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# Improved criteria of detecting multipartite entanglement structure

Kai Wu  School of Science, Jimei University, Xiamen 361021,China  Zhihua Chen
School of Science, Jimei University, Xiamen 361021,China  Zhen-Peng Xu  School
of Physics and Optoelectronics Engineering, Anhui University, Hefei 230601,
China  Zhihao Ma  School of Mathematical Sciences, MOE-LSC, Shanghai Jiao Tong
University, Shanghai 200240, China  Shanghai Seres Information Technology Co.,
Ltd, Shanghai 200040, China  Shenzhen Institute for Quantum Science and
Engineering, Southern University of Science and Technology, Shenzhen 518055,
China  Shao-Ming Fei  School of Mathematical Sciences, Capital Normal
University, Beijing 100048, China  Max Planck Institute for Mathematics in the
Sciences, 04103 Leipzig, Germany

###### Abstract

Multipartite entanglement is one of the crucial resources in quantum
information processing tasks such as quantum metrology, quantum computing and
quantum communications. It is essential to verify not only the multipartite
entanglement, but also the entanglement structure in both fundamental theories
and the applications of quantum information technologies. However, it is
proved to be challenging to detect the entanglement structures, including
entanglement depth, entanglement intactness and entanglement stretchability,
especially for general states and large-scale quantum systems. By using the
partitions of the tensor product space we propose a systematic method to
construct powerful entanglement witnesses which identify better the
multipartite entanglement structures. Besides, an efficient algorithm using
semi-definite programming and a gradient descent algorithm are designed to
detect entanglement structure from the inner polytope of the convex set
containing all the states with the same entanglement structure. We demonstrate
by detailed examples that our criteria perform better than other known ones.
Our results may be applied to many quantum information processing tasks.

##  I Introduction

Multipartite quantum entanglement is crucial in quantum communication and
information processing tasks including quantum computing, quantum cryptography
and quantum teleportation [1]. Very different from the bipartite case,
multipartite quantum entanglement has a rich structure. Two important concepts
have been introduced to characterize such multipartite entanglement structure,
i.e., the entanglement producibility and entanglement partitionability [2, 3].
Intuitively for pure states, the partitionability refers to the number of
subsystems separable from each other in the multipartite system, while the
entanglement producibility, refers to the size of the largest entangled
subsystem. Recently, the entanglement stretchability has been coined [4],
which stands for the difference between the entanglement producibility and
entanglement partitionability.

Quantum entanglement structure has many applications, e.g., in spin chains and
quantum metrology [5, 3, 6, 7]. However, as the number of subsystems grows,
verifying the entanglement structure becomes a hard problem [2, 3]. Though one
can detect theoretically the entanglement structure via state tomography, it
becomes infeasible in practice due to the exponential growth of the dimensions
of the underlying Hilbert space. Significant efforts have been devoted in this
direction to develop feasible techniques, such as the utilization of more
convenient inequalities given by spin-squeezing and the entries of density
matrices, local uncertainty relations, Wigner-Yanase skew information, Fisher
information and machine learning[1, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 6, 21, 17, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Gradually,
entanglement structure has also been detected experimentally [32, 33, 34, 35,
36, 37, 38, 39] for special cases, like for Dicke states [33].

Recently, a systematic approach based on Young diagrams and the partitions of
the tensor product space has been proposed to classify the entanglement
structures[4, 40]. Based on these works, we propose a systematic method to
construct linear criteria to verify the quantum entanglement structures in
this paper, where the number of measurements needed grows polynomially with
the system size in experiments.

On the other hand, an algorithm based on the adaptive polytope approximation
has been proposed to certify quantum separability of bipartite and
multipartite quantum systems [41], each iteration is a semidefinite
programming (SDP). This algorithm updates each polytope in every iteration to
ultimately achieve a satisfactory result, and depends on well-chosen initial
polytopes so as to obtain a good enough result. Inspired by this algorithm, we
proposed a new algorithm by employing the gradient descent method, which does
not depend very much on the initial polytopes in obtaining a good enough
result.

