# Certification of genuine multipartite entanglement in spin ensembles with
measurements of total angular momentum

Khoi-Nguyen Huynh-Vu  College of Engineering and Computer Science,
VinUniversity, Hanoi, Vietnam  Lin Htoo Zaw  Centre for Quantum Technologies,
National University of Singapore, 3 Science Drive 2, Singapore 117543  Valerio
Scarani  Centre for Quantum Technologies, National University of Singapore, 3
Science Drive 2, Singapore 117543  Department of Physics, National University
of Singapore, 2 Science Drive 3, Singapore 117542

###### Abstract

We introduce entanglement witnesses for spin ensembles which detect genuine
multipartite entanglement using only measurements of the total angular
momentum. States that are missed by most other angular-momentum-based
witnesses for spin ensembles, which include Greenberger-Horne-Zeilinger states
and certain superpositions of Dicke states, can be effectively detected by our
witness. The protocol involves estimating the probability that the total
angular momentum is positive along equally-spaced directions on a plane.
Alternatively, one could measure along a single direction at different times,
under the assumption that the total spins undergoes a uniform precession.
Genuine multipartite entanglement is detected when the observed score exceeds
a separable bound. Exact analytical expressions for the separable bound are
obtained for spin ensembles $j_{1}\otimes j_{2}\otimes\dots\otimes j_{N}$ such
that the total spin is a half-integer, and numerical results are reported for
the other cases. Finally, we conjecture an expression for the separable bound
when the total spin is not known, which is well supported by the numerical
results.

##  I Introduction

Entanglement is an important resource that enables many quantum protocols [1,
2, 3]. For this reason, experimental certification of entanglement in real
quantum devices is extremely important for both practical and foundational
purposes. Consequently, many methods for detecting entanglement have been
proposed and studied since the early days of quantum information [4].
Entanglement witnesses can be constructed analytically or numerically [5, 6],
and it is even possible to certify entanglement without full characterization
of the measurement device [7, 8].

However, there are some challenges when implementing entanglement witnesses in
practice. Each experimental setup has measurements and operations that are
native to that system. In general, arbitrarily constructed witnesses would
require measurements in bases different from the native ones. So, some
operations on the state need to be performed prior to measurement, which
impacts the fidelity of the measurement. Alternatively, full or partial
tomography must be done to extract the expectation value of the witness from
the tomographic data, but tomography can be intractable for large systems. As
such, when taking pragmatic reasons into account, witnesses that depend only
on measurements native to the experimental setup might be more suitable in
some cases.

In this paper, we focus on the _spin ensemble_ , a collection of particles
labelled by $n$, each with a fixed spin $j_{n}$. Experimental settings in
which spin ensembles appear include ultracold atoms in optical lattices [9],
and spin defects [10] and donors [11] in solid state materials. Measurements
of angular momentum $\vec{J}^{(j_{n})}$ are natural in such systems. We take
that any component (e.g. $J_{x}$) or function of $\vec{J}$ (e.g.
$\lvert\vec{J}\rvert^{2}$) can be easily measured.

Several novel witnesses that utilize only angular momentum measurements have
been introduced for spin ensembles. A mainstay are the spin-squeezing
inequalities [12, 1, 13], which are based on variances of different angular
momentum components, and are built upon the uncertainty relations of spin
observables. Following the seminal paper by Sørensen  _et al._ [14],
generalized spin-squeezing criteria have also been proposed to detect the
entanglement of two to three spin-half particles [15, 16], approximate many-
body singlet states [17], and symmetric Dicke states [18]. Families of such
multipartite witnesses have also been characterized for spin-half particles
[19, 20], and later, for more general spin ensembles [21, 22].

Another notable approach involves the energy observable, which for spin
ensembles are some function of the angular momentum operators [23]. If the
ground state of the Hamiltonian is known to be entangled, and the measured
energy is below a certain threshold, then the system itself must be entangled.
Energy-based witnesses have been applied to various spin models, including XY-
[24, 25, 26], XYZ- [27], Heisenberg- [24], and more general spin chains [28,
29, 30].

Since these witnesses act upon spin ensembles, the witnessed correlations are
that of entanglement between many particles. Such multipartite entanglement
takes on a more complex character in comparison to bipartite entanglement.
Firstly, the entanglement might be dependent on the partition chosen: For
example,
$\propto(\ket{\uparrow\downarrow}_{1,2}-\ket{\downarrow\uparrow}_{1,2})\otimes\ket{\downarrow}_{3}$
is entangled over the $\\{1\\}{\text{-}}\\{2,3\\}$ partition, but separable
over the $\\{1,2\\}{\text{-}}\\{3\\}$ partition. As such, the stronger notion
of _genuine multipartite entanglement_ (GME) has to be introduced: The
entanglement present in a state is called GME if and only if it is inseparable
over _every possible bipartition_ of the system [31]. Secondly, there are
inequivalent types of maximally-entangled states that cannot be interconverted
using only local operations and classical communication: For three spin-half
particles, these are the W state
$\ket{\text{W}_{3}}\propto\ket{\uparrow\downarrow\downarrow}+\ket{\downarrow\uparrow\downarrow}+\ket{\downarrow\downarrow\uparrow}$
and the Greenberger-Horne-Zeilinger (GHZ) state
$\ket{\text{GHZ}_{3}}\propto\ket{\uparrow}^{\otimes
3}+\ket{\downarrow}^{\otimes 3}$ [32].

In terms of the different types of multipartite entanglement, only some of the
aforementioned angular-momentum-based witnesses can detect GME [33, 34, 25,
35, 36], and those that do are mostly effective at detecting Dicke states
[37], a generalization of W states. Notably missing are angular-momentum-based
witnesses that can detect $N$-partite GHZ states
$\ket{\text{GHZ}_{N}}\propto\ket{\uparrow}^{\otimes
N}+\ket{\downarrow}^{\otimes N}$ beyond the tripartite case. Since GHZ states
are resources for distributed computing [38], and are also useful in quantum
secret sharing protocols [39], angular-momentum-based witnesses that can
detect GHZ states are desirable.

The primary contribution of this paper is to fill this gap. We introduce a
witness of GME that requires only measurements of the total angular momentum
of the spin ensemble, and can detect states that are missed by existing
angular-momentum-based criteria—in particular, our witness can detect
$N$-partite GHZ states.

##  II The Precession Protocol

###  II.1 The Protocol as a Nonclassicality test

We first present the protocol as a nonclassicality test. It consists of many
independent rounds. Given a positive integer $K$, in each round, an integer
$k\in\\{0,1,...,K-1\\}$ is drawn with uniform probability. The total angular
momentum is then measured along the direction

$\displaystyle J_{k}$ $\displaystyle\coloneqq e^{-i(2\pi
k/K)J_{z}/\hbar}J_{x}e^{i(2\pi k/K)J_{z}/\hbar}$ $\displaystyle=\cos(2\pi
k/K)J_{x}+\sin(2\pi k/K)J_{y},$ (1)

where the $x$–$y$ plane of the measurement directions can be chosen at one’s
convenience. After many such rounds, the average probability that $J_{k}$ was
found to be positive is calculated as the score

$P_{K}\coloneqq\frac{1}{K}\sum_{k=0}^{K-1}\bqty{\Pr(J_{k}>0)+\frac{1}{2}\Pr(J_{k}=0)}.$
(2)

This protocol is based on a nonclassicality test known as the “precession
protocol” or “Tsirelson protocol” [40, 41, 42], a witness of nonclassicality
for single systems. While the “precession” in the former expression comes from
a dynamical assumption in those works that we do not use here (see Sec.
II.3.2), for the sake of a name, we shall continue to refer to this protocol
as to the precession protocol.

The upper bound $P_{K}\leq\mathbf{P}_{K}^{c}$ predicted by classical theory is
$\mathbf{P}_{K}^{c}=1/2$ for $K$ even and $\mathbf{P}_{K}^{c}=(1+1/K)/2$ for
$K$ odd [40, 41]. Meanwhile, the expected score for a quantum system in the
state $\rho$ is given by $P_{K}=\tr(\rho Q_{K})$, with

$Q_{K}\coloneqq\frac{1}{K}\sum_{k=0}^{K-1}\operatorname{pos}(J_{k}).$ (3)

Here, $\operatorname{pos}(J_{k})$ is defined on the eigenstates
$\ket{j,m}_{k}$ of $J_{k}$, such that $J_{k}\ket{j,m}_{k}=\hbar
m\ket{j,m}_{k}$ and
$2\operatorname{pos}(J_{k})\ket{j,m}_{k}=[1+\operatorname{sgn}(m)]\ket{j,m}_{k}$,
with the usual convention $\operatorname{sgn}(0)=0$. There exists quantum
states that violate the classical bound. In particular, the maximal
eigenstates of $Q_{K}$—the states that achieve the largest quantum score—are
known in some cases [41]. Therefore, upon performing the protocol, the
observation $P_{K}>\mathbf{P}_{K}^{c}$ detects the nonclassicality of the
measured system.

###  II.2 The Protocol as an Entanglement Witness

Figure 1: (a) We consider a spin ensemble of $N$ particles. Each particle has
a fixed spin $j_{n}$ with angular momentum $\vec{J}^{(j_{n})}$. (b) The
protocol is performed on the total angular momentum
$\vec{J}=\sum_{n=1}^{N}\vec{J}^{(j_{n})}$ of the spin ensemble
$\\{j_{n}\\}_{n=1}^{N}$, which is the sum of the individual angular momentum
of each particle. (c) In general, the entanglement of a spin ensemble depends
on how the ensemble is partitioned. The system might be entangled over one
bipartition, but separable over another. As such, we are interested in the
stronger notion of _genuine multipartite entanglement_ (GME): a state is GME
if it is entangled over all possible bipartitions of the system. Our
entanglement witness detects GME if the score of the precession protocol is
found to be larger than a separable bound.

We will now apply the precession protocol to the total angular momentum
$\vec{J}$ of a composite system. Concretely, let us consider a spin ensemble
consisting of $N$ particles with fixed spins $j_{1},j_{2},\dots,j_{N}$. With a
slight abuse of notation, the angular momentum along the $k$ direction of the
$n$th spin is denoted by

$J_{k}^{(j_{n})}=\mathbbm{1}^{(j_{1})}\otimes\dots\otimes\mathbbm{1}^{(j_{n-1})}\otimes
J_{x}^{(j_{n})}\otimes\mathbbm{1}^{(j_{n+1})}\otimes\dots\otimes\mathbbm{1}^{(j_{N})},$
(4)

such that $J_{k}^{(j)}$ for a fixed spin is given by

$J_{k}^{(j)}=\sum_{m=-j}^{j}\hbar m\ket{j,m}_{k}\bra{j,m},$ (5)

where $\ket{j,m}_{k}$ is a simultaneous eigenstate of
$\lvert{\vec{J}}\rvert^{2}$ and $J_{k}$ with eigenvalues $\hbar^{2}j(j+1)$ and
$\hbar m$ respectively. It is related to the usual eigenstate $\ket{j,m}$ of
$\lvert{\vec{J}}\rvert^{2}$ and $J_{z}$ by $\ket{j,m}_{k}=e^{-i(2\pi
k/K)J_{y}/\hbar}e^{-i(\pi/2)J_{y}/\hbar}\ket{j,m}$.

With this, we can consider an implementation of the precession protocol with
$J_{k}=\sum_{n=1}^{N}J_{k}^{(j_{n})}$, and ask whether a composite system
performs better in this protocol when the particles are entangled—in other
words, whether we can define a _precession-based entanglement witness (PEW)_.
Previous works suggest it to be so: for two fixed spins, the state conjectured
to be maximally-violating is always entangled [41]; for two harmonic
oscillators, violation by a linear combination of the original modes implies
entanglement of the latter in a wide parameter range [43].

Indeed, we shall show that a sufficiently large violation of the classical
bound by the spin ensemble implies GME. Our main results are: an exact
analytical bound when $\sum_{n=1}^{N}j_{n}=K/2$, for all odd $K\geq 3$ (Result
1); and a conjectured bound for all cases (Conjecture 3), with strong
numerical evidence for $\sum_{n=1}^{N}j_{n}\leq 15$ and $K\leq 21$ (Result 4).

###  II.3 Remarks on the Implementation of the Protocol

####  II.3.1 Collective measurements are possible but not necessary

Our PEW belongs to the class of angular-momentum-based entanglement witnesses.
Indeed, it can be implemented by measuring only collective observables, namely
the suitable components of the total $\vec{J}$.

That being said, collective measurements are not necessary. Because
$J_{k}=\sum_{n=1}^{N}J_{k}^{(j_{n})}$, any measurement of $J_{k}$ can be done
by first locally measuring $J_{k}^{(j_{n})}$ for each spin, or
$\sum_{n\in\\{n_{1},n_{2},\dots\\}}J_{k}^{(j_{n})}$ for a subset of spins;
then manually adding up the measurement outcomes afterwards. This feature is
shared with some other witnesses [24, 18], but is contrasted with those that
require measurements in a nonlocal basis [17].

####  II.3.2 Measuring along only one direction under the assumption of
dynamics

In the earlier works about the precession protocol as a nonclassicality test
[40, 41, 42], the protocol was presented as a measurement of a single
observable at different times, under the assumption that, during the
certification process, that observable undergoes a uniform precession.
Specifically, if $\vec{J}=(J_{x},J_{y},J_{z})$ is known to precess uniformly
about the $z$ axis with period $T\coloneqq 2\pi/\omega$ as

$\vec{J}(t)=\pmqty{undef},$ (6)

then indeed $J_{k}(0)=J_{x}(t_{k})$ for $t_{k}=(k/K)T$. In this dynamical
implementation, upon drawing $k$, one would wait till time $t_{k}$ and then
measure $J_{x}$. In some systems, having to measure only in one direction
could be an advantage, coming at the cost of the assumption on the dynamics.
Hence, this offers another way of performing the precession protocol, rather
than measuring at a fixed time along one among several directions as we have
presented in this paper.

This alternate method can be useful in witnessing the entanglement of
postquench states, which is a common problem in the study of spin ensembles
[26, 44]. Here, “ _quenching_ ” refers to a sudden change of some parameters
of the Hamiltonian. By starting with a highly-entangling Hamiltonian, then
quenching to the free Hamiltonian $H=-\omega J_{z}$, the dynamical assumption
of uniform precession holds, and the precession protocol can be performed.

This approach fits naturally with recent efforts in optimal quantum control
that seeks to engineer driven Hamiltonians that achieve target states with
GME, like the GHZ state [45, 46]. Our protocol provides a way to certify the
success of the quantum control procedure after it has been implemented.

##  III Main Results: Separable Bounds

In this section, we detail the results with which we will eventually arrive at
Results 1 & 4 and Conjecture 3.

Since we are interested in detecting GME states, we have to consider _every
possible bipartition_ of the spin ensemble. Let us therefore formalize the
notion of a bipartition of a spin ensemble $\\{j_{n}\\}_{n=1}^{N}$. When
partitioning the spin ensemble into two subsets, if the first subset is
$\mathbf{J}=\\{j_{n_{1}},j_{n_{2}},\dots,j_{n_{L}}\\}$ containing $1\leq L<N$
spins, then the second subset will contain the remaining spins
$\mathbf{J}^{\complement}:=\\{j_{n}\\}_{n=1}^{N}\setminus\mathbf{J}$.
Meanwhile, for all possible bipartitions, the first subset can be any
$\mathbf{J}\in
2^{\\{j_{n}\\}_{n=1}^{N}}\setminus\\{\varnothing,\\{j_{n}\\}_{n=1}^{N}\\}$,
where $2^{\\{j_{n}\\}_{n=1}^{N}}$ is the power set of $\\{j_{n}\\}_{n=1}^{N}$,
and we exclude $\varnothing$ and its complement to ensure that at least one
spin is present in each partition.

In addition, $\mathcal{H}^{(j_{n})}$ denotes the Hilbert space spanned by
$\\{\ket{j_{n},m}\\}_{m=-j_{n}}^{j_{n}}$. We will also loosely call both
$\sum_{m}\psi_{m}\ket{j_{n},m}$ and
$\sum_{m,m^{\prime}}\rho_{m,m^{\prime}}\outerproduct{j_{n},m}{j_{n},m^{\prime}}\succeq
0$, pure and mixed states respectively, as “states that belong in
$\mathcal{H}^{(j_{n})}$”.

With these notations, a state $\rho_{\mathbf{J},\mathbf{J}^{\complement}}$ of
a spin ensemble is separable over the
$\mathbf{J}$-$\mathbf{J}^{\complement}$-bipartition if
$\rho_{\mathbf{J},\mathbf{J}^{\complement}}=\sum_{k}p_{k}\rho_{\mathbf{J},k}\otimes\rho_{\mathbf{J}^{\complement},k}$,
where $\rho_{\mathbf{J},k}$ (respectively $\rho_{\mathbf{J}^{\complement},k}$)
is a state within the subspace $\bigotimes_{j\in\mathbf{J}}\mathcal{H}^{(j)}$
(respectively
$\bigotimes_{j^{\prime}\in\mathbf{J}^{\complement}}\mathcal{H}^{(j^{\prime})}$).
Conversely, $\rho_{\text{GME}}$ is GME if
$\rho_{\text{GME}}\neq\sum_{k}p_{k}\rho_{\mathbf{J},k}\otimes\rho_{\mathbf{J}^{\complement},k}$
for all bipartitions $\mathbf{J}$.

Finally, we define the separable bound
$\mathbf{P}_{K}^{\text{sep}}(\\{j_{n}\\}_{n=1}^{N})$ as the maximum score
achieved by a state of the spin ensemble $\\{j_{n}\\}_{n=1}^{N}$ that is
separable over _some_ bipartition, upon performing the precession protocol on
the total angular momentum of the spin ensemble:

$\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{j_{n}}_{n=1}^{N}}:=\max_{\mathbf{J}}\max_{\rho_{\mathbf{J},\mathbf{J}^{\complement}}}\tr(\rho_{\mathbf{J},\mathbf{J}^{\complement}}Q_{K}).$
(7)

Here, $Q_{K}$ is as defined in Eq. (3) with
$\vec{J}=\sum_{n=1}^{N}\vec{J}^{(j_{n})}$, while the maximizations are over
all $\mathbf{J}\in
2^{\\{j_{n}\\}_{n=1}^{N}}\setminus\\{\varnothing,\\{j_{n}\\}_{n=1}^{N}\\}$ and
$\rho_{\mathbf{J},\mathbf{J}^{\complement}}=\sum_{k}p_{k}\rho_{\mathbf{J},k}\otimes\rho_{\mathbf{J}^{\complement},k}$,
where $\mathbf{J}^{\complement}=2^{\\{j_{n}\\}_{n=1}^{N}}\setminus\mathbf{J}$.

