quantum state tomography, we find that the highest
fidelity of the bipartite source compared to the generalized EPR state is
attained at $\theta=0.2734\pm 0.0001$, which is $0.9809\pm 0.0001$. We find
that the highest fidelity of the tripartite source compared to the generalized
GHZ state is given at $\theta=0.2741\pm 0.0014$, which is $0.9605\pm 0.0011$.
However, when measuring the standard Svetlichny inequality, we obtain a result
of $2.7353\pm 0.0172$, which does not exceed the classical bound 4, so the GMN
of the tripartite source cannot be verified.

In the hybrid network, each of the three leaf nodes receives one photon, and
we employ waveplates and polarizing beam splitter (PBS) to conduct single-
qubit Pauli measurements according to their inputs. The central node receives
one photon from each source and performs partial PBSM, resulting in three
outputs. The PBSM is realized with PBS and several pseudo-number-resolving
detectors. The response of different groups of detectors distinguishes the
three outputs (see Supplemental Material for more details). To characterize
the setup, we perform the measurement tomography on our setup, and the result
shows a fidelity of $0.9581\pm 0.0005$ compared to the ideal PBSM.

Figure 3: The experimental results of two nonlocal tests. The tripartite
correlation generated by the generalized GHZ state cannot be certified as
genuine multipartite nonlocality by violating the standard Svetlichny
inequality when $\theta\leq\pi/8$. In the hybrid network, the genuine
multipartite nonlocality and strong full network nonlocality can be
simultaneously verified by our network Bell inequality, regardless of
$\theta$. The error bars are deduced from the photon statistical error of the
raw data.

We experimentally performed 8 sets of measurement settings, each collecting
approximately 5300 six-photon coincidence events over 60,000 seconds. By
normalizing the raw data we can obtain the distribution $P(a,b,c,d|x,y,u)$
(see Supplemental Material for more details). Therefore, we find $\langle
S_{4}^{0}\rangle/\langle C^{0}\rangle=4.4056\pm 0.0868$, $\langle
S_{4}^{1}\rangle/\langle C^{1}\rangle=4.4443\pm 0.0865$ and $\langle
S_{4}^{2}\rangle/\langle C^{2}\rangle=4$. By calculating the marginal
probability we find $\langle C^{0}\rangle=0.0815\pm 0.0010$, $\langle
C^{1}\rangle=0.0815\pm 0.0011$ and $\langle C^{2}\rangle=0.0870\pm 0.0015$.
The average result of three inequalities consequently yielded $\langle
S_{4}^{0}\rangle+\langle S_{4}^{1}\rangle+\langle S_{4}^{2}\rangle=4.0693\pm
0.0101$, surpassing the classical bound by more than 6 standard deviations
(see Fig. 3). Thus, our results simultaneously witness GMN of the tripartite
source and FNN# of the quantum network.

Conclusion.–The present scheme is robust against noise. Consider a noisy
network consisting of two Werner states Werner (1989):
$\rho_{v}=v_{1}|GHZ(\theta)\rangle\langle
GHZ(\theta)|+\frac{1-v_{1}}{8}\mathbbm{1}_{8}$ and
$\varrho_{v}=v_{2}|EPR(\theta)\rangle\langle
EPR(\theta)|+\frac{1-v_{2}}{4}\mathbbm{1}_{4}$, where $v_{i}\in(0,1]$ denotes
the noise parameters of two entangled states and $\mathbbm{1}_{k}$ is the
identity matrix with rank $k$. The linear correlator (5) implies a total noise
visibility of $v_{1}v_{2}\geq\frac{1}{\sqrt{2}}$, which further allows a
single constant noise visibility of $v_{1}\geq\frac{1}{\sqrt{2}}$ for
witnessing all generalized GHZ pure states. This implies that any nontrivial
quantum correlations activated from entanglement swapping show the non-
classicality of both quantum entanglement, going beyond the standard Bell test
by using the MABK inequalities, Svetlichny inequality Mermin (1990); Ardehali
(1992); Belinskii and Klyshko (1993); Svetlichny (1987); Ajoy and Rungta
(2010) or the Hardy inequality Chen  _et al._ (2014); Yu  _et al._ (2012).
Similar results hold for witnessing all generalized EPR states beyond the CHSH
test Clauser  _et al._ (1969). This means that the present network method
shows a new possibility for witnessing noisy entanglement beyond Bell
experiments with single entanglement. The present scheme generally does not
require the consistency of two entangled sources. The extended setting up,
i.e., $\cos\theta_{1}|000\rangle+\sin\theta_{1}|111\rangle$ and
$\cos\theta_{2}|00\rangle+\sin\theta_{2}|11\rangle$ with
$\theta_{1}\not=\theta_{2}$, requires a generalized Bell-basis measurement
performed by Charlie. Our results provide new insight into quantum networks
and intrigue further studies to explore the non-classicality of quantum
networks with different resources.