##  II PRELIMINARY

A partition of a set is to classify each element into one and only one subset.
For two partitions $\gamma$ and $\xi$, we say $\gamma$ is a refinement of
$\xi$, denoted as $\gamma\preceq\xi$, if any subset in $\gamma$ in a subset of
the one in $\xi$. For a given pure state $|\psi\rangle$ in the form that
$|\psi\rangle=\bigotimes_{i}|\psi_{X_{i}}\rangle$ where $\\{X_{i}\\}$ is a
partition of the set of all parties, its entanglement structure can be
represented by this partition. If an entanglement structure is representable
by a partition $\gamma$ and $\gamma$ is a refinement of $\xi$, then it is also
representable by the partition $\xi$.

Denote $|S|$ the size of the set $S$. For a given pure state
$|\psi\rangle=\bigotimes_{i}|\psi_{X_{i}}\rangle$ and $\max|X_{i}|\leq k$,
where $\\{X_{i}\\}$ is a partition of the set of all parties, $|\psi\rangle$
is said to be

  1. 1.

$k$-producible, if $\max|X_{i}|\leq k$;

  2. 2.

$k$-partitionable, if $|\\{X_{i}\\}|\leq k$;

  3. 3.

$k$-stretchable, if $\max|X_{i}|-|\\{X_{i}\\}|\leq k$.

A mixed state $\rho$ is called $k$-producible if it can be decomposed as a
convex combination of $k$-producible pure states. Similarly, the definition of
$k$-partitionable, or $k$-stretchable can be extended to mixed states.

Figure 1:  Seven partitions of $\\{1,2,3,4\\}$ are $\gamma_{1}=1|2|3|4$,
$\gamma_{2}=1|2|3,4$, $\gamma_{3}=1,3|2|4$, $\gamma_{4}=1,2|3|4$,
$\gamma_{5}=1,3,4|2$, $\gamma_{6}=1,2|3,4$ and $\gamma_{7}=1,2,3|4$. The
arrows in the diagram denote the refinement relationship, denoted by
$\preceq$. For the partition set $\Gamma$ consists of all seven partitions,
$\Gamma^{\max}=\\{\gamma_{5},\gamma_{6},\gamma_{7}\\}$.

Given a collection $\Gamma$ of partitions, the set of maximal elements under
the partial order of refinement $\preceq$ is denoted $\Gamma^{\max}$. To
continue, we introduce further notations:

$\displaystyle\Gamma_{k-\text{part}}=\\{\xi=\\{X_{i}\\}:|\\{X_{i}\\}|\leq
k\\},$ (1)
$\displaystyle\Gamma_{k-\text{prod}}=\\{\xi=\\{X_{i}\\}:\max|X_{i}|\leq k\\},$
(2)
$\displaystyle\Gamma_{k-\text{str}}=\\{\xi=\\{X_{i}\\}:\max|X_{i}|-|\\{X_{i}\\}|\leq
k\\}.$ (3)

Consequently, $\Gamma_{k-\text{prod}}^{\max}$, $\Gamma_{k-\text{part}}^{\max}$
and $\Gamma_{k-\text{str}}^{\max}$ can be defined. If the entanglement
structure of a pure quantum state is representable by a partition within
$\Gamma$, it is necessarily also representable by a partition within
$\Gamma^{\max}$. An example is depicted in Fig. 1 with $\Gamma$ to be the set
of all the partitions of $\\{1,2,3,4\\}$.