By definition, a state separable over some bipartition must achieve the score
$P_{K}\leq\mathbf{P}_{K}^{\text{sep}}(\\{j_{n}\\}_{n=1}^{N})$. The negation of
this statement implies that if the separable bound is violated, the system is
inseparable over every bipartition. Therefore, our PEW detects GME states if
the score $P_{K}>\mathbf{P}_{K}^{\text{sep}}(\\{j_{n}\\}_{n=1}^{N})$ is
achieved when performing the precession protocol on the total angular momentum
of the spin ensemble $\\{j_{n}\\}_{n=1}^{N}$.

###  III.1 Relation Between the Multipartite and Bipartite Separable Bounds

By taking advantage of the decomposition of $Q_{K}$ in terms of the
irreducible blocks of the angular momentum observable, the separable bounds of
large spin ensembles can be related to separable bounds of two-spin systems.

For a particular bipartition $\mathbf{J}:=\\{j_{n_{l}}\\}_{l=1}^{L}$, the
usual rules for the addition of angular momentum applied to the total angular
momentum $\vec{J}=\sum_{n=1}^{N}\vec{J}^{(j_{n})}$ results in the
decomposition [47]

$\displaystyle\vec{J}$
$\displaystyle=\sum_{l=1}^{L}\vec{J}^{(j_{n_{l}})}+\sum_{l^{\prime}=L+1}^{N}\vec{J}^{(j_{n_{l^{\prime}}})}$
(8)
$\displaystyle=\bigoplus_{\tilde{\jmath}_{1}=\absolutevalue{j_{n_{1}}-j_{n_{2}}}}^{j_{n_{1}}+j_{n_{2}}}\;\bigoplus_{\tilde{\jmath}_{2}=\absolutevalue{\tilde{\jmath}_{1}-j_{n_{3}}}}^{\tilde{\jmath}_{1}+j_{n_{3}}}\dots\bigoplus_{\tilde{\jmath}=\absolutevalue{\tilde{\jmath}_{L-2}-j_{n_{L}}}}^{\tilde{\jmath}_{L-2}+j_{n_{L}}}\vec{J}^{(\tilde{\jmath})}$
$\displaystyle\qquad{}+{}\bigoplus_{\tilde{\jmath}^{\prime}_{1}=\absolutevalue{j_{n_{L+1}}-j_{n_{L+2}}}}^{j_{n_{L+1}}+j_{n_{L+2}}}\dots\bigoplus_{\tilde{\jmath}^{\prime}=\absolutevalue{\tilde{\jmath}^{\prime}_{N-L-2}-j_{n_{N}}}}^{\tilde{\jmath}_{N-L-2}+j_{n_{N}}}\vec{J}^{(\tilde{\jmath}^{\prime})},$
$\displaystyle\eqqcolon\bigoplus_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J})}\bigoplus_{\tilde{\jmath}^{\prime}\in\mathcal{J}(\mathbf{J}^{\complement})}\bqty{\vec{J}^{(\tilde{\jmath})}\otimes\mathbbm{1}^{(\tilde{\jmath}^{\prime})}+\mathbbm{1}^{(\tilde{\jmath})}\otimes\vec{J}^{(\tilde{\jmath}^{\prime})}},$

where
$\mathcal{J}(\\{j_{n_{1}},j_{n_{2}},\dots,j_{n_{L}}\\}):=\\{\tilde{\jmath}:\absolutevalue{\tilde{\jmath}_{L-2}-j_{n_{L}}}\leq\tilde{\jmath}\leq\tilde{\jmath}_{L-2}+j_{n_{L}},\absolutevalue{\tilde{\jmath}_{L-3}-j_{n_{L-1}}}\leq\tilde{\jmath}_{L-2}\leq\tilde{\jmath}_{L-3}+j_{n_{L-1}},\dots,\absolutevalue{j_{n_{1}}-j_{n_{2}}}\leq\tilde{\jmath}_{1}\leq
j_{n_{1}}+j_{n_{2}}\\}$ is the set of irreducible spins of the block
decomposition of $\sum_{l=1}^{L}\vec{J}^{(j_{n_{l}})}$, which does not depend
on the order of the elements of $\\{j_{n_{l}}\\}_{l=1}^{L}$ in its definition.
The implicit tensor product with the identity has been made visible in Eq.
(8), which makes explicit that each term in the square brackets has nonzero
support only in the
$\mathcal{H}^{(\tilde{\jmath})}\otimes\mathcal{H}^{(\tilde{\jmath}^{\prime})}$
subspace.

It follows that, for an observable $f(\vec{J})$ that is a function of the
total angular momentum,

$\displaystyle\tr[\rho_{\mathbf{J},\mathbf{J}^{\complement}}f(\vec{J})]$ (9)
$\displaystyle\quad{}={}\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}\tilde{\jmath}\in\mathcal{J}(\mathbf{J}),\\\
\tilde{\jmath}^{\prime}\in\mathcal{J}(\mathbf{J}^{\complement})\end{subarray}}\\!\\!\\!\\!\\!\tr[\rho_{\mathbf{J},\mathbf{J}^{\complement}}f\pqty{\mathbbm{1}^{(\tilde{\jmath})}\otimes\vec{J}^{(\tilde{\jmath}^{\prime})}+\vec{J}^{(\tilde{\jmath})}\otimes\mathbbm{1}^{(\tilde{\jmath}^{\prime})}}].$

In addition, using the convexity of the set
$\\{\rho_{\mathbf{J},\mathbf{J}^{\complement}}\\}$,

$\displaystyle\max_{\rho_{\mathbf{J},\mathbf{J}^{\complement}}}\tr[\rho_{\mathbf{J},\mathbf{J}^{\complement}}f(\vec{J})]=\max_{\lvert{\Psi_{\mathbf{J}},\Psi_{\mathbf{J}^{\complement}}}\rangle}\langle{\Psi_{\mathbf{J}},\Psi_{\mathbf{J}^{\complement}}}\rvert
f(\vec{J})\lvert{\Psi_{\mathbf{J}},\Psi_{\mathbf{J}^{\complement}}}\rangle,$
(10)

where the maximization is over
$\lvert{\Psi_{\mathbf{J}},\Psi_{\mathbf{J}^{\complement}}}\rangle=\lvert{\Psi_{\mathbf{J}}}\rangle\otimes\lvert{\Psi_{\mathbf{J}^{\complement}}}\rangle$,
with
$\lvert{\Psi_{\mathbf{J}}}\rangle\in\bigotimes_{j\in\mathbf{J}}\mathcal{H}^{(j)}$
and
$\lvert{\Psi_{\mathbf{J}^{\complement}}}\rangle\in\bigotimes_{j^{\prime}\in\mathbf{J}^{\complement}}\mathcal{H}^{(j^{\prime})}$,
which are pure states separable over the
$\mathbf{J}$-$\mathbf{J}^{\complement}$-bipartition.

In relation to
$\bigotimes_{j\in\mathbf{J}}\mathcal{H}^{(j)}=\bigoplus_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J})}\mathcal{H}^{(\tilde{\jmath})}$,
$\lvert{\Psi_{\mathbf{J}}}\rangle$ can be rewritten as
$\lvert{\Psi_{\mathbf{J}}}\rangle=\bigoplus_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J})}\sqrt{p_{\tilde{\jmath}}}\ket{\psi_{\tilde{\jmath}}}$
with independent parameters
$\vec{p}=(p_{\tilde{\jmath}})_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J})}$ and
$\\{\ket{\psi_{\tilde{\jmath}}}\\}_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J})}$,
where $0\leq p_{\tilde{\jmath}}\leq 1$,
$\sum_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J})}p_{\tilde{\jmath}}=1$, and
$\ket{\psi_{\tilde{\jmath}}}\in\mathcal{H}^{(\tilde{\jmath})}$.

Similarly with
$\lvert{\Psi_{\mathbf{J}^{\complement}}}\rangle=\bigoplus_{\tilde{\jmath}^{\prime}\in\mathcal{J}(\mathbf{J}^{\complement})}\sqrt{p_{\tilde{\jmath}^{\prime}}^{\prime}}\ket{\psi_{\tilde{\jmath}^{\prime}}}$,
we have

$\lvert{\Psi_{\mathbf{J}},\Psi_{\mathbf{J}^{\complement}}}\rangle=\\!\\!\\!\\!\\!\bigoplus_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J}),\tilde{\jmath}^{\prime}\in\mathcal{J}(\mathbf{J}^{\complement})}\\!\\!\\!\\!\\!\sqrt{p_{\tilde{\jmath}}p_{\tilde{\jmath}^{\prime}}^{\prime}}\ket{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}},$
(11)

where
$\ket{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}=\ket{\psi_{\tilde{\jmath}}}\otimes\ket{\psi_{\tilde{\jmath}^{\prime}}}$.
Therefore,

$\displaystyle\langle{\Psi_{\mathbf{J}},\Psi_{\mathbf{J}^{\complement}}}\rvert
f\pqty{\mathbbm{1}^{(\tilde{\jmath})}\otimes\vec{J}^{(\tilde{\jmath}^{\prime})}+\vec{J}^{(\tilde{\jmath})}\otimes\mathbbm{1}^{(\tilde{\jmath}^{\prime})}}\lvert{\Psi_{\mathbf{J}},\Psi_{\mathbf{J}^{\complement}}}\rangle$
(12)
$\displaystyle\;{}={}p_{\tilde{\jmath}}p_{\tilde{\jmath}^{\prime}}^{\prime}\bra{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}f\pqty{\mathbbm{1}^{(\tilde{\jmath})}\otimes\vec{J}^{(\tilde{\jmath}^{\prime})}+\vec{J}^{(\tilde{\jmath})}\otimes\mathbbm{1}^{(\tilde{\jmath}^{\prime})}}\ket{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}.$

This parameterization of
$\lvert{\Psi_{\mathbf{J}},\Psi_{\mathbf{J}^{\complement}}}\rangle$ in terms of
$\vec{p}$, $\vec{p}^{\prime}$,
$\\{\ket{\psi_{\tilde{\jmath}}}\\}_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J})}$,
and
$\\{\ket{\psi_{\tilde{\jmath}^{\prime}}}\\}_{\tilde{\jmath}^{\prime}\in\mathcal{J}(\mathbf{J}^{\complement})}$
simplifies Eq. (10) to

$\displaystyle\max_{\rho_{\mathbf{J},\mathbf{J}^{\complement}}}\tr[\rho_{\mathbf{J},\mathbf{J}^{\complement}}f(\vec{J})]$
$\displaystyle=\max_{\vec{p},\vec{p}^{\prime}}\sum_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J}),\tilde{\jmath}^{\prime}\in\mathcal{J}(\mathbf{J}^{\complement})}p_{\tilde{\jmath}}p_{\tilde{\jmath}^{\prime}}^{\prime}\bqty{\max_{\ket{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}}\bra{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}f\pqty{\mathbbm{1}^{(\tilde{\jmath})}\otimes\vec{J}^{(\tilde{\jmath}^{\prime})}+\vec{J}^{(\tilde{\jmath})}\otimes\mathbbm{1}^{(\tilde{\jmath}^{\prime})}}\ket{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}}$
(13)
$\displaystyle=\max_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J}),\tilde{\jmath}^{\prime}\in\mathcal{J}(\mathbf{J}^{\complement})}\bqty{\max_{\ket{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}}\bra{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}f\pqty{\mathbbm{1}^{(\tilde{\jmath})}\otimes\vec{J}^{(\tilde{\jmath}^{\prime})}+\vec{J}^{(\tilde{\jmath})}\otimes\mathbbm{1}^{(\tilde{\jmath}^{\prime})}}\ket{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}}$
$\displaystyle\therefore\max_{\rho_{\mathbf{J},\mathbf{J}^{\complement}}}\tr[\rho_{\mathbf{J},\mathbf{J}^{\complement}}\;f\pqty{\sum_{j\in\mathbf{J}\cup\mathbf{J}^{\complement}}\vec{J}^{(j)}}]$
$\displaystyle=\max_{\tilde{\jmath}\in\mathcal{J}(\mathbf{J}),\tilde{\jmath}^{\prime}\in\mathcal{J}(\mathbf{J}^{\complement})}\Bqty{\max_{\rho_{\\{\tilde{\jmath}\\},\\{\tilde{\jmath}^{\prime}\\}}}\tr[\rho_{\\{\tilde{\jmath}\\},\\{\tilde{\jmath}^{\prime}\\}}\;f\pqty{\sum_{j\in\\{\tilde{\jmath}\\}\cup\\{\tilde{\jmath}^{\prime}\\}}\vec{J}^{(\tilde{\jmath})}}]},$

where the second line follows from the maximization of the convex sum in the
first, and the last line similarly follows from the convexity of separable
states $\rho_{\\{\tilde{\jmath}\\},\\{\tilde{\jmath}^{\prime}\\}}$ in the
subspace
$\mathcal{H}^{(\tilde{\jmath})}\otimes\mathcal{H}^{(\tilde{\jmath}^{\prime})}$.

Notably, each term in the braces in Eq. (13) is a maximization over separable
states on the subspace of the tensor product of just the two fixed spins
$\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$, and the observable is itself
defined within that subspace. With $f(\vec{J})=Q_{K}$ in Eq. (13) substituted
into Eq. (7), the GME and bipartite separable bounds can be related:

$\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{j_{n}}_{n=1}^{N}}=\max_{\mathbf{J}}\max_{\begin{subarray}{c}\tilde{\jmath}\in\mathcal{J}(\mathbf{J}),\\\
\tilde{\jmath}^{\prime}\in\mathcal{J}(\mathbf{J}^{\complement})\end{subarray}}\mathbf{P}_{K}^{\text{sep}}\pqty{\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}},$
(14)

with $\mathbf{J}\in
2^{\\{j_{n}\\}_{n=1}^{N}}\setminus\\{\varnothing,\\{j_{n}\\}_{n=1}^{N}\\}$.

To evaluate $\mathbf{P}_{K}^{\text{sep}}({\\{j_{n}\\}_{n=1}^{N}})$, one first
identifies the spins $\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$ that
appear in Eq. (14), evaluate
$\mathbf{P}_{k}^{\text{sep}}\pqty{\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}}$
for each pair, and pick out the maximum value. This simplifies the process of
calculating the multipartite separable bound, as it reduces it to the
calculation of several bipartite separable bounds.

A possible hassle might come from iterating over the possible subsets
$\mathbf{J}$, or from the tedium of working out the set of irreducible spins
$\mathcal{J}(\mathbf{J})$. Closed-form expressions of $\mathbf{J}$ and
$\mathcal{J}(\mathbf{J})$ are known, and can be worked out algorithmically
with a computer [48]. Alternatively, from the necessity that
$\tilde{\jmath},\tilde{\jmath}^{\prime}$ must be nonnegative, and that
$\tilde{\jmath}+\tilde{\jmath}^{\prime}\leq\sum_{n=1}^{N}j_{n}$ since they
arise from the block decomposition of $\sum_{n=1}^{N}\vec{J}^{(j_{n})}$, a
convenient upper bound is

$\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{j_{n}}_{n=1}^{N}}\leq\max_{\begin{subarray}{c}0\leq\tilde{\jmath}+\tilde{\jmath}^{\prime}\leq\sum_{n=1}^{N}j_{n},\\\
\tilde{\jmath},\tilde{\jmath}^{\prime}\in\frac{1}{2}\mathbb{Z}^{+}_{0}\end{subarray}}\mathbf{P}_{K}^{\text{sep}}\pqty{\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}},$
(15)

where $\frac{1}{2}\mathbb{Z}^{+}_{0}$ is the set of positive integers and
half-integers.

###  III.2 Trivial Separable Bounds for $\sum_{n=1}^{N}j_{n}<K/2$

For the special case $\sum_{n=1}^{N}j_{n}<K/2$, the separable bound can be
calculated directly. By performing a similar block decomposition to Eq. (8),
the total angular momentum of the spin ensemble is

$\vec{J}=\sum_{n=1}^{N}\vec{J}^{(j_{n})}=\bigoplus_{j\in\mathcal{J}\pqty{\\{j_{n}\\}_{n=1}^{N}}}^{\sum_{n=1}^{N}j_{n}}\vec{J}^{(j)}.$
(16)

As before, the observable $Q_{K}$, which describes the precession protocol
performed on the total angular momentum, will also inherit this block diagonal
structure:

$Q_{K}=\bigoplus_{j\in\mathcal{J}\pqty{\\{j_{n}\\}_{n=1}^{N}}}^{\sum_{n=1}^{N}j_{n}}\underbrace{\frac{1}{K}\sum_{k=0}^{K-1}\operatorname{pos}[J_{k}^{(j)}]}_{\eqqcolon
Q_{K}^{(j)}}.$ (17)

The eigendecomposition of $Q_{K}^{(j)}$ is known for some values of $j$ [41].
In particular,

$Q_{K}^{(j<K/2)}=\frac{1}{2}\mathbbm{1}^{(j)}.$ (18)

Since it must be that $j\leq\sum_{n=1}^{N}j_{n}$ for every
$j\in\mathcal{J}({\\{j_{n}\\}_{n=1}^{N}})$,

$Q_{K}=\bigoplus_{j\in\mathcal{J}\pqty{\\{j_{n}\\}_{n=1}^{N}}}^{\sum_{n=1}^{N}j_{n}<K/2}\frac{1}{2}\mathbbm{1}^{(j)}=\frac{1}{2}\mathbbm{1}.$
(19)

Hence, for a spin ensemble such that $\sum_{n=1}^{N}j_{n}<K/2$, $P_{K}=1/2$
for _every_ state of the system. This also trivially gives
$\mathbf{P}_{K}^{\text{sep}}(\\{j_{n}\\}_{n=1}^{N})=1/2$, but since there are
no states that can violate this separable bound, performing the precession
protocol in this case does not reveal anything about the entanglement of the
system.

###  III.3 Separable Bounds for $\sum_{n=1}^{N}j_{n}=K/2$

For $\sum_{n=1}^{N}j_{n}=K/2$, the upper bound in Eq. (15) requires the values
of $\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
for every $\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$ such that
$\tilde{\jmath}+\tilde{\jmath}^{\prime}\leq K/2$. Since the separable bounds
for $\tilde{\jmath}+\tilde{\jmath}^{\prime}<K/2$ have already been evaluated
in the previous section, we would only need to additionally work out
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
for $\tilde{\jmath}+\tilde{\jmath}^{\prime}=K/2$.