Experimentally, we verified FNN#, which can rule out network correlations
generated by at least one biseparable source, in a high-fidelity photonic
hybrid network consisting of tripartite and bipartite sources, employing
entanglement swapping. Simultaneously, the results verified the GMN exhibited
by the generalized GHZ states, even if it cannot be directly verified via the
standard Svetlichny inequality Svetlichny (1987). Thus, two kinds of complex
nonlocalities are verified in a single experiment. This approach can also be
viewed as a network-assisted device-independent certification protocol for
genuine multipartite entanglement Zhang  _et al._ (2022); Šupić  _et al._
(2023). Finally, we note that our results are subject to common loopholes in
nonlocal experiments: locality loophole, source independence loophole,
detection loophole, and freedom-of-choice loophole. Further experiments
conducted on larger spatial scales Sun  _et al._ (2019); Gu  _et al._ (2023)
and utilizing herald setups Cao  _et al._ (2024); Chen  _et al._ (2024) offer
possible routes for closing these loopholes.

Acknowledgments.– This work was supported by the National Natural Science
Foundation of China (Nos.62172341, 62075208, 11821404, 11734015), the
Fundamental Research Funds for the Central Universities (No. YD2030002015),
Sichuan Natural Science Foundation
(Nos.2023NSFSC0447,2024NSFSC1365,2024NSFSC1375), Innovation Program for
Quantum Science and Technology (No. 2021ZD0301604), Interdisciplinary Research
of Southwest Jiaotong University China (No.2682022KJ004). This work was
partially carried out at the USTC Center for Micro and Nanoscale Research and
Fabrication.

