# Entanglement classification and _non-k_ -separability certification via
Greenberger-Horne-Zeilinger-class fidelity

Marcin Płodzień  ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute
of Science and Technology, 08860 Castelldefels (Barcelona), Spain  Jan
Chwedeńczuk  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
Grzegorz Rajchel-Mieldzioć  NASK National Research Institute, ul. Kolska 12,
01-045 Warszawa, Poland

###### Abstract

Many-body quantum systems can be characterised using the notions of _k_
-separability and entanglement depth. A quantum state is _k_ -separable if it
can be expressed as a mixture of _k_ entangled subsystems, and its
entanglement depth is given by the size of the largest entangled subsystem. In
this paper we propose a multipartite entanglement measure that satisfies the
following criteria: (i) it can be used with both pure and mixed states; (ii)
it is encoded in a single element of the density matrix, so it does not
require knowledge of the full spectrum of the density matrix; (iii) it can be
applied to large systems; and (iv) it can be experimentally verified. The
proposed method allows the certification of _non-k_ -separability of a given
quantum state. We show that the proposed method successfully classifies three-
qubit systems into known stochastic local operations and classical
communication (SLOCC) classes, namely bipartite, W-, and GHZ-type
entanglement. Furthermore, we characterise the _non-k_ -separability in known
nine SLOCC classes of four-qubit states, absolutely maximally entangled states
for five and six qubits and for arbitrary size qubit Dicke states.

††preprint:

##  I Introduction

One of the important consequences of the quantum description of physical
systems is the notion of entanglement [horodecki2009entanglement] which is the
necessary resource for the Einstein-Podolsky-Rosen steering [uola2020steering]
and the Bell correlations [bell, brunner2014bell, bell_rmp]. None of these
correlations has a classical counterpart. The highlighted role of entanglement
requires a systematic characterisation and classification of quantum states in
terms of some entanglement measure [PhysRevLett.78.2275, Horodecki2001,
Plenio_2007]. Nevertheless, the tools for entanglement classification are
usually difficult to implement in realistic experimental settings and are not
universal due to the large number of entanglement classes [Guhne_2009]. The
problem of assigning a multipartite state to some entanglement class is hard
even for three qubits [Bengtsson_2006, Sabin_2008, e20100745, DeChiara_2018,
PhysRevA.92.042329, Srivastava_2024, Xie2024-du], and is even more complicated
for four qubits, where the number of entanglement classes is already infinite
[Verstraete_2002, 10.1063/1.3511477, 10.1063/1.4946895]. Hence the variety of
multipartite entanglement measures, such as based on local
[PhysRevA.95.022301] or on the collective measurements [PhysRevA.69.052327].
Research has focused on limiting the experimental effort
[PhysRevLett.117.210504] and recently a method called _entanglement detection
length_ has been proposed to witness many-body entanglement with a minimal
number of observables [shi2024entanglement]. A technique has been developed
for detecting multipartite entanglement based on a constant number of
measurements, independent of the number of qubits, suitable for the
Greenberger-Horne-Zeilinger (GHZ) states, cluster- or Dicke states
[PhysRevLett.112.155304, PhysRevLett.103.020504]. The urgency for a versatile
entanglement certification method is underlined by the fact that the Dicke
states have been created in experiments with few qubits in different
configurations such as photonic systems [PhysRevLett.103.020504,
PhysRevA.83.013821, PhysRevLett.103.020503, Zhao:15, Wang2016, Mu:20],
ultracold atoms [PhysRevLett.112.155304, doi:10.1126/science.1208798,
Hamley2012, doi:10.1073/pnas.1715105115] and quantum circuits [9951196,
Narisada2023]. A Hilbert-Schmidt distance between a given state and the
closest separable state is another important entanglement witness
[PhysRevA.102.012409]. The other methods to distinguish entanglement of
quantum states use local unitary optimisations [Rudnicki_2011, Rudnicki_2012].
The next measures are entanglement of formation [10.5555/2011326.2011329,
PhysRevLett.80.2245, Luo2008, Patrick_M_Hayden_2001, Adam_W_Majewski_2002,
FEI2003333, Zhao_Hui_2007, Gao2008, PhysRevA.102.042408], quantum discord
[https://doi.org/10.1002/andp.200051211-1204, L_Henderson_2001,
PhysRevLett.88.017901, PhysRevA.77.042303, PhysRevA.79.042325,
PhysRevLett.105.030501, PhysRevA.88.014302, PhysRevLett.112.210401,
Bera_2018], and concurrence [PhysRevLett.78.5022, 10.1063/1.2795840,
PhysRevA.105.052441, Bhaskara2017, PhysRevA.64.042315, PhysRevA.86.062303].

A separable class of witnesses to multipartite entanglement is related to
quantum metrology [Toth_2014]. The prominent example is the quantum Fisher
information (QFI), which sets the upper limit for the precision of parameter
estimation [braunstein1994statistical]. Broadly speaking, an increasing value
of the QFI implies increasing depth of multipartite entanglement, decreasing
$k$-separability, and a better precision of a quantum sensor taking this state
as input [hyllus2012fisher, toth2012multipartite, PhysRevLett.126.080502]. The
QFI can be difficult to calculate and even harder to measure as it requires
full information about the spectrum of a many-body state. However other
related quantities are easier to access, such as the classical Fisher
information, the spin-squeezing, or other witnesses based on two-body
correlators that carry some information about the multipartite entanglement
[kitagawa1993squeezed, wineland1994squeezed, PhysRevLett.95.120502,
PhysRevA.74.052319, esteve2008squeezing, PhysRevA.84.022107, smerzi_ob,
DeChiara2011, PhysRevA.67.022112, PhysRevA.97.020301, PhysRevLett.107.240502,
PhysRevLett.86.4431, Tura2014, Tura2014a, Tura2015, PhysRevA.100.032307,
Baccarri2019, PhysRevA.79.042334, PhysRevLett.99.250405, PhysRevA.95.032330,
PhysRevA.89.032307, Tura2014, marty2017multiparticle, Vitagliano_2017,
PhysRevLett.127.010401, PRXQuantum.2.030329, PhysRevA.100.032307, Frowis2017,
PhysRevLett.123.100507, ROIK2022128270, PhysRevLett.116.093602,
MullerRigat2023certifyingquantum, Fadel_2023, Vitagliano2023numberphase,
PRXQuantum.2.030329, PhysRevLett.131.070201, Poggi2024measurementinduced].

Another method is based on the measurement of macroscopic observables such as
magnetic susceptibility [Wiesniak_2005], heat capacity [PhysRevB.78.064108],
energy [brukner2004macroscopic, PhysRevA.70.062113, PhysRevA.71.010301], mean
spin and spin correlations [PhysRevA.72.032309, Frydryszak2017,
KUZMAK2020126579], with magnetic resonance spectroscopy [PhysRevA.100.022330,
Lazarev2020], entropic correlations [PhysRevResearch.2.043152], out-of-time-
order correlations [Lewis-Swan2019, PhysRevA.106.042429, PRXQuantum.2.020339]
or the energy of a thermal state in equilibrium [PhysRevResearch.5.013158].
Next, there is a family of methods based on SLOCC (stochastic local operations
and classical communications) and tensor properties [PhysRevA.62.062314,
PhysRevA.109.032424, PhysRevLett.110.030501, 10.5555/2011734.2011739,
PhysRevLett.111.110502, PhysRevA.84.062306, Zangi_2017,
PhysRevLett.111.060502, Palazuelos2022genuinemultipartite]. These methods can
generally be divided into subgroups [Guhne_2009]: eigenvalue criteria
[Horodecki_1996, Peres_1996, Chen_2003, Rudolph_2005, PhysRevLett.106.190502],
entanglement witnesses [Guhne_2006, TERHAL2002313, PhysRevA.63.044304,
Acin_2001_mixed, PhysRevLett.92.087902, Horodecki_2003, PhysRevA.77.062304,
PhysRevX.8.021072, PhysRevLett.127.040401, Rico_2024], geometric and
robustness criteria [Vidal_1999, PhysRevA.68.042307, 10.1063/1.1497700,
doi:10.1142/S1230161222500111, 9605268, Guo_2022, quantum2010004,
PhysRevLett.127.040403, PhysRevResearch.4.023059], and operational measures
[Klyachko2006, Plenio_2007, PhysRevLett.115.150502]. Entanglement has been
also classified with the help of mathematical tools from group theory
[PhysRevA.74.022318], topology [PhysRevD.107.126005], and algebraic geometry
[PhysRevResearch.2.043003]. Finally, in recent years the problem of
entanglement classification in many-body systems has been approached from a
machine learning point of view [PhysRevA.98.012315, Harney_2020, Harney_2021,
Chen_2021, s22186767, ayachi2022general,
https://doi.org/10.1002/qute.202200025, Vintskevich_2023, Giordano_2022,
Asif2023, Sanavio2023, pawlowski2023identification,
doi:10.1126/sciadv.add7131, palmieri2023enhancing, PhysRevA.109.022405,
dawid2023modern, Wiesniak2023, urena2024entanglement].

Here we propose a method for detecting and classifying many-body entanglement
using a correlator originally proposed for detecting Bell correlations and the
Einstein-Podolsky-Rosen steering [cavalcanti2007bell, he2010bell,
cavalcanti2011unified, spiny.milosz, PhysRevLett.126.210506]. Among the
advantages of this method is the possibility to quantify the entanglement with
a single element of the density matrix, which is directly accessible in an
experiment without the need for full tomography [garttner2017measuring,
PhysRevLett.120.040402, PRXQuantum.2.010307]. It has already been shown to
detect the entanglement depth in the multipartite setting [spiny.milosz,
PhysRevA.102.013328, PhysRevLett.129.250402, PhysRevResearch.6.023050]. This
correlator belongs to a broader family considered in Ref. [zukowski2002bell],
where a general framework for certifying many-body Bell correlations was
formulated. Its other variants include the Mermin form of Bell inequalities
[Mermin_1990, PhysRevLett.100.200407, 10.1119/1.12594], the $N$-body
inequalities [zukowski2002bell, cavalcanti2007bell, Reid2012,
cavalcanti2011unified] or the Ardehali inequalities [Ardehali_1992]. Some of
these inequalities have been used to distinguish W and GHZ states
[Brunner_2012]. Recently, several other techniques have been developed that
tackle the problem of certifying entanglement without using all possible
$N$-body correlations [Friis_2018, Frerot_2022].

Our method allows SLOCC classification of three- and four-qubit entanglement
classes for both pure and mixed states. We use the proposed setup to
characterise the $k$-separability of all remaining qubit absolutely maximally
entangled (AME) states, and to classify the $k$-separability of Dicke states
of any number of qubits. All characterisations are performed by inspecting the
single element of the density matrix, corresponding to the GHZ coherence in
the optimally chosen basis.

The paper is structured as follows: In Sec. II we introduce the framework used
for characterisation and classification of _non-k_ -separability. Next, in
Sec. III we use the proposed method to characterise _non-k_ -separability in
many-body qubit systems. In Sec. III.1 we focus on three-qubit quantum states
and show its relation with SLOCC classification. In Sec. III.2 we characterize
_non-k_ -separability of four-qubit states from nine SLOCC classes; in Sec.
III.3 we characterise Absolutely Maximally Entangled (AME) states and in Sec.
III.4 we characterise $k$-separability of Dicke states for any number of
qubits and any magnetisation sector. In Sec. IV we comment on the protocol for
measuring _non-k_ -separability. Finally, we conclude in Sec. V. Some
additional details are presented in Appendices.

##  II Methods

###  II.1 Many-body correlator

Consider a many-body system containing $N$ parties. Each of $N$ 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 $k\in\\{1\ldots
N\\}$. At this point, it is not assumed that these observables are quantum.
The correlation function (also referred to as the correlator) of these results
is here defined as

$\displaystyle\mathcal{E}_{N}=\left|\langle\sigma_{+}^{(1)}\dots\sigma_{+}^{(N)}\rangle\right|^{2},$
(1)

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

$\displaystyle\langle\sigma_{+}^{(1)}\dots\sigma_{+}^{(N)}\rangle=\int\\!\\!d\lambda\,p(\lambda)\prod_{k=1}^{N}\sigma_{+}^{(k)}(\lambda).$
(2)

Using the Cauchy-Schwarz inequality for complex integrals together with the
fact that $\left|\sigma_{+}^{(k)}(\lambda)\right|^{2}=1/2$ for all $k$, we see
that the ${\cal E}_{N}$ is bounded from above:

$\displaystyle\mathcal{E}_{N}$
$\displaystyle=\left|\int\\!\\!d\lambda\,p(\lambda)\prod_{k=1}^{N}\sigma_{+}^{(k)}(\lambda)\right|^{2}$
$\displaystyle\leqslant\int\\!\\!d\lambda\,p(\lambda)\left|\sigma_{+}^{(k)}(\lambda)\right|^{2}=2^{-N}.$
(3)

We thus conclude that

${\cal E}_{N}\leqslant 2^{-N}$ (4)

is the $N$-body Bell inequality, because its violation defies the postulates
of local realism expressed in Eq. (2).