Hence, it is adequate to first restrict ourselves to the
$\mathcal{H}^{(\tilde{\jmath})}\otimes\mathcal{H}^{(\tilde{\jmath}^{\prime})}$
subspace. A block decomposition similar to the one in Eq. (17) gives

$Q_{K}=\\!\\!\\!\\!\\!\bigoplus_{j=\absolutevalue{\tilde{\jmath}-\tilde{\jmath}^{\prime}}}^{\tilde{\jmath}+\tilde{\jmath}^{\prime}=K/2}\\!\\!\\!\\!\\!Q_{K}^{(j)}=\frac{1}{2}\pqty{\mathbbm{1}-\mathbbm{1}^{(j=K/2)}}\oplus
Q_{K}^{(j=K/2)},$ (20)

where we have used $Q_{K}^{(j<K/2)}=\mathbbm{1}^{(j<K/2)}/2$. On the other
hand, the eigendecomposition of $Q_{K}^{(j=K/2)}$ is [41]

$\displaystyle Q_{K}^{(j=K/2)}$
$\displaystyle=\frac{1}{2}\mathbbm{1}^{(j=K/2)}+\frac{2^{-(K-1)}}{2}\pmqty{undef}$
(21)
$\displaystyle\qquad{}\times{}\pqty\Big{\outerproduct{\mathbf{P}_{+K}}{\mathbf{P}_{+K}}-\outerproduct{\mathbf{P}_{-K}}{\mathbf{P}_{-K}}},$

where $\ket{\mathbf{P}_{\pm
K}}=(\ket{\frac{K}{2},\frac{K}{2}}\pm(-1)^{(K-1)/2}\ket{\frac{K}{2},-\frac{K}{2}})/\sqrt{2}$.
Placing this back into Eq. (20),

$\displaystyle Q_{K}$
$\displaystyle=\frac{1}{2}\mathbbm{1}+\frac{2^{-(K-1)}}{2}\pmqty{undef}$ (22)
$\displaystyle\qquad{}\times{}\pqty\Big{\outerproduct{\mathbf{P}_{+K}}{\mathbf{P}_{+K}}-\outerproduct{\mathbf{P}_{-K}}{\mathbf{P}_{-K}}}.$

Meanwhile, observing that the only way for
$-\tilde{\jmath}\leq\tilde{m}\leq\tilde{\jmath}$ and
$-\tilde{\jmath}^{\prime}\leq\tilde{m}^{\prime}\leq\tilde{\jmath}^{\prime}$ to
satisfy $\tilde{m}+\tilde{m}^{\prime}=\pm K/2$ is to have
$\tilde{m}=\pm\tilde{\jmath}$ and
$\tilde{m}^{\prime}=\pm\tilde{\jmath}^{\prime}$, $\ket{\mathbf{P}_{\pm K}}$ is
expressed in the
$\\{\ket{\tilde{\jmath},\tilde{m}}\otimes\ket{\tilde{\jmath}^{\prime},\tilde{m}^{\prime}}\\}_{\tilde{m},\tilde{m}^{\prime}=-\tilde{\jmath},-\tilde{\jmath}^{\prime}}^{\tilde{\jmath},\tilde{\jmath}^{\prime}}$
basis as

$\displaystyle\ket{\mathbf{P}_{\pm K}}=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{(}\ket{\tilde{\jmath},\tilde{\jmath}}\otimes\ket{\tilde{\jmath}^{\prime},\tilde{\jmath}^{\prime}}$
(23)
$\displaystyle\quad\qquad{}\pm{}(-1)^{(K-1)/2}\ket{\tilde{\jmath},-\tilde{\jmath}}\otimes\ket{\tilde{\jmath}^{\prime},-\tilde{\jmath}^{\prime}}\Big{)}.$

With this, we are now able to evaluate

$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})=\max_{\\!\lvert{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}\rangle\\!}\langle{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}\rvert
Q_{K}\lvert{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}\rangle,$
(24)

where
$\lvert{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}\rangle=\lvert{\psi_{\tilde{\jmath}}}\rangle\otimes\lvert{\psi_{\tilde{\jmath}^{\prime}}}\rangle$
with $\lvert\psi_{\tilde{\jmath}}\rangle\in\mathcal{H}^{(\tilde{\jmath})}$ and
$\lvert\psi_{\tilde{\jmath}^{\prime}}\rangle\in\mathcal{H}^{(\tilde{\jmath}^{\prime})}$.
Let us first parameterize the latter state as
$\lvert{\psi_{\tilde{\jmath}^{\prime}}}\rangle=\sum_{\tilde{m}^{\prime}=-\tilde{\jmath}^{\prime}}^{\tilde{\jmath}^{\prime}}c_{\tilde{m}^{\prime}}\ket{\tilde{\jmath}^{\prime},\tilde{m}^{\prime}}$.
Then,

$\displaystyle\pqty{\mathbbm{1}^{(\tilde{\jmath})}\otimes\langle{\psi_{\tilde{\jmath}^{\prime}}}\rvert}Q_{K}\pqty{\mathbbm{1}^{(\tilde{\jmath})}\otimes\lvert{\psi_{\tilde{\jmath}^{\prime}}}\rangle}$
(25)
$\displaystyle\quad\qquad{}\widehat{=}{}\frac{1}{2}\mathbbm{1}^{(\tilde{\jmath})}+(-1)^{\frac{K-1}{2}}\frac{2^{-(K-1)}}{4}\pmqty{undef}$
$\displaystyle\quad\qquad\qquad{}\times{}\pmqty{undef},$

where the operator is represented as a matrix in the
$\\{\ket{\tilde{\jmath},\tilde{m}}\\}_{\tilde{m}=-\tilde{\jmath}}^{\tilde{\jmath}}$
basis. Since
$\max_{\lvert{\psi_{\tilde{\jmath}}}\rangle}\langle{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}\rvert
Q_{K}\lvert{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}\rangle$ _for
a fixed_ $\lvert{\psi_{\tilde{\jmath}^{\prime}}}\rangle$ is the largest
eigenvalue of the operator in Eq. (25), Eq. (24) can be evaluated by first
diagonalizing Eq. (25) using standard analytical methods, then maximizing over
$\lvert{\psi_{\tilde{\jmath}^{\prime}}}\rangle$. Therefore,

$\displaystyle\mathbf{P}_{K}^{K\text{-sep}}:$
$\displaystyle=\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})\;\,\text{for
$\tilde{\jmath}+\tilde{\jmath}^{\prime}=K/2$}$ (26)
$\displaystyle=\max_{\lvert{\psi_{\tilde{\jmath}^{\prime}}}\rangle}\pqty{\max_{\lvert{\psi_{\tilde{\jmath}}}\rangle}\langle{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}\rvert
Q_{K}\lvert{\psi_{\tilde{\jmath}},\psi_{\tilde{\jmath}^{\prime}}}\rangle}$
$\displaystyle=\\!\\!\max_{\\{c_{\tilde{m}^{\prime}}\\}_{\tilde{m}^{\prime}}}\\!\\!\bqty{\tfrac{1}{2}+\tfrac{2^{-(K-1)}}{4}\spmqty{undef}\pqty{\absolutevalue{c_{-\tilde{\jmath}^{\prime}}}^{2}+\absolutevalue{c_{+\tilde{\jmath}^{\prime}}}^{2}}}$
$\displaystyle=\frac{1}{2}\bqty{1+2^{-K}\pmqty{undef}}.$

Turning our attention back to the spin ensemble $\\{j_{n}\\}_{n=1}^{N}$ such
that $\sum_{n=1}^{N}j_{n}=K/2$, Eq. (15) gives

$\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{j_{n}}_{n=1}^{N}}\leq\max\Bqty{1/2,\mathbf{P}_{K}^{K\text{-sep}}}=\mathbf{P}_{K}^{K\text{-sep}},$
(27)

where $1/2$ appears in the maximization from the contribution of
$\mathbf{P}_{K}^{\text{sep}}(\tilde{\jmath},\tilde{\jmath}^{\prime})$ for the
case $\tilde{\jmath}+\tilde{\jmath}^{\prime}<K/2$, while
$\mathbf{P}_{K}^{K\text{-sep}}$ covers the
$\tilde{\jmath}+\tilde{\jmath}^{\prime}=K/2$ case.

From Eq. (27), Result 1 follows immediately.

######  Result 1.

Consider an $N$-partite spin ensemble $\\{j_{n}\\}_{n=1}^{N}$ such that
$\sum_{n=1}^{N}j_{n}=K/2$ for odd $K\geq 3$. Perform the precession protocol
with $K$ measurements on the total angular momentum of the system. If the
score $P_{K}>\mathbf{P}_{K}^{K\text{-sep}}$ is obtained, where
$\mathbf{P}_{K}^{K\text{-sep}}=\frac{1}{2}\bqty{1+2^{-K}\spmqty{undef}},$ (28)
the spin ensemble is GME.

Note that $\mathbf{P}_{K\leq 5}^{K\text{-sep}}<\mathbf{P}_{K}^{c}$ and
$\mathbf{P}_{K>5}^{K\text{-sep}}>\mathbf{P}_{K}^{c}$. That is, violating the
classical bound is sufficient for detecting entanglement when
$\sum_{n=1}^{N}j_{n}\leq 5/2$, while a larger violation is required when
$\sum_{n=1}^{N}j_{n}>5/2$. Regardless, the violation only needs to be at most
$\sim 3.9\%$ larger than the classical bound in the latter case: a loose upper
bound is $\mathbf{P}_{K}^{K\text{-sep}}<1.0391\mathbf{P}_{K}^{c}$ for all $K$.

###  III.4 Separable Bounds for General $\sum_{n=1}^{N}j_{n}$, Fixed $K$

For spin ensembles with general $\sum_{n=1}^{N}j_{n}$, the values of
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
for $\tilde{\jmath}+\tilde{\jmath}^{\prime}>K/2$ also comes into play in Eq.
(14). However, these bipartite separable bounds cannot be solved analytically,
and we have to instead rely on numerical methods.

Two numerical methods were used to evaluate the bipartite separable bounds:
the first is a variant of separability power iteration, which provides lower
bounds [6]; the second is semidefinite programming, whose global convergence
is guaranteed, which provides conservative upper bounds [49]. Implementation
of the numerical methods are detailed in Appendix D. By using both techniques,
we are able to ascertain that the true value $\mathbf{P}_{K}^{\text{sep}*}$
falls within a range
$\mathbf{P}_{K}^{\text{sep}}-\delta\mathbf{P}_{K}^{\text{sep}}\leq\mathbf{P}_{K}^{\text{sep}*}\leq\mathbf{P}_{K}^{\text{sep}}+\delta\mathbf{P}_{K}^{\text{sep}}$.
The numerical errors $\delta\mathbf{P}_{K}^{\text{sep}}$ are plotted in
Appendix A.

####  III.4.1 Numerical results for general
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}$, fixed $K$

Figure 2: Heatmap of
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
against $\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$, for (a) $K=3$ and (b)
$K=5$. Note that the values
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})=1/2$
for $\tilde{\jmath}+\tilde{\jmath}^{\prime}<K/2$ are neither plotted here nor
for the subsequent figures. The separable bounds are large when
$\min(\tilde{\jmath},\tilde{\jmath}^{\prime})$ is small, and
$\mathbf{P}_{K}^{\text{sep}}$ decreases as
$\min(\tilde{\jmath},\tilde{\jmath}^{\prime})$ increases. The maximum value of
$\mathbf{P}_{K}^{\text{sep}}$ in these cases occur at
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1/2,(K+1)/2\\}$. The colored
arrows mark out the direction along which the line cuts in Fig. 4 are taken.

The plots of
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
against $\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$ are shown in Fig. 2 for
$K=3$ and $K=5$. Notice that the separable bound is large for
$\min(\tilde{\jmath},\tilde{\jmath}^{\prime})<2$, and takes on smaller values
as $\min(\tilde{\jmath},\tilde{\jmath}^{\prime})$ increases.

This behavior—that the separable bound is large when $\min(\tilde{\jmath}$,
$\tilde{\jmath}^{\prime})$ is small—can be found not just for $K=7$ and $K=9$,
as plotted in Fig. 3, but up to $K=21$, as plotted in Fig. A1. From the
numerical results, we can find the values of
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}$ where the maximum occurs. As
the gap between the largest and second largest value is larger than the
numerical error, we can be certain that the largest computed value is indeed
the maximum, leading us to the following result:

######  Result 2.

The maximum separable bound over
$0\leq\tilde{\jmath},\tilde{\jmath}^{\prime}\leq 15$ for fixed $3\leq K\leq
21$ occurs at

$\underset{\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}}{\operatorname{argmax}}\,\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})=\begin{cases}\Bqty{\frac{1}{2},\frac{K+1}{2}}&\text{for
$K\leq 7$,}\\\\[4.30554pt] \Bqty{1,\frac{K}{2}}&\text{for $K\geq
7$,}\end{cases}$ (29)

where $\operatorname{argmax}_{x}f(x)=\mathbf{x}$ such that
$f(\mathbf{x})=\max_{x}f(x)$.

Equation (29) is multivalued for $K=7$ because, up to numerical precision,
$\mathbf{P}_{7}^{\text{sep}}(\\{1/2,4\\})=\mathbf{P}_{7}^{\text{sep}}(\\{1,7/2\\})$.

The numerical results for various values of $K$, especially when placed side-
by-side as in Fig. A1, strongly suggest that these behaviors should hold for
any value of $K$. As such, for a given $K$, it is worth identifying some
trends about the values of $\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}$ at
which the maximum separable bound occurs.

Figure 3: Heatmap of
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
against $\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$, for (a) $K=7$ and (b)
$K=9$. Similarly to Fig. 2, the separable bounds are large for small values of
$\min(\tilde{\jmath},\tilde{\jmath}^{\prime})$. For $K=7$,
$\mathbf{P}_{K}^{\text{sep}}$ is maximal at both
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1/2,(K+1)/2\\}$ and
$\\{1,K/2\\}$. For $K=9$, the maximum value occurs at
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1,K/2\\}$. The colored arrows
mark out the line cuts plotted in Fig. 5.

For $K=3$, $5$, and $7$, the maximum values of $\mathbf{P}_{K}^{\text{sep}}$
that appear in Fig. 2 and Fig. 3(a) occur at
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1/2,(K+1)/2\\}$. For a closer
inspection, line cuts taken along $\tilde{\jmath}^{\prime}<2$ are shown in
Figs. 4 & 5. We find that for large $\tilde{\jmath}$, the values of
$\mathbf{P}_{K}^{\text{sep}}$ cluster around a range of values that are
strictly smaller than the previously-identified maximum score. As such, we can
be reasonably sure that the maximum of $\mathbf{P}_{K}^{\text{sep}}$ over
$0\leq\tilde{\jmath},\tilde{\jmath}^{\prime}\leq\infty$ occurs at
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1/2,(K+1)/2\\}$ for $K=3$,
$5$, and $7$.

Figure 4: Line cuts from Fig. 2 of (a)
$\mathbf{P}_{3}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
and (b)
$\mathbf{P}_{5}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
along fixed values of $\tilde{\jmath}^{\prime}$, plotted against a larger
range of $\tilde{\jmath}$. For this and subsequent line cuts, separable bounds
for $\tilde{\jmath}<\tilde{\jmath}^{\prime}$ are not shown; they can be
obtained by swapping $\tilde{\jmath}\leftrightarrow\tilde{\jmath}^{\prime}$.
Note also that error bars
$\pm\delta\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
have been drawn for every point, but most are too small to be visible. As
$\tilde{\jmath}$ increases, $\mathbf{P}_{K}^{\text{sep}}$ clusters around a
range of values that are strictly smaller than
$\mathbf{P}_{K}^{\text{sep}}(\\{1/2,(K+1)/2\\})$. Due to this converging
behavior for large $\tilde{\jmath}$, we conjecture that the maximum value of
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
over $0\leq\tilde{\jmath},\tilde{\jmath}^{\prime}\leq\infty$ occurs at
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1/2,(K+1)/2\\}$ for $K=3$ and
$K=5$. Figure 5: Line cuts from Fig. 3(a) of
$\mathbf{P}_{7}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
along fixed values of $\tilde{\jmath}^{\prime}$. Error bars
$\pm\delta\mathbf{P}_{7}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
have been drawn for every point, but most are too small to be visible. Up to
numerical precision,
$\mathbf{P}_{7}^{\text{sep}}(\\{1/2,4\\})=\mathbf{P}_{7}^{\text{sep}}(\\{1,7/2\\})$,
and $\mathbf{P}_{7}^{\text{sep}}$ clusters around a range strictly smaller
than
$\mathbf{P}_{7}^{\text{sep}}(\\{1/2,(K+1)/2\\})=\mathbf{P}_{7}^{\text{sep}}(\\{1,K/2\\})$.
We conjecture that the
$\mathbf{P}_{7}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$ is
maximal at both
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1/2,(K+1)/2\\}$ and
$\\{1,K/2\\}$.

Meanwhile, for $7\leq K\leq 21$, Figs. 3 & A1 clearly show that the maximums
occur at $\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1,K/2\\}$, and the
separable bound decreases sharply as $\tilde{\jmath}$ and
$\tilde{\jmath}^{\prime}$ increases. We conjecture that this pattern also
holds for larger values of $K$, and in conjunction with the previous
conjecture, we have the following:

######  Conjecture 1.

The maximum separable bound over
$0\leq\tilde{\jmath},\tilde{\jmath}^{\prime}\leq\infty$ for fixed $K$ occurs
at

$\underset{\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}}{\operatorname{argmax}}\,\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})=\begin{cases}\Bqty{\frac{1}{2},\frac{K+1}{2}}&\text{for
$K\leq 7$,}\\\\[4.30554pt] \Bqty{1,\frac{K}{2}}&\text{for $K\geq
7$.}\end{cases}$ (30)

Having found the probable spins at which the maximum occurs, we shall now
attempt to find the maximum separable bounds themselves. For
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1/2,(K+1)/2\\}$, the
separable bound can be exactly solved to be

$\displaystyle\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{\tfrac{1}{2},\tfrac{K+1}{2}}}$
(31) $\displaystyle\quad{}={}\begin{cases}\frac{23}{32}\qquad\text{if $K=3$,
otherwise}\\\\[4.30554pt]
\frac{1}{2}\bqty{1+\frac{2^{-(K+1)}}{K+1}\spmqty{undef}\pqty{\scriptstyle
K+\sqrt{3K^{2}+18K+16}}}.\end{cases}$

The derivation of Eq. (31) is given in Appendix C.

We were unable to apply the same analytical methods to
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1,K/2\\}$, but we have a
guess for the expression of the separable bound in that case.

######  Conjecture 2.

The separable bound for
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1,K/2\\}$ with $K\geq 7$ is

$\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{1,\tfrac{K}{2}}}=\frac{1}{2}\bqty{1+2^{-(K-1)}\binom{K-2}{\frac{K-1}{2}}\frac{K-1}{K+1}}.$
(32)

Note that $\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{1,\tfrac{K}{2}}}=1/2$ for
$K=3$ and $5$, hence their exclusion from the conjecture. While we were unable
to prove Eq. (32), it is in excellent agreement with the numerics up to
$K=101$, as shown in Fig. 6. Furthermore, Eqs. (31) & (32) are equal for
$K=7$, which is consistent with the two maximums in Fig. 5.

Figure 6: The conjectured and numerically-calculated separable bounds for
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1,K/2\\}$. The numerical
error $\delta\mathbf{P}_{K}^{\text{sep}}<10^{-11}$ is too small to be plotted.
This is in excellent agreement with Eq. (32) for all plotted values of $7\leq
K\leq 101$.

Additionally, by taking the numerical precision into account, Eq. (32) can be
turned into a reliable upper bound for Fig. 6.