## References

  * Bell (1964) J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika 1, 195 (1964).
  * Einstein  _et al._ (1935) A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 47, 777 (1935).
  * Gisin (1991) N. Gisin, Bell’s inequality holds for all non-product states, Phys. Lett. A 154, 201 (1991).
  * Gisin and Peres (1992) N. Gisin and A. Peres, Maximal violation of bell’s inequality for arbitrarily large spin, Phys. Lett. A 162, 15 (1992).
  * Giustina  _et al._ (2015) M. Giustina, M. A. M. Versteegh, S. Wengerowsky, J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlechner, J. Kofler, J.-A. Larsson, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, J. Beyer, T. Gerrits, A. E. Lita, L. K. Shalm, S. W. Nam, T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, Significant-loophole-free test of bell’s theorem with entangled photons, Phys. Rev. Lett. 115, 250401 (2015).
  * Shalm  _et al._ (2015) L. K. Shalm, E. Meyer-Scott, B. G. Christensen, P. Bierhorst, M. A. Wayne, M. J. Stevens, T. Gerrits, S. Glancy, D. R. Hamel, M. S. Allman, K. J. Coakley, S. D. Dyer, C. Hodge, A. E. Lita, V. B. Verma, C. Lambrocco, E. Tortorici, A. L. Migdall, Y. Zhang, D. R. Kumor, W. H. Farr, F. Marsili, M. D. Shaw, J. A. Stern, C. Abellán, W. Amaya, V. Pruneri, T. Jennewein, M. W. Mitchell, P. G. Kwiat, J. C. Bienfang, R. P. Mirin, E. Knill, and S. W. Nam, Strong loophole-free test of local realism, Phys. Rev. Lett. 115, 250402 (2015).
  * Hensen  _et al._ (2015) B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson, Loophole-free bell inequality violation using electron spins separated by 1.3 kilometres, Nature 526, 682 (2015).
  * Clauser  _et al._ (1969) J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880 (1969).
  * Acín  _et al._ (2007) A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Device-independent security of quantum cryptography against collective attacks, Phys. Rev. Lett. 98, 230501 (2007).
  * Pironio  _et al._ (2010) S. Pironio, A. Acín, S. Massar, A. Boyer de La Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Random numbers certified by Bell’s theorem, Nature 464, 1021 (2010).
  * Svetlichny (1987) G. Svetlichny, Distinguishing three-body from two-body nonseparability by a Bell-type inequality, Phys. Rev. D 35, 3066 (1987).
  * Popescu and Rohrlich (1992) S. Popescu and D. Rohrlich, Generic quantum nonlocality, Phys. Lett. A 166, 293 (1992).
  * Mermin (1990) N. D. Mermin, Extreme quantum entanglement in a superposition of macroscopically distinct states, Phys. Rev. Lett. 65, 1838 (1990).
  * Ardehali (1992) M. Ardehali, Bell inequalities with a magnitude of violation that grows exponentially with the number of particles, Phys. Rev. A 46, 5375 (1992).
  * Belinskii and Klyshko (1993) A. V. Belinskii and D. N. Klyshko, Interference of light and bell’s theorem, Phys. Usp. 36, 653 (1993).
  * Chen  _et al._ (2014) Q. Chen, S. Yu, C. Zhang, C. H. Lai, and C. H. Oh, Test of genuine multipartite nonlocality without inequalities, Phys. Rev. Lett. 112, 140404 (2014), arXiv:1305.4472 .
  * Yu  _et al._ (2012) S. Yu, Q. Chen, C. Zhang, C. H. Lai, and C. H. Oh, All entangled pure states violate a single bell’s inequality, Phys. Rev. Lett. 109, 120402 (2012).
  * Scarani and Gisin (2001) V. Scarani and N. Gisin, J. Phys. A: Math. Gen. 34, 6043 (2001).
  * Greenberger  _et al._ (1989) D. M. Greenberger, M. A. Horne, and A. Zeilinger, Bell’s theorem, quantum theory, and conceptions of the universe (Springer Dordrecht, Kluwer, Dordrecht, 1989) Chap. 10: Going Beyond Bell’s Theorem, pp. 69–72.
  * Żukowski  _et al._ (2002) M. Żukowski, C. Brukner, W. Laskowski, and M. Wieśniak, Do all pure entangled states violate Bell’s inequalities for correlation functions?, Phys. Rev. Lett. 88, 210402 (2002).
  * Schwemmer  _et al._ (2015) C. Schwemmer, L. Knips, M. C. Tran, A. de Rosier, W. Laskowski, T. Paterek, and H. Weinfurter, Genuine multipartite entanglement without multipartite correlations, Phys. Rev. Lett. 114, 180501 (2015).
  * Augusiak  _et al._ (2015) R. Augusiak, M. Demianowicz, J. Tura, and A. Acín, Entanglement and nonlocality are inequivalent for any number of parties, Phys. Rev. Lett. 115, 030404 (2015).
  * Contreras-Tejada  _et al._ (2021a) P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Genuine multipartite nonlocality is intrinsic to quantum networks, Phys. Rev. Lett. 126, 040501 (2021a).
  * Contreras-Tejada  _et al._ (2022) P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Asymptotic survival of genuine multipartite entanglement in noisy quantum networks depends on the topology, Phys. Rev. Lett. 128, 220501 (2022).
  * Gallego  _et al._ (2012) R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, Operational framework for nonlocality, Phys. Rev. Lett. 109, 070401 (2012).
  * Coiteux-Roy  _et al._ (2021) X. Coiteux-Roy, E. Wolfe, and M.-O. Renou, No bipartite-nonlocal causal theory can explain nature’s correlations, Phys. Rev. Lett. 127, 200401 (2021).
  * Branciard  _et al._ (2010) C. Branciard, N. Gisin, and S. Pironio, Characterizing the nonlocal correlations created via entanglement swapping, Phys. Rev. Lett. 104, 170401 (2010).
  * Rosset  _et al._ (2016) D. Rosset, C. Branciard, T. J. Barnea, G. Pütz, N. Brunner, and N. Gisin, Nonlinear bell inequalities tailored for quantum networks, Phys. Rev. Lett. 116, 010403 (2016).
  * Chaves (2016) R. Chaves, Polynomial bell inequalities, Phys. Rev. Lett. 116, 010402 (2016).