The provided inequality is valid for systems that yield binary outcomes of
local, single-particle observables - such conditions are realized in the case
of the $N$-qubit quantum system described by density matrix $\hat{\varrho}$.
The quantum-mechanical equivalent of Eq. (1) is defined as

$\displaystyle{\cal E}=\left|{\rm
Tr}\bigg{[}{\hat{\varrho}\bigotimes_{k=1}^{N}\hat{\sigma}_{+}^{(k)}}\bigg{]}\right|^{2}.$
(5)

Assuming quantum-mechanical behavior of each parties of the system, i.e.,
$\big{|}\langle\hat{\sigma}_{+}^{(k)}\rangle\big{|}^{2}\leq 4^{-1}$, we obtain
that for a separable state of $N$ qubits the following holds: ${\cal E}\leq
4^{-N}$

Based on the above considerations, to witness and classify the multi-qubit
entanglement pf an $N$-qubit density matrix $\hat{\varrho}$, we use the
generalized $N$-body correlator

$\displaystyle\mathcal{Q}=\log_{4}\big{[}4^{N}\big{|}{\rm
Tr}\big{[}\hat{\varrho}\,\bigotimes_{k=1}^{N}\hat{\sigma}_{+,n_{k}}^{(k)}\big{]}\big{|}^{2}\big{]},$
(6)

where $\hat{\sigma}_{+,n_{k}}^{(k)}$ is the single-qubit operator that is
rising the projection of the spin along the axis $n_{k}$, i.e.,

$\displaystyle\hat{\sigma}_{+,n_{k}}^{(k)}=\frac{1}{2}(\hat{\sigma}^{(k)}_{m_{k}}+i\hat{\sigma}^{(k)}_{l_{k}}).$
(7)

where $\vec{r}_{k}=(n_{k},m_{k},l_{k})$ are three mutually orthogonal
components of a Bloch vector. Axis $n_{k}$ defines the rotated version of the
operator
$\hat{\sigma}_{+}^{(k)}=1/2(\hat{\sigma}_{x}^{(k)}+i\hat{\sigma}_{y}^{(k)})$
after the action of local unitary matrices, i.e.,
$\hat{\sigma}_{+,n_{k}}^{(k)}=\hat{R}_{k}^{\dagger}(\vec{\theta}_{k})\hat{\sigma}_{+}^{(k)}\hat{R}_{k}(\vec{\theta}_{k}),$
where
$\hat{R}_{k}(\vec{\theta}_{k})=e^{-i\vec{\theta}_{k}\cdot\vec{\sigma}_{k}}$,
$\vec{\theta}_{k}=\\{\theta^{(k)}_{x},\theta^{(k)}_{y},\theta^{(k)}_{z}\\}$,
$\vec{\sigma}_{k}=\\{\hat{\sigma}_{x}^{(k)},\hat{\sigma}_{y}^{(k)},\hat{\sigma}_{z}^{(k)}\\}$.

Figure 1: Pictorial representation of $N=12$ qubit quantum states. (a) _6_
-separable state with entanglement depth $d_{e}=4$, (b) _4_ -separable state
with entanglement depth $d_{e}=4$, (c) _2_ -separable quantum state with
entanglement depth $d_{e}=6$, and (d) genuinely entangled quantum state with
entanglement depth $d_{e}=12$, not-_2_ -separable.

###  II.2 Quantum state structure characterization

In the following we show how the correlator $\mathcal{Q}$ carries extensive
information on the entanglement of a given state $\hat{\varrho}$
[spiny.milosz, PhysRevLett.129.250402, PhysRevResearch.6.023050,
plodzien2024inherent].

Let us consider the density operator of $N$ qubits as a product of $k$
entangled states, each with $n_{l}$ qubits, $\sum_{l=1}^{k}n_{l}=N$, i.e.,

$\hat{\varrho}=\int d\lambda
p(\lambda)\bigotimes_{l=1}^{k}\hat{\varrho}_{n_{l}}(\lambda)\equiv\hat{\varrho}_{n_{1}\dots
n_{k}}.$ (8)

Such states are called $k$-separable [10.5555/2011464.2011472, Guhne_2010,
PhysRevA.78.032101, Gao2011, Gao_2013, Szalay_2019, HONG2021127347], and its
entanglement depth is defined as $d_{e}=\max\\{n_{1}\dots n_{k}\\}$, see Fig.
1.

For all $N$-separable states, $k=N$, $\hat{\varrho}_{n_{1}\dots n_{k}}$,
$n_{l}=1$, $l\in\\{1\dots N\\}$, the correlator $\mathcal{Q}$ takes non-
positive values

$\displaystyle\mathcal{Q}\leqslant 0,$ (9)

and violation of this inequality indicates entanglement in the system. On the
other hand, the maximal value is obtained for a _non-2_ -separable state
$\hat{\varrho}_{N}=|\psi_{\rm ghz}\rangle\langle\psi_{\rm ghz}|$, being a pure
Greenberg-Horne-Zeilinger state $|\psi_{\rm
ghz}\rangle=\frac{1}{\sqrt{2}}\left(|1\rangle^{\otimes N}+|0\rangle^{\otimes
N}\right)$,

$\mathcal{Q}=N-1.$ (10)

For states being a product of a single qubit and entangled state of the
remaining $N-1$ qubits, $\hat{\varrho}_{1,N-1}$, the correlator is bounded by

$\mathcal{Q}\leqslant N-2,$ (11)

and its violation indicates genuine entanglement, i.e. state is _non-2_
-separable; the same bound is valid for every _2_ -separable state, i.e.
$\hat{\varrho}_{n,N-n}$.

Bounds for $\mathcal{Q}$ can be generalized to states being a product of $n$
single qubits and $(N-n)$-entangled state, $\hat{\varrho}_{n_{1}\dots
n_{n},N-n}$, $n_{l}=1$, $l\in\\{1\dots n\\}$:

$\displaystyle N-1-(n+1)<\mathcal{Q}\leqslant N-1-n,$ (12)

and the correlator $\mathcal{Q}$ indicates existence of entanglement depth
$d_{e}=N-n$.

Next, for even number of qubits $N=2M$, when $\hat{\varrho}$ is a product of
$N/2$ pairs of entangled qubits, $\hat{\varrho}_{n_{1}\dots n_{k}}$,
$n_{l}=2$, $l=\\{1\dots N/2\\}$, the correlator is bounded from above as

$\mathcal{Q}\leqslant\frac{N}{2}=M,$ (13)

and its violation certifies existence of entanglement depth $d_{e}=3$.

Finally, considering $N=3M$ qubits, $\hat{\varrho}_{n_{1}\dots n_{M}}$,
$n_{l}=3$, $l\in\\{1,\dots,M\\}$, the bound is

$\mathcal{Q}\leqslant\frac{2}{3}N=2M,$ (14)

and its violation indicates entanglement depth $d_{e}=4$.

In general, for $k$-separable states the correlator $\mathcal{Q}$ is bounded
from above

$\displaystyle\mathcal{Q}\leqslant N-k,$ (15)

and violation of this inequality witnesses a _non-k_ -separability, i.e.
$\hat{\varrho}$ is a product of at most $(k-1)$-entangled states.

| $N=3$ | $N=4$ | $N=5$ | $N=6$
---|---|---|---|---
$0<\mathcal{Q}\leqslant 1$ | $k<3$ | $k<4$ | $k<5$ | $k<6$
$1<\mathcal{Q}\leqslant 2$ | gen. ent. | $k<3$ | $k<4$ | $k<5$
$2<\mathcal{Q}\leqslant 3$ | — | gen. ent. | $k<3$ | $k<4$
$3<\mathcal{Q}\leqslant 4$ | — | — | gen. ent. | $k<3$
$4<\mathcal{Q}\leqslant 5$ | — | — | — | gen. ent.
Table 1: The values of $\mathcal{Q}$ for $N=3,4,5$, and 6 that signal the
_non-k_ -separability, i.e., $N$ qubit entangled state can only be represented
as product of at most $(k-1)$ entangled states. Minimal _non-k_ -separability
($k=2$) signifies genuine entanglement (gen. ent.)

As an examples, let us consider $N=5$ and $N=6$. For $N=5$ qubits, the density
operator $\hat{\varrho}$ can take the form $\hat{\varrho}_{1,4}$,
$\hat{\varrho}_{2,3}$ or $\hat{\varrho}_{1,2,2}$. The respective value of the
correlator are bounded by: $\mathcal{Q}_{1,4}\leqslant 3$ and its violation
certifies genuine entanglement $d=5$, $\mathcal{Q}_{2,3}\leq 3$ and its
violation certifies entanglement depth $d_{e}=4$;
$\mathcal{Q}_{1,2,2}\leqslant 2$ and its violation certifies entanglement
depth $d_{e}=3$. For $N=6$, the entangled state $\hat{\varrho}$ can have the
following structure: $\hat{\varrho}_{1,5}$, $\hat{\varrho}_{2,4}$,
$\hat{\varrho}_{3,3}$, $\hat{\varrho}_{2,2,2}$. The corresponding bounds are:
$\mathcal{Q}_{1,5}\leqslant 4$, $\mathcal{Q}_{2,4}\leqslant 4$,
$\mathcal{Q}_{3,3}\leqslant 4$, and $\mathcal{Q}_{2,2,2}\leqslant 3$. These
observations, adapted to the $N=3,4,5,6$ are summarized in Tab. 1.

###  II.3 Extraction of $\mathcal{Q}$

The correlator $\mathcal{Q}$ can be expressed in terms of a single element of
the density matrix , i.e., ${\cal C}\equiv\langle\tilde{0}|^{\otimes
N}\hat{\varrho}|\tilde{1}\rangle^{\otimes N}$, such that determines the GHZ-
type coherence between all the spins $|\tilde{0}\rangle^{\otimes N}$ up and
all down $|\tilde{1}\rangle^{\otimes N}$ in the optimal local bases, namely

$\displaystyle\mathcal{Q}=N+\log_{4}(\left|{\cal C}\right|^{2}).$ (16)

This observation will be relevant in the forthcoming sections of this work.
Importantly for its entanglement measure interpretation, $\mathcal{Q}$ is
convex in the input states, see App. A.

The freedom of choice of the orientations $\vec{r}_{k}$ is of fundamental
importance, as it allows to adapt the correlator to the geometry of the state
that is scrutinized [plodzien2024inherent]. To see this, take two examples of
three-qubit GHZ states,

$\displaystyle|\psi_{\rm
ghz}^{zzz}\rangle=\frac{1}{\sqrt{2}}\left(|0_{z},0_{z},0_{z}\rangle+|1_{z},1_{z},1_{z}\rangle\right)$
(17a) $\displaystyle|\psi_{\rm
ghz}^{xyz}\rangle=\frac{1}{\sqrt{2}}\left(|0_{x},0_{y},0_{z}\rangle+|1_{x},1_{y},1_{z}\rangle\right),$
(17b)

where, for instance, $|0_{x}\rangle$ denotes the eigenstate of
$\hat{\sigma}_{x}$ with the spin pointing up. Only the correct choice of the
three axes: all operators rising the spin in the $z$-direction in the former,
and in the $x$, $y$ and $z$ directions in the latter case, allow to maximize
the value of $\mathcal{Q}$. In what follows, we always display results for the
correlator, optimized over the choice of the separable basis. What is crucial
from our perspective of classification of entanglement, the value of
correlations is invariant under local unitary operations. Hence the proposed
method for entanglement classification respects SLOCC entanglement classes,
see App. B.

As we stated above, our scheme uses local unitary operations, i.e., local
rotations, to transform the state into the optimal basis from the perspective
of the GHZ-fidelity. A similar setup was already considered from a more
general perspective: instead of local unitary operations, the authors of Refs.
[Kaniewski_2016, Li_2019] consider all local operations, because they do not
introduce entanglement. For such a setup there exist lower bounds
[Kaniewski_2016] on this type of GHZ-fidelity; however, these cannot be used
in our case where GHZ-fidelity is necessarily smaller or equal due to a
restricted set of operations.

##  III Results

The number of entanglement classes grows rapidly with the number of parties
and the local dimension. For pure states of two qubits, there are only two
entanglement classes, with a Bell state as an extremal representative of one
and separable states forming the other. Increasing the number of qubits to
three yields two disjoint genuinely multipartite entanglement classes,
together forming six different classes. Interestingly, the growth of the
number of classes is not even exponential; already for four qubits, there are
infinitely many entanglement classes, which can be divided into nine
[Verstraete_2002, Zangi_2017, Vintskevich_2023] or forty-nine groups [Li_2009,
Giordano_2022].

For mixed states $\hat{\varrho}$, the situation is even more complicated. The
usual method of assigning $\hat{\varrho}$ to some class requires the knowledge
if it can be written as a convex sum of pure states from a given SLOCC class.
Notwithstanding, for a complete characterization of all mixed states, one
requires also to extend it to pure states of lower entanglement, that is, non-
genuine one [Acin_2001_mixed].

In the following sections we illustrate the power of our method of
entanglement classification for $N=3$ qubits states in three SLOCC classes,
$N=4$ qubits states in nine SLOCC classes. Next, we characterize
$k$-separability for $N=5,6$ qubits AME states, and Dicke states of an
arbitrary size.

###  III.1 Three-qubit entanglement classification

We start by the first non-trivial example which requires classification of
entanglement. First, we review well-known results concerning classification
and then we show the effectiveness of our setup.