######  Result 3.

The separable bound for
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1,K/2\\}$ with $7\leq K\leq
101$ is upper bounded by
$\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{1,\tfrac{K}{2}}}+\delta\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{1,\tfrac{K}{2}}}$,
where $\delta\mathbf{P}_{K}^{\text{sep}}<10^{-11}$ and

$\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{1,\tfrac{K}{2}}}=\frac{1}{2}\bqty{1+2^{-(K-1)}\binom{K-2}{\frac{K-1}{2}}\frac{K-1}{K+1}}.$
(33)

We emphasize that Result 3, while numerical, is a definite result, as the
global convergence of semidefinite programs guarantees that the computed
values are reliable upper bounds.

####  III.4.2 Results and conjectures on separable bounds for general
$\sum_{n=1}^{N}j_{n}$, fixed $K$

Finally, we can assemble the results and conjectures from the preceding
sections to find a separable bound for a spin ensemble $\\{j_{n}\\}_{n=1}^{N}$
with general $\sum_{n=1}^{N}j_{n}$ and fixed $K$. Using the upper bounds on
the bipartite separable bounds from Results 2 & 3, an upper bound on the
multipartite separable bound can be found with Eq. (15).

######  Result 4.

Consider an $N$-partite spin ensemble $\\{j_{n}\\}_{n=1}^{N}$ such that
$\sum_{n=1}^{N}j_{n}\leq 15$. Perform the precession protocol with odd $3\leq
K\leq 21$ on the total angular momentum of the system. If the score
$P_{K}>\mathbf{P}_{K}^{\text{{conj}}}+10^{-11}$ is obtained, where
$\mathbf{P}_{K}^{\text{{conj}}}:=\begin{cases}\frac{23}{32}&\text{if
$K=3$,}\\\\[4.30554pt] \frac{69+\sqrt{181}}{128}&\text{if
$K=5$,}\\\\[4.30554pt]
\frac{1}{2}\bqty{1+2^{-(K-1)}\spmqty{undef}\frac{K-1}{K+1}}&\text{otherwise,}\end{cases}$
the spin ensemble is GME. Meanwhile, when the makeup of the spin ensemble is
completely unknown, the only possible bound for the total spin is
$\sum_{n=1}^{N}j_{n}\leq\infty$, and the maximization in Eq. (15) would be
over all possible $0\leq\tilde{\jmath},\tilde{\jmath}^{\prime}\leq\infty$.
While there are no analytical results in this case, Conjectures 1 & 2 can
likewise be extended into a conjecture about the multipartite separable bound.

######  Conjecture 3.

Consider a spin ensemble. Perform the precession protocol with odd $K\geq 3$
on the total angular momentum of the system. If the score
$P_{K}>\mathbf{P}_{K}^{\text{{conj}}}$ is obtained, then the spin ensemble is
GME.

##  IV Comparison to Other Protocols

###  IV.1 Precession protocol as a quantum circuit

Figure 7: The detection of GHZ entanglement as a quantum circuit, for (a) the
precession protocol and (b) the fidelity-based witness. Here, $\theta_{k}=2\pi
k/K$, $R_{z}(\theta)=\exp(-i\theta\sigma_{z}/2)$ is a $z$-rotation gate,
$H=(\sigma_{x}+\sigma_{z})/\sqrt{2}$ is the Hadamard gate, and $z_{n}$ is the
outcome of $\sigma_{z}$ on the $n$th qubit. For both witnesses, a measurement
is performed for each $\theta_{k}$ with $k\in\\{1,2,\dots,K\\}$, a collective
quantity is calculated based on the outcomes $\\{z_{n}\\}_{n=1}^{K}$, and
entanglement is detected when the average of this quantity over $k$ violates
some separable bound. Our PEW requires $K$ measurement settings while the
fidelity-based witness requires $K+1$ settings, and as such the former does
not provide a significant improvement over the latter. Therefore, one might be
better off using the fidelity-based witness in the context of quantum
computation.

As the precession protocol can be performed with local spin measurements and
classical post-processing, our PEW can also be implemented as a quantum
circuit as shown in Fig. 7(a). In this context, it is interesting to point out
the similarities with the fidelity-based witness that is commonly used in the
field of quantum computation [50], as is shown in Fig. 7(b):

(1) Both witnesses involve $K$ equally-distributed measurement settings:
$\sigma_{\theta_{k}}$ for our PEW and $\sigma_{\theta_{k}/2}$ for the
fidelity-based witness, where
$\sigma_{\theta}=\cos\theta\sigma_{x}+\sin\theta\sigma_{y}$,
$\sigma_{s}=J_{s}^{(j=1/2)}/2\hbar$ are the usual Pauli operators, and
$\theta_{k}=2\pi k/K$ for $k\in\\{1,2,\dots,K\\}$.

(2) For each measurement setting $k$, a collective quantity is calculated from
$\\{z_{n}\\}_{n=1}^{K}$, where $z_{n}$ is the measurement outcome of
$\sigma_{\theta}$ for the $n$th qubit. This is the majority
$\Delta_{k}\coloneqq\operatorname{maj}(\\{z_{n}\\}_{n=1}^{K})$ for our PEW,
where $\operatorname{maj}(\mathbf{S})$ returns the most commonly-occurring
value of the set $\mathbf{S}$, and the parity
$\tilde{\Delta}_{k}\coloneqq(-1)^{k}\prod_{n=1}^{N}z_{n}$ for the fidelity-
based witness.

(3) Finally, entanglement is detected in both witnesses when the average of
the collective quantity over all measurement settings violates some separable
bound. This is
$(1+\frac{1}{K}\sum_{k=1}^{K}\Delta_{k})/2>\mathbf{P}_{K}^{K\text{-sep}}$ for
our PEW, and
$(1-\frac{1}{K}\sum_{k=1}^{K}\tilde{\Delta}_{k})/2<\tilde{\Delta}_{0}/2$ for
the fidelity-based witness, which requires an additional quantity
$\tilde{\Delta}_{0}\coloneqq\langle(\lvert{\frac{K}{2},\frac{K}{2}}\rangle\\!\langle{\frac{K}{2},\frac{K}{2}}\rvert+\lvert{\frac{K}{2},-\frac{K}{2}}\rangle\\!\langle{\frac{K}{2},-\frac{K}{2}}\rvert)\rangle$
that can be computed by measuring all qubits in the $\sigma_{z}$ basis.

As such, our PEW, which requires measuring in $K$ different local bases, does
not offer a significant improvement over the fidelity-based witness, which
requires $K+1$ different local bases. Therefore, the fidelity-based witness
might be a better choice if only individual addressing is available, as is
usual in quantum computation.

###  IV.2 Nondetection of Highly-symmetric States

A $\pi$ rotation around the $z$ axis, which acts as a parity operator for the
eigenstates of $J_{z}$, has the action $e^{-i\pi J_{z}/\hbar}J_{k}e^{i\pi
J_{z}/\hbar}=-J_{k}$ on the $x$–$y$ plane, which implies $e^{-i\pi
J_{z}/\hbar}Q_{K}e^{i\pi J_{z}/\hbar}=\mathbbm{1}-Q_{K}$. Hence, for any state
$\rho$ such that $\commutator{e^{i\pi J_{z}/\hbar}}{\rho}=0$,

$\tr(\rho Q_{K})=\tr(e^{-i\pi J_{z}/\hbar}\rho e^{i\pi
J_{z}/\hbar}Q_{K})=1-\tr(\rho Q_{K}).$ (34)

As such, $\tr(\rho
Q_{K})=1/2<\mathbf{P}_{K}^{K\text{-sep}}<\mathbf{P}_{K}^{\text{conj}}$ for any
odd and finite $K$, so $\rho$ does not violate the separable bound.

If $\commutator{e^{i\pi J_{\hat{n}}/\hbar}}{\rho}\neq 0$ for another direction
$\hat{n}$, the protocol can be performed with measurements in the plane
perpendicular to $\hat{n}$, where there might possibly still be a violation of
the separable bound.

However, if $\rho$ commutes with the parity operator along every direction,
then it will certainly not be detected by our witness. Hence, an important
family of states that is missed by our protocol are the Werner states and
their multipartite extensions [51, 52].

###  IV.3 Nondetection of Dicke States

Another well-studied family of entangled states are the Dicke states

$\ket{D_{l}^{n}}:={\binom{n}{l}}^{-\frac{1}{2}}\sum_{\absolutevalue{\mathcal{P}}}\mathcal{P}\pqty{\ket{\uparrow}^{\otimes
l}\otimes\ket{\downarrow}^{\otimes(n-l)}},$ (35)

where the sum is over all permutations $\mathcal{P}$ of the $n$ spin-half
states, with $\ket{\uparrow}\coloneqq\lvert{\frac{1}{2},\frac{1}{2}}\rangle$
and $\ket{\downarrow}\coloneqq\lvert{\frac{1}{2},-\frac{1}{2}}\rangle$. These
include W states as members, and can be detected effectively using just spin
angular momentum measurements [16, 20, 22, 18, 53, 54, 55]. Any of the
referenced witnesses would be a better choice for detecting the entanglement
of such states, as our witness is unable to detect the entanglement of Dicke
states in general.

We note an exception: the tripartite Dicke/W state

$\ket{\text{W}_{3}}:=\ket{D_{1}^{3}}=\frac{1}{\sqrt{3}}\pqty{\ket{\uparrow\downarrow\downarrow}+\ket{\downarrow\uparrow\downarrow}+\ket{\downarrow\downarrow\uparrow}}$
(36)

achieves
$\expectationvalue{Q_{3}}{\text{W}_{3}}=11/16>\mathbf{P}_{3}^{\text{conj}}>\mathbf{P}_{3}^{3\text{-sep}}$
when performing the precession protocol with measurements in the $y$–$z$
plane. Hence, $\ket{\text{W}_{3}}$ is detected by our PEW, although this does
not work beyond the tripartite case.

###  IV.4 States Detected by Our PEW

It follows directly from Eq. (22) that the precession protocol with $K$
measurements can be used to detect the entanglement of the $K$-partite GHZ
state, defined as

$\ket{\text{GHZ}_{K}}:=\frac{1}{\sqrt{2}}\pqty{\ket{\uparrow}^{\otimes
K}+(-1)^{\frac{K-1}{2}}\ket{\downarrow}^{\otimes K}}.$ (37)

As mentioned in the introduction, GHZ states fail to be detected by almost
every other angular-momentum-based witness mentioned in this paper [35, 36,
17, 19, 20, 21, 22, 24]. The only exception is $\ket{\text{GHZ}_{3}}$, whose
detection using angular momentum measurements is described in both Refs. [15]
& [35]; but the reported methods do not work beyond the tripartite case.
Because of their paradigmatic status, we have devoted our companion letter
[56] to a comprehensive study of the detection of the GHZ states by our PEW.
But they are just one family of several that our PEW can detect, while
possessing certain symmetries that prevent their detection by previous
angular-momentum-based witness.

For a spin ensemble $\\{j_{n}\\}_{n=1}^{N}$ such that $\sum_{n=1}^{N}j_{n}$ is
a half integer and $\sum_{n=1}^{N}j_{n}>K/2$, the $j=K/2$ subspace will be in
general degenerate. Hence, the space of states that are eigenvalues of
$Q_{K}^{(j=K/2)}$ with the same eigenvalue as $\ket{\text{GHZ}_{K}}$ will have
a dimension larger than one, and any state that lives in this subspace will
violate the separable bound and hence will be detected. A specific example for
a spin-half ensemble with $N=7$ particles that is detected by the precession
protocol with $K=5$ is

$\displaystyle\ket{\Phi_{5}}$
$\displaystyle\coloneqq\frac{1}{2}\sqrt{\frac{7}{3}}\pqty{\ket{\uparrow}\otimes\ket{\downarrow}^{\otimes
6}-\ket{\downarrow}\otimes\ket{\uparrow}^{\otimes 6}}$ (38)
$\displaystyle\qquad{}-{}\frac{1}{2\sqrt{3}}\pqty{\ket{D_{1}^{7}}-\ket{D_{6}^{7}}}.$

More generally, any state that lives in the subspace spanned by the
eigenstates of $Q_{K}$ whose eigenvalues are larger than the separable bound
will be detected by our PEW. An example of this is

$\ket{\Phi_{3}}\coloneqq\frac{\sqrt{3}}{2}\ket{\mathrm{GHZ}_{9}}-\frac{1}{2\sqrt{2}}\pqty{\ket{D_{3}^{9}}+\ket{D_{6}^{9}}},$
(39)

which is also an eigenstate of $Q_{3}$ with an eigenvalue that is different
from $\bra{\text{GHZ}_{3}}Q_{3}\ket{\text{GHZ}_{3}}$, but which nonetheless
satisfies $\bra{\Phi_{3}}Q_{3}\ket{\Phi_{3}}>\mathbf{P}_{3}^{\text{conj}}$.
Hence $\ket{\Phi_{3}}$, and indeed any superposition of $\ket{\text{GHZ}_{3}}$
and $\ket{\Phi_{3}}$, will be detected by the precession protocol with $K=3$.
We show in Appendix E that all of the discussed states—$\ket{\text{GHZ}_{K}}$,
$\ket{\Phi_{5}}$, and $\ket{\Phi_{3}}$—are missed by all the other previously
reported angular-momentum-based criteria, so these states can be detected only
by our PEW.

The complete characterization of detectable states is still an open problem.
This is partly because the eigendecomposition of each block $Q_{K}^{(j)}$ is
only known analytically for certain values of $j$, and partly because the
space of detectable states depends on the degeneracy of each $j$, which in
turn depends on the exact makeup of the ensemble. Regardless, with the above
sufficient conditions, $Q_{K}$ can be constructed numerically to find the
subspace of states that will be detected by our PEW.

##  V Variations on the protocol

###  V.1 Improved Separable Bound in the Presence of More Information

The reported results and conjectures heavily utilizes Eq. (14), which only
takes into account the value of $\sum_{n=1}^{N}j_{n}$ and the score achieved
when performing the precession protocol on the total angular momentum of the
spin ensemble. No specific makeup of the spin ensemble $\\{j_{n}\\}_{n=1}^{N}$
was assumed.

Of course, more information can be obtained by characterizing the system in
more detail or by performing other measurements. For example, if the system is
a collection of bosons, the ensemble will be one of integer spins, and any
fixed spin $\tilde{\jmath}$ that appears in the block decomposition of the
total angular momentum must be a positive integer
$\tilde{\jmath}\in\mathbb{Z}_{0}^{+}$. Another example would be if the
magnitude $\lvert\vec{J}\rvert^{2}$ of the total angular momentum is measured
in addition to performing the precession protocol, and was found take on
possible values $j(j+1)$ for some
$j\in\\{j_{\text{min}},\dots,j_{\text{max}}\\}$. Then, the only spins
$\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$ that are compatible with the
observed measurements of
$\lvert\vec{J}\rvert^{2}=\lvert\vec{J}^{(\tilde{\jmath})}+\vec{J}^{(\tilde{\jmath}^{\prime})}\rvert^{2}$
are those such that $\lvert\tilde{\jmath}-\tilde{\jmath}^{\prime}\rvert\leq
j_{\text{min}}$ and $\tilde{\jmath}+\tilde{\jmath}^{\prime}\geq
j_{\text{max}}$.

Just like the above two examples, any additional information about the system
can restrict the possible spin pairs that appear in the maximization of Eq.
(14). As this results in a maximization over a restricted range of arguments,
the separable bound can be possibly lowered, requiring less violation to
detect entanglement.

###  V.2 Constructing Other Witnesses Using Sec. III.1

A relation between the multipartite and bipartite separable bounds derived in
Sec. III.1 was used in the process of constructing our PEW. On closer
inspection, this relation is not specific to our PEW, and is in fact
applicable to any function $f(\vec{J})$ of angular momentum.

To illustrate this, consider the commonly-used witness
$\lvert\vec{J}\rvert^{2}$: given
$g^{\text{sep}}:=\min_{\rho_{\text{sep}}}{\tr}(\rho_{\text{sep}}\lvert\vec{J}\rvert^{2})$
minimized over separable states $\rho_{\text{sep}}$,
$\langle\lvert\vec{J}\rvert^{2}\rangle<g^{\text{sep}}$ implies entanglement
[17, 24]. This witness corresponds to setting
$f(\vec{J})=-\lvert\vec{J}\rvert^{2}$ in Eq. (13), and the bipartite separable
bound can be worked out to be

$\displaystyle g^{\text{sep}}(\\{j_{1},j_{2}\\})$ (40)
$\displaystyle\quad{}={}\min_{\ket{\psi_{j_{1}},\psi_{j_{2}}}}\bra{\psi_{j_{1}},\psi_{j_{2}}}\absolutevalue{\vec{J}^{(j_{1})}+\vec{J}^{(j_{2})}}^{2}\ket{\psi_{j_{1}},\psi_{j_{2}}}$
$\displaystyle\quad{}={}\min_{\ket{\psi_{j_{1}},\psi_{j_{2}}}}\Bigg{(}\overbrace{\bra{\psi_{j_{1}}}{\absolutevalue{\vec{J}^{(j_{1})}}^{2}}\ket{\psi_{j_{1}}}}^{\hbar^{2}(j_{1}^{2}+j_{1})}+\overbrace{\bra{\psi_{j_{2}}}{\absolutevalue{\vec{J}^{(j_{2})}}^{2}}\ket{\psi_{j_{2}}}}^{\hbar^{2}(j_{2}^{2}+j_{2})}$
$\displaystyle\hskip
60.00009pt{}+{}2\underbrace{\bra{\psi_{j_{1}}}\vec{J}^{(j_{1})}\ket{\psi_{j_{1}}}\cdot\bra{\psi_{j_{2}}}\vec{J}^{(j_{2})}\ket{\psi_{j_{2}}}}_{\begin{aligned}
&\scriptstyle\geq-\sqrt{\absolutevalue{\bra{\psi_{j_{1}}}\vec{J}^{(j_{1})}\ket{\psi_{j_{1}}}}^{2}\absolutevalue{\bra{\psi_{j_{2}}}\vec{J}^{(j_{2})}\ket{\psi_{j_{2}}}}^{2}}\\\
&\scriptstyle=-\hbar^{2}j_{1}j_{2}\end{aligned}}\Bigg{)}$
$\displaystyle\quad{}\geq{}\hbar^{2}[(j_{1}-j_{2})^{2}+(j_{1}+j_{2})],$

with equality
$g^{\text{sep}}(\\{j_{1},j_{2}\\})=\hbar^{2}[(j_{1}-j_{2})^{2}+(j_{1}+j_{2})]$
as setting $\ket{\psi_{j_{n}}}=\ket{j_{n},(-1)^{n}j_{n}}$ saturates Eq. (40).
Substituting this into Eq. (13) gives
$g^{\text{sep}}(\\{j_{n}\\}_{n=1}^{N})=\hbar^{2}\min_{\tilde{\jmath},\tilde{\jmath}^{\prime}}[(\tilde{\jmath}-\tilde{\jmath}^{\prime})^{2}+(\tilde{\jmath}+\tilde{\jmath}^{\prime})]$
for all bipartitions $\mathbf{J}$-$\mathbf{J}^{\complement}$,
$\tilde{\jmath}\in\mathcal{J}(\mathbf{J})$, and
$\tilde{\jmath}^{\prime}\in\mathcal{J}(\mathbf{J}^{\complement})$.