  * Luo (2018) M.-X. Luo, Computationally efficient nonlinear Bell inequalities for quantum networks, Phys. Rev. Lett. 120, 140402 (2018).
  * Renou  _et al._ (2019) M.-O. Renou, E. Bäumer, S. Boreiri, N. Brunner, N. Gisin, and S. Beigi, Genuine quantum nonlocality in the triangle network, Phys. Rev. Lett. 123, 140401 (2019).
  * Pozas-Kerstjens  _et al._ (2019) A. Pozas-Kerstjens, R. Rabelo, L. Rudnicki, R. Chaves, D. Cavalcanti, M. Navascués, and A. Acín, Bounding the sets of classical and quantum correlations in networks, Phys. Rev. Lett. 123, 140503 (2019).
  * Šupić  _et al._ (2020) I. Šupić, J.-D. Bancal, and N. Brunner, Quantum nonlocality in networks can be demonstrated with an arbitrarily small level of independence between the sources, Phys. Rev. Lett. 125, 240403 (2020).
  * Contreras-Tejada  _et al._ (2021b) P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Genuine multipartite nonlocality is intrinsic to quantum networks, Phys. Rev. Lett. 126, 040501 (2021b).
  * Tavakoli  _et al._ (2021) A. Tavakoli, N. Gisin, and C. Branciard, Bilocal Bell inequalities violated by the quantum Elegant Joint Measurement, Phys. Rev. Lett. 126, 220401 (2021).
  * Renou and Beigi (2022) M.-O. Renou and S. Beigi, Nonlocality for generic networks, Phys. Rev. Lett. 128, 060401 (2022).
  * Pozas-Kerstjens  _et al._ (2023) A. Pozas-Kerstjens, N. Gisin, and M.-O. Renou, Proofs of network quantum nonlocality in continuous families of distributions, Phys. Rev. Lett. 130, 090201 (2023).
  * Gatto Lamas and Chitambar (2023) A. Gatto Lamas and E. Chitambar, Multipartite nonlocality in clifford networks, Phys. Rev. Lett. 130, 240802 (2023).
  * Tavakoli  _et al._ (2022) A. Tavakoli, A. Pozas-Kerstjens, M.-X. Luo, and M.-O. Renou, Bell nonlocality in networks, Rep. Prog. Phys. 85, 056001 (2022).
  * Pozas-Kerstjens  _et al._ (2022) A. Pozas-Kerstjens, N. Gisin, and A. Tavakoli, Full network nonlocality, Phys. Rev. Lett. 128, 010403 (2022).
  * Carvacho  _et al._ (2017) G. Carvacho, F. Andreoli, L. Santodonato, M. Bentivegna, R. Chaves, and F. Sciarrino, Experimental violation of local causality in a quantum network, Nat. Commun. 8, 14775 (2017).
  * Saunders  _et al._ (2017) D. J. Saunders, A. J. Bennet, C. Branciard, and G. J. Pryde, Experimental demonstration of nonbilocal quantum correlations, Sci. Adv. 3, e1602743 (2017).
  * Andreoli  _et al._ (2017) F. Andreoli, G. Carvacho, L. Santodonato, M. Bentivegna, R. Chaves, and F. Sciarrino, Experimental bilocality violation without shared reference frames, Phys. Rev. A 95, 062315 (2017).
  * Sun  _et al._ (2019) Q.-C. Sun, Y.-F. Jiang, B. Bai, W. Zhang, H. Li, X. Jiang, J. Zhang, L. You, X. Chen, Z. Wang, Q. Zhang, J. Fan, and J.-W. Pan, Experimental demonstration of non-bilocality with truly independent sources and strict locality constraints, Nat. Photon. 13, 687 (2019).
  * Poderini  _et al._ (2020) D. Poderini, I. Agresti, G. Marchese, E. Polino, T. Giordani, A. Suprano, M. Valeri, G. Milani, N. Spagnolo, G. Carvacho, R. Chaves, and F. Sciarrino, Experimental violation of n-locality in a star quantum network, Nat, Commun. 11, 2467 (2020).
  * Huang  _et al._ (2022) C.-X. Huang, X.-M. Hu, Y. Guo, C. Zhang, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, N. Gisin, C. Branciard, and A. Tavakoli, Entanglement swapping and quantum correlations via symmetric joint measurements, Phys. Rev. Lett. 129, 030502 (2022).
  * Gu  _et al._ (2023) X.-M. Gu, L. Huang, A. Pozas-Kerstjens, Y.-F. Jiang, D. Wu, B. Bai, Q.-C. Sun, M.-C. Chen, J. Zhang, S. Yu, Q. Zhang, C.-Y. Lu, and J.-W. Pan, Experimental full network nonlocality with independent sources and strict locality constraints, Phys. Rev. Lett. 130, 190201 (2023).
  * Wang  _et al._ (2023) N.-N. Wang, A. Pozas-Kerstjens, C. Zhang, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, N. Gisin, and A. Tavakoli, Certification of non-classicality in all links of a photonic star network without assuming quantum mechanics, Nat. Commun. 14, 2153 (2023).
  * Żukowski  _et al._ (1993) M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “event-ready-detectors” bell experiment via entanglement swapping, Phys. Rev. Lett. 71, 4287 (1993).
  * Werner (1989) R. F. Werner, Quantum states with einstein-podolsky-rosen correlations admitting a hidden-variable model, Phys. Rev. A 40, 4277 (1989).
  * Ajoy and Rungta (2010) A. Ajoy and P. Rungta, Svetlichny’s inequality and genuine tripartite nonlocality in three-qubit pure states, Phys. Rev. A 81, 052334 (2010).
  * Zhang  _et al._ (2022) C. Zhang, W.-H. Zhang, P. Sekatski, J.-D. Bancal, M. Zwerger, P. Yin, G.-C. Li, X.-X. Peng, L. Chen, Y.-J. Han, J.-S. Xu, Y.-F. Huang, G. Chen, C.-F. Li, and G.-C. Guo, Certification of genuine multipartite entanglement with general and robust device-independent witnesses, Phys. Rev. Lett. 129, 190503 (2022).
  * Šupić  _et al._ (2023) I. Šupić, J. Bowles, M.-O. Renou, A. Acín, and M. J. Hoban, Quantum networks self-test all entangled states, Nat. Phys. 19, 670 (2023).
  * Cao  _et al._ (2024) H. Cao, L. M. Hansen, F. Giorgino, L. Carosini, P. Zahálka, F. Zilk, J. C. Loredo, and P. Walther, Photonic source of heralded greenberger-horne-zeilinger states, Phys. Rev. Lett. 132, 130604 (2024).
  * Chen  _et al._ (2024) S. Chen, L.-C. Peng, Y.-P. Guo, X.-M. Gu, X. Ding, R.-Z. Liu, J.-Y. Zhao, X. You, J. Qin, Y.-F. Wang, Y.-M. He, J. J. Renema, Y.-H. Huo, H. Wang, C.-Y. Lu, and J.-W. Pan, Heralded three-photon entanglement from a single-photon source on a photonic chip, Phys. Rev. Lett. 132, 130603 (2024).