####  III.1.1 Pure states

In three-qubit system, there are two types of entanglement: genuine tripartite
entanglement and the one in which one qubit is separated from the rest. In the
genuine entanglement case, there are two inequivalent (under SLOCC) types of
states: GHZ and W, defined in App. C 111The former class encompasses the
majority of states in the Haar-measure sense; a state, drawn at random, will
belong to the GHZ class with probability 1.. One of the key differences
between both genuinely entangled states GHZ and W is their tensor rank, see
App. C. Therefore, in total there are six classes: two genuine, three two-
partite entangled (AB-C, AC-B, BC-A) and one separable class.

In the following we analyze the correlator $\mathcal{Q}$ for three-qubit SLOCC
entanglement classes both for pure and mixed states. To do so, we generate the
set of three-qubit quantum states as follows

  * •

separable class: $3$ Haar-random unitary matrices of size $2\times 2$ acting
on $|000\rangle$, yielding
$\hat{U}_{A}\otimes\hat{U}_{B}\otimes\hat{U}_{C}|000\rangle$,

  * •

two-party entangled classes: $2$ random unitary matrices: one of size $4\times
4$ and another of size $2\times 2$ to entangle only two subsystems, e.g.,
$\hat{U}_{AB}\otimes\hat{U}_{C}|000\rangle$,

  * •

W class: we apply the parametrization
$|\psi\rangle=\lambda_{0}|000\rangle+\lambda_{1}|100\rangle+\lambda_{2}|101\rangle+\lambda_{3}|110\rangle$
from Ref. [Acin_2001, Acin_2001_mixed] with
$\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{3}$ sampled uniformly,

  * •

GHZ class: a random state belongs to this class with probability 1
[Acin_2001]; therefore, we act with a random unitary matrix of size $8\times
8$ on any pure state: $\hat{U}_{ABC}|000\rangle$.

Figure 2: Probability distribution of $\mathcal{Q}$ for three-qubit pure
states from SLOCC classes: bi-separable-, W-, and GHZ-type (from top to
bottom). The green dashed lines correspond to the maximal values of
$\mathcal{Q}$ for each class. Red dashed lines presents numerically obtained
probability distribution function.

We generate $100$ states of a maximal representatives of each class, and
$20000$ states for non-maximal representatives of each class. Next, we
calculate the optimized Bell correlator $\mathcal{Q}$ for each state.

In Fig. 2 we show the distribution of outcomes, separately for each class.
First, we focus on the maximal representatives of each class. In this case,
correlator $\mathcal{Q}$ takes always same value, i.e. $\mathcal{Q}=1$ for bi-
separable states, $\mathcal{Q}=1.6$ for W states, and $\mathcal{Q}=2$ for GHZ
states – denoted as vertical green lines in Fig. 2. This clearly indicates
that correlator $\mathcal{Q}$ does not depend on local rotations, and
distinguish between SLOCC classes properly. The bound $\mathcal{Q}\leqslant
1.6$ serves as a GHZ-class witness. When $\mathcal{Q}>1.6$ for either pure or
mixed states, state $\hat{\varrho}$ does not belong to the W-class. Values of
the correlator $\mathcal{Q}$ indicates the genuine entanglement of states W
and GHZ, in accordance with Tab. 1.

Next, let us consider the value of $\mathcal{Q}$ for non-maximal
representatives of each class. Frequencies of $\mathcal{Q}$ for each class are
presented as histogram in Fig. 2. Interestingly, for class W the most probable
is $\mathcal{Q}=1$. For the GHZ class the most probable is $\mathcal{Q}=1.65$.
From the purely numerical results, we observe that most probable value of
$\mathcal{Q}$ for W class is the the maximal value of $\mathcal{Q}$ for bi-
separable class, while most probable value of $\mathcal{Q}$ for GHZ class is
close to the maximal value of $\mathcal{Q}$ for the W class.

Finally, the following question arises: how precisely does $\mathcal{Q}$
distinguish between the classes. To address this point we use the Wootters’
distance, which for probability distributions $p$ and $q$ of a random variable
$n$ is defined as

$\displaystyle d(p,q)=1-\sum_{n}\sqrt{p_{n}q_{n}},$ (18)

where the measure $\sum_{n}\sqrt{p_{n}q_{n}}$ is known as the Bhattacharyya
coefficient. Two probabilities are distant if $d(p,q)\simeq 1$ and naturally
$d(p,p)=0$. We calculate the Wootters’ distance, pairwise, for the
probabilities shown in Fig 2 (red lines). The outcome, shown in Table 2,
informs about the success chance of discrimination between the classes. The
GHZ and the bi-separable states are the most distant, confirming the result of
Fig. 2. As expected, the W-class that lies in between the two has a
significant overlap with the both.

To assign a state to one of two given classes one can process it through the
random SLOCC many times, calculating the correlator for each case. This
procedure will give one of the two shapes from Fig. 2. The degree of
distinguishability can be controlled with the number of samples. This protocol
can be thought of as a black-box device, with random SLOCC class states on the
input that, as an output, produces the correct classification provided
sufficiently many samples.

Ent. class | B-S | W | GHZ
---|---|---|---
B-S | 0 | 0.27 | 0.93
W | 0.27 | 0 | 0.57
GHZ | 0.93 | 0.57 | 0
Table 2: The table of pairwise Wootters’ distance between the probability
distributions of the optimized correlators $\mathcal{Q}$ for three-qubit
entanglement classes: biseparable (B-S), W, and GHZ.

####  III.1.2 Mixed states

In the following, we study how $\mathcal{Q}$ is vulnerable to noise, i.e., a
given state $|\psi\rangle$ with a given value of $\mathcal{Q}$ belonging to
some entanglement class, is passed through a decoherence channel. We take into
account only states which give the maximal value of $\mathcal{Q}$ for each
class (see the vertical dashed green lines in Fig. 2). We consider two types
of decoherence, where the first represents a global depolarizing channel

$\displaystyle\Lambda_{p}[\hat{\varrho}]=(1-p)\hat{\varrho}+\frac{p}{8}\hat{\mathds{1}}.$
(19)

As it does not depend on the orientation of the optimal correlator, the value
of $\mathcal{Q}$ is analytical and reads

$\mathcal{Q}(p)=\mathcal{Q}+2\log_{4}(1-p).$ (20)

This result is drawn in Fig. 3 with solid lines.

The other source of noise is the dephasing acting on randomly chosen qubit
$i\in\\{1,2,3\\}$

$\Delta_{p}[\hat{\varrho}]=(1-\frac{p}{2})\hat{\varrho}+\frac{p}{2}\hat{\sigma}^{(i)}_{z}\hat{\varrho}\hat{\sigma}^{(i)}_{z}.$
(21)

For every $p$ we generate $100$ random states of maximal representatives of
each class, and we obtain three values of the optimized $\mathcal{Q}$,
depending on which qubit the dephasing acts. These samples give a range of
$\mathcal{Q}$’s, which together with the average value is shown as a shaded
area for each class separately, see Fig. 3. We observe that for reasonably
small noise levels ($p<0.3$ for the dephasing and $p<0.25$ for the
depolarizing noise), the maximal representatives of the SLOCC entanglement
classes for three qubits are still mutually distinguishable using the Bell
correlators metric.

Figure 3: The value of the correlator $\mathcal{Q}$ for extremal
representatives from each of three-qubit entanglement class, upon which we act
with dephasing (dashed lines) and depolarizing (solid lines) noise. The red
lines correspond to the noisy GHZ states, the blue one to W states, and the
green line depicts the decay of correlations for two-qubit Bell states.

####  III.1.3 Application: dynamical generation of entanglement

Next, we apply this correlator to classify the three-qubit entangled states
generated in the following dynamical protocol. Qubits are initialized in the
product state of eigenstate of the $\hat{\sigma}_{x}^{(k)}$ operators, namely

$\displaystyle|\psi(0)\rangle=\left(\frac{|0\rangle+|1\rangle}{\sqrt{2}}\right)^{\otimes
3}$ (22)

and undergo the evolution

$\displaystyle|\psi(t)\rangle=e^{-it\hat{H}}|\psi(0)\rangle$ (23)

generated by either of the two Hamiltonians

$\displaystyle\hat{H}_{\rm oat}$
$\displaystyle=\sum_{i>j=1}^{3}\hat{\sigma}^{(i)}_{z}\hat{\sigma}^{(j)}_{z}$
(24a) $\displaystyle\hat{H}_{\rm tact}$
$\displaystyle=\sum_{i>j=1}^{3}\left(\hat{\sigma}^{(i)}_{z}\hat{\sigma}^{(j)}_{z}-\hat{\sigma}^{(i)}_{x}\hat{\sigma}^{(j)}_{x}\right).$
(24b)

Here, the two subscripts stand for one-axis- or two-axis-counter-twisting,
respectively. Note that according to Eq. (12) and the Tab. 1 both these
Hamiltonians generate entanglement at all times, and genuinely 3-partite
entanglement at most instants, see Fig. 4. Moreover, the correlator
$\mathcal{Q}$ provides information that only the OAT protocol generates the
GHZ-class state with $\mathcal{Q}=2$ at maximum.

Figure 4: Dynamical generation of the many-body entanglement in spin squeezing
protocols in three-qubit system, i.e., $\mathcal{Q}$ versus the time $t$. The
red solid line corresponds to the one-axis twisting protocol (OAT), while the
blue dashed line corresponds to the two-axis counter twisting protocol (TACT).
Both protocols generate genuine entanglement at the same critical time
$t_{c}$, when $\mathcal{Q}(t_{c})$ exceeds unity. The OAT protocol generates
the most entangled state, namely the GHZ state, for which the correlator takes
the maximal value $\mathcal{Q}=2$.

###  III.2 Four-qubit entanglement classification

The efforts to classify the multipartite entanglement, relevant in the context
of experimental advances [Sackett2000], have been recently expanded to the
four-qubit case [Ghahi_2016, Vintskevich_2023, Wieczorek_2009, Kiesel_2007],
where nine groups can be distinguished [Verstraete_2002]. The main group is
formed by those states that are characterized by the generalized GHZ-type
entanglement. The other groups are formed by three- and two-qubit entangled
states.

For each of the nine classes, we maximize the correlator $\mathcal{Q}$ by
picking the maximally entangled representatives of each group (Appendix D and
Ref. [Vintskevich_2023]). As shown in Table 3, the highest value of the
correlator is reached by the GHZ state ($|G\rangle$ group), and
$|E_{1}\rangle$ and $|E_{2}\rangle$ groups, while the smallest $\mathcal{Q}$
is for group $|E_{6}\rangle$.

An important question is about $k$-separability of a resulting mixed quantum
state after tracing out one of the qubits. The correlator $\mathcal{Q}$
provides an information about $k$-separability of the final three-qubit state.
When $\mathcal{Q}>0$ the remaining quantum state is entangled. Naturally, the
entanglement in states $|G\rangle$ is washed out by tracing out any of the
qubits. This however is not the case with classes $|E_{3}\rangle$,
$|E_{4}\rangle$, $|E_{6}\rangle$, $|E_{7}\rangle$, and $|E_{8}\rangle$, see
Table 3. Moreover, when $\mathcal{Q}>1$, the mixed states remains genuinely
entangled. In particular state $|E_{7}\rangle$ remains genuinely entangled
after tracing out the first qubit, $\mathcal{Q}=1.32$.

State Discarded | None | $i=1$ | $i=2$ | $i=3$ | $i=4$
---|---|---|---|---|---
$|G\rangle$ | 3 | $-$ | $-$ | $-$ | $-$
$|E_{1}\rangle$ | 2.98 | $-$ | $-$ | $-$ | $-$
$|E_{2}\rangle$ | 2.98 | $-$ | $-$ | $-$ | $-$
$|E_{3}\rangle$ | 2.41 | $-$ | $-$ | $0.43$ | $0.43$
$|E_{4}\rangle$ | 2.5 | $0.55$ | $-$ | $0.68$ | $-$
$|E_{5}\rangle$ | 2.98 | $-$ | $-$ | $-$ | $-$
$|E_{6}\rangle$ | 1.54 | $1$ | $-$ | $-$ | $1$
$|E_{7}\rangle$ | 1.76 | $1.32$ | $1$ | $1$ | $1$
$|E_{8}\rangle$ | 2 | $2$ | $-$ | $-$ | $-$
Table 3: Value of the correlator $\mathcal{Q}$ for maximally entangled four-
qubit states in nine entanglement classes (see App. D). The first column
corresponds to the value of the correlator of the full state, while the latter
four show $\mathcal{Q}$ after discarding (tracing out) one qubit. Null values
corresponds to negative lack of entanglement. Observe that the state
exhibiting the highest entanglement with respect to four-qubit correlator
($|G\rangle$) does not possess any three-qubit entanglement. Conversely, the
states with lower four-qubit entanglement preserve high values of the three-
qubit correlator. This is in agreement with similar observations concerning
distribution of entanglement among different divisions [Verstraete_2002].

###  III.3 non-$k$-separability of AME states for $N=5$ and $6$ qubits

A pure state on $N$ qu$d$its is called an absolutely maximally entangled
AME($N$,$d$) state if all its reduced density matrices after discarding at
least half of the subsystems are maximally mixed [Helwig_2012]. Interestingly,
the question of the existence of an AME state of a given number of parties and
local dimensions is still open. There are a few methods for (dis)proving their
existence but none is universal [Table_AME, Scott_2004]. Recently there were
several developments in the search for AME states, including the question of
their local orbit, connected with local unitary equivalence [Huber_2017,
Rather_2022, Rather_2023]. The usefulness of AME states extends also to dense
coding, quantum secret sharing, and quantum error correction codes
[Scott_2004, Goyeneche_2015].