As long as the singlet subspace of the chosen spin ensemble is nondegenerate,
at least one of $\tilde{\jmath}$ or $\tilde{\jmath}^{\prime}$ will be larger
than zero, so $g^{\text{sep}}(\\{j_{n}\\}_{n=1}^{N})>0$. Meanwhile, if the
singlet subspace exists, the corresponding singlet state achieves
$\langle\lvert\vec{J}\rvert^{2}\rangle=0<g^{\text{sep}}(\\{j_{n}\\}_{n=1}^{N})$,
and hence will be detected to be GME. This method is particularly useful for
witnessing GME of unequal spins with the observable $\lvert\vec{J}\rvert^{2}$,
which, to the best of our knowledge, has yet to be explored. As an example,
take $\\{j_{n}\\}_{n=1}^{4}=\\{\frac{1}{2},\frac{1}{2},1,2\\}$. Minimizing
over all possible pairs $\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}$ that
arise, we find that $g^{\text{sep}}(\\{j_{n}\\}_{n})=1$. At the same time, the
chosen spins permit a singlet subspace for
$j=\lvert(j_{1}+j_{2}+j_{3})-j_{4}\rvert=\lvert\frac{1}{2}+\frac{1}{2}+1-2\rvert=0$,
so the GME of the corresponding singlet state can be detected.

More generally, Eq. (13) can be used to construct other GME witnesses by
choosing other functions of $f(\vec{J})$. As demonstrated here, this approach
allows us to extend existing bipartite witnesses into GME witnesses, which is
especially convenient if an analytical form of the bipartite separable bound
is known.

###  V.3 Bipartite Separable Bounds When
$\tilde{\jmath}=\tilde{\jmath}^{\prime}$ for Fixed $K$

An interesting observation can be made about the behaviour of the separable
bound along the $\tilde{\jmath}=\tilde{\jmath}^{\prime}$ diagonal. In Figs. 2,
3, & A1, the separable bounds along these diagonals are smaller in magnitude
than the surrounding values, and seem to stabilize to a limit for large
$\tilde{\jmath}$.

Figure 8: Line cuts along the diagonals
$\tilde{\jmath}=\tilde{\jmath}^{\prime}$ along Figs. 2–3 & A1. The vertical
axis has been scaled to fit every case, and the dashed lines are the classical
bounds $\mathbf{P}_{K}^{c}$. The unfilled points mark those where only the
lower bound $\mathbf{P}_{K}^{\text{sep}}-\delta\mathbf{P}_{K}^{\text{sep}}$
could be computed. For the filled points, the error bars
$\delta\mathbf{P}_{K}^{\text{sep}}$ are smaller than the marker size. We find
that
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
approaches $\mathbf{P}_{K}^{c}$ from below as $\tilde{\jmath}$ increases.

Indeed, isolating these diagonals in Fig. 8 reveals that
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}=\tilde{\jmath}\\})$
approaches the classical bound $\mathbf{P}_{K}^{c}$ from below as
$\tilde{\jmath}$ increases.

This behaviour is in accordance with two previously-known results. Firstly,
the observable $\operatorname{pos}(J_{x}^{(j)})$ has the limit
$\lim_{j\to\infty}\operatorname{pos}(J_{x}^{(j)})=\operatorname{pos}(a+a^{\dagger})$
for the annihilation operator $a$ of a harmonic oscillator. Secondly, for two
harmonic oscillators with annihilation operators $a_{1}$ and $a_{2}$
respectively, if the precession protocol is performed on the normal mode
$a_{+}\propto a_{1}+a_{2}$, $\mathbf{P}_{K}^{c}$ is the maximum score
achievable by a state separable over the $\\{a_{1},a_{2}\\}$ modes.

Therefore, as
$J_{x}=J_{x}^{(\tilde{\jmath})}+J_{x}^{(\tilde{\jmath}^{\prime})}$ for
$\tilde{\jmath}=\tilde{\jmath}^{\prime}$ appears analogous to $a_{+}\propto
a_{1}+a_{2}$, it would be expected that
$\lim_{\tilde{\jmath}\to\infty}\operatorname{pos}(J_{x})\overset{?}{=}\operatorname{pos}(a_{+}+a_{+}^{\dagger})$.
If so, the separable bound should similarly have the limit
$\lim_{\tilde{\jmath}\to\infty}\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}=\tilde{\jmath}\\})\overset{?}{=}\mathbf{P}_{K}^{c}$.
We have not managed to prove this, but Fig. 8 certainly supports this
expectation, and insinuates the following:

######  Conjecture 4.

Consider a bipartite spin ensemble
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}$, where
$\tilde{\jmath}=\tilde{\jmath}^{\prime}$. Perform the precession protocol with
odd $K\geq 3$ on the total angular momentum of the system. If the classical
bound $\mathbf{P}_{K}^{c}$ is violated, then the two spins are entangled.
While this conjecture is limited to the bipartite witness, it hints at a
possible link between the classical bound of the precession protocol,
entanglement, and the harmonic oscillator limit.

##  VI Conclusion

In this work, we have introduced entanglement witnesses that detect genuine
multipartite entanglement of a spin ensemble, which require only measurements
of the total angular momentum along several equally-spaced directions. We have
reported analytical expressions for the separable bound when the total spin is
a half-integer. For the other cases, reliable numerical values and a
conjectured expression is reported for the separable bound, where the latter
is well supported by the former.

Of similar angular-momentum-based witnesses, only some detect genuine
multipartite entanglement; of those that do, most are effective at detecting
Dicke-like states, but none can detect Greenberger-Horne-Zeilinger states
beyond the tripartite case. This gap is filled by our PEW, which is effective
at detecting precisely those states. We have also shown that our witness can
detect other GME states, which share similar symmetries to the GHZ state, that
are missed by existing criteria.

An possible extension would be to consider the effects of anharmonicities when
performing the precession protocol under the assumption of dynamics. A popular
experimental platform where spin ensembles naturally arise are spin defects or
donors in solid-state materials [10, 11], where quadrupole interactions are
present in the Hamiltonian as terms quadratic in angular momentum, which
perturbs the dynamics of the system away from a perfectly uniform precession.
Such anharmonic effects have been studied for continuous variable systems
[42], and many of those findings could be extended to the case of spin angular
momentum.

Finally, we once again highlight the generality of the result in Sec. III.1.
By choosing different functions of angular momentum to be used with Eq. (13),
it provides a generic recipe for constructing witnesses of genuine
multipartite entanglement out of witnesses of bipartite entanglement. This can
be applicable to many other entanglement witnesses completely unrelated to the
precession protocol, as we have shown with an example in the previous section.
This provides another avenue for future study: Since our PEW is mostly
effective when the total spin is a half-integer, these techniques might allow
us to construct a similar witness that will also be effective for integral
total spin.

## Acknowledgments

This work is supported by the National Research Foundation, Singapore, and
A*STAR under its CQT Bridging Grant. We also thank the National University of
Singapore Information Technology for the use of their high performance
computing resources.

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##  Appendix A Additional Figures

Some additional figures are gathered in this appendix. Figure A1 plots
$\mathbf{P}_{K}^{\text{sep}}$ for $11\leq K\leq 21$, which complements Figs. 2
& 3; Figure A2 plots the discrepancy $\delta\mathbf{P}_{K}^{\text{sep}}$
between the numerically-computed lower and upper bounds of
$\mathbf{P}_{K}^{\text{sep}}$, as detailed in Appendix. D; Figure A3 plots the
lower bound of $\mathbf{P}_{K}^{\text{sep}}$ for a wider range of
$\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$, demonstrating that the
expected behaviour of $\mathbf{P}_{K}^{\text{sep}}$ appears to hold for larger
values of $\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$, providing further
support for Conjecture 3.

Figure A1: Heatmap of
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$
against $\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$, for (a) $K=11$, (b)
$K=13$, (c) $K=15$, (d) $K=17$, (e) $K=19$, and (f) $K=21$. The values
$\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})=1/2$
for $\tilde{\jmath}+\tilde{\jmath}^{\prime}<K/2$ are not plotted here. The
separable bounds are large when $\min(\tilde{\jmath},\tilde{\jmath}^{\prime})$
is small, and $\mathbf{P}_{K}^{\text{sep}}$ decreases as
$\min(\tilde{\jmath},\tilde{\jmath}^{\prime})$ increases. The maximum value of
$\mathbf{P}_{K}^{\text{sep}}$ in these cases occur at
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1,K/2\\}$. Figure A2: (a)–(j)
Heatmap of
$\delta\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$,
the deviations between the lower and upper bounds of
$\mathbf{P}_{K}^{\text{sep}}$ computed using SPI and SDP respectively, for
different values of $K$. The errors are worse for large $j$ as we used larger
tolerances in those cases. This is because the value of
$\mathbf{P}_{K}^{\text{sep}}$ is small, so the lower bounds for small $j$ is
larger than the upper bound of large $j$ even with the larger tolerances for
the latter. (k) Box plots of the errors against the number of measurements.
The worst-case errors are of order $10^{-3}$, but most lie within
$\delta\mathbf{P}_{K}^{\text{sep}}\leq 10^{-5}$. Figure A3: Heatmap of
$\mathbf{P}_{K}^{\text{SPI}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})=\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})-\delta\mathbf{P}_{K}^{\text{sep}}(\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\})$,
the lower bound of $\mathbf{P}_{K}^{\text{sep}}$, computed for larger values
of $\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$. We were unable to compute
the upper bounds for $\tilde{\jmath},\tilde{\jmath}^{\prime}>15$ due to
constraints of computational resources. However, from the small deviations
between the upper and lower bounds for
$\tilde{\jmath},\tilde{\jmath}^{\prime}\leq 15$ as seen in Fig. A2, we expect
that $\mathbf{P}_{K}^{\text{SPI}}$ should also be rather close to the actual
value $\mathbf{P}_{K}^{\text{sep}}$. We find that the behaviour of
$\mathbf{P}_{K}^{\text{SPI}}$ continues as expected, which is that is reduces
in magnitude as $\tilde{\jmath}$ and $\tilde{\jmath}^{\prime}$ increases,
which lends further credence to Conjecture 3.

##  Appendix B Key Technical Aspects of the Precession Protocol

We review some key technical aspects of the precession protocol used in this
paper. For the full study with details and derivations, refer to Ref. [42].

Consider the protocol performed with a single fixed spin $j$ such that the
expected score for the state $\rho$ given by $P_{K}={\tr}(\rho Q_{K}^{(j)})$,
where

$Q_{K}^{(j)}=\frac{1}{K}\sum_{k=0}^{K-1}\operatorname{pos}(J_{k}^{(j)})=\frac{1}{2}\bqty{\mathbbm{1}+\frac{1}{K}\sum_{k=0}^{K-1}e^{-i(2\pi
k/K)J_{z}^{(j)}/\hbar}\operatorname{sgn}(J_{x}^{(j)})e^{i(2\pi
k/K)J_{z}^{(j)}/\hbar}}.$ (41)

Here,
$2\operatorname{pos}(J_{k}^{(j)})\ket{j,m}_{k}=\ket{j,m}_{k}+\operatorname{sgn}(m)\ket{j,m}_{k}$
and $J_{k}^{(j)}=e^{-i(2\pi k/K)J_{z}^{(j)}/\hbar}J_{x}^{(j)}e^{i(2\pi
k/K)J_{z}^{(j)}/\hbar}$ were used. Some key properties of $Q_{K}$ follow.

(1) $Q_{K}^{(j)}$ is block diagonal. Define the projector

$\Pi_{K}^{(j,\bar{m})}:=\sum_{k=0}^{\lfloor({j-\bar{m}})/{K}\rfloor}\outerproduct{j,\bar{m}+kK}{j,\bar{m}+kK},$
(42)

which projects onto the subspace of the eigenstates of $J_{z}^{(j)}$ whose
eigenvalues are $\bar{m}$ modulo $K$. Then, $Q_{K}^{(j)}$ has the block
decomposition

$Q_{K}^{(j)}=\frac{1}{2}\bqty{\mathbbm{1}+\bigoplus_{\bar{m}=-j}^{\min(j,-j+K-1)}\Pi_{K}^{(j,\bar{m})}\operatorname{sgn}(J_{x}^{(j)})\Pi_{K}^{(j,\bar{m})}}.$
(43)

Consequently, if $j<K/2$, then $\bar{m}\leq K/2-1$ and hence
$\lfloor({j-\bar{m}})/{K}\rfloor=0$, which implies that
$2Q_{K}^{(j<K/2)}=\mathbbm{1}+\sum_{m=-j}^{j}\outerproduct{j,m}{j,m}\operatorname{sgn}(J_{x}^{(j)})\outerproduct{j,m}{j,m}$
is diagonal with respect to the eigenstates of $J_{z}$.

(2) The matrix elements
$\bra{j,m}\operatorname{sgn}(J_{x}^{(j)})\ket{j,m^{\prime}}$ are

$\displaystyle\matrixelement{j,m}{\operatorname{sgn}(J_{x}^{(j)})}{j,m^{\prime}}$
$\displaystyle=\begin{cases}0&\text{if $(m-m^{\prime})\bmod 2=0$,}\\\
\frac{(-1)^{\pqty{m^{\prime}-m-1}/{2}}2^{-(2j-1)}}{m^{\prime}-m}\sqrt{\pmqty{undef}\pmqty{undef}}\\\
\qquad\qquad\qquad\qquad{}\times{}\sqrt{\pmqty{undef}\pmqty{undef}}&\text{otherwise.}\\\
\qquad\qquad\qquad\qquad{}\times{}\sqrt{(j+m)^{(j+m)\bmod
2}(j+m^{\prime})^{(j+m^{\prime})\bmod 2}}\\\
\qquad\qquad\qquad\qquad{}\times{}\sqrt{(j-m)^{(j-m)\bmod
2}(j-m^{\prime})^{(j-m^{\prime})\bmod 2}}\end{cases}$ (44)

In particular, the diagonal elements of $\operatorname{sgn}(J_{x}^{(j)})$ are
zero. From the previous result for $j<K/2$, we therefore have
$2Q_{K}^{(j<K/2)}=\mathbbm{1}^{(j<K/2)}$.

With the above two properties, $Q_{K}^{(j)}$ can be constructed for any given
$j$, and its eigenvectors and eigenvalues can be found using standard
numerical techniques. For some cases, this can even be done analytically. A
special case we use in this paper is $K/2\leq j<K$, where

$\Pi_{K}^{(j,\bar{m})}\operatorname{sgn}(J_{x}^{(j)})\Pi_{K}^{(j,\bar{m})}=\begin{cases}0&\text{if
$j+\bar{m}\leq\lfloor j-K/2\rfloor$,}\\\
\Psi^{(j,\bar{m})}\pqty{\lvert{\Psi_{+}^{(j,\bar{m})}}\rangle\\!\langle{\Psi_{+}^{(j,\bar{m})}}\rvert-\lvert{\Psi_{-}^{(j,\bar{m})}}\rangle\\!\langle{\Psi_{-}^{(j,\bar{m})}}\rvert}&\text{otherwise;}\end{cases}$
(45)

with
$\Psi^{(j,\bar{m})}=\lvert\matrixelement{j,\bar{m}}{\operatorname{sgn}(J_{x}^{(j)})}{j,\bar{m}+K}\rvert$
and
$\ket{\Psi_{\pm\bar{m}}}=(\ket{j,\bar{m}}\pm(-1)^{(K-1)/2}\ket{j,\bar{m}+K})/\sqrt{2}$.

##  Appendix C Separable Bound for
$\\{\tilde{\jmath},\tilde{\jmath}^{\prime}\\}=\\{1/2,(K+1)/2\\}$

Given two spins with $\\{j_{1},j_{2}\\}=\\{1/2,(K+1)/2\\}$, their total
angular momentum can take values of $j\in\\{K/2,K/2+1\\}$. Therefore, defining
$\Delta
Q_{K}:=\mathbbm{1}-2Q_{K}=\frac{1}{K}\sum_{k=0}^{K-1}\operatorname{sgn}(J_{k})$
for concision,

$\displaystyle\Delta Q_{K}$
$\displaystyle=\bqty{\Pi_{K}^{\pqty{\frac{K}{2},-\frac{K}{2}}}\operatorname{sgn}(J_{x})\Pi_{K}^{\pqty{\frac{K}{2},-\frac{K}{2}}}}\oplus\bqty{\Pi_{K}^{\pqty{\frac{K}{2}+1,-\frac{K}{2}}}\operatorname{sgn}(J_{x})\Pi_{K}^{\pqty{\frac{K}{2}+1,-\frac{K}{2}}}}$
(46)
$\displaystyle\qquad\qquad{}\oplus{}\bqty{\Pi_{K}^{\pqty{\frac{K}{2}+1,-\frac{K}{2}-1}}\operatorname{sgn}(J_{x})\Pi_{K}^{\pqty{\frac{K}{2}+1,-\frac{K}{2}-1}}}\oplus\bqty{\Pi_{K}^{\pqty{\frac{K}{2}+1,-\frac{K}{2}+1}}\operatorname{sgn}(J_{x})\Pi_{K}^{\pqty{\frac{K}{2}+1,-\frac{K}{2}+1}}}.$

With
$\Pi_{K}^{(j,\bar{m})}\operatorname{sgn}(J_{x})\Pi_{K}^{(j,\bar{m})}=\Psi^{(j,\bar{m})}(\lvert{\Psi_{+}^{(j,\bar{m})}}\rangle\\!\langle{\Psi_{+}^{(j,\bar{m})}}\rvert-\lvert{\Psi_{-}^{(j,\bar{m})}}\rangle\\!\langle{\Psi_{-}^{(j,\bar{m})}}\rvert)$
from Eq. (45) and direct computation of $\Psi^{(j,\bar{m})}$ from Eq. (44), we
obtain

$\displaystyle\Psi^{(\frac{K}{2},-\frac{K}{2})}$
$\displaystyle=2^{-(K-1)}\pmqty{undef},$
$\displaystyle\Psi^{(\frac{K}{2}+1,-\frac{K}{2})}$
$\displaystyle=\frac{\Psi^{(\frac{K}{2},-\frac{K}{2})}}{K+1},$
$\displaystyle\Psi^{(\frac{K}{2}+1,-\frac{K}{2}\mp 1)}$
$\displaystyle=\sqrt{\frac{K+2}{2(K+1)}}\Psi^{(\frac{K}{2},-\frac{K}{2})}.$
(47)