†††Deceased

Centrality dependence of multistrange baryon production
in high-energy heavy-ion collisions

G.H. Arakelyan†1, C. Merino2$\star$ and Yu.M. Shabelski3

1 A.Alikhanyan National Scientific Laboratory (Yerevan Physics Institute)

Yerevan, Armenia

2 Dpto. de Física de Partículas, Facultade de Física

Instit. Galego de Física de Altas Enerxías (IGFAE)

Universidade de Santiago de Compostela, Galiza, Spain

3 Petersburg Nuclear Physics Institute, NCR Kurchatov Institute

Gatchina, St.Petersburg, Russia

*<EMAIL_ADDRESS>
51st International Symposium on Multiparticle Dynamics (ISMD2022)
Pitlochry, Scottish Highlands, 1-5 August 2022 10.21468/SciPostPhysProc.?

## Abstract

We consider the experimental data on the production of strange
$\Lambda^{\prime}$s, and multistrange baryons ($\Xi$, $\Omega$), and
antibaryons, on nuclear targets, at the energy region from SPS up to LHC, in
the framework of the Quark-Gluon String Model. One remarkable result of this
analysis is the significant dependence on the centrality of the collision of
the experimental $\overline{\Xi}^{+}/\overline{\Lambda}$ and
$\overline{\Omega}^{+}/\overline{\Lambda}$ ratios in heavy-ion collisions, at
SPS energies.