Here, we consider only the qubit case, $d=2$. In the case of $N=2,3$ qubits,
the AME is the corresponding GHZ state. Surprisingly, there are no four-qubit
AME states [Higuchi_2000], and there are no AME states for $N>6$ [Scott_2004,
Huber_2017]; but they do exist for $N=5,6$ [Facchi_2008, Goyeneche_2015],
taking the form of

$\displaystyle|\text{AME}(N,2)\rangle=\frac{1}{2^{N/2}}\sum_{i=0}^{2^{N}-1}\alpha^{(N)}_{i}|i\rangle,$
(25)

where the values of binary ($\pm 1$) coefficients $\alpha^{(5)}_{i}$ and
$\alpha^{(6)}_{i}$ are given in Appendix E. For ${\rm AME}(5,2)$ the
correlator takes $\mathcal{Q}=1.7$ and for ${\rm AME}(6,2)$ it takes value
$\mathcal{Q}=2.4$. These values are about half of maximal values for $N=5$ and
$N=6$ qubits ($\mathcal{Q}=4$ and $\mathcal{Q}=5$ respectively, see Table 1.).

The type of entanglement of AME states cannot be explained via the provided
measure; the correlator for those two AME states can be explained with only
three-partite and four-partite correlations, respectively. In this sense,
AME’s are the strongest entangled states that exist. The correlator
$\mathcal{Q}$ can be thus thought of as an alternative to the usual
description of AME-type entanglement, understood as the sum of entanglement
entropies across every balanced bipartition.

###  III.4 non-$k$-separability of Dicke states, any $N$

As the last illustration of $k$-separability characterization with the
correlator $\mathcal{Q}$ we use the Dicke states. A set of $N$-qubit Dicke
states $|J,m\rangle$ is defined as a eigenbasis of collective multi-qubit spin
operators $\hat{J}^{2}=\hat{J}^{2}_{x}+\hat{J}^{2}_{y}+\hat{J}^{2}_{z}$ and
$\hat{J}_{z}$, where
$\hat{J}_{\tau}=\frac{1}{2}\sum_{i=1}^{N}\hat{\sigma}^{(i)}_{\tau}$,
$\tau=x,y,z$; $[\hat{J}^{2},\hat{J}_{z}]=0$,

$\begin{split}\hat{J}^{2}|J,m\rangle&=J(J+1)|J,m\rangle\\\
\hat{J}_{z}|J,m\rangle&=m|J,m\rangle,\end{split}$ (26)

with $J=N/2$, $m\in\\{-J\dots J\\}$. The special case are balanced Dicke
states, $m=0$,

$|D_{n}\rangle=\binom{N}{\frac{N}{2}}^{-\frac{1}{2}}\sum_{k}{\cal
P}_{k}(|0\rangle^{\otimes\frac{N}{2}}\otimes|1\rangle^{\otimes\frac{N}{2}}),$
(27)

summed over all distinct permutations ${\cal P}_{k}$. Such symmetric Dicke
states posses strong multipartite entanglement [PhysRevLett.112.155304,
Campbell_2009, PhysRevLett.107.180502, PhysRevA.94.042333, Demianowicz_2021,
Aloy_2021, Marconi2021entangledsymmetric, Tura2018separabilityof,
PhysRevA.95.042128, PhysRevA.104.022426], and are useful for quantum metrology
[Toth:07, hyllus2012fisher, toth2012multipartite].

For Dicke states the correlator $\mathcal{Q}$ can be obtain analytically
[plodzien2024inherent] both for balanced $m=0$ case, and for $m\neq 0$:

$\mathcal{Q}=2\log_{4}\binom{N}{m+\frac{N}{2}}.$ (28)

For the large number of qubits, $N\to\infty$, the above formula can be
approximated as

$\mathcal{Q}=N-\log_{4}N-4\frac{m^{2}}{N}\log_{4}e+\log_{4}\frac{2}{\pi}.$
(29)

The maximal value of the correlator $\mathcal{Q}$ is for balanced Dicke states
with $m=0$. Moreover, the expression in Eq. (29) and inequality Eq. (12) allow
classification of Dicke states $|J,m\rangle$ with respect to their _non-k_
-separability, one of the results of this work.

##  IV Measurements

As discussed in the previous section, many-qubit entanglement can be inferred
from the single element of the density matrix $\hat{\varrho}$ related to GHZ
fidelity. The remaining question concerns the identification of an
experimental protocol capable of accessing this quantity.

The measurement of the off-diagonal GHZ element of the density matrix in $N=4$
qubit system has been performed experimentally with trapped ions
[Sackett2000]. Another approach is to use the quantum state tomography, e.g.,
with classical shadows, and rotate the state to find the optimal basis in
which the GHZ coherence element is maximal [PhysRevResearch.6.023050]. An
alternative experimental protocol, not relying on the full tomography, is
based on the multiple quantum coherences measurements. This technique provides
extensive information about the structure of a many-body state, and allows
intertwining of the quantum correlators with other physical quantities like
the out-of-time-order correlations [garttner2017measuring,
PhysRevLett.120.040402].

##  V Conclusions

In this work we have provided a many-body quantum correlator that captures the
_non_ -$k$-separability of a given pure or mixed quantum state. By
construction, the proposed correlator respects the SLOCC entanglement
classification; we used it to classify three entanglement classes in a three-
qubit system, and to characterise nine entanglement classes in a four-qubit
system. We then characterised _non_ -$k$-separability for qubit AME states,
and Dicke states of an arbitrary number of qubits. The proposed method is not
limited by the system size, and is experimentally accessible.

Like any other method, the one presented in this paper is not able to
perfectly classify all entangled states. The strength of the method lies
rather in its simplicity. This comes at a price as for each state one has to
choose a suitable basis by local unitary rotations, such that maximises the
generalised GHZ fidelity.

Our setup is not specifically designed for pure states, and can be adapted to
mixed states as well. Therefore, the technique is able to statistically
distinguish between states even in the more realistic, mixed case scenarios
with experimental imperfections, including particle losses. In many
experimental setups it is not necessary to study the most general form of
entanglement possible but rather to confirm that the losses do not hinder the
state preparation. For such setups, our framework fits perfectly.

## Acknowledgement

We thank Guillem Müller-Rigat, Anna Sanpera, Geza Toth, and Karol Życzkowski
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.

##  Appendix A Convexity of correlator $\mathcal{Q}$

Entanglement cannot be increased while mixing the quantum states
[PhysRevLett.78.2275, Horodecki2001, Plenio_2007], therefore, any entanglement
measure should be convex. As such we require that for any two states
$\hat{\varrho}_{1}$, $\hat{\varrho}_{2}$, the correlator $\mathcal{Q}$
fulfills

$\mathcal{Q}(\lambda\hat{\varrho}_{1}+(1-\lambda)\hat{\varrho}_{2})\leqslant\lambda\mathcal{Q}(\hat{\varrho}_{1})+(1-\lambda)\mathcal{Q}(\hat{\varrho}_{2}).$
(30)

The correlator $\mathcal{Q}$, Eq.(16), for a linear combination of two states
reads:

$\begin{split}\mathcal{Q}\big{(}\lambda&\hat{\varrho}_{1}+(1-\lambda)\hat{\varrho}_{2}\big{)}=\\\
&=N+\log_{4}|\langle\tilde{0}|^{\otimes
N}(\lambda\hat{\varrho}_{1}+(1-\lambda)\hat{\varrho}_{2})|\tilde{1}\rangle^{\otimes
N}|^{2}.\end{split}$ (31)

Let us now focus on the logarithm. It inherently involves maximization over a
choice of basis, which can be upper bounded by two independent optimizations

$\begin{split}2\log_{4}\bigg{(}\max_{\\{|\tilde{0}\rangle,|\tilde{1}\rangle\\}}&|\langle\tilde{0}|^{\otimes
N}(\lambda\hat{\varrho}_{1}+(1-\lambda)\hat{\varrho}_{2})|\tilde{1}\rangle^{\otimes
N}|\bigg{)}\leq\\\ \leq
2\log_{4}\bigg{(}&\lambda\max_{\\{|\tilde{0}\rangle,|\tilde{1}\rangle\\}}|\langle\tilde{0}|^{\otimes
N}\hat{\varrho}_{1}|\tilde{1}\rangle^{\otimes N}|\,\,\,+\\\
&+(1-\lambda)\max_{\\{|\tilde{0}\rangle,|\tilde{1}\rangle\\}}|\langle\tilde{0}|^{\otimes
N}\hat{\varrho}_{2}|\tilde{1}\rangle^{\otimes N}|\bigg{)}.\end{split}$ (32)

Using the concavity of the logarithm, we can bound the latter expression by

$\begin{split}\lambda\log_{4}\bigg{(}\max_{\\{|\tilde{0}\rangle,|\tilde{1}\rangle\\}}&|\langle\tilde{0}|^{\otimes
N}\hat{\varrho}_{1}|\tilde{1}\rangle^{\otimes N}|^{2}\bigg{)}+\\\
+&(1-\lambda)\log_{4}\bigg{(}\max_{\\{|\tilde{0}\rangle,|\tilde{1}\rangle\\}}|\langle\tilde{0}|^{\otimes
N}\hat{\varrho}_{2}|\tilde{1}\rangle^{\otimes N}|^{2}\bigg{)},\end{split}$
(33)

from which we directly obtain the expressions for
$\mathcal{Q}(\hat{\varrho}_{1})$ and $\mathcal{Q}(\hat{\varrho}_{2})$,
yielding Eq. (30), what proves the convexity of the correlator $\mathcal{Q}$.

##  Appendix B SLOCC entanglement classes

The two pure multipartite quantum states $|\psi\rangle$ and $|\phi\rangle$
belong to the same SLOCC (stochastic local operations and classical
communication) entanglement class if and only if it is possible to convert
$|\psi\rangle\mapsto|\phi\rangle$ and $|\phi\rangle\mapsto|\psi\rangle$ with
non-zero probabilities using local operations and classical communication,
i.e., if there exist a set of invertible operators $\hat{O}_{k}$ (so that the
following condition is symmetric) such that
$|\phi\rangle=\bigotimes_{k=1}^{N}\hat{O}_{k}|\psi\rangle$, where $N$ is the
number of constituents.

The number of entanglement classes grows rapidly with the number of parties
and the local dimension. For pure states of two qubits, there are only two
entanglement classes, with a Bell state as an extremal representative of one
and separable states forming the other. Increasing the number of qubits to
three yields two disjoint genuinely multipartite entanglement classes,
together forming six different classes. Interestingly, the growth of the
number of classes is not even exponential; already for four qubits, there are
infinitely many entanglement classes, which can be divided into nine groups
[Verstraete_2002].

##  Appendix C Tensor rank and multipartite entanglement classification

For two-particle pure states
$|\psi\rangle=\sum_{i_{1}i_{2}}\alpha_{i_{1}i_{2}}|i_{1}i_{2}\rangle$,
classification and quantification of bipartite pure entanglement stems from
the singular value decomposition (SVD). This procedure allows to tansform the
matrix of the coefficients $[\hat{\alpha}]_{i_{1}i_{2}}=\alpha_{i_{1}i_{2}}$
to give the Schmidt decomposition of the state
$|\psi\rangle=\sum_{i}\sqrt{\lambda_{i}}|\chi_{i},\xi_{i}\rangle.$ From the
vector of the singular values $\Lambda=[\lambda_{i}]_{i}$ it is possible to
infer all the information about entanglement, such as the entanglement
entropy$-\sum_{i}\lambda_{i}\ln{\lambda_{i}}$, or the Schmidt rank (number of
non-zero $\lambda_{i}$s) [Bengtsson_2006]. For example, one can convert one
two-partite state $|\psi\rangle$ into another $|\phi\rangle$ via SLOCC if and
only if the Schmidt rank of the latter is not larger.

When $N\geqslant 3$, the situation is more complicated because the
coefficients of the state $|\psi\rangle=\sum_{i_{1}\ldots
i_{N}}\alpha_{i_{1}\ldots i_{N}}|i_{1}\ldots i_{N}\rangle$, form a tensor, for
which, in contrast to matrices, SVD is not always possible and various
generalizations of higher-order SVDs were proposed (for a list of their
applications in quantum information see Ref. [Bruzda_2023]). One way is to use
the tensor rank, which is the minimal number of separable states into which
one can decompose a given state. We use this notion to exemplify the
difference between two _most entangled_ states of three qubits, namely between
$|\text{GHZ}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$, which has a
rank 2 and
$|\text{W}\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)$,
which has a rank 3.

Surprisingly, there are states of tensor rank 2 that approximate the W state
to an arbitrary precision [Acin_2001, Vrana_2015], but the converse is not
true – the GHZ state does not have states of tensor rank 3 in its neighborhood
inside the Hilbert space. It is again in direct contrast with the bipartite
case, as matrices of lower rank always have matrices of higher rank in the
neighborhood, while the converse does not hold. Therefore, we say that the
border rank of the W state is $2$. What is more, the generic rank of a matrix
is the maximal one, but for tensors of three indices 2-level each (so three-
qubit pure states) this is not the case; the maximal tensor rank is 3 and the
generic is 2. As such, the generic pure state of three qubits belongs to the
GHZ SLOCC-class, not to the W class.