We are interested in finding the separable bound

$\displaystyle\mathbf{P}_{K}^{\text{sep}}({\\{j_{1},j_{2}}\\})$
$\displaystyle=\max_{\ket{\psi_{j_{1}}},\ket{\psi_{j_{2}}}}(\bra{\psi_{j_{1}}}\otimes\bra{\psi_{j_{2}}})Q_{K}(\ket{\psi_{j_{1}}}\otimes\ket{\psi_{j_{2}}})$
(48)
$\displaystyle=\frac{1}{2}+\frac{1}{2}\max_{\ket{\psi_{j_{1}}}}\Bqty{\max_{\ket{\psi_{j_{2}}}}\bra{\psi_{j_{2}}}\bqty{\pqty{\bra{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})}}\Delta
Q_{K}\pqty{\ket{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})}}}\ket{\psi_{j_{2}}}}.$

By rewriting it as above, the maximization can be thought of as a two-step
process: we first find the maximum eigenvalue of
$(\bra{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})})\Delta
Q_{K}(\ket{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})})$ for a fixed
$\ket{\psi_{j_{1}}}$, then maximize this eigenvalue over all states
$\ket{\psi_{j_{1}}}$. Parameterizing a generic state in
$\mathcal{H}^{(j_{1})}$ as
$\ket{\psi_{j_{1}}}:=\cos(\frac{\theta}{2})\ket{\frac{1}{2},-\frac{1}{2}}+e^{i\phi}\sin(\frac{\theta}{2})\ket{\frac{1}{2},\frac{1}{2}}$,
and using the Clebsch–Gordan coefficients, we find

$\displaystyle\pqty{\bra{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})}}\lvert{\Psi^{(\frac{K}{2},-\frac{K}{2})}_{\pm}}\rangle$
$\displaystyle=\cos(\frac{\theta}{2})\sqrt{\frac{K+1}{2(K+2)}}\ket{\tfrac{K+1}{2},-\tfrac{K+1}{2}}-\frac{e^{-i\phi}\sin(\frac{\theta}{2})}{\sqrt{2(K+2)}}\ket{\tfrac{K+1}{2},-\tfrac{K-1}{2}}$
(49a)
$\displaystyle\quad\qquad{}\pm{}(-1)^{\frac{K-1}{2}}\pqty{\frac{\cos(\frac{\theta}{2})}{\sqrt{2(K+2)}}\ket{\tfrac{K+1}{2},\tfrac{K-1}{2}}-e^{-i\phi}\sin(\frac{\theta}{2})\sqrt{\frac{K+1}{2(K+2)}}\ket{\tfrac{K+1}{2},\tfrac{K+1}{2}}},$
$\displaystyle\pqty{\bra{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})}}\lvert{\Psi^{(\frac{K}{2}+1,-\frac{K}{2})}_{\pm}}\rangle$
$\displaystyle=\frac{\cos(\frac{\theta}{2})}{\sqrt{2(K+2)}}\ket{\tfrac{K+1}{2},\tfrac{K+1}{2}}+e^{-i\phi}\sin(\frac{\theta}{2})\sqrt{\frac{K+1}{2(K+2)}}\ket{\tfrac{K+1}{2},\tfrac{K-1}{2}}$
(49b)
$\displaystyle\quad\qquad{}\pm{}(-1)^{\frac{K-1}{2}}\pqty{\cos(\frac{\theta}{2})\sqrt{\frac{K+1}{2(K+2)}}\ket{\tfrac{K+1}{2},-\tfrac{K-1}{2}}+\frac{e^{-i\phi}\sin(\frac{\theta}{2})}{\sqrt{2(K+2)}}\ket{\tfrac{K+1}{2},-\tfrac{K+1}{2}}},$
$\displaystyle\pqty{\bra{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})}}\lvert{\Psi^{(\frac{K}{2}+1,-\frac{K}{2}-1)}_{\pm}}\rangle$
$\displaystyle=\cos(\frac{\theta}{2})\sqrt{\frac{K}{2(K+2)}}\ket{\tfrac{K+1}{2},\tfrac{K-3}{2}}+\frac{e^{-i\phi}\sin(\frac{\theta}{2})}{\sqrt{K+2}}\ket{\tfrac{K+1}{2},\tfrac{K-1}{2}}$
(49c)
$\displaystyle\quad\qquad{}\pm{}(-1)^{\frac{K-1}{2}}\frac{e^{-i\phi}\sin(\frac{\theta}{2})}{\sqrt{2}}\ket{\tfrac{K+1}{2},-\tfrac{K+1}{2}},$
$\displaystyle\pqty{\bra{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})}}\lvert{\Psi^{(\frac{K}{2}+1,-\frac{K}{2}+1)}_{\pm}}\rangle$
$\displaystyle=\frac{\cos(\frac{\theta}{2})}{\sqrt{K+2}}\ket{\tfrac{K+1}{2},-\tfrac{K-1}{2}}+e^{-i\phi}\sin(\frac{\theta}{2})\sqrt{\frac{K}{2(K+2)}}\ket{\tfrac{K+1}{2},-\tfrac{K-3}{2}}$
(49d)
$\displaystyle\quad\qquad{}\pm{}(-1)^{\frac{K-1}{2}}\frac{\cos(\frac{\theta}{2})}{\sqrt{2}}\ket{\tfrac{K+1}{2},\tfrac{K+1}{2}}.$

Notice therefore that $(\bra{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})})\Delta
Q_{K}(\ket{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})})$ has support only in the
subspace spanned by $\\{\ket{\tfrac{K+1}{2},-\tfrac{K+1}{2}}$,
$\ket{\tfrac{K+1}{2},-\tfrac{K-1}{2}}$,
$\ket{\tfrac{K+1}{2},-\tfrac{K-3}{2}}$, $\ket{\tfrac{K+1}{2},\tfrac{K-3}{2}}$,
$\ket{\tfrac{K+1}{2},\tfrac{K-1}{2}}$,
$\ket{\tfrac{K+1}{2},\tfrac{K+1}{2}}\\}$. For $K>3$, in this six-dimensional
subspace,

$\frac{(\bra{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})})\Delta
Q_{K}(\ket{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})})}{\Psi^{(\frac{K}{2},-\frac{K}{2})}}\widehat{=}\sqrt{\frac{K/2}{K+1}}\left(\begin{matrix}0&0&0&e^{-i\phi}\sin(\theta)&1&-\sqrt{\frac{K/2}{K+1}}e^{i\phi}\sin(\theta)\\\
0&0&0&0&0&1\\\ 0&0&0&0&0&e^{-i\phi}\sin(\theta)\\\
e^{i\phi}\sin(\theta)&0&0&0&0&0\\\ 1&0&0&0&0&0\\\
-\sqrt{\frac{K/2}{K+1}}e^{i\phi}\sin(\theta)&1&e^{i\phi}\sin(\theta)&0&0&0\end{matrix}\right).$
(50)

This matrix is highly sparse, and its eigenvalues can be found by directly
solving the characteristic equation
$\det[(\bra{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})})\Delta
Q_{K}(\ket{\psi_{j_{1}}}\otimes\mathbbm{1}^{(j_{2})})-\lambda\mathbbm{1}]=0$,
which is given by

$\pqty{\frac{\lambda}{\Psi^{(\frac{K}{2},-\frac{K}{2})}}}^{2}\Bqty{\pqty{\frac{\lambda}{\Psi^{(\frac{K}{2},-\frac{K}{2})}}}^{4}(K+1)^{2}-\pqty{\frac{\lambda}{\Psi^{(\frac{K}{2},-\frac{K}{2})}}}^{2}\bqty{\frac{2K+1}{4}K\sin^{2}(\theta)+2(K+1)}+\pqty{\frac{K\sin^{2}(\theta)}{8}+1}^{2}}=0.$
(51)

Removing the $\lambda=0$ solutions, this equation is quadratic in
$\lambda^{2}$, with the solutions

$\pqty{\frac{\lambda(\theta)}{\Psi^{(\frac{K}{2},-\frac{K}{2})}}}^{2}=\pqty{\frac{K\sin(\theta)\pm\sqrt{(3K+1)K\sin^{2}(\theta)+16(K+1)}}{4(K+1)}}^{2}.$
(52)

Amongst the four solutions for $\lambda(\theta)$, whose the dependence on
$\ket{\psi_{j_{1}}}$ that is present through $\theta$ is now made explicit, we
clearly have

$\frac{\lambda_{\text{max}}(\theta)}{\Psi^{(\frac{K}{2},-\frac{K}{2})}}=\frac{K\absolutevalue{\sin(\theta)}+\sqrt{(3K+1)K\sin^{2}(\theta)+16(K+1)}}{4(K+1)}.$
(53)

To maximize this over $\theta$, we only need to consider the range
$\theta\in[0,\pi/2]$, as $\lambda_{\text{max}}(\theta)$ repeats outside this
range. Furthermore, since $\lambda_{\text{max}}(\theta)$ is monotonically
increasing with $\theta$ in this range, we arrive at

$\max_{\theta}\lambda_{\text{max}}(\theta)=\frac{2^{-(K+1)}}{K+1}\pmqty{undef}\pqty{K+\sqrt{3K^{2}+18K+16}}.$
(54)

Note that the above is true only for $K>3$. When $K=3$,
$\tfrac{K-3}{2}=-\tfrac{K-3}{2}=0$, so the subspace is only five-dimensional.
Nonetheless, repeating the above steps lead to a similar characteristic
equation quadratic in $\lambda^{2}$, with

$\max_{\theta}\lambda_{\text{max}}(\theta)=\max_{\theta}\frac{3\absolutevalue{\sin(\theta)}+\sqrt{57\sin^{2}(\theta)+64}}{32}=\frac{7}{16}.$
(55)

As such, recalling that $\max_{\theta}\lambda_{\text{max}}(\theta)$ is the
separability bound for $\Delta Q_{K}$ and $Q_{K}=(\mathbbm{1}+\Delta
Q_{K})/2$, we arrive at

$\mathbf{P}_{K}^{\text{sep}}\pqty{\Bqty{\tfrac{1}{2},\tfrac{K+1}{2}}}=\frac{1}{2}(1+\lambda_{K}),\quad\text{where}\quad\lambda_{K}=\begin{cases}\dfrac{7}{16}&\text{if
$K=3$,}\\\\[12.91663pt]
\displaystyle\frac{2^{-(K+1)}}{K+1}\pmqty{undef}\pqty{K+\sqrt{3K^{2}+18K+16}}&\text{otherwise.}\end{cases}$
(56)

##  Appendix D Numerical methods

Two numerical methods were used to find the lower and upper bounds of
$\mathbf{P}^{\text{sep}}_{K}(\\{j_{1},j_{2}\\})$ for general $j_{1},j_{2}$,
which allows us to obtain reliable values of the separable bound to within
numerical precision. We use a variant of the separability power iteration
(SPI) [6] to obtain a lower bound $\mathbf{P}_{K}^{\text{SPI}}$, and
semidefinite programming (SDP) [57] to obtain an upper bound
$\mathbf{P}_{K}^{\text{SDP}}$. Note that the methods, while numerical, returns
reliable lower and upper bounds respectively. The implementation details of
these two numerical techniques will be covered in the following sections.

After obtaining the two values, we can further define
$\mathbf{P}_{K}^{\text{sep}}:=(\mathbf{P}_{K}^{\text{SDP}}+\mathbf{P}_{K}^{\text{SPI}})/2$
and
$\delta\mathbf{P}_{K}^{\text{sep}}:=(\mathbf{P}_{K}^{\text{SDP}}-\mathbf{P}_{K}^{\text{SPI}})/2$.
It is guaranteed that the actual value of the separability bound lies within
the range
$[\mathbf{P}_{K}^{\text{sep}}-\delta\mathbf{P}_{K}^{\text{sep}},\mathbf{P}_{K}^{\text{sep}}+\delta\mathbf{P}_{K}^{\text{sep}}]$:
henceforth, we will treat $\mathbf{P}_{K}^{\text{sep}}$ as an estimate for the
true value of the separable bound, while $\delta\mathbf{P}_{K}^{\text{sep}}$
will be treated as a numerical error due to numerical precision and possible
gaps between the two bounds.

For all studied cases, the gaps between the results obtained through these two
numerical methods are small, where the largest deviation is of the order
$\delta\mathbf{P}_{K}^{\text{sep}}\sim 10^{-3}$. These deviations are plotted
for odd $K\leq 21$ in Fig. A2.

###  D.1 Lower Bounds: Separability Power Iteration

The lower bound can be found by noticing that
$\mathbf{P}^{\text{sep}}_{K}(\\{j_{1},j_{2}\\})$ is the global maximum of the
product numerical range $\\{\langle\psi_{j_{1}},\psi_{j_{2}}\rvert
Q_{K}\lvert\psi_{j_{1}},\psi_{j_{2}}\rangle:\lvert\psi_{j_{1}},\psi_{j_{2}}\rangle=\lvert\psi_{j_{1}}\rangle\otimes\lvert\psi_{j_{2}}\rangle,\lvert\psi_{j}\rangle\in\mathcal{H}^{(j)}\\}$.
As such, any particular choice of $\lvert\psi_{j_{1}},\psi_{j_{2}}\rangle$
must necessarily give a lower bound $\langle\psi_{j_{1}},\psi_{j_{2}}\rvert
Q_{K}\lvert\psi_{j_{1}},\psi_{j_{2}}\rangle\leq\mathbf{P}^{\text{sep}}_{K}(\\{j_{1},j_{2}\\})$.

By maximizing $\langle\psi_{j_{1}},\psi_{j_{2}}\rvert
Q_{K}\lvert\psi_{j_{1}},\psi_{j_{2}}\rangle$ over
$\lvert\psi_{j_{1}},\psi_{j_{2}}\rangle$ using any method, even if only a
local maximum is reached, we can still obtain a lower bound that is as large
as possible. As mentioned, we use a variant of the separability power
iteration (SPI) [6] to obtain the lower bound $\mathbf{P}_{K}^{\text{SPI}}$.
We detail the technique in Algorithm 1.

Algorithm 1 Bipartite variant of SPI

$\ket{a_{0}}\in\mathcal{H}^{(j_{1})}$, $0<\varepsilon$

$p_{\text{prev}}\leftarrow-\infty$

$p_{\text{curr}}\leftarrow\infty$

$\ket{a}\leftarrow\ket{a_{0}}$

$\ket{b}\leftarrow\ket{b_{0}}\in\mathcal{H}^{(j_{2})}$ $\triangleright$ Any
$\ket{b_{0}}\in\mathcal{H}^{(j_{2})}$ will do
$|p_{\text{curr}}-p_{\text{prev}}|\geq\varepsilon$

$\ket{b}\leftarrow\operatorname{argmax}_{\tilde{\ket{b}}\in\mathcal{H}^{(j_{2})}}\tilde{\bra{b}}\operatorname{tr}_{j_{1}}\\!\\!\left[Q_{K}\left(\outerproduct{a}{a}\otimes\mathbbm{1}^{(j_{2})}\right)\right]\tilde{\ket{b}}$

$\ket{a}\leftarrow\operatorname{argmax}_{\ket{\tilde{a}}\in\mathcal{H}^{(j_{1})}}\bra{\tilde{a}}\operatorname{tr}_{j_{2}}\\!\\!\left[Q_{K}\left(\mathbbm{1}^{(j_{1})}\otimes\outerproduct{b}{b}\right)\right]\ket{\tilde{a}}$

$p_{\text{prev}}\leftarrow p_{\text{curr}}$

$p_{\text{curr}}\leftarrow\bra{a,b}Q_{K}\ket{a,b}$

$p_{\text{curr}}$

Here $\tr_{j_{n}}$ is the partial trace over $\mathcal{H}^{(j_{n})}$, while
the steps requiring $\operatorname{argmax}$ are eigenvalue problems that can
be solved with standard numerical libraries.

Algorithm 1 halts when the convergence condition $\lvert
p_{\text{curr}}-p_{\text{prev}}\rvert<\varepsilon$ for a target precision
$\varepsilon$ is met. It can be shown that the algorithm is always convergent,
and returns a local maximum for the given starting vector
$\ket{a_{0}}\in\mathcal{H}^{(j_{1})}$. Denote the vectors obtained in the
$n$_th_ round of Algorithm 1 by $\ket{a_{i}}$ and $\ket{b_{i}}$, and let
$p_{i}=\bra{a_{i},b_{i}}Q_{K}\ket{a_{i},b_{i}}$. Then,

$\displaystyle p_{i+1}$
$\displaystyle=\bra{a_{i+1},b_{i+1}}Q_{K}\ket{a_{i+1},b_{i+1}}$
$\displaystyle=\bra{a_{i+1}}\tr_{j_{2}}\\!\\!\bqty{Q_{K}\pqty{\mathbbm{1}^{(j_{1})}\otimes\outerproduct{b_{i+1}}{b_{i+1}}}}\ket{a_{i+1}}$
$\displaystyle\geq\bra{a_{i}}\tr_{j_{2}}\\!\\!\bqty{Q_{K}\pqty{\mathbbm{1}^{(j_{1})}\otimes\outerproduct{b_{i+1}}{b_{i+1}}}}\ket{a_{i}}$
(57)
$\displaystyle=\bra{b_{i+1}}\tr_{j_{1}}\\!\\!\bqty{Q_{K}\pqty{\outerproduct{a_{i}}{a_{i}}\otimes\mathbbm{1}^{(j_{2})}}}\ket{b_{i+1}}$
$\displaystyle\geq\bra{a_{i},b_{i}}Q_{K}\ket{a_{i},b_{i}}=p_{i}$

This proves that $\\{p_{i}\\}_{i\geq 0}$ is monotonically increasing.
Moreover, since $Q_{K}$ is an observable that describes a probability and is
therefore bounded from above, the sequence $\\{p_{i}\\}_{i\geq 0}$, which is a
sequence of expectations of $Q_{K}$, is also bounded. Therefore, the monotonic
sequence $\\{p_{i}\\}_{i\geq 0}$ converges, and hence Algorithm 1 will
eventually halt for any starting vector $\ket{a_{0}}\in\mathcal{H}^{(j_{1})}$.

Since the local maximum $p$ we obtain depends on our choice of $\ket{a_{0}}$,
we perform the algorithm with many different starting vectors
$\\{\ket{a_{0}},\ket{a_{0}^{\prime}},\ket{a_{0}^{\prime\prime}},\dots\\}$ to
obtain a set of local maximums $\\{p,p^{\prime},p^{\prime\prime},\dots\\}$.
Following Ref. [6], we chose
$\\{\ket{a_{0}},\ket{a_{0}^{\prime}},\ket{a_{0}^{\prime\prime}},\dots\\}$ to
be the eigenvectors of generalized Gell-Mann matrices of dimension $2j_{1}+1$,
which spans the full operator space in $\mathcal{H}^{(j_{1})}$. Finally, we
define
$\mathbf{P}_{K}^{\text{SPI}}:=\max\\{p,p^{\prime},p^{\prime\prime},\dots\\}$,
which provides us with the best lower bound
$\mathbf{P}_{K}^{\text{SPI}}\leq\mathbf{P}_{K}^{\text{sep}}$ that was found
from the many runs of the algorithm.