###### Contents

  1. 1 Introduction
  2. 2 Comparison of QGSM results with experimental data
  3. 3 Conclusion

##  1 Introduction

We compare [1] the results obtained in the Quark-Gluon String Model (QGSM)
formalism, with the corresponding experimental data on yields of $\Lambda$,
$\Xi$, and $\Omega$ baryons, and the corresponding antibaryons, in nucleus-
nucleus collisions, for a wide energy region going from SPS up to LHC. We also
consider the ratios of multistrange to strange antihyperon production in
nucleus-nucleus collisions with different centralities, in the same energy
range.

The QGSM [2] is based on the Dual Topological Unitarization, Regge
phenomenology, and nonperturbative notions of QCD. In QGSM, high energy
interactions are considered as proceeding via the exchange of one or several
Pomerons. The cut of at least some of those Pomerons determines the inelastic
scattering amplitude of the particle production processes. In the case of
interaction with a nuclear target, the Multiple Scattering Theory (Gribov-
Glauber Theory) is used. At very high energies, the contribution of enhanced
Reggeon diagrams leads to the suppression of the inclusive density of
secondaries into the central (midrapidity) region. The QGSM provides a
succesful description of multiparticle production in hadron-hadron, hadron-
nucleus, and nucleus-nucleus collisions, for a wide energy region.

The production of multistrange hyperons, $\Xi^{-}$ (dss), and $\Omega^{-}$
(sss), has special interest in high energy particle and nuclear physics. The
production of each additional strange quark featuring in the secondary
baryons, i.e., the production rate of secondary $B(qqs)$ over secondary
$B(qqq)$, then of $B(qss)$ over $B(qqs)$, and, finally, of $B(sss)$ over
$B(qss)$, is affected by one universal strangeness suppression factor,
$\lambda_{s}$:

$\lambda_{s}=\frac{B(qqs)}{B(qqq)}=\frac{B(qss)}{B(qqs)}=\frac{B(sss)}{B(qss)}\;,$
(1)

together with some simple quark combinatorics.

Let us define:

$R(\overline{\Xi}^{+}/\overline{\Lambda})=\frac{dn}{dy}(A+B\to\overline{\Xi}^{+}+X)/\frac{dn}{dy}(A+B\to\overline{\Lambda}+X)\;,$
(2)
$R(\overline{\Omega}^{+}/\overline{\Lambda})=\frac{dn}{dy}(A+B\to\overline{\Omega}^{+}+X)/\frac{dn}{dy}(A+B\to\overline{\Lambda}+X)\;.$
(3)

The ratios in eqs. (2) and (3) can be calculated in the QGSM as a function of
the strangeness suppression parameter $\lambda_{s}$, and the corresponding
QGSM results reasonably describe the experimental data for a large energy
range, when a relatively small number of incident nucleons participate in the
collision (nucleon-nucleus collisions, or peripheral nucleus-nucleus
collisions), by thoroughly using the value $\lambda_{s}$=0.32 to fit those
experimental values. The ratio of yields of different particles should not
depend, in principle, on the centrality of the collision [1].

##  2 Comparison of QGSM results with experimental data

In Fig. 1 we show the comparison of the QGSM prediction with the experimental
data [3] on the dependence of the ratios
$\overline{\Omega}^{+}$/$\overline{\Lambda}$ (left panel) and
$\overline{\Xi}^{+}$ to $\overline{\Lambda}$ (right panel), on the number of
wounded nucleons, $N_{\omega}$, in Pb+Pb collisions at 158 GeV/c per nucleon.
Small values of $N_{w}$ correspond to peripheral collisions (large impact
parameter), while large values of $N_{w}$ corresponds to central collisions
(small impact parameter).