##  Appendix D Entanglement classes for four qubits

Here we list nine representatives of entanglement classes for four qubits
[Verstraete_2002, Zangi_2017, Vintskevich_2023]:

$\begin{split}|G\rangle&=\frac{a+b}{2}\big{(}|0000\rangle+|1111\rangle\big{)}+\frac{a-d}{2}\big{(}|0011\rangle+|1100\rangle\big{)}+\frac{b+c}{2}\big{(}|0101\rangle+|1010\rangle\big{)}+\frac{b-c}{2}\big{(}|0110\rangle+|1001\rangle\big{)},\\\
|E_{1}\rangle&=\frac{a+b}{2}\big{(}|0000\rangle+|1111\rangle\big{)}+\frac{a-b}{2}\big{(}|0011\rangle+|1100\rangle\big{)}+c\big{(}|0101\rangle+|1010\rangle\big{)}+|0110\rangle,\\\
|E_{2}\rangle&=a\big{(}|0000\rangle+|1111\rangle\big{)}+b\big{(}|0101\rangle+|1010\rangle\big{)}+|0110\rangle+|0011\rangle,\\\
|E_{3}\rangle&=a\big{(}|0000\rangle+|1111\rangle\big{)}+\frac{a+b}{2}\big{(}|0101\rangle+|1010\rangle\big{)}+\frac{a-b}{2}\big{(}|0110\rangle+|1001\rangle\big{)}+\frac{i}{\sqrt{2}}\big{(}|0001\rangle+|0010\rangle+|0111\rangle+|1011\rangle\big{)},\\\
|E_{4}\rangle&=a\big{(}|0000\rangle+|0101\rangle+|1010\rangle+|1111\rangle\big{)}+i|0001\rangle+|0110\rangle-i|1011\rangle,\\\
|E_{5}\rangle&=a\big{(}|0000\rangle+|1111\rangle\big{)}+|0011\rangle+|0101\rangle+|0110\rangle,\\\
|E_{6}\rangle&=|0000\rangle+|0101\rangle+|1000\rangle+|1110\rangle,\\\
|E_{7}\rangle&=|0000\rangle+|1011\rangle+|1101\rangle+|1110\rangle,\\\
|E_{8}\rangle&=|0000\rangle+|0111\rangle.\end{split}$ (34)

##  Appendix E Absolutely Maximally Entangled states AME(5,2) and AME(6,2)

The AME coefficients [Facchi_2008, Goyeneche_2015] for $N=5$ are

$\begin{split}\alpha^{(5)}&=\\{1,1,1,1,1,-1,-1,1,1,-1,-1,1,\\\
&1,1,1,1,1,-1,-1,1,-1,1,-1,-1,\\\ &1,-1,1,-1,-1,1,1\\}\end{split}$ (35)

while for $N=6$

$\begin{split}\alpha^{(6)}&=\\{-1,-1,-1,1,-1,1,1,1,-1,-1,-1,1,\\\
&1,-1,-1,-1,-1,-1,1,-1,-1,1,-1,-1,\\\ &1,1,-1,1,-1,1,-1,-1,-1,1,-1,-1,-1,\\\
&-1,1,-1,1,-1,1,1,-1,-1,1,-1,1,-1,-1,\\\
&-1,1,1,1,-1,1,-1,-1,-1,-1,-1,-1,1\\}.\end{split}$ (36)