###  D.2 Upper Bounds: Semidefinite Programming

The upper bound is due to the Peres–Horodecki criterion, which states that the
set of separable states
$\\{\rho_{\text{SEP}}:\rho_{\text{SEP}}=\sum_{k}p_{k}\rho_{j_{1},k}\otimes\rho_{j_{2},k}\\}$
is a subset of the set of positive-partial-transpose states
$\\{\rho_{\text{PPT}}:\rho_{\text{PPT}}^{\Gamma_{2}}\succeq 0\\}$ [58, 59].
Here, $\rho_{\text{PPT}}^{\Gamma_{2}}$ is the partial transposition of
$\rho_{\text{PPT}}$ whose matrix elements satisfy

$\bra{\psi_{j_{1}},\psi_{j_{2}}}\rho_{\text{PPT}}^{\Gamma_{2}}\ket{\phi_{j_{1}},\phi_{j_{2}}}=\bra{\psi_{j_{1}},\phi_{j_{2}}}\rho_{\text{PPT}}\ket{\phi_{j_{1}},\psi_{j_{2}}}.$
(58)

As $\mathbf{P}_{K}^{\text{sep}}$ involves a maximization over a subset of the
positive-partial-tranpose states,
$\max_{\rho_{\text{PTT}}}\tr(\rho_{\text{PTT}}Q_{K})$ is an upper bound for
$\mathbf{P}_{K}^{\text{sep}}$. In turn, a reliable upper bound on
$\max_{\rho_{\text{PTT}}}\tr(\rho_{\text{PTT}}Q_{K})$ can be obtained using
semidefinite programming (SDP) [57], which we shall denote as
$\mathbf{P}_{K}^{\text{SDP}}$. Being an upper bound of the upper bound for
$\mathbf{P}_{K}^{\text{sep}}$, it is of course the case that
$\mathbf{P}_{K}^{\text{SDP}}\geq\mathbf{P}_{K}^{\text{sep}}$.

Semidefinite programs are linear optimization problems of the form

$\begin{array}[]{rl}\displaystyle\max_{X\succeq 0}&\displaystyle\tr(CX)\\\
\displaystyle\text{subject to}&\displaystyle\tr(A_{l}X)=b_{l}\text{ for
$l=1,2,\dots$}\\\ &\displaystyle X\succeq 0,\end{array}$ (59)

where $C$ and $A_{l}$ are Hermitian operators. We refer to the above as the
primal problem, and there is a related dual problem given by

$\begin{array}[]{rl}\displaystyle\min_{y_{1},y_{2},\dots}&\sum_{l}b_{l}y_{l}\eqqcolon\vec{b}\cdot\vec{y}\\\
\text{subject to}&\sum_{l}y_{l}A_{l}\succeq C.\end{array}$ (60)

The weak duality theorem states that the dual objective always upper bounds
the primal one: that is, $\vec{b}\cdot\vec{y}\geq\tr(CX)$. Weak duality
therefore ensures that the global optimum of both the primal and dual problems
are guaranteed to be between the gap given by the two objectives. As such,
this is used as a convergence condition when numerically solving SDPs: local
optimizations of the primal and dual problems are performed until
$|\vec{b}\cdot\vec{y}-\tr(CX)|<\varepsilon$ for a target precision
$\varepsilon$.

For our purposes, we are interested in the optimization

$\begin{array}[]{rl}\displaystyle\max_{\rho_{\text{PPT}}}&\displaystyle\tr(\rho_{\text{PPT}}Q_{K})\\\
\text{ subject to }&\displaystyle\rho_{\text{PPT}}\succeq 0\\\
&\displaystyle\rho_{\text{PPT}}^{\Gamma_{2}}\succeq 0.\end{array}$ (61)

By setting $X=\rho_{\text{PPT}}\oplus\rho_{\text{PPT}}^{\Gamma_{2}}$,
$C=Q_{K}\oplus 0$, and, for a Hermitian operator basis $\\{O_{l}\\}_{l}$ of
$\mathcal{H}^{(j_{1})}\otimes\mathcal{H}^{(j_{2})}$,
$A_{l}=O_{l}\oplus(-O_{l}^{\Gamma_{2}})$, Eq. (61) can be rewritten into the
form given in Eq. (59). As such, our problem is an SDP, which can be solved by
any SDP solver—we used JuMP [60] and COSMO [61] for the reported results. The
objective value of the solution to the dual problem gives us a reliable upper
bound
$\mathbf{P}_{K}^{\text{SDP}}\geq\max_{\rho_{\text{PPT}}}\tr(\rho_{\text{PPT}}Q_{K})\geq\mathbf{P}^{\text{sep}}_{K}$.

##  Appendix E Comparison to Other Criteria

In this Appendix, we explicitly show that the states given as examples in Sec.
IV.4 cannot be detected by other angular-momentum-based criteria. For
reference, these are

$\displaystyle\ket{\text{GHZ}_{K}}$
$\displaystyle=\frac{1}{\sqrt{2}}\pqty{\bigotimes_{n=1}^{N}\ket{j_{n},j_{n}}+(-1)^{\frac{K-1}{2}}\bigotimes_{n=1}^{N}\ket{j_{n},-j_{n}}},$
(62) $\displaystyle\ket{\Phi_{5}}$
$\displaystyle=\frac{1}{2}\sqrt{\frac{7}{3}}\pqty{\ket{\uparrow}\otimes\ket{\downarrow}^{\otimes
6}-\ket{\downarrow}\otimes\ket{\uparrow}^{\otimes 6}}$
$\displaystyle\qquad{}-{}\frac{1}{2\sqrt{3}}\pqty{\ket{D_{1}^{7}}-\ket{D_{6}^{7}}},$
$\displaystyle\ket{\Phi_{3}}$
$\displaystyle=\frac{\sqrt{3}}{2}\ket{\mathrm{GHZ}_{9}}-\frac{1}{2\sqrt{2}}\pqty{\ket{D_{3}^{9}}+\ket{D_{6}^{9}}}.$

The subscript in the label denotes the number of measurements $K$ with which
to perform the precession protocol.

It can be verified that these example states are, by construction, eigenstates
of $Q_{K}^{(j)}$, where $Q_{K}=\bigoplus_{j}Q_{K}^{(j)}$ and $Q_{K}^{(j)}$
acts on an irreducible subspace of $\lvert\vec{J}\rvert^{2}$.

In the rest of this section, we will use
$\ket{\Psi_{K}}\in\\{\ket{\text{GHZ}_{K}},\ket{\Phi_{3}},\ket{\Phi_{5}}\\}$ to
discuss properties common to all three states. As we will be discussing them
in some generality, these discussions will also hold true for similar states
with the same properties—for example, other simultaneous eigenstates of
$Q_{K}$ and $\lvert\vec{J}\rvert^{2}$.

###  E.1 Some Properties of These Example States

To derive some general properties of $\ket{\Psi_{K}}$, we highlight two
symmetries of $Q_{K}$: $e^{-i2\pi J_{z}/\hbar K}Q_{K}e^{i2\pi J_{z}/\hbar
K}=Q_{K}$ and $e^{-i\pi J_{x}/\hbar}Q_{K}e^{i\pi J_{x}/\hbar}=Q_{K}$.

These can be verified with direct calculation, but also seen more intuitively:
the chosen rotations applied on $Q_{K}$ merely reshuffles, up to the modulus
of $T$, the times $\\{t_{k}\\}_{k=0}^{K-1}$ over which the average of
$\operatorname{pos}(J_{k})$ is calculated, which leaves $Q_{K}$ invariant.

####  E.1.1 All example states have
$\expectationvalue{\sum_{j\in\mathbf{J}}\vec{J}^{(j)}}=0$

From the previous discussion, we know that $Q_{K}$ commutes with both $e^{i\pi
J_{x}/\hbar}$, and therefore the two observables can be simultaneously
diagonalized. Indeed, the given example states can be verified to also be
eigenstates of $e^{-i\pi J_{x}/\hbar}$ with eigenvalues $\pm(-1)^{j}$.

This leads us to

$\displaystyle\bra{\Psi_{K}}J_{z}^{(j_{n})}\ket{\Psi_{K}}$
$\displaystyle=\bra{\Psi_{K}}e^{-i\pi J_{x}/\hbar}J_{z}^{(j_{n})}e^{i\pi
J_{x}/\hbar}\ket{\Psi_{K}}$ (63)
$\displaystyle=-\bra{\Psi_{K}}J_{z}^{(j_{n})}\ket{\Psi_{K}},$

which implies $\langle{J_{z}^{(j_{n})}}\rangle=0$, and hence
$\langle{\sum_{j\in\mathbf{J}}J_{z}^{(j)}}\rangle=0$. Replacing $J_{z}\to
J_{y}$ also gives $\langle{\sum_{j\in\mathbf{J}}J_{y}^{(j)}}\rangle=0$.

Meanwhile, the commutation and hence simultaneous diagonalizability of $Q_{K}$
and $e^{i2\pi J_{z}/\hbar K}$ similarly lets us choose the example states to
be eigenstates of $e^{-i2\pi J_{z}/K}$ with eigenvalue $e^{-i2\pi m/K}$ for
some $m$. Then, for any integer $k$,

$\displaystyle\bra{\Psi_{K}}{J_{x}^{(j_{n})}}\ket{\Psi_{K}}$ (64)
$\displaystyle\quad{}={}\bra{\Psi_{K}}\pqty{e^{i2\pi
J_{z}/K\hbar}}^{k}J_{x}^{(j_{n})}\pqty{e^{-i2\pi
J_{z}/K\hbar}}^{k}\ket{\Psi_{K}}$ $\displaystyle\quad{}={}\cos(\frac{2\pi
k}{K})\expectationvalue{J_{x}^{(j_{n})}}+\sin(\frac{2\pi
k}{K})\expectationvalue{J_{y}^{(j_{n})}}.$

Taking the sum over $k\in\\{0,1,\dots,K-1\\}$ and $j\in\mathbf{J}$ on both
sides gives $\langle{\sum_{j\in\mathbf{J}}J_{x}^{(j)}}\rangle=0$.

Since $e^{-i2\pi J_{z}/\hbar K}$ commutes with $J_{z}^{(j_{n})}$, we also have

$\displaystyle 0$
$\displaystyle=\bra{\Psi_{K}}{J_{z}^{(j_{n-1})}J_{x}^{(j_{n})}J_{z}^{(j_{n+1})}}\ket{\Psi_{K}}$
(65)
$\displaystyle=\bra{\Psi_{K}}{J_{x}^{(j_{n})}J_{z}^{(j_{n+1})}}\ket{\Psi_{K}}$
$\displaystyle=\bra{\Psi_{K}}{J_{z}^{(j_{n-1})}J_{x}^{(j_{n})}}\ket{\Psi_{K}}$

for the example states.

Relatedly, with $e^{i2\pi J_{z}/\hbar K}J_{\pm}e^{-i2\pi/\hbar K}=e^{\mp
i2\pi/K}J_{\pm}$ for $J_{\pm}\coloneqq J_{x}\pm iJ_{y}$, similar steps imply

$\bra{\Psi_{K}}{J_{\pm}^{2}}\ket{\Psi_{K}}=0.$ (66)

These properties will be useful in later sections.

####  E.1.2 Variances of total angular momentum of example states

The variance $(\Delta J_{s})^{2}\coloneqq\langle J_{s}^{2}\rangle-\langle
J_{s}\rangle^{2}$ for $s\in\\{x,y,z\\}$ are all related to the quantity
$\langle J_{z}^{2}\rangle$ for these example states. Starting with $(\Delta
J_{s})^{2}=\langle J_{s}^{2}\rangle$ from $\langle J_{s}\rangle=0$, and using
$J_{+}J_{-}+J_{-}J_{+}=2\lvert\vec{J}\rvert^{2}-2J_{z}^{2}$, we have

$\displaystyle J_{x}^{2}$
$\displaystyle=\frac{1}{4}\pqty{J_{+}+J_{-}}^{2}=\frac{1}{4}\pqty{J_{+}^{2}+J_{-}^{2}+2\lvert\vec{J}\rvert^{2}-2J_{z}^{2}},$
(67) $\displaystyle J_{y}^{2}$
$\displaystyle=-\frac{1}{4}\pqty{J_{+}-J_{-}}^{2}=\frac{1}{4}\pqty{-J_{+}^{2}-J_{-}^{2}+2\lvert\vec{J}\rvert^{2}-2J_{z}^{2}}.$

Since $\langle{J_{\pm}^{2}}\rangle=0$, we end up with

$\expectationvalue{J_{x}^{2}}=\expectationvalue{J_{y}^{2}}=\frac{1}{2}\expectationvalue{\lvert\vec{J}\rvert^{2}}-\frac{1}{2}\expectationvalue{J_{z}^{2}}$
(68)

for these states.

###  E.2 Nondetection by Other Criteria

Given the above properties, we can now systematically apply each criteria to
the example states.

####  E.2.1 Spin squeezing inequalities

Spin squeezing inequalities are GME witnesses that can be expressed in terms
of the sum or product of variances of spin operators [35]. For a spin ensemble
$\\{j_{n}\\}_{n=1}^{N}$, these inequalities take the general form

$\displaystyle(\Delta u)^{2}+(\Delta v)^{2}$ $\displaystyle\geq\hbar\min
S_{B},$ (69) $\displaystyle\Delta u\Delta v$
$\displaystyle\geq\frac{\hbar}{2}\min S_{B},$ (70)

where $u\coloneqq\sum_{n=1}^{N}h_{n}J_{x}^{(j_{n})}$,
$v\coloneqq\sum_{n=1}^{N}g_{n}J_{y}^{(j_{n})}$, and $h_{n}$ and $g_{n}$ are
real numbers. Meanwhile, the set $S_{B}$ is defined as

$\displaystyle S_{B}$
$\displaystyle\coloneqq\Bigg{\\{}\left|\sum_{j_{\alpha}\in\mathbf{J}}h_{\alpha}g_{\alpha}\expectationvalue{J_{z}^{(j_{\alpha})}}\right|+\left|\sum_{j_{\beta}\in\mathbf{J}^{\complement}}h_{\beta}g_{\beta}\expectationvalue{J_{z}^{(j_{\beta})}}\right|$
$\displaystyle\quad\qquad:\text{$\mathbf{J}$-$\mathbf{J}^{\complement}$ is a
bipartition of $\\{j_{n}\\}_{n=1}^{N}$}\Bigg{\\}}.$ (71)

For the tripartite case, the authors also derived a spin version of an
inequality first introduced for continuous variable systems [62]:

$\min\Bqty{\sum_{n=1}^{3}B_{n},\>\sum_{n=1}^{3}S_{n}}\geq\hbar\absolutevalue{\sum_{n=1}^{3}\expectationvalue{J_{z}^{(j_{n})}}},$
(72)

where

$\displaystyle B_{n}$
$\displaystyle\coloneqq\bqty{\Delta\left(J_{y}^{(j_{n^{\prime}})}+J_{y}^{(j_{n^{\prime\prime}})}+g_{n}J_{y}^{(j_{n})}\right)}^{2}$
(73)
$\displaystyle\qquad\quad{}+{}\bqty{\Delta\left(J_{x}^{(j_{n^{\prime}})}-J_{x}^{(j_{n^{\prime\prime}})}\right)}^{2},$
$\displaystyle S_{n}$ $\displaystyle\coloneqq
2\Delta\left(J_{y}^{(j_{n^{\prime}})}+J_{y}^{(j_{n^{\prime\prime}})}+g_{n}J_{y}^{(j_{n})}\right)$
(74)
$\displaystyle\qquad\quad{}\times{}\Delta\left(J_{x}^{(j_{n^{\prime}})}-J_{x}^{(j_{n^{\prime\prime}})}\right),$

and $(n,n^{\prime},n^{\prime\prime})$ is a permutation of $(1,2,3)$. Violating
any of Eqs. (69), (70), or (72) witnesses GME.

From Appendix E.1.1, we know that $\langle{J_{z}^{(j_{n})}}\rangle=0$ for all
states in Eq. (62), so all inequalities mentioned above become trivial, and no
violation occurs. Hence, these inequalities cannot certify the GME of our
example states.

Equation (69) was later improved by replacing $S_{B}$ on the right-hand side
of the inequality with [36]

$\displaystyle\tilde{S}_{B}\coloneqq$
$\displaystyle\left\\{\left|\sum_{j_{\alpha}\in\mathbf{J}}h_{\alpha}g_{\alpha}\expectationvalue{J_{z}^{(j_{\alpha})}}\right|+\left|\sum_{j_{\beta}\in\mathbf{J}^{\complement}}h_{\beta}g_{\beta}\expectationvalue{J_{z}^{(j_{\beta})}}\right|\right.$
$\displaystyle\qquad{}+{}W^{2}_{\mathbf{J},\mathbf{J}^{\complement}}\left.:\text{$\mathbf{J}$-$\mathbf{J}^{\complement}$
is a bipartition of $\\{j_{n}\\}_{n=1}^{N}$}\right\\},$ (75)

where

$\displaystyle W_{\mathbf{J},\mathbf{J}^{\complement}}$
$\displaystyle\coloneqq\sqrt{\Delta^{2}u_{\mathbf{J}}+\Delta^{2}v_{\mathbf{J}}-\left|\sum_{j_{\alpha}\in\mathbf{J}}h_{\alpha}g_{\alpha}\expectationvalue{J_{z}^{(j_{\alpha})}}\right|}$
$\displaystyle\qquad{}-{}\sqrt{\Delta^{2}u_{\mathbf{J}^{\complement}}+\Delta^{2}v_{\mathbf{J}^{\complement}}-\left|\sum_{j_{\beta}\in\mathbf{J}^{\complement}}h_{\beta}g_{\beta}\expectationvalue{J_{z}^{(j_{\beta})}}\right|}$
(76)

for
$u_{\mathbf{J}}\coloneqq\sum_{j_{\alpha}\in\mathbf{J}}h_{\alpha}J_{x}^{(j_{\alpha})}$
and
$v_{\mathbf{J}}\coloneqq\sum_{j_{\alpha}\in\mathbf{J}}g_{\alpha}J_{y}^{(j_{\alpha})}$,
with $u_{\mathbf{J}^{\complement}}$ and $v_{\mathbf{J}^{\complement}}$
similarly defined. Again with $\langle{J_{z}^{(j_{n})}}\rangle=0$, the
inequality becomes

$\displaystyle\expectationvalue{(u_{\mathbf{J}}+u_{\mathbf{J}^{\complement}})^{2}}+\expectationvalue{(v_{\mathbf{J}}+v_{\mathbf{J}^{\complement}})^{2}}$
$\displaystyle\qquad{}\geq{}\left(\sqrt{\expectationvalue{u_{\mathbf{J}}^{2}}+\expectationvalue{v_{\mathbf{J}}^{2}}}-\sqrt{\langle{u_{\mathbf{J}^{\complement}}^{2}}\rangle+\langle{v_{\mathbf{J}^{\complement}}^{2}}\rangle}\right)^{2},$
(77)

which is also trivially true, so the stronger form of this witness cannot
detect the entanglement of the states under consideration.