##  III Detection of entanglement structure

Consider a direct product quantum state pairs $(|m\rangle,|n\rangle)$ with
$|m\rangle=\bigotimes_{i=1}^{N}|m_{i}\rangle$,
$|n\rangle=\bigotimes_{i=1}^{N}|n_{i}\rangle$, where $|m_{i}\rangle$ and
$|n_{i}\rangle$ are the standard orthogonal basis of $\mathcal{H}_{i}$. We
have the following result, which pertains to an inequality involving a off-
diagonal element of a density matrix and two of its diagonal elements:

######  Proposition 1.

For any pure state
$|\psi\rangle=\bigotimes_{X_{i}\in\gamma}|\psi_{X_{i}}\rangle$ and $\forall
X_{i}\in\gamma$, we have

$\displaystyle\begin{aligned} |\langle
m|\psi\rangle\langle\psi|n\rangle|\leq&\frac{1}{2}[(\langle
m_{\bar{X}_{i}}|\langle
n_{X_{i}}|)|\psi\rangle\langle\psi|(|m_{\bar{X}_{i}}\rangle|n_{X_{i}}\rangle)+\\\
&(\langle m_{X_{i}}|\langle
n_{\bar{X}_{i}}|)|\psi\rangle\langle\psi|(|m_{X_{i}}\rangle|n_{\bar{X}_{i}}\rangle)],\end{aligned}$
(4)

where $\bar{X}_{i}$ stands for the complement set of the set $X_{i}$.

The proof is given in Appendix I. Violation of the inequality in Eq. (4) means
that $|\psi\rangle\neq\bigotimes_{X_{i}\in\gamma}|\psi_{X_{i}}\rangle$.

Based on the inequality in Eq. (4), we continue to construct further criteria
involving more off-diagonal elements.

###  III.1 Criteria based on the outer polytopes

Firstly, we take a $3$-qubit pure quantum state
$\rho=|\psi\rangle\langle\psi|$ to illustrate the whole procedure. Consider
the set of off-diagonal elements $P=\\{\rho_{2,3},\rho_{2,5},\rho_{3,5}\\}$
and the partition $12|3$. If
$|\psi\rangle=|\psi_{12}\rangle\otimes|\psi_{3}\rangle$, a sequence of
inequalities can be derived from Proposition 1 as follows,

$\displaystyle\begin{cases}|\rho_{2,3}|-\frac{1}{2}\rho_{1,1}-\frac{1}{2}\rho_{4,4}\leq
0,\\\ |\rho_{2,5}|-\frac{1}{2}\rho_{1,1}-\frac{1}{2}\rho_{6,6}\leq 0,\\\
|\rho_{3,5}|-\frac{1}{2}\rho_{3,3}-\frac{1}{2}\rho_{5,5}\leq 0.\end{cases}$
(5)

Besides, any diagonal element of the state $\rho$ is non-negative and the sum
of the appeared diagonal elements is no more than $1$. These inequalities
define a series of closed half-spaces, whose intersection results in a convex
polytope [42], denoted by $\Theta_{12|3}(P)$. Thus, if $|\psi\rangle$ can be
decomposed as the tensor product $|\psi_{12}\rangle\otimes|\psi_{3}\rangle$,
we must have the corresponding elements of the density operator
$\rho=|\psi\rangle\langle\psi|$ to be in the polytope $\Theta_{12|3}(P)$.
Consequently, any mixed state of the same entanglement structure is also
contained in $\Theta_{12|3}(P)$.

Similarly, we could construct another two polytopes $\Theta_{1|23}(P)$ and
$\Theta_{2|13}(P)$ for the partitions $1|23$ and $2|13$, respectively. Denote
$\Theta_{2-\text{prod}}(P)$ the convex hull of all the points in
$\Theta_{12|3}(P)$, $\Theta_{1|23}(P)$ and $\Theta_{2|13}(P)$. Any mixed
$2$-producible states must be in $\Theta_{2-\text{prod}}(P)$.