## References

  * Horodecki  _et al._ [2009] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  * Uola  _et al._ [2020] R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, Quantum steering, Rev. Mod. Phys. 92, 015001 (2020).
  * Bell [1964] J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics 1, 195 (1964).
  * Brunner  _et al._ [2014] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  * Bell [1966] J. S. Bell, On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447 (1966).
  * Vedral  _et al._ [1997] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying entanglement, Phys. Rev. Lett. 78, 2275 (1997).
  * Horodecki [2001] M. Horodecki, Entanglement measures, Quantum Information and Computation 1, 3–26 (2001).
  * Plenio and Virmani [2007] M. Plenio and S. Virmani, An introduction to entanglement measures, Quantum Information and Computation 7, 1 (2007).
  * Gühne and Tóth [2009] O. Gühne and G. Tóth, Entanglement detection, Physics Reports 474, 1 (2009).
  * Bengtsson and Życzkowski [2006] I. Bengtsson and K. Życzkowski,  _Geometry of Quantum States: An Introduction to Quantum Entanglement_ (Cambridge University Press, 2006).
  * Sabín and García-Alcaine [2008] C. Sabín and G. García-Alcaine, A classification of entanglement in three-qubit systems, The European Physical Journal D 48, 435 (2008).
  * Enríquez  _et al._ [2018] M. Enríquez, F. Delgado, and K. Życzkowski, Entanglement of three-qubit random pure states, Entropy 20, 10.3390/e20100745 (2018).
  * Chiara and Sanpera [2018] G. D. Chiara and A. Sanpera, Genuine quantum correlations in quantum many-body systems: a review of recent progress, Reports on Progress in Physics 81, 074002 (2018).
  * Szalay [2015] S. Szalay, Multipartite entanglement measures, Phys. Rev. A 92, 042329 (2015).
  * Srivastava  _et al._ [2024] A. Srivastava, G. Müller-Rigat, M. Lewenstein, and G. Rajchel-Mieldzioć, Introduction to quantum entanglement in many-body systems (2024), arXiv:2402.09523 [quant-ph] .
  * Xie  _et al._ [2024] S. Xie, D. Younis, Y. Mei, and J. H. Eberly, Multipartite entanglement: A journey through geometry, Entropy 26, 217 (2024).
  * Verstraete  _et al._ [2002] F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde, Four qubits can be entangled in nine different ways, Physical Review A 65, 10.1103/physreva.65.052112 (2002).
  * Gour and Wallach [2010] G. Gour and N. R. Wallach, All maximally entangled four-qubit states, Journal of Mathematical Physics 51, 112201 (2010), https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/1.3511477/14694696/112201_1_online.pdf .
  * Spee  _et al._ [2016] C. Spee, J. I. de Vicente, and B. Kraus, The maximally entangled set of 4-qubit states, Journal of Mathematical Physics 57, 052201 (2016), https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/1.4946895/15923401/052201_1_online.pdf .
  * Sadhukhan  _et al._ [2017] D. Sadhukhan, S. S. Roy, A. K. Pal, D. Rakshit, A. Sen(De), and U. Sen, Multipartite entanglement accumulation in quantum states: Localizable generalized geometric measure, Phys. Rev. A 95, 022301 (2017).
  * Tóth [2004] G. Tóth, Entanglement detection in optical lattices of bosonic atoms with collective measurements, Phys. Rev. A 69, 052327 (2004).
  * Knips  _et al._ [2016] L. Knips, C. Schwemmer, N. Klein, M. Wieśniak, and H. Weinfurter, Multipartite entanglement detection with minimal effort, Phys. Rev. Lett. 117, 210504 (2016).
  * Shi  _et al._ [2024] F. Shi, L. Chen, G. Chiribella, and Q. Zhao, Entanglement detection length of multipartite quantum states (2024), arXiv:2401.03367 [quant-ph] .
  * Lücke  _et al._ [2014] B. Lücke, J. Peise, G. Vitagliano, J. Arlt, L. Santos, G. Tóth, and C. Klempt, Detecting multiparticle entanglement of dicke states, Phys. Rev. Lett. 112, 155304 (2014).
  * Wieczorek  _et al._ [2009a] W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, Experimental entanglement of a six-photon symmetric Dicke state, Phys. Rev. Lett. 103, 020504 (2009a).
  * Vanderbruggen  _et al._ [2011] T. Vanderbruggen, S. Bernon, A. Bertoldi, A. Landragin, and P. Bouyer, Spin-squeezing and Dicke-state preparation by heterodyne measurement, Phys. Rev. A 83, 013821 (2011).
  * Prevedel  _et al._ [2009] R. Prevedel, G. Cronenberg, M. S. Tame, M. Paternostro, P. Walther, M. S. Kim, and A. Zeilinger, Experimental realization of dicke states of up to six qubits for multiparty quantum networking, Phys. Rev. Lett. 103, 020503 (2009).
  * Zhao  _et al._ [2015] Y.-Y. Zhao, Y.-C. Wu, G.-Y. Xiang, C.-F. Li, and G.-C. Guo, Experimental violation of the local realism for four-qubit dicke state, Opt. Express 23, 30491 (2015).
  * Wang  _et al._ [2016] M.-Y. Wang, F.-L. Yan, and T. Gao, Deterministic distribution of four-photon Dicke state over an arbitrary collective-noise channel with cross-Kerr nonlinearity, Scientific Reports 6, 29853 (2016).
  * Mu  _et al._ [2020] F. Mu, Y. Gao, H. Yin, and G. Wang, Dicke state generation via selective interactions in a Dicke-Stark model, Opt. Express 28, 39574 (2020).
  * Lücke  _et al._ [2011] B. Lücke, M. Scherer, J. Kruse, L. Pezzé, F. Deuretzbacher, P. Hyllus, O. Topic, J. Peise, W. Ertmer, J. Arlt, L. Santos, A. Smerzi, and C. Klempt, Twin matter waves for interferometry beyond the classical limit, Science 334, 773 (2011), https://www.science.org/doi/pdf/10.1126/science.1208798 .
  * Hamley  _et al._ [2012] C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Bookjans, and M. S. Chapman, Spin-nematic squeezed vacuum in a quantum gas, Nature Physics 8, 305 (2012).
  * Zou  _et al._ [2018] Y.-Q. Zou, L.-N. Wu, Q. Liu, X.-Y. Luo, S.-F. Guo, J.-H. Cao, M. K. Tey, and L. You, Beating the classical precision limit with spin-1 dicke states of more than 10,000 atoms, Proceedings of the National Academy of Sciences 115, 6381 (2018), https://www.pnas.org/doi/pdf/10.1073/pnas.1715105115 .
  * Bärtschi and Eidenbenz [2022] A. Bärtschi and S. Eidenbenz, Short-depth circuits for dicke state preparation, in  _2022 IEEE International Conference on Quantum Computing and Engineering (QCE)_ (2022) pp. 87–96.
  * Narisada  _et al._ [2023] S. Narisada, S. Beppu, F. S., and S. Kiyomoto, Concrete quantum circuits to prepare generalized dicke states on a quantum mac, in  _Proceedings of the 9th International Conference on Information Systems Security and Privacy (ICISSP 2_ (2023) pp. 329–338.
  * Pandya  _et al._ [2020] P. Pandya, O. Sakarya, and M. Wieśniak, Hilbert-schmidt distance and entanglement witnessing, Phys. Rev. A 102, 012409 (2020).
  * Rudnicki  _et al._ [2011] Ł. Rudnicki, P. Horodecki, and K. Życzkowski, Collective uncertainty entanglement test, Physical Review Letters 107, 10.1103/physrevlett.107.150502 (2011).
  * Rudnicki  _et al._ [2012] Ł. Rudnicki, Z. Puchała, P. Horodecki, and K. Życzkowski, Collectibility for mixed quantum states, Physical Review A 86, 10.1103/physreva.86.062329 (2012).
  * Wootters [2001] W. K. Wootters, Entanglement of formation and concurrence, Quantum Info. Comput. 1, 27–44 (2001).
  * Wootters [1998] W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits, Phys. Rev. Lett. 80, 2245 (1998).
  * Luo [2008a] S. Luo, Entanglement measures based on observable correlations, Theoretical and Mathematical Physics 155, 896 (2008a).
  * Hayden  _et al._ [2001] P. M. Hayden, M. Horodecki, and B. M. Terhal, The asymptotic entanglement cost of preparing a quantum state, Journal of Physics A: Mathematical and General 34, 6891 (2001).
  * Majewski [2001] A. W. Majewski, On entanglement of formation, Journal of Physics A: Mathematical and General 35, 123 (2001).
  * Fei  _et al._ [2003] S.-M. Fei, J. Jost, X. Li-Jost, and G.-F. Wang, Entanglement of formation for a class of quantum states, Physics Letters A 310, 333 (2003).
  * Hui and Zhi-Xi [2007] Z. Hui and W. Zhi-Xi, Entanglement of formation for quantum states, Communications in Theoretical Physics 47, 45 (2007).
  * Gao  _et al._ [2008] X. Gao, A. Sergio, K. Chen, S. Fei, and X. Li-Jost, Entanglement of formation and concurrence for mixed states, Frontiers of Computer Science in China 2, 114–128 (2008).
  * Onoe  _et al._ [2020] S. Onoe, S. Tserkis, A. P. Lund, and T. C. Ralph, Multipartite Gaussian entanglement of formation, Phys. Rev. A 102, 042408 (2020).
  * Zurek [2000] W. Zurek, Einselection and decoherence from an information theory perspective, Annalen der Physik 512, 855 (2000), https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.200051211-1204 .
  * Henderson and Vedral [2001] L. Henderson and V. Vedral, Classical, quantum and total correlations, Journal of Physics A: Mathematical and General 34, 6899 (2001).
  * Ollivier and Zurek [2001] H. Ollivier and W. H. Zurek, Quantum discord: A measure of the quantumness of correlations, Phys. Rev. Lett. 88, 017901 (2001).
  * Luo [2008b] S. Luo, Quantum discord for two-qubit systems, Phys. Rev. A 77, 042303 (2008b).
  * Datta and Gharibian [2009] A. Datta and S. Gharibian, Signatures of nonclassicality in mixed-state quantum computation, Phys. Rev. A 79, 042325 (2009).
  * Adesso and Datta [2010] G. Adesso and A. Datta, Quantum versus classical correlations in Gaussian states, Phys. Rev. Lett. 105, 030501 (2010).
  * Huang [2013] Y. Huang, Quantum discord for two-qubit $x$ states: Analytical formula with very small worst-case error, Phys. Rev. A 88, 014302 (2013).
  * Girolami  _et al._ [2014] D. Girolami, A. M. Souza, V. Giovannetti, T. Tufarelli, J. G. Filgueiras, R. S. Sarthour, D. O. Soares-Pinto, I. S. Oliveira, and G. Adesso, Quantum discord determines the interferometric power of quantum states, Phys. Rev. Lett. 112, 210401 (2014).
  * Bera  _et al._ [2017] A. Bera, T. Das, D. Sadhukhan, S. S. Roy, A. Sen(De), and U. Sen, Quantum discord and its allies: a review of recent progress, Reports on Progress in Physics 81, 024001 (2017).
  * Hill and Wootters [1997] S. A. Hill and W. K. Wootters, Entanglement of a pair of quantum bits, Phys. Rev. Lett. 78, 5022 (1997).
  * Hildebrand [2007] R. Hildebrand, Concurrence revisited, Journal of Mathematical Physics 48, 102108 (2007), https://pubs.aip.org/aip/jmp/article-pdf/doi/10.1063/1.2795840/16056058/102108_1_online.pdf .
  * Swain  _et al._ [2022] S. N. Swain, V. S. Bhaskara, and P. K. Panigrahi, Generalized entanglement measure for continuous-variable systems, Phys. Rev. A 105, 052441 (2022).
  * Bhaskara and Panigrahi [2017] V. S. Bhaskara and P. K. Panigrahi, Generalized concurrence measure for faithful quantification of multiparticle pure state entanglement using Lagrange’s identity and wedge product, Quantum Information Processing 16, 118 (2017).
  * Rungta  _et al._ [2001] P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Milburn, Universal state inversion and concurrence in arbitrary dimensions, Phys. Rev. A 64, 042315 (2001).
  * Hashemi Rafsanjani  _et al._ [2012] S. M. Hashemi Rafsanjani, M. Huber, C. J. Broadbent, and J. H. Eberly, Genuinely multipartite concurrence of $n$-qubit $x$ matrices, Phys. Rev. A 86, 062303 (2012).
  * Tóth and Apellaniz [2014] G. Tóth and I. Apellaniz, Quantum metrology from a quantum information science perspective, Journal of Physics A: Mathematical and Theoretical 47, 424006 (2014).
  * Braunstein and Caves [1994] S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  * Hyllus  _et al._ [2012] P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezzé, and A. Smerzi, Fisher information and multiparticle entanglement, Phys. Rev. A 85, 022321 (2012).
  * Tóth [2012] G. Tóth, Multipartite entanglement and high-precision metrology, Phys. Rev. A 85, 022322 (2012).
  * Ren  _et al._ [2021] Z. Ren, W. Li, A. Smerzi, and M. Gessner, Metrological detection of multipartite entanglement from young diagrams, Phys. Rev. Lett. 126, 080502 (2021).
  * Kitagawa and Ueda [1993] M. Kitagawa and M. Ueda, Squeezed spin states, Phys. Rev. A 47, 5138 (1993).
  * Wineland  _et al._ [1994] D. Wineland, J. Bollinger, W. Itano, and D. Heinzen, Squeezed atomic states and projection noise in spectroscopy, Phys. Rev. A 50, 67 (1994).
  * Korbicz  _et al._ [2005] J. K. Korbicz, J. I. Cirac, and M. Lewenstein, Spin squeezing inequalities and entanglement of $n$ qubit states, Phys. Rev. Lett. 95, 120502 (2005).
  * Korbicz  _et al._ [2006] J. K. Korbicz, O. Gühne, M. Lewenstein, H. Häffner, C. F. Roos, and R. Blatt, Generalized spin-squeezing inequalities in $n$-qubit systems: Theory and experiment, Phys. Rev. A 74, 052319 (2006).
  * Esteve  _et al._ [2008] J. Esteve, C. Gross, A. Weller, S. Giovanazzi, and M. Oberthaler, Squeezing and entanglement in a bose–einstein condensate, Nature 455, 1216 (2008).
  * He  _et al._ [2011] Q. Y. He, S.-G. Peng, P. D. Drummond, and M. D. Reid, Planar quantum squeezing and atom interferometry, Phys. Rev. A 84, 022107 (2011).
  * Strobel  _et al._ [2014] H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezzé, A. Smerzi, and M. K. Oberthaler, Fisher information and entanglement of non-Gaussian spin states, Science 345, 424 (2014).
  * De Chiara and Sanpera [2011] G. De Chiara and A. Sanpera, Detection of entanglement in ultracold lattice gases, Journal of Low Temperature Physics 165, 292 (2011).
  * Stockton  _et al._ [2003] J. K. Stockton, J. M. Geremia, A. C. Doherty, and H. Mabuchi, Characterizing the entanglement of symmetric many-particle spin-$\frac{1}{2}$ systems, Phys. Rev. A 67, 022112 (2003).
  * Vitagliano  _et al._ [2018] G. Vitagliano, G. Colangelo, F. Martin Ciurana, M. W. Mitchell, R. J. Sewell, and G. Tóth, Entanglement and extreme planar spin squeezing, Phys. Rev. A 97, 020301 (2018).
  * Vitagliano  _et al._ [2011] G. Vitagliano, P. Hyllus, I. n. L. Egusquiza, and G. Tóth, Spin squeezing inequalities for arbitrary spin, Phys. Rev. Lett. 107, 240502 (2011).
  * Sørensen and Mølmer [2001] A. S. Sørensen and K. Mølmer, Entanglement and extreme spin squeezing, Phys. Rev. Lett. 86, 4431 (2001).
  * Tura  _et al._ [2014a] J. Tura, R. Augusiak, A. B. Sainz, T. Vértesi, M. Lewenstein, and A. Acín, Detecting nonlocality in many-body quantum states, Science 344, 1256 (2014a), https://www.science.org/doi/pdf/10.1126/science.1247715 .
  * Tura  _et al._ [2014b] J. Tura, A. B. Sainz, T. Vértesi, A. Acín, M. Lewenstein, and R. Augusiak, Translationally invariant multipartite bell inequalities involving only two-body correlators, Journal of Physics A: Mathematical and Theoretical 47, 424024 (2014b).