Figure 1: Experimental SPS ratios $\overline{\Omega}^{+}$/$\overline{\Lambda}$
(left panel), and of $\overline{\Xi}^{+}$/$\overline{\Lambda}$ (right panel),
in Pb+Pb collisions, as function of the number of wounded nucleons, $N_{w}$,
compared with the corresponding QGSM results. The full lines show the result
of QGSM calculation obtained with the value $\lambda_{s}$=0.32 at the left
end, and with larger values of $\lambda_{s}$ at its right end, while the
dashed lines show the result of QGSM calculation obtained with a constant
value $\lambda_{s}$=0.32, disregarding of the value of the number of wounded
nucleons (centrality).

At small values of $N_{w}$ the ratio is practically equal to that in the cases
of p+Be and p+Pb collisions, and they all can be correctly described by the
QGSM by using a value of the strangeness suppression parameter,
$\lambda_{s}$=0.32. However, the experimental ratio increases rather fast with
the increasing value of $N_{w}$, i.e. when we move from peripheral to central
Pb+Pb collisions, being central heavy-nuclei collisions the first case in
which a different (larger than $\lambda_{s}$=0.32) value of the strangeness
suppression parameter is needed to fit the result of the QGSM calculation to
the experimental data on the ratios $R(\overline{\Xi}^{+}/\overline{\Lambda})$
and $R(\overline{\Omega}^{+}/\overline{\Lambda})$. This unexpected behaviour
for the case of central heavy-nuclei collisions is shown by the full line in
Fig. 1, that it has been calculated with the value $\lambda_{s}$=0.32 at its
left end, and with larger values of $\lambda_{s}$ at its right end (see ref.
[1] for details). The results of QGSM calculations with a constant value
$\lambda_{s}$=0.32, disregarding of the value of the number of wounded
nucleons (centrality), $N_{\omega}$, are shown in Fig. 1 by dashed lines.

This behaviour in which the value of the strangeness suppression factor
$\lambda_{s}$ increases with the value of $N_{w}$ (centrality), indicates (see
[1] for details) that the simple quark combinatorial rules are not valid for
central collisions of heavy nuclei.

Now we consider the experimental data on midrapidity densities of hyperons in
Au+Au collisions measured by the STAR Collaboration [4, 5, 6] at RHIC
energies.

In Fig. 2 we present the comparison of the QGSM predictions with experimental
data on the $N_{\omega}$ dependence of the ratios
$\overline{\Omega}^{+}$/$\overline{\Lambda}$ (left panel), and
$\overline{\Xi}^{+}$/$\overline{\Lambda}$ (right panel), measured by the STAR
Collaboration in the midrapidity region, at $\sqrt{s}$ = 62.4 GeV. Similarly
as in Fig. 1, the left end of the full line here was calculated with a value
$\lambda_{s}$=0.32, while for the right end larger values of $\lambda_{s}$
were used [1]. The dashed line was calculated with a constant value of
$\lambda_{s}$=0.32. We see here that the value of $\lambda_{s}$ that correctly
describes multistrange hyperon production at RHIC energies is larger than the
one for the case of $\Lambda$ and $\overline{\Lambda}$ production, though this
difference between those two values of $\lambda_{s}$ is not so large as it is
for collisions at lower energies. This seems to indicate that the difference
in the values of the parameter $\lambda_{s}$, for multistrange hyperon, and
for $\Lambda$ and $\overline{\Lambda}$ production, decreases with the growing
of the initial energy of the collision.

Figure 2: Experimental STAR Collaboration ratios
$\overline{\Omega}^{+}$/$\overline{\Lambda}$ (left panel), and
$\overline{\Xi}^{+}$ to $\overline{\Lambda}$ (right panel), in Au+Au
collisions at $\sqrt{s}$ = 62.4 GeV, as function of the number of wounded
nucleons, $N_{w}$, compared with the corresponding QGSM results. The full
lines show the result of QGSM calculation obtained with the value
$\lambda_{s}$=0.32 at the left end, and with larger values of $\lambda_{s}$ at
its right end, while the dashed lines show the result of QGSM calculation
obtained with a constant value $\lambda_{s}$=0.32, disregarding of the value
of the number of wounded nucleons (centrality).