In general, given a partition set $\Gamma$ and a collection $P$ of non-
diagonal elements from a density matrix, Proposition 1 can be employed to
derive inequalities that define a convex polytope $\Theta_{\gamma}(P)$ for
each maximum partition $\gamma$ within $\Gamma$, i.e., each partition in
$\Gamma^{\max}$. The smallest convex polytope that encompasses all
$\Theta_{\gamma}(P)$ is referred to as $\Theta_{\Gamma}(P)$. The set $\Gamma$
can be either $\Gamma_{k-\text{prod}}$, $\Gamma_{k-\text{part}}$, or
$\Gamma_{k-\text{str}}$, each capturing a distinct aspect of the entanglement
structure. The explicit result is summarized as follows.

######  Theorem 2.

Let $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$ be an $N$-partite
state with $\sum_{i}p_{i}=1$ and $p_{i}>0$. If there exists $\xi_{i}\in\Gamma$
such that
$|\psi_{i}\rangle=\bigotimes_{X_{j}\in\xi_{i}}|\psi_{i,{X_{j}}}\rangle$ for
all $|\psi_{i}\rangle$, we have

$\rho\in\Theta_{\Gamma}(P),$ (6)

where $\Theta_{\Gamma}(P)$ is the polytope constructed from a subset of all
possible inequalities that can be formulated from $\Gamma^{\max}$.

In practice, one can employ the Polyhedra.jl package [43] in Julia to carry
out operations on polytope such as intersection and convex hull of several
polytopes. An example with the mixed $4$-qubit W-state $\rho_{W_{4}}$ is
presented in Section III.D.

###  III.2 Criteria based on the hyperplanes

Due to the fact that the computational complexity of the polytope-based
criteria increases fastly as the dimension grows, we propose a simplified
criterion based on hyperplanes, which offers a more friendly computational
approach.

Similar as in Subsection A, we still consider the set of non-diagonal elements
$P=\\{\rho_{2,3},\rho_{2,5},\rho_{3,5}\\}$ of a $3$-qubit pure quantum state
$\rho=|\psi\rangle\langle\psi|$, and the partitions in $\\{12|3,1|23,2|13\\}$.
By simply summing up all the inequalities within $\Theta_{12|3}(P)$ in Eq.
(5), a unified inequality is created. Similar inequalities can be obtained by
applying this method to $\Theta_{2|13}(P)$ and $\Theta_{1|23}(P)$. Those three
new inequalities are as follows:

$\displaystyle\begin{cases}\sum\limits_{\rho_{m,n}\in
P}|\rho_{m,n}|\leq\frac{1}{2}(2\rho_{1,1}+\rho_{6,6}+\rho_{7,7}+\rho_{2,2}+\rho_{3,3}),\\\
\sum\limits_{\rho_{m,n}\in
P}|\rho_{m,n}|\leq\frac{1}{2}(2\rho_{1,1}+\rho_{4,4}+\rho_{7,7}+\rho_{2,2}+\rho_{5,5}),\\\
\sum\limits_{\rho_{m,n}\in
P}|\rho_{m,n}|\leq\frac{1}{2}(2\rho_{1,1}+\rho_{4,4}+\rho_{6,6}+\rho_{3,3}+\rho_{5,5}).\end{cases}$
(7)

The right-hand side of the set of inequalities can be formulated as
$M\cdot\vec{\beta}^{T}$, where the row vector $\vec{\beta}$ contains all
diagonal elements of $\rho$, and $M$ is a coefficient matrix. More explicitly,

$\vec{\beta}=[\rho_{1,1},\rho_{2,2},\rho_{3,3},\rho_{4,4},\rho_{5,5},\rho_{6,6},\rho_{7,7},\rho_{8,8}],$
$M=\begin{bmatrix}1&1/2&1/2&0&0&1/2&1/2&0\\\ 1&1/2&0&1/2&1/2&0&1/2&0\\\
1&0&1/2&1/2&1/2&1/2&0&0\\\ \end{bmatrix}.$