  * Tura  _et al._ [2015] J. Tura, R. Augusiak, A. Sainz, B. Lücke, C. Klempt, M. Lewenstein, and A. Acín, Nonlocality in many-body quantum systems detected with two-body correlators, Annals of Physics 362, 370 (2015).
  * Tura  _et al._ [2019] J. Tura, A. Aloy, F. Baccari, A. Acín, M. Lewenstein, and R. Augusiak, Optimization of device-independent witnesses of entanglement depth from two-body correlators, Phys. Rev. A 100, 032307 (2019).
  * Baccari  _et al._ [2019] F. Baccari, J. Tura, M. Fadel, A. Aloy, J.-D. Bancal, N. Sangouard, M. Lewenstein, A. Acín, and R. Augusiak, Bell correlation depth in many-body systems, Phys. Rev. A 100, 022121 (2019).
  * Tóth  _et al._ [2009] G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, Spin squeezing and entanglement, Phys. Rev. A 79, 042334 (2009).
  * Tóth  _et al._ [2007] G. Tóth, C. Knapp, O. Gühne, and H. J. Briegel, Optimal spin squeezing inequalities detect bound entanglement in spin models, Phys. Rev. Lett. 99, 250405 (2007).
  * Apellaniz  _et al._ [2017] I. Apellaniz, M. Kleinmann, O. Gühne, and G. Tóth, Optimal witnessing of the quantum fisher information with few measurements, Phys. Rev. A 95, 032330 (2017).
  * Vitagliano  _et al._ [2014] G. Vitagliano, I. Apellaniz, I. n. L. Egusquiza, and G. Tóth, Spin squeezing and entanglement for an arbitrary spin, Phys. Rev. A 89, 032307 (2014).
  * Marty  _et al._ [2017] O. Marty, M. Cramer, G. Vitagliano, G. Toth, and M. B. Plenio, Multiparticle entanglement criteria for nonsymmetric collective variances (2017), arXiv:1708.06986 [quant-ph] .
  * Vitagliano  _et al._ [2017] G. Vitagliano, I. Apellaniz, M. Kleinmann, B. Lücke, C. Klempt, and G. Tóth, Entanglement and extreme spin squeezing of unpolarized states, New Journal of Physics 19, 013027 (2017).
  * Fadel  _et al._ [2021] M. Fadel, A. Usui, M. Huber, N. Friis, and G. Vitagliano, Entanglement quantification in atomic ensembles, Phys. Rev. Lett. 127, 010401 (2021).
  * Müller-Rigat  _et al._ [2021] G. Müller-Rigat, A. Aloy, M. Lewenstein, and I. Frérot, Inferring nonlinear many-body bell inequalities from average two-body correlations: Systematic approach for arbitrary spin-$j$ ensembles, PRX Quantum 2, 030329 (2021).
  * Fröwis  _et al._ [2017] F. Fröwis, P. C. Strassmann, A. Tiranov, C. Gut, J. Lavoie, N. Brunner, F. Bussières, M. Afzelius, and N. Gisin, Experimental certification of millions of genuinely entangled atoms in a solid, Nature Communications 8, 907 (2017).
  * Aloy  _et al._ [2019] A. Aloy, J. Tura, F. Baccari, A. Acín, M. Lewenstein, and R. Augusiak, Device-independent witnesses of entanglement depth from two-body correlators, Phys. Rev. Lett. 123, 100507 (2019).
  * Roik  _et al._ [2022] J. Roik, K. Bartkiewicz, A. Černoch, and K. Lemr, Entanglement quantification from collective measurements processed by machine learning, Physics Letters A 446, 128270 (2022).
  * Cox  _et al._ [2016] K. C. Cox, G. P. Greve, J. M. Weiner, and J. K. Thompson, Deterministic squeezed states with collective measurements and feedback, Phys. Rev. Lett. 116, 093602 (2016).
  * Müller-Rigat  _et al._ [2023] G. Müller-Rigat, A. K. Srivastava, S. Kurdziałek, G. Rajchel-Mieldzioć, M. Lewenstein, and I. Frérot, Certifying the quantum Fisher information from a given set of mean values: a semidefinite programming approach, Quantum 7, 1152 (2023).
  * Fadel  _et al._ [2023] M. Fadel, B. Yadin, Y. Mao, T. Byrnes, and M. Gessner, Multiparameter quantum metrology and mode entanglement with spatially split nonclassical spin ensembles, New Journal of Physics 25, 073006 (2023).
  * Vitagliano  _et al._ [2023] G. Vitagliano, M. Fadel, I. Apellaniz, M. Kleinmann, B. Lücke, C. Klempt, and G. Tóth, Number-phase uncertainty relations and bipartite entanglement detection in spin ensembles, Quantum 7, 914 (2023).
  * Guo  _et al._ [2023] J. Guo, J. Tura, Q. He, and M. Fadel, Detecting bell correlations in multipartite non-Gaussian spin states, Phys. Rev. Lett. 131, 070201 (2023).
  * Poggi and Muñoz-Arias [2024] P. M. Poggi and M. H. Muñoz-Arias, Measurement-induced multipartite-entanglement regimes in collective spin systems, Quantum 8, 1229 (2024).
  * Wieśniak  _et al._ [2005] M. Wieśniak, V. Vedral, and Časlav Brukner, Magnetic susceptibility as a macroscopic entanglement witness, New Journal of Physics 7, 258 (2005).
  * Wieśniak  _et al._ [2008] M. Wieśniak, V. Vedral, and i. c. v. Brukner, Heat capacity as an indicator of entanglement, Phys. Rev. B 78, 064108 (2008).
  * Brukner and Vedral [2004] C. Brukner and V. Vedral, Macroscopic thermodynamical witnesses of quantum entanglement (2004), arXiv:quant-ph/0406040 [quant-ph] .
  * Dowling  _et al._ [2004] M. R. Dowling, A. C. Doherty, and S. D. Bartlett, Energy as an entanglement witness for quantum many-body systems, Phys. Rev. A 70, 062113 (2004).
  * Tóth [2005] G. Tóth, Entanglement witnesses in spin models, Phys. Rev. A 71, 010301 (2005).
  * Wu  _et al._ [2005] L.-A. Wu, S. Bandyopadhyay, M. S. Sarandy, and D. A. Lidar, Entanglement observables and witnesses for interacting quantum spin systems, Phys. Rev. A 72, 032309 (2005).
  * Frydryszak  _et al._ [2017] A. M. Frydryszak, M. I. Samar, and V. M. Tkachuk, Quantifying geometric measure of entanglement by mean value of spin and spin correlations with application to physical systems, The European Physical Journal D 71, 233 (2017).
  * Kuzmak and Tkachuk [2020] A. Kuzmak and V. Tkachuk, Detecting entanglement by the mean value of spin on a quantum computer, Physics Letters A 384, 126579 (2020).
  * Doronin  _et al._ [2019] S. I. Doronin, E. B. Fel’dman, and I. D. Lazarev, Many-particle entanglement in multiple quantum nuclear-magnetic-resonance spectroscopy, Phys. Rev. A 100, 022330 (2019).
  * Lazarev and Fel’dman [2020] I. D. Lazarev and E. B. Fel’dman, Many-spin entanglement in multiple quantum nmr with a dipolar ordered initial state, Journal of Experimental and Theoretical Physics 131, 723 (2020).
  * Schneeloch  _et al._ [2020] J. Schneeloch, C. C. Tison, M. L. Fanto, S. Ray, and P. M. Alsing, Quantifying tripartite entanglement with entropic correlations, Phys. Rev. Res. 2, 043152 (2020).
  * Lewis-Swan  _et al._ [2019] R. J. Lewis-Swan, A. Safavi-Naini, J. J. Bollinger, and A. M. Rey, Unifying scrambling, thermalization and entanglement through measurement of fidelity out-of-time-order correlators in the Dicke model, Nature Communications 10, 1581 (2019).
  * Blocher  _et al._ [2022] P. D. Blocher, S. Asaad, V. Mourik, M. A. I. Johnson, A. Morello, and K. Mølmer, Measuring out-of-time-ordered correlation functions without reversing time evolution, Phys. Rev. A 106, 042429 (2022).
  * Harrow  _et al._ [2021] A. W. Harrow, L. Kong, Z.-W. Liu, S. Mehraban, and P. W. Shor, Separation of out-of-time-ordered correlation and entanglement, PRX Quantum 2, 020339 (2021).
  * Iglói and Tóth [2023] F. Iglói and G. Tóth, Entanglement witnesses in the $xy$ chain: Thermal equilibrium and postquench nonequilibrium states, Phys. Rev. Res. 5, 013158 (2023).
  * Dür  _et al._ [2000] W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000).
  * Burchardt  _et al._ [2024] A. Burchardt, G. m. c. M. Quinta, and R. André, Entanglement classification via a single entanglement measure, Phys. Rev. A 109, 032424 (2024).
  * Huber and de Vicente [2013] M. Huber and J. I. de Vicente, Structure of multidimensional entanglement in multipartite systems, Phys. Rev. Lett. 110, 030501 (2013).
  * De Vicente [2007] J. I. De Vicente, Separability criteria based on the bloch representation of density matrices, Quantum Info. Comput. 7, 624–638 (2007).
  * de Vicente  _et al._ [2013] J. I. de Vicente, C. Spee, and B. Kraus, Maximally entangled set of multipartite quantum states, Phys. Rev. Lett. 111, 110502 (2013).
  * de Vicente and Huber [2011] J. I. de Vicente and M. Huber, Multipartite entanglement detection from correlation tensors, Phys. Rev. A 84, 062306 (2011).
  * Zangi  _et al._ [2017] S. M. Zangi, J.-L. Li, and C.-F. Qiao, Entanglement classification of four-partite states under the SLOCC, Journal of Physics A: Mathematical and Theoretical 50, 325301 (2017).
  * Gour and Wallach [2013] G. Gour and N. R. Wallach, Classification of multipartite entanglement of all finite dimensionality, Phys. Rev. Lett. 111, 060502 (2013).
  * Palazuelos and Vicente [2022] C. Palazuelos and J. I. d. Vicente, Genuine multipartite entanglement of quantum states in the multiple-copy scenario, Quantum 6, 735 (2022).
  * Horodecki  _et al._ [1996] M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: necessary and sufficient conditions, Physics Letters A 223, 1 (1996).
  * Peres [1996] A. Peres, Separability criterion for density matrices, Physical Review Letters 77, 1413 (1996).
  * Chen and Wu [2003] K. Chen and L.-A. Wu, A matrix realignment method for recognizing entanglement, Quantum Inf. Comput. 3, 193 (2003).
  * Rudolph [2005] O. Rudolph, Further results on the cross norm criterion for separability, Quantum Information Processing 4, 219 (2005).
  * Jungnitsch  _et al._ [2011] B. Jungnitsch, T. Moroder, and O. Gühne, Taming multiparticle entanglement, Phys. Rev. Lett. 106, 190502 (2011).
  * Gühne  _et al._ [2006] O. Gühne, M. Mechler, G. Tóth, and P. Adam, Entanglement criteria based on local uncertainty relations are strictly stronger than the computable cross norm criterion, Physical Review A 74, 10.1103/physreva.74.010301 (2006).
  * Terhal [2002] B. M. Terhal, Detecting quantum entanglement, Theoretical Computer Science 287, 313 (2002), natural Computing.
  * Lewenstein  _et al._ [2001] M. Lewenstein, B. Kraus, P. Horodecki, and J. I. Cirac, Characterization of separable states and entanglement witnesses, Phys. Rev. A 63, 044304 (2001).
  * Acín  _et al._ [2001a] A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, Classification of mixed three-qubit states, Physical Review Letters 87, 10.1103/physrevlett.87.040401 (2001a).
  * Bourennane  _et al._ [2004] M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bruß, M. Lewenstein, and A. Sanpera, Experimental detection of multipartite entanglement using witness operators, Phys. Rev. Lett. 92, 087902 (2004).
  * Horodecki [2003] P. Horodecki, From limits of quantum operations to multicopy entanglement witnesses and state-spectrum estimation, Physical Review A 68, 10.1103/physreva.68.052101 (2003).
  * Blasone  _et al._ [2008] M. Blasone, F. Dell’Anno, S. De Siena, and F. Illuminati, Hierarchies of geometric entanglement, Phys. Rev. A 77, 062304 (2008).
  * Lu  _et al._ [2018a] H. Lu, Q. Zhao, Z.-D. Li, X.-F. Yin, X. Yuan, J.-C. Hung, L.-K. Chen, L. Li, N.-L. Liu, C.-Z. Peng, Y.-C. Liang, X. Ma, Y.-A. Chen, and J.-W. Pan, Entanglement structure: Entanglement partitioning in multipartite systems and its experimental detection using optimizable witnesses, Phys. Rev. X 8, 021072 (2018a).
  * Frérot and Roscilde [2021] I. Frérot and T. Roscilde, Optimal entanglement witnesses: A scalable data-driven approach, Phys. Rev. Lett. 127, 040401 (2021).
  * Rico and Huber [2024] A. Rico and F. Huber, Entanglement detection with trace polynomials, Physical Review Letters 132, 10.1103/physrevlett.132.070202 (2024).
  * Vidal and Tarrach [1999] G. Vidal and R. Tarrach, Robustness of entanglement, Physical Review A 59, 141 (1999).
  * Wei and Goldbart [2003] T.-C. Wei and P. M. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states, Phys. Rev. A 68, 042307 (2003).
  * Meyer and Wallach [2002] D. A. Meyer and N. R. Wallach, Global entanglement in multiparticle systems, Journal of Mathematical Physics 43, 4273 (2002), https://pubs.aip.org/aip/jmp/article-pdf/43/9/4273/19183190/4273_1_online.pdf .
  * Lewenstein  _et al._ [2022] M. Lewenstein, G. Müller-Rigat, J. Tura, and A. Sanpera, Linear maps as sufficient criteria for entanglement depth and compatibility in many-body systems, Open Systems & Information Dynamics 29, 2250011 (2022), https://doi.org/10.1142/S1230161222500111 .
  * Susulovska and Gnatenko [2021] N. A. Susulovska and K. P. Gnatenko, Quantifying geometric measure of entanglement of multi-qubit graph states on the ibm’s quantum computer, in  _2021 IEEE International Conference on Quantum Computing and Engineering (QCE)_ (2021) pp. 465–466.
  * Guo  _et al._ [2022] Y. Guo, Y. Jia, X. Li, and L. Huang, Genuine multipartite entanglement measure, Journal of Physics A: Mathematical and Theoretical 55, 145303 (2022).
  * Wieśniak  _et al._ [2020] M. Wieśniak, P. Pandya, O. Sakarya, and B. Woloncewicz, Distance between bound entangled states from unextendible product bases and separable states, Quantum Reports 2, 49 (2020).
  * Xie and Eberly [2021] S. Xie and J. H. Eberly, Triangle measure of tripartite entanglement, Phys. Rev. Lett. 127, 040403 (2021).
  * Li and Shang [2022] Y. Li and J. Shang, Geometric mean of bipartite concurrences as a genuine multipartite entanglement measure, Phys. Rev. Res. 4, 023059 (2022).
  * Klyachko  _et al._ [2006] A. A. Klyachko, B. Öztop, and A. S. Shumovsky, Operational measure of entanglement for pure states of bipartite systems, Acta Physica Hungarica Series B, Quantum Electronics 26, 311 (2006).
  * Schwaiger  _et al._ [2015] K. Schwaiger, D. Sauerwein, M. Cuquet, J. I. de Vicente, and B. Kraus, Operational multipartite entanglement measures, Phys. Rev. Lett. 115, 150502 (2015).
  * Korbicz and Lewenstein [2006] J. K. Korbicz and M. Lewenstein, Group-theoretical approach to entanglement, Phys. Rev. A 74, 022318 (2006).
  * Melnikov [2023] D. Melnikov, Entanglement classification from a topological perspective, Phys. Rev. D 107, 126005 (2023).
  * Gharahi  _et al._ [2020] M. Gharahi, S. Mancini, and G. Ottaviani, Fine-structure classification of multiqubit entanglement by algebraic geometry, Phys. Rev. Res. 2, 043003 (2020).
  * Lu  _et al._ [2018b] S. Lu, S. Huang, K. Li, J. Li, J. Chen, D. Lu, Z. Ji, Y. Shen, D. Zhou, and B. Zeng, Separability-entanglement classifier via machine learning, Phys. Rev. A 98, 012315 (2018b).
  * Harney  _et al._ [2020] C. Harney, S. Pirandola, A. Ferraro, and M. Paternostro, Entanglement classification via neural network quantum states, New Journal of Physics 22, 045001 (2020).