####  E.2.2 Spin-half chain criterion

Another GME witness proposed for an $N$-partite spin-half chain involves the
violation of the inequalities [17]

$\displaystyle\expectationvalue{\sum_{n=1}^{N}\tilde{\sigma}_{x}^{(j_{n})}}$
$\displaystyle\leq\frac{N}{2},$
$\displaystyle\sum_{n=1}^{N}\expectationvalue{\tilde{\sigma}_{x}^{(j_{n})}}^{2}$
$\displaystyle\leq\frac{N}{2}$ (78)

for
$\tilde{\sigma}_{x}^{(j_{n})}=\sigma_{z}^{(n-1)}\sigma_{x}^{(j_{n})}\sigma_{z}^{(j_{n+1})}$
and $\sigma_{z}^{(j_{0})}=\sigma_{z}^{j_{N+1}}=\mathbbm{1}$, where
$\sigma_{s}^{(j_{n})}\coloneqq 2J_{s}^{(j_{n})}/\hbar$.

From Eq. (65), $\langle{\tilde{\sigma}_{x}^{(j_{n})}}\rangle=0$ for all states
under consideration, so the above inequalities are trivially satisfied.
Therefore, they cannot be detected by these criteria.

####  E.2.3 Generalized spin squeezing inequalities

Generalized spin squeezing inequalities were first introduced for spin-half
ensembles [19, 20], and later extended to larger spins [21, 22]. For an
ensemble containing $N$ spin-$j$ particles, they take the form

$\displaystyle D_{1}$ $\displaystyle=(\Delta J_{x})^{2}+(\Delta
J_{y})^{2}+(\Delta J_{z})^{2}-N\hbar^{2}j,$ (79) $\displaystyle D_{2}^{(qrs)}$
$\displaystyle=(N-1)(\tilde{\Delta}J_{s})^{2}-\left(\expectationvalue{\tilde{J}_{q}^{2}}+\expectationvalue{\tilde{J}_{r}^{2}}\right)+N(N-1)\hbar^{2}j^{2},$
(80) $\displaystyle D_{3}^{(qrs)}$
$\displaystyle=(N-1)\bqty{(\tilde{\Delta}J_{q})^{2}+(\tilde{\Delta}J_{r})^{2}}-\expectationvalue{\tilde{J}_{s}^{2}}+N(N-1)\hbar^{2}j^{2},$
(81)

where $(q,r,s)$ is a permutation of $(x,y,z)$, and

$\expectationvalue{\tilde{J}_{s}^{2}}\coloneqq\expectationvalue{J_{s}^{2}}-\sum_{n=1}^{N}\expectationvalue{\left(J_{s}^{(j_{n})}\right)^{2}}=2\sum_{n\neq
n^{\prime}}\expectationvalue{J_{s}^{(j_{n})}J_{s}^{(j_{n^{\prime}})}},\qquad\qquad\left(\tilde{\Delta}J_{s}\right)^{2}\coloneqq\expectationvalue{\tilde{J}_{s}^{2}}-\expectationvalue{J_{s}}^{2}.$
(82)

Entanglement is certified by the negativity of any of the following quantities
in Eqs. (79)–(81).

Since $\langle{\vec{J}}\rangle=0$ for the example states, which gives $(\Delta
J_{s})^{2}=J_{s}^{2}$ and
$(\tilde{\Delta}J_{k})^{2}=\langle{(\tilde{J}_{k})^{2}}\rangle$, we have the
simplification

$\displaystyle D_{1}$
$\displaystyle=\expectationvalue{J_{x}^{2}}+\expectationvalue{J_{y}^{2}}+\expectationvalue{J_{z}^{2}}-N\hbar^{2}j=\expectationvalue{\lvert\vec{J}\rvert^{2}}-N\hbar^{2}j$
(83) $\displaystyle D_{2}^{(qrs)}$
$\displaystyle=(N-1)\expectationvalue{\tilde{J}_{s}^{2}}-\left(\expectationvalue{\tilde{J}_{q}^{2}}+\expectationvalue{\tilde{J}_{r}^{2}}\right)+N(N-1)\hbar^{2}j^{2}$
$\displaystyle=N\expectationvalue{\tilde{J}_{s}^{2}}+N(N-1)\hbar^{2}j^{2}-\sum_{r\in\\{x,y,z\\}}\expectationvalue{\tilde{J}_{r}^{2}}$
(84)
$\displaystyle=N\expectationvalue{\tilde{J}_{s}^{2}}+N(N-1)\hbar^{2}j^{2}-\expectationvalue{\lvert{\vec{J}}\rvert^{2}}+\underbrace{\sum_{n=1}^{N}\expectationvalue{\lvert{\vec{J}^{(j_{n})}}\rvert^{2}}}_{N\hbar^{2}j(j+1)}$
$\displaystyle=N\expectationvalue{\tilde{J}_{s}^{2}}+N\hbar^{2}j(Nj+1)-\expectationvalue{\lvert{\vec{J}}\rvert^{2}},$
$\displaystyle D_{3}^{(qrs)}$
$\displaystyle=(N-1)\pqty{\expectationvalue{\tilde{J}_{q}^{2}}+\expectationvalue{\tilde{J}_{r}^{2}}}-\expectationvalue{\tilde{J}_{s}^{2}}+N(N-1)\hbar^{2}j^{2}$
(85)
$\displaystyle=N\pqty{\expectationvalue{\tilde{J}_{q}^{2}}+\expectationvalue{\tilde{J}_{r}^{2}}}+N\hbar^{2}j(Nj+1)-\expectationvalue{\lvert{\vec{J}}\rvert^{2}},$

which depends only on $\langle{\tilde{J}_{s}^{2}}\rangle$ and
$\lvert\vec{J}\rvert^{2}$. Using $\lvert\vec{J}\rvert^{2}\leq
N\hbar^{2}j(Nj+1)$, we also have convenient lower bounds

$D_{2}^{(qrs)}\geq N\expectationvalue{\tilde{J}_{s}^{2}},\qquad
D_{3}^{(qrs)}\geq
N\pqty{\expectationvalue{\tilde{J}_{q}^{2}}+\expectationvalue{\tilde{J}_{r}^{2}}}.$
(86)

Hence, the nonnegativity of $\langle{\tilde{J}_{s}^{2}}\rangle$ for all
$s\in\\{x,y,z\\}$ is sufficient to imply the nonnegativity of $D_{2}^{(qrs)}$
and $D_{3}^{(qrs)}$.

We can now compute $D_{2}^{(qrs)}$ and $D_{3}^{(qrs)}$ for each of our example
states. For the GHZ-like state defined in Eq. (62), which for the $N$-partite
spin-$j$ chain is written as $\ket{\text{GHZ}_{K}}\propto\ket{j,j}^{\otimes
N}+(-1)^{\frac{K-1}{2}}\ket{j,-j}^{\otimes N}$, we have

$\displaystyle\bra{\text{GHZ}_{K}}\lvert{\vec{J}}\rvert^{2}\ket{\text{GHZ}_{K}}$
$\displaystyle=N\hbar^{2}j(Nj+1)$ (87)
$\displaystyle\bra{\text{GHZ}_{K}}{\tilde{J}_{s}^{2}}\ket{\text{GHZ}_{K}}$
$\displaystyle=\sum_{\mu\in\\{+j,-j\\}}\left(\sum_{m\neq
n}\bra{j,\mu}J_{s}^{(j_{m})}\ket{j,\mu}\bra{j,\mu}J_{s}^{(j_{n})}\ket{j,\mu}\right)$
$\displaystyle=\begin{cases}0,&\text{if $s\in\\{x,y\\}$}\\\
N(N-1)\hbar^{2}j^{2}&\text{otherwise.}\end{cases}$

Since
$\bra{\text{GHZ}_{K}}\lvert{\vec{J}}\rvert^{2}\ket{\text{GHZ}_{K}}-N\hbar^{2}j=N^{2}\hbar^{2}j^{2}\geq
0$ and $\bra{\text{GHZ}_{K}}{\tilde{J}_{s}^{2}}\ket{\text{GHZ}_{K}}\geq 0$ for
all $s$, none of the quantities in Eqs. (79)–(81) are negative.

We turn to the example states $\ket{\Phi_{5}}$ and $\ket{\Phi_{3}}$, which are
states of an ensemble of $7$ and $9$ spin-half particles respectively. Since
$(J_{s}^{(j_{n}=1/2)})^{2}=\hbar^{2}\mathbbm{1}/4$ for all $s$, we have
$\sum_{n=1}^{N}\langle({J_{s}^{(j_{n}=1/2)}})^{2}\rangle=N\hbar^{2}/4$.
Meanwhile, by direct computation, we also have

$\bra{\Phi_{5}}\lvert\vec{J}\rvert^{2}\ket{\Phi_{5}}=\frac{35}{4}\hbar^{2},\quad\bra{\Phi_{5}}J_{z}^{2}\ket{\Phi_{5}}=\frac{25}{4}\hbar^{2},\quad\bra{\Phi_{3}}\lvert\vec{J}\rvert^{2}\ket{\Phi_{3}}=\frac{99}{4}\hbar^{2},\quad\bra{\Phi_{3}}J_{z}^{2}\ket{\Phi_{3}}=\frac{63}{4}\hbar^{2}.$
(88)

From this, we already have
$\bra{\Phi_{5}}\lvert\vec{J}\rvert^{2}\ket{\Phi_{5}}-7\hbar^{2}/2\geq 0$ and
$\bra{\Phi_{3}}\lvert\vec{J}\rvert^{2}\ket{\Phi_{3}}-9\hbar^{2}/2\geq 0$,
hence $D_{1}\geq 0$ for these two states. From Eq. (68), we can further work
out

$\bra{\Phi_{5}}J_{x}^{2}\ket{\Phi_{5}}=\bra{\Phi_{5}}J_{y}^{2}\ket{\Phi_{5}}=\frac{\hbar^{2}}{2}\pqty{\frac{35}{4}-\frac{25}{4}}=\frac{5}{4}\hbar^{2},\quad\bra{\Phi_{3}}J_{x}^{2}\ket{\Phi_{3}}=\bra{\Phi_{3}}J_{y}^{2}\ket{\Phi_{3}}=\frac{\hbar^{2}}{2}\pqty{\frac{99}{4}-\frac{63}{4}}=\frac{18}{4}\hbar^{2}.$
(89)

This gives us

$\bra{\Phi_{5}}\tilde{J}_{s}^{2}\ket{\Phi_{5}}=\begin{cases}-\frac{\hbar^{2}}{2}&\text{if
$s\in\\{x,y\\}$}\\\
\frac{9}{2}\hbar^{2}&\text{otherwise,}\end{cases}\qquad\bra{\Phi_{3}}\tilde{J}_{s}^{2}\ket{\Phi_{3}}=\begin{cases}\frac{11}{4}\hbar^{2}&\text{if
$s\in\\{x,y\\}$}\\\ \frac{27}{2}\hbar^{2}&\text{otherwise.}\end{cases}$ (90)

Since $\bra{\Phi_{3}}\tilde{J}_{s}^{2}\ket{\Phi_{3}}\geq 0$, this already
implies that the entanglement of $\ket{\Phi_{3}}$ cannot be witnessed by this
criteria. For $\ket{\Phi_{5}}$, as
$\bra{\Phi_{5}}\tilde{J}_{z}^{2}\ket{\Phi_{5}}\geq 0$ and
$\bra{\Phi_{5}}\tilde{J}_{z}^{2}\ket{\Phi_{5}}+\bra{\Phi_{5}}\tilde{J}_{s}^{2}\ket{\Phi_{5}}\geq
0$ for $s\in\\{x,y\\}$, it is clear that $D_{2}^{(zxy)}$, $D_{3}^{(zxy)}$ and
$D_{3}^{(yzx)}$ will be nonnegative. With further calculations, the other
quantities can be found to be

$D_{2}^{(xyz)}=D_{2}^{(yzx)}=\frac{7}{4}\hbar^{2},\qquad D_{3}^{(xyz)}=0.$
(91)

Therefore, none of the example states can be detected by the generalized spin
inequalities.

####  E.2.4 Energy-based witnesses

For various spin models with a governing Hamiltonian $H$, the minimum energies
$E^{\text{sep}}$ achievable by separable states are known [24]. With that
separable bound, a spin ensemble is detected to be entangled when
$\expectationvalue{H}<E^{\text{sep}}$.

As an example, we consider two specific spin models from the cited paper. In
both spin models, $N$ spins are arranged on the vertices of a $d$-dimensional
cubic lattice with periodic boundary conditions. Every spin pair connected by
an edge is taken to be interacting.

_Heisenberg lattice_. For an anti-ferromagnetic Heisenberg Hamiltonian

$H_{\text{H}}\coloneqq\sum_{s\in\\{x,y,z\\}}\sum_{\langle{j_{n},j_{l}}\rangle}\sigma_{s}^{(j_{n})}\sigma_{s}^{(j_{l})}+B\sum_{n=1}^{N}\sigma_{z}^{(j_{n})},$
(92)

where $\langle{j_{n},j_{l}}\rangle$ are interacting spin pairs and
$\sigma_{s}^{(j_{n})}\coloneqq 2J_{s}^{(j_{n})}/\hbar$, the separable bound
for the energy is

$E^{\text{sep}}_{\text{H}}=\begin{cases}-dN\bqty{\frac{1}{8}\pqty{\frac{B}{d}}^{2}+1}&\text{if
$\left|\dfrac{B}{d}\right|\leq 4$},\\\
-dN\pqty{\absolutevalue{\frac{B}{d}}-1}&\text{otherwise.}\end{cases}$ (93)

Meanwhile, since $\langle\vec{J}\rangle=0$ implies
$\langle{\vec{\sigma}}\rangle=0$ for all example states, the last term of Eq.
(92) vanishes. Hence, we are only left with the
$\sum_{\langle{j_{n},j_{l}}\rangle}\langle{\sigma_{s}^{(j_{n})}\sigma_{s}^{(j_{l})}}\rangle$
terms.

For $\ket{\text{GHZ}_{K}}$ and $\ket{\Phi_{3}}$, notice that they are totally
symmetric under any permutation of the individual spins. So, the expectation
value $\langle{\sigma_{s}^{(j_{n})}\sigma_{s}^{(j_{l})}}\rangle$ is the same
for all interacting spin pairs, and since there are $Nd$ of them,

$\sum_{\langle
j_{n},j_{l}\rangle}\bra{\Psi_{K}}\sigma_{s}^{(j_{n})}\sigma_{s}^{(j_{l})}\ket{\Psi_{K}}=Nd\bra{\Psi_{K}}\sigma_{s}^{(j_{1})}\sigma_{s}^{(j_{2})}\ket{\Psi_{K}}$
(94)

for these two states.

Computing the expectation value
$\langle{\sigma_{s}^{(j_{1})}\sigma_{s}^{(j_{2})}}\rangle$ gives us

$\displaystyle\sum_{\langle
j_{n},j_{l}\rangle}\bra{\text{GHZ}_{K}}\sigma_{s}^{(j_{n})}\sigma_{s}^{(j_{l})}\ket{\text{GHZ}_{K}}$
$\displaystyle=\begin{cases}\frac{Nd}{2}&\text{if $s\in\\{x,y\\}$}\\\
0&\text{otherwise,}\end{cases}$ (95) $\displaystyle\sum_{\langle
j_{n},j_{l}\rangle}\bra{\Phi_{3}}\sigma_{s}^{(j_{n})}\sigma_{s}^{(j_{l})}\ket{\Phi_{3}}$
$\displaystyle=\begin{cases}\frac{Nd}{8}&\text{if $s\in\\{x,y\\}$}\\\
\frac{3Nd}{4}&\text{otherwise.}\end{cases}$ (96)

For the state $\ket{\Phi_{5}}$, since the only possible cubic lattice with $7$
spins is a spin chain, we only need to consider the $d=1$ case. Direct
computation gives

$\displaystyle\sum_{\langle
j_{n},j_{l}\rangle}\bra{\Phi_{5}}\sigma_{s}^{(j_{n})}\sigma_{s}^{(j_{l})}\ket{\Phi_{5}}$
$\displaystyle=\begin{cases}-\frac{1}{3}&\text{if $s\in\\{x,y\\}$}\\\
3&\text{otherwise.}\end{cases}$ (97)

The positivity of the energies for $\ket{\text{GHZ}_{K}}$ and $\ket{\Phi_{3}}$
is evident from Eqs. (95) & (96). Meanwhile, Eq. (97) also gives

$\bra{\Phi_{5}}H_{\text{H}}\ket{\Phi_{5}}=\frac{7}{3}>0.$ (98)

Therefore, since
$E^{\text{sep}}_{\text{H}}<0<\bra{\Psi_{K}}H_{\text{H}}\ket{\Psi_{K}}$, this
method is unable to detect the entanglement of all example states.

_XY model_. For the XY Hamiltonian

$H_{\text{XY}}\coloneqq\sum_{s\in\\{x,y\\}}I_{s}\sum_{\langle{n,l}\rangle}\sigma_{s}^{(j_{n})}\sigma_{s}^{(j_{l})}+B\sum_{n=1}^{N}\sigma_{z}^{(k)},$
(99)

where $\langle{j_{n},j_{l}}\rangle$ are interacting spin pairs and
$\sigma_{s}^{(j_{n})}\coloneqq 2J_{s}^{(j_{n})}/\hbar$, the separable bound
for the energy is

$\displaystyle
E_{\text{XY}}^{\text{sep}}=\begin{cases}-dNI\pqty{1+\frac{B^{2}}{4}}&\text{if
$b\leq 2$}\\\ -dNIb&\text{otherwise,}\end{cases}$ (100)

with $I=\max\\{|I_{x}|,|I_{y}|\\}$ and $b={|B|}/{(Id)}$.

Similar to the case of the Heisenberg lattice, since
$E_{\text{XY}}^{\text{sep}}<0$, Eqs. (95) & (96) already rule out the
possibility of detecting the entanglement of $\ket{\text{GHZ}_{K}}$ and
$\ket{\Phi_{3}}$ using this method.

For the remaining state, Eq. (97) gives

$\displaystyle\bra{\Phi_{5}}H_{\mathrm{XY}}\ket{\Phi_{5}}=-\frac{1}{3}(I_{x}+I_{y})\geq-\frac{2I}{3}.$
(101)

As previously mentioned, the only cubic lattice configuration for $7$ spins is
one-dimensional. Consequently, Eq. (100) implies that
$E_{\text{XY}}^{\text{sep}}\leq-7I<\bra{\Phi_{5}}H_{\mathrm{XY}}\ket{\Phi_{5}}$.
Therefore, this method also fails to detect the entanglement of
$\ket{\Phi_{5}}$.