By applying a column-wise maximum operation to matrix $M$, we obtain a vector
$\vec{\alpha}$:

$\vec{\alpha}=\begin{bmatrix}1&1/2&1/2&1/2&1/2&1/2&1/2&0\end{bmatrix}.$

Consequently, we have a new inequality

$\sum\limits_{\rho_{m,n}\in P}|\rho_{m,n}|\leq\vec{\alpha}.\vec{\beta}^{T},$
(8)

which is a relaxation of all the three inequalities in Eq. (7). More
explicitly, the new derived criterion is

$\displaystyle|\rho_{2,3}|+|\rho_{2,5}|+|\rho_{3,5}|$ $\displaystyle\leq\ $
$\displaystyle\rho_{1,1}+(\rho_{2,2}+\rho_{3,3}+\rho_{4,4}+\rho_{5,5}+\rho_{6,6}+\rho_{7,7})/2.$
(9)

In general, for a given set of non-diagonal elements denoted as $P$ and the
partition set $\Gamma$, we can only consider all the partitions in
$\Gamma^{\max}$ and carry out the same procedure to obtain a new criterion.
The main steps are summarized as follows.

  1. 1.

For each partition $\gamma\in\Gamma^{\max}$, and $\rho_{m,n}\in P$, we obtain
from Proposition 1 the inequalities that

$|\rho_{m,n}|\leq(\rho_{d_{\gamma,1},d_{\gamma,1}}+\rho_{d_{\gamma,2},d_{\gamma,2}})/2,$
(10)

where $d_{\gamma,1}$ and $d_{\gamma,2}$ depend on $m,n$ and the partition
$\gamma$. We remark that there could be different inequalities for the same
pair of $(m,n)$.

  2. 2.

By choosing one inequality as in Eq. (10) for each $\rho_{m,n}\in P$ and a
given partition $\gamma$, we obtain

$\sum_{\rho_{m,n}\in P}|\rho_{m,n}|\leq\vec{m}_{\gamma}.\vec{\beta}^{T},$ (11)

where $\vec{m}_{\gamma}$ corresponds to a row in the matrix $M$.

  3. 3.

By applying a column-wise maximum operation to matrix $M$, we obtain a vector
$\vec{\alpha}$:

$\sum_{\rho_{m,n}\in P}|\rho_{m,n}|\leq\vec{\alpha}.\vec{\beta}^{T},$ (12)

The partition set $\Gamma$ can correspond to either $\Gamma_{k-\text{prod}}$,
$\Gamma_{k-\text{part}}$ or $\Gamma_{k-\text{str}}$.

Since there could be different options at the second step, the goal to
construct the vector $\vec{\alpha}$ is to minimize the sum of all elements in
vector $\vec{\alpha}$. To achieve this goal, we can follow the principle to
have as many common non-zero columns as possible for each row in the matrix
during the construction of $M$, i.e., the choice of inequality for each pair
$(m,n)$. In other words, we aim to reduce the number of appeared non-zero
columns. This strategy helps us to effectively utilize the structure of matrix
$M$ while keeping the sum of the elements in vector a minimal. This approach
is beneficial for our optimization process.

Further elaboration on the optimization process is provided in Appendix II.
The greedy algorithm is implemented to obtain the optimal inequalities, see
the codes listed in [44].

The left hand side of the inequality (12) can be replaced by its lower bound
$\sum\limits_{\rho_{m,n}\in P}\frac{1}{2}(\rho_{m,n}+\rho_{n,m})$, which can
be reresented by the expectation value of the observable
$W_{P}=\frac{1}{2}|m\rangle\langle n|+|m\rangle\langle n|$. On the other hand,
the right hand side of (12) can be represented the expectation value of the
observable $W=diag\\{\vec{\alpha\\}}$. Our hyperplane-based criteria is
experimentally friendly, which can be implemented by measuring the two
observables without state tomography. The two observables can be realized by
using local observables and the number of the required local observables
scales polynomially with the system size. In comparison, to realize full state
tomography, the number of the required local observables grows exponentially.