  * Harney  _et al._ [2021] C. Harney, M. Paternostro, and S. Pirandola, Mixed state entanglement classification using artificial neural networks, New Journal of Physics 23, 063033 (2021).
  * Chen  _et al._ [2021] C. Chen, C. Ren, H. Lin, and H. Lu, Entanglement structure detection via machine learning, Quantum Science and Technology 6, 035017 (2021).
  * Rizvi  _et al._ [2022] S. M. A. Rizvi, N. Asif, M. S. Ulum, T. Q. Duong, and H. Shin, Multiclass classification of metrologically resourceful tripartite quantum states with deep neural networks, Sensors 22, 10.3390/s22186767 (2022).
  * Ayachi and Baz [2022] F. E. Ayachi and M. E. Baz, General classification of entanglement using machine learning (2022), arXiv:2210.07711 [quant-ph] .
  * Tian  _et al._ [2022] Y. Tian, L. Che, X. Long, C. Ren, and D. Lu, Machine learning experimental multipartite entanglement structure, Advanced Quantum Technologies 5, 2200025 (2022), https://onlinelibrary.wiley.com/doi/pdf/10.1002/qute.202200025 .
  * Vintskevich  _et al._ [2023] S. Vintskevich, N. Bao, A. Nomerotski, P. Stankus, and D. Grigoriev, Classification of four-qubit entangled states via machine learning, Physical Review A 107, 10.1103/physreva.107.032421 (2023).
  * Giordano and Martin-Delgado [2022] S. Giordano and M. Martin-Delgado, Reinforcement-learning generation of four-qubit entangled states, Physical Review Research 4, 10.1103/physrevresearch.4.043056 (2022).
  * Asif  _et al._ [2023] N. Asif, U. Khalid, A. Khan, T. Q. Duong, and H. Shin, Entanglement detection with artificial neural networks, Scientific Reports 13, 1562 (2023).
  * Sanavio  _et al._ [2023] C. Sanavio, E. Tignone, and E. Ercolessi, Entanglement classification via witness operators generated by support vector machine, The European Physical Journal Plus 138, 936 (2023).
  * Pawłowski and Krawczyk [2023] J. Pawłowski and M. Krawczyk, Identification of quantum entanglement with siamese convolutional neural networks and semi-supervised learning (2023), arXiv:2210.07410 [quant-ph] .
  * Koutný  _et al._ [2023] D. Koutný, L. Ginés, M. Moczała-Dusanowska, S. Höfling, C. Schneider, A. Predojević, and M. Ježek, Deep learning of quantum entanglement from incomplete measurements, Science Advances 9, eadd7131 (2023), https://www.science.org/doi/pdf/10.1126/sciadv.add7131 .
  * Palmieri  _et al._ [2023] A. M. Palmieri, G. Müller-Rigat, A. K. Srivastava, M. Lewenstein, G. Rajchel-Mieldzioć, and M. Płodzień, Enhancing quantum state tomography via resource-efficient attention-based neural networks (2023), arXiv:2309.10616 [quant-ph] .
  * Krawczyk  _et al._ [2024] M. Krawczyk, J. Pawłowski, M. M. Maśka, and K. Roszak, Data-driven criteria for quantum correlations, Phys. Rev. A 109, 022405 (2024).
  * Dawid  _et al._ [2023] A. Dawid, J. Arnold, B. Requena, A. Gresch, M. Płodzień, K. Donatella, K. A. Nicoli, P. Stornati, R. Koch, M. Büttner, R. Okuła, G. Muñoz-Gil, R. A. Vargas-Hernández, A. Cervera-Lierta, J. Carrasquilla, V. Dunjko, M. Gabrié, P. Huembeli, E. van Nieuwenburg, F. Vicentini, L. Wang, S. J. Wetzel, G. Carleo, E. Greplová, R. Krems, F. Marquardt, M. Tomza, M. Lewenstein, and A. Dauphin, Modern applications of machine learning in quantum sciences (2023), arXiv:2204.04198 [quant-ph] .
  * Wieśniak [2023] M. Wieśniak, Performance comparison of Gilbert’s algorithm and machine learning in classifying Bell-diagonal two-qutrit entanglement, Scientific Reports 13, 19500 (2023).
  * Ureña  _et al._ [2024] J. Ureña, A. Sojo, J. Bermejo, and D. Manzano, Entanglement detection with classical deep neural networks (2024), arXiv:2304.05946 [quant-ph] .
  * Cavalcanti  _et al._ [2007] E. G. Cavalcanti, C. J. Foster, M. D. Reid, and P. D. Drummond, Bell inequalities for continuous-variable correlations, Phys. Rev. Lett. 99, 210405 (2007).
  * He  _et al._ [2010] Q. He, E. Cavalcanti, M. Reid, and P. Drummond, Bell inequalities for continuous-variable measurements, Phys. Rev. A 81, 062106 (2010).
  * Cavalcanti  _et al._ [2011] E. Cavalcanti, Q. He, M. Reid, and H. Wiseman, Unified criteria for multipartite quantum nonlocality, Phys. Rev. A 84, 032115 (2011).
  * Niezgoda  _et al._ [2020] A. Niezgoda, M. Panfil, and J. Chwedeńczuk, Quantum correlations in spin chains, Phys. Rev. A 102, 042206 (2020).
  * Niezgoda and Chwedeńczuk [2021] A. Niezgoda and J. Chwedeńczuk, Many-body nonlocality as a resource for quantum-enhanced metrology, Phys. Rev. Lett. 126, 210506 (2021).
  * Gärttner  _et al._ [2017] M. Gärttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Measuring out-of-time-order correlations and multiple quantum spectra in a trapped-ion quantum magnet, Nature Physics 13, 781 (2017).
  * Gärttner  _et al._ [2018] M. Gärttner, P. Hauke, and A. M. Rey, Relating out-of-time-order correlations to entanglement via multiple-quantum coherences, Phys. Rev. Lett. 120, 040402 (2018).
  * Struchalin  _et al._ [2021] G. Struchalin, Y. A. Zagorovskii, E. Kovlakov, S. Straupe, and S. Kulik, Experimental estimation of quantum state properties from classical shadows, PRX Quantum 2, 010307 (2021).
  * Płodzień  _et al._ [2020] M. Płodzień, M. Kościelski, E. Witkowska, and A. Sinatra, Producing and storing spin-squeezed states and Greenberger-Horne-Zeilinger states in a one-dimensional optical lattice, Phys. Rev. A 102, 013328 (2020).
  * Płodzień  _et al._ [2022] M. Płodzień, M. Lewenstein, E. Witkowska, and J. Chwedeńczuk, One-axis twisting as a method of generating many-body bell correlations, Phys. Rev. Lett. 129, 250402 (2022).
  * Płodzień  _et al._ [2024] M. Płodzień, T. Wasak, E. Witkowska, M. Lewenstein, and J. Chwedeńczuk, Generation of scalable many-body bell correlations in spin chains with short-range two-body interactions, Phys. Rev. Res. 6, 023050 (2024).
  * Żukowski and Brukner [2002] M. Żukowski and Č. Brukner, Bell’s theorem for general n-qubit states, Phys. Rev. Lett. 88, 210401 (2002).
  * Mermin [1990] N. Mermin, Extreme quantum entanglement in a superposition of macroscopically distinct states, Physical Review Letters 65, 1838 (1990).
  * Schmid  _et al._ [2008] C. Schmid, N. Kiesel, W. Laskowski, W. Wieczorek, M. Żukowski, and H. Weinfurter, Discriminating multipartite entangled states, Phys. Rev. Lett. 100, 200407 (2008).
  * Mermin [1981] N. D. Mermin, Bringing home the atomic world: Quantum mysteries for anybody, American Journal of Physics 49, 940 (1981), https://pubs.aip.org/aapt/ajp/article-pdf/49/10/940/11864850/940_1_online.pdf .
  * Reid  _et al._ [2012] M. D. Reid, Q.-Y. He, and P. D. Drummond, Entanglement and nonlocality in multi-particle systems, Frontiers of Physics 7, 72 (2012).
  * Ardehali [1992] M. Ardehali, Bell inequalities with a magnitude of violation that grows exponentially with the number of particles, Physical Review A 46, 5375 (1992).
  * Brunner  _et al._ [2012] N. Brunner, J. Sharam, and T. Vértesi, Testing the structure of multipartite entanglement with Bell inequalities, Physical Review Letters 108, 10.1103/physrevlett.108.110501 (2012).
  * Friis  _et al._ [2018] N. Friis, O. Marty, C. Maier, C. Hempel, M. Holzäpfel, P. Jurcevic, M. Plenio, M. Huber, C. Roos, R. Blatt, and B. Lanyon, Observation of entangled states of a fully controlled 20-qubit system, Physical Review X 8, 10.1103/physrevx.8.021012 (2018).
  * Frérot  _et al._ [2022] I. Frérot, F. Baccari, and A. Acín, Unveiling quantum entanglement in many-body systems from partial information, PRX Quantum 3, 10.1103/prxquantum.3.010342 (2022).
  * Płodzień  _et al._ [2024] M. Płodzień, J. Chwedeńczuk, and M. Lewenstein, Inherent quantum resources in the stationary spin chains (2024), arXiv:2405.16974 [quant-ph] .
  * Gabriel  _et al._ [2010] A. Gabriel, B. C. Hiesmayr, and M. Huber, Criterion for k-separability in mixed multipartite states, Quantum Info. Comput. 10, 829–836 (2010).
  * Gühne and Seevinck [2010] O. Gühne and M. Seevinck, Separability criteria for genuine multiparticle entanglement, New Journal of Physics 12, 053002 (2010).
  * Seevinck and Uffink [2008] M. Seevinck and J. Uffink, Partial separability and entanglement criteria for multiqubit quantum states, Phys. Rev. A 78, 032101 (2008).
  * Gao and Hong [2011] T. Gao and Y. Hong, Separability criteria for several classes of n-partite quantum states, The European Physical Journal D 61, 765–771 (2011).
  * Gao  _et al._ [2013] T. Gao, Y. Hong, Y. Lu, and F. Yan, Efficient k-separability criteria for mixed multipartite quantum states, Europhysics Letters 104, 20007 (2013).
  * Szalay [2019] S. Szalay, k-stretchability of entanglement, and the duality of k-separability and k-producibility, Quantum 3, 204 (2019).
  * Hong  _et al._ [2021] Y. Hong, T. Gao, and F. Yan, Detection of k-partite entanglement and k-nonseparability of multipartite quantum states, Physics Letters A 401, 127347 (2021).
  * Kaniewski [2016] J. Kaniewski, Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities, Physical Review Letters 117, 10.1103/physrevlett.117.070402 (2016).
  * Li  _et al._ [2019] J. Li, T. Liu, S. Wang, C. Jebarathinam, and Q. Wang, Experimental violation of Mermin steering inequality by three-photon entangled states with nontrivial GHZ-fidelity, Optics Express 27, 13559 (2019).
  * Li  _et al._ [2009] D. Li, X. Li, H. Huang, and X. Li, SLOCC classification for nine families of four-qubits, Quantum Information and Computation 9, 778 (2009).
  * Note [1] The former class encompasses the majority of states in the Haar-measure sense; a state, drawn at random, will belong to the GHZ class with probability 1.
  * Acín  _et al._ [2001b] A. Acín, A. Andrianov, E. Jané, and R. Tarrach, Three-qubit pure-state canonical forms, Journal of Physics A: Mathematical and General 34, 6725 (2001b).
  * Sackett  _et al._ [2000] C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, D. J. Wineland, and C. Monroe, Experimental entanglement of four particles, Nature 404, 256 (2000).
  * Ghahi and Akhtarshenas [2016] M. Ghahi and S. Akhtarshenas, Entangled graphs: a classification of four-qubit entanglement, The European Physical Journal D 70, 10.1140/epjd/e2016-60729-1 (2016).
  * Wieczorek  _et al._ [2009b] W. Wieczorek, N. Kiesel, C. Schmid, and H. Weinfurter, Multiqubit entanglement engineering via projective measurements, Physical Review A 79, 10.1103/physreva.79.022311 (2009b).
  * Kiesel  _et al._ [2007] N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter, Experimental observation of four-photon entangled Dicke state with high fidelity, Physical Review Letters 98, 10.1103/physrevlett.98.063604 (2007).
  * Helwig  _et al._ [2012] W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.-K. Lo, Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A 86, 052335 (2012).
  * [211] F. Huber and N. Wyderka, Table of Absolutely Maximally Entangled states.
  * Scott [2004] A. J. Scott, Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions, Phys. Rev. A 69, 052330 (2004).
  * Huber  _et al._ [2017] F. Huber, O. Gühne, and J. Siewert, Absolutely maximally entangled states of seven qubits do not exist, Phys. Rev. Lett. 118, 200502 (2017).
  * Rather  _et al._ [2022] S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć, A. Lakshminarayan, and K. Życzkowski, Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem, Physical Review Letters 128, 10.1103/physrevlett.128.080507 (2022).
  * Rather  _et al._ [2023] S. A. Rather, N. Ramadas, V. Kodiyalam, and A. Lakshminarayan, Absolutely maximally entangled state equivalence and the construction of infinite quantum solutions to the problem of 36 officers of Euler, Physical Review A 108, 10.1103/physreva.108.032412 (2023).
  * Goyeneche  _et al._ [2015] D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. Życzkowski, Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices, Phys. Rev. A 92, 032316 (2015).
  * Higuchi and Sudbery [2000] A. Higuchi and A. Sudbery, How entangled can two couples get?, Physics Letters A 273, 213 (2000).
  * Facchi  _et al._ [2008] P. Facchi, G. Florio, G. Parisi, and S. Pascazio, Maximally multipartite entangled states, Physical Review A 77, 10.1103/physreva.77.060304 (2008).
  * Campbell  _et al._ [2009] S. Campbell, M. S. Tame, and M. Paternostro, Characterizing multipartite symmetric Dicke states under the effects of noise, New Journal of Physics 11, 073039 (2009).
  * Duan [2011] L.-M. Duan, Entanglement detection in the vicinity of arbitrary dicke states, Phys. Rev. Lett. 107, 180502 (2011).
  * Chen  _et al._ [2016] J.-Y. Chen, Z. Ji, N. Yu, and B. Zeng, Entanglement depth for symmetric states, Phys. Rev. A 94, 042333 (2016).
  * Demianowicz  _et al._ [2021] M. Demianowicz, G. Rajchel-Mieldzioć, and R. Augusiak, Simple sufficient condition for subspace to be completely or genuinely entangled, New Journal of Physics 23, 103016 (2021).
  * Aloy  _et al._ [2021] A. Aloy, M. Fadel, and J. Tura, The quantum marginal problem for symmetric states: applications to variational optimization, nonlocality and self-testing, New Journal of Physics 23, 033026 (2021).
  * Marconi  _et al._ [2021] C. Marconi, A. Aloy, J. Tura, and A. Sanpera, Entangled symmetric states and copositive matrices, Quantum 5, 561 (2021).
  * Tura  _et al._ [2018] J. Tura, A. Aloy, R. Quesada, M. Lewenstein, and A. Sanpera, Separability of diagonal symmetric states: a quadratic conic optimization problem, Quantum 2, 45 (2018).
  * Quesada  _et al._ [2017] R. Quesada, S. Rana, and A. Sanpera, Entanglement and nonlocality in diagonal symmetric states of $n$ qubits, Phys. Rev. A 95, 042128 (2017).
  * Burchardt  _et al._ [2021] A. Burchardt, J. Czartowski, and K. Życzkowski, Entanglement in highly symmetric multipartite quantum states, Phys. Rev. A 104, 022426 (2021).
  * Tóth [2007] G. Tóth, Detection of multipartite entanglement in the vicinity of symmetric Dicke states, J. Opt. Soc. Am. B 24, 275 (2007).
  * Bruzda  _et al._ [2023] W. Bruzda, S. Friedland, and K. Życzkowski, Rank of a tensor and quantum entanglement, Linear and Multilinear Algebra , 1 (2023).
  * Vrana and Christandl [2015] P. Vrana and M. Christandl, Asymptotic entanglement transformation between W and GHZ states, Journal of Mathematical Physics 56, 10.1063/1.4908106 (2015).