In Table 1 we consider the experimental data on ${\Xi}^{-}$,
$\overline{\Xi}^{+}$, ${\Omega}^{-}$, $\overline{\Omega}^{+}$ production in
central Pb+Pb collisions at $\sqrt{s}$=2.76 Tev measured by the ALICE
Collaboration [7], at the CERN LHC. Now, the strangeness suppression parameter
$\lambda_{s}$ for ${\Xi}^{-}$ and $\overline{\Xi}^{+}$ production becomes
smaller than at RHIC energies, taking the standard value $\lambda_{s}$=0.32.
In the case of ${\Omega}^{-}$ and $\overline{\Omega}^{+}$ production, the
value of $\lambda_{s}$ also decreases with respect to the RHIC energy range.
Thus we see that the unusually large values of $\lambda_{s}$ for central Pb+Pb
collisions at 158 GeV/c per nucleon, monotonically decrease with the increase
of the initial energy of the collision.

Table 1: Experimental data on dn/dy of ${\Xi}^{-}$, and of $\Xi^{-}$, $\overline{\Xi}^{+}$, ${\Omega}^{-}$, and $\overline{\Omega}^{+}$ production in central Pb+Pb collisions at $\sqrt{s}$=2.76 TeV per nucleon by the ALICE Collaboration [7], compared with the corresponding QGSM results. Process |  $\sqrt{s}$ (TeV) | Centrality | dn/dy (Exp. Data) | dn/dy (QGSM) | $\lambda_{s}$
---|---|---|---|---|---
Pb+Pb $\to\Xi^{-}$ | 2.76 | 0$-$10% | $3.34\pm 0.06\pm 0.24$ | 3.357 | 0.32
Pb+Pb $\to\overline{\Xi}^{+}$ | 2.76 | 0$-$10% | $3.28\pm 0.06\pm 0.23$ | 3.317 |
Pb+Pb $\to\Omega^{-}$ | 2.76 | 0$-$10% | $0.58\pm 0.04\pm 0.09$ | 0.606 | 0.38
Pb+Pb $\to\overline{\Omega}^{+}$ | 2.76 | 0$-$10% | $0.60\pm 0.05\pm 0.09$ | 0.601 |

##  3 Conclusion

It is shown that the experimental data on the production of hyperons in
central collisions of heavy nuclei present a very significant centrality
dependence at CERN-SPS energies $\sqrt{s}$ = 17.3 GeV. This dependence
decreases with the growth of the initial energy of the collision (at RHIC and
LHC energies). As for today, there is no any consistent theoretical
explanation of this, in principle unexpected, experimental fact, but it must
be surely connected to some intrinsic dynamical difference between the central
and the peripheral interactions. Our aim is now to investigate more deeply the
dynamics behind this specific behaviour of the central heavy-nuclei
collisions.

## References

  * [1] G.H. Arakelyan, C. Merino, and Yu.M. Shabelski, _M_ ultistrange hyperon production on nuclear targets, _Phys. Rev._ D105, 114013 (2022) [hep-ph/2112.01096].
  * [2] A.B. Kaidalov and K.A. Ter-Martirosian, _P_ omeron as quark-gluon strings and multiple hadron production at SPS collider energies, _Phys. Lett._ B117, 247 (1982).
  * [3] F. Antinori et al., NA57 Collaboration, _E_ nhancement of hyperon production at central rapidity in 158-A-GeV/c Pb-Pb collisions, _J. Phys._ G32, 427 (2006). [nucl-ex/0601021].
  * [4] J. Adams et al., STAR Collaboration, _M_ ultistrange baryon production in Au-Au collisions at $\sqrt{s_{NN}}$=130 GeV, _Phys. Rev. Lett._ 92, 182301 (2004) [nucl-ex/0307024].
  * [5] J. Adams et al., STAR Collaboration, _S_ caling properties of hyperon production in Au+Au collisiions at $\sqrt{s_{NN}}$=200 GeV, _Phys. Rev. Lett._ 98, 062301 (2007) [nucl-ex/0606014].
  * [6] M.M. Aggarwal et al., STAR Collaboration, _S_ trange and Multistrange particle production in Au+Au collisions at $\sqrt{s_{NN}}$=62.4 GeV, _Phys. Rev._ C83, 024901 (2011) [nucl-ex/1010.0142].
  * [7] B. Abelev et al., ALICE Collaboration, _M_ ulyistrange baryon production at midrapidity in Pb-Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV, _Phys. Lett._ B728, 216 (2014), _Phys. Lett._ B734, 409 (2014) (erratum) [nucl-ex/1307.5543].