###  III.3 Criteria based on the inner polytopes

An SDP-based method was also developed to create evolving polytopes, whose
nodes are different kinds of separable states, for confirming quantum
entanglement [41]. Besides, a modified version of Gilbert’s algorithm was
recently used to solve the convex membership problem [45, 46]. The performance
of this SDP-based algorithm depends on the choice of the initial polytope, and
the current algorithm generates the initial polytope randomly. As the
complexity increases with the increasing system, inspired by the previous
works[41, 45, 46], we propose a systematic approach in the following to select
the initial polytope so as to guarantee the efficiency and performance.

We introduce a new algorithm to certify the entanglement structure of the
mixed state $\rho(t)=t\rho+(1-t)\mathbb{I}_{d}/d$ for given $\rho$.
Especially, we are interested in the minimal $t$ such that $\rho(t)$ still
have the targeted entanglement structure. Our algorithm involves two stages as
illustrated in Fig. 2. Different from [41], initially, we employ gradient
descent at the first stage, which is followed by SDP at the second stage. The
gradient descent is conducted within the Pytorch framework, while the SDP is
facilitated through the PICOS [47] library. The whole procedure can be
summarized as follows.

  1. 1.

Initialize an arbitrary polytope $\mathcal{P}$ determined by a set of states
$\\{\varrho_{i}\\}$ of the entanglement structure corresponding to a partition
$\gamma$, and a set of probabilities $\\{p_{i}\\}$, where $\sum_{i}p_{i}=1$.
An example is illustrated in Fig. 2(a).

  2. 2.

Employing the gradient descent algorithm, we minimize the geometric distance
from the state $\varrho=\sum_{i}p_{i}\varrho_{i}$ to the line segment $l$
between $\rho$ and the maximally mixed state, as shown in Fig. 2(b).

A predefined threshold $r$ serves as the criterion for progression to the next
step, ensuring a systematic approach to state optimization.

  3. 3.

Using gradient descent to minimize the geometric distance between the state
$\varrho$ and $\rho$, ensuring that the polytope’s boundaries progressively
converge to $\rho$, as illustrated in Fig. 2(c). The optimization process is
directed by a loss function
$\lVert\rho-\varrho\rVert=\sqrt{\textrm{tr}\left[(\rho-\varrho)^{{\dagger}}(\rho-\varrho)\right]}$,
with the initial polytope configuration determined by the results of step 2.
If the distance, as defined in step 2, exceeds the predetermined threshold
$r$, the algorithm reverts to step 2 to ensure that the distance between
$\varrho$ and the line segment $l$ remains within the specified limit.

  4. 4.

Finally, we use SDP to approximate the optimal $t_{0}$. The initial points are
chosen as the results in step 3, but we choose one subsystem from each vertex
of $\mathcal{P}$ as the unknown one, which is optimized using SDP. Other
subsystems except this one from each vertex are denoted as a new set
$\mathcal{P}^{\prime}$. We do this for each subsystem iteratively. The details
of the SDP are as follows:

$\displaystyle\begin{aligned} \max&\ \ t\\\ \mathrm{w.r.t.}&\ \
t,\tau_{i}\succcurlyeq 0\\\ \mathrm{s.t.}&\ \
\rho(t)=\sum_{\varrho_{i}\in\mathcal{P}^{\prime}}\varrho_{i}\otimes\tau_{i},\end{aligned}$
(13)

where the trace of $\tau_{i}$ is not necessarily one.

Figure 2:  (a) Initialise an arbitrary polytope
$\mathcal{P}=\\{\varrho_{i}\\}$ and $\varrho=\sum_{i}P_{i}\varrho_{i}$. (b)
Minimize the distance between the quantum state $\varrho$ and the segment $l$.
(c) Minimize the distance between the quantum state $\varrho$ and $\rho$.

While step 2 permits the convergence of $\varrho$ towards the line segment
$l$, the intersection of the polytope $\mathcal{P}$ with $l$, and consequently
the feasibility of SDP, cannot be assured. To improve on this point, we
optimize the quantum subsystems corresponding to the largest partition of each
node via SDP at each iteration. For fully separable states and certain
entanglement structures where SDP is infeasible, we can also construct an
initial polytope using only states derived from mutually-unbiased bases
(MUBs), which not only covers a substantial volume but also encapsulate the
maximally mixed state. More explicitly for the case that the local dimension
is $2$, the constraint condition in Eq. (13) transforms into the following
expression:

$\rho(t)=\sum_{\varrho_{i}\in(\mathcal{P}^{\prime}\bigcup\mathcal{P}_{N-1})}\varrho_{i}\otimes\tau_{i},$

where ${\cal P}_{N-1}$ is defined as follows:

$\big{\\{}\bigotimes_{i=1}^{N-1}\rho_{i},\rho_{i}\in\big{\\{}\frac{(\mathbb{I}_{2}\pm\sigma_{x})}{2},\frac{(\mathbb{I}_{2}\pm\sigma_{y})}{2},\frac{(\mathbb{I}_{2}\pm\sigma_{z})}{2}\big{\\}}\big{\\}}.$
(14)

Since $\mathcal{P}_{N-1}$ contains the maximally mixed state, the feasibility
of SDP can be guaranteed.

###  III.4 Applications of method with inner polytopes

Firstly, we study the full separability of the states
$\rho(t)=t\rho+(1-t)\mathbb{I}_{d}/d$, where $\rho$ is any state up to six
qubits. The results are computed by the algorithm based on the inner polytope.
We compare the results with the known ones in the literature in Table 1.

Table 1:  Comparison of the full separability computed by our method and the ones by adaptive polytopes [41]. The “-” in the table signifies that [41] either did not provide the results or the code provided by [41] was unable to perform the calculations.  Quantum States | Our result | Lower bound of [41]
---|---|---
GHZ(6 Qubits) | 0.030303 | -
W(6 Qubits) | 0.02345 | -
Cluster state (6 Qubits) | 0.030303 | -
GHZ(5 Qubits) | 0.05882 | 0.05878
W(5 Qubits) | 0.047084 | 0.046956
Cluster state (5 Qubits) | 0.05882 | -
GHZ(4 Qubits) | 0.1111 | 0.1111
W(4 Qubits) | 0.0926 | 0.0926
Cluster state (4 Qubits) | 0.1111 | 0.1111
Dicke (4 Qubits, 2 ex.) | 0.085708 | 0.08571

For the $4$ qubit, $5$-qubit and $6$-qubit noisy GHZ states, the results
computed by our algorithm based on the inner polytope give the necessary and
sufficient values of the full separability, which are $1/9$, $1/17$ and
$1/33$, respectively [14]. Our method performs better compared with the
algorithm in [41] in most cases.

Then, we utilize our algorithm based on the inner polytope to compute the
entanglement parititionability and the entanglement producibility of $\rho(t)$
with $\rho$ to be 4-qubit W state, 4-qubit and 5-qubit GHZ states in Tables 2,
3 and 4 as follows.

Table 2:  Comparison of the partitionability and the producibility computed by our algorithm and the known ones in the literature for the mixture of 4-qubit W state and white noise.  Entanglement Structure | Our result | Other known results
---|---|---
3-part | 0.247 | 0.243[45]
2-prod | 0.247 | 0.245[45]
2-part | 0.4705 | 0.474[8]
Table 3:  Comparison of the partitionability and the producibility computed by our algorithm and the known ones in the literature for the mixture of 4-qubit GHZ state and white noise.  Entanglement Structure | Our result | Other known results
---|---|---
3-part | 0.19997 | 0.198[45]
2-prod | 0.27251 | 0.269[45]
2-part | 0.46456 | 0.467[8](exact)

Table 2 shows that our results based on inner polytope perform better than the
results in [45] in most cases for $4$-qubit noisy W state. Table 3 shows the
bound corresponding to $3$-partitionablility is $0.19997$ for $4$-qubit noisy
GHZ state, which approaches the necessary and sufficient value $p=0.2$ [14].
Besides, the bound $0.2725$ for $2$-produciblity is better than the one in
[45].

Table 4:  Comparison of the partitionability and the producibility computed by our algorithm and the known ones in the literature for the mixture of 5-qubit GHZ state and white noise.  Entanglement Structure | Our result | Other known Results
---|---|---
4-part | 0.09434 | 0.0943 [14]
2-prod | 0.2375 | 0.19[45]
3-part | 0.2377 | 0.2380[14]
3-prod | 0.3846 | 0.278[45]
2-part | 0.48378 | 0.4839[14]

Table 4 is for $5$-qubit noisy GHZ state. Our results of
$2/3/4$-partitionablity are very close to the optimal known values given in
[14]. For the producibility, our results are better compared with the ones in
[45]. From [17] the exact bound is no more than $0.238$ for $2$-producibility
of $\rho(t)$, which is close to our bound $0.2375$.

Besides, our algorithm can provide the exact decomposition of
$k$-partitionable (producible, or stretchable) states. By calculation the
fidelity between the state $\rho(t)$ and the decomposition of the state
obtained by our algorithm is $99.999\%$. Thus the precision of our algorithm
is guaranteed.

Furthermore, we can detect the entanglement structure of any state $\rho$
mixed with any fully separable states. For example, we consider
$\rho(t)=t|W\rangle\langle W|+(1-t)\rho_{f}$, where
$|W\rangle=(|0001\rangle+|0010\rangle+|0100\rangle+|1000\rangle)/2$ and
$\rho_{f}=\bigotimes\limits_{i=1}^{4}\tau_{i}$ with
$\tau_{i}=(3|0\rangle\langle 0|+|1\rangle\langle 1|)/4$. Our algorithm shows
that $\rho(t)$ is fully separable when $t<0.1468$ and 2-partitionable if
$t<0.58$, while $\rho(t)$ is fully separable when $t<0.1464$ and
2-partitionable for $t<0.5623$ in [45].

To guarantee the universality of our algorithm, in our algorithm the initial
states are optimized by the gradient conjugate method. We only optimize one
partite states, given that the other partite states are initialized. In this
way, the runtime of our algorithm could be slightly longer than that in [41].

###  III.5 Applications of method with hyperplane

To illustrate our results by the witness based on the hyperplane, we give the
following examples.

Example 1. Consider the family of $4$-qubit quantum states,

$\rho_{W_{4}}=(1-t)\frac{\mathbb{I}}{2^{4}}+t|W_{4}\rangle\langle W_{4}|,$

with
$|W_{4}\rangle=\frac{1}{2}(|0001\rangle+|0010\rangle+|0100\rangle+|1000\rangle)$.

We select $P$ as the one given in Appendix III and obtain that if $\rho$ is
$2$-producible or $3$-partitionable, $0$-stretchable or $(-1)$-stretchable,
the entries of $\rho$ fulfill the following inequality,

$\frac{1}{2}|\textrm{Tr}[W_{P}\cdot\rho]|\leq\textrm{Tr}[W_{2-\text{prod}}\cdot\rho],$
(15)

where $W_{P}$ select the targeted off-diagonal elements of $\rho$, and
$W_{2-\text{prod}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal matrix
with $a_{1}=2$ and
$a_{2}=a_{3}=a_{4}=a_{5}=a_{6}=a_{7}=a_{9}=a_{10}=a_{11}=a_{13}=1/2$.

If $\rho$ is $3$-producible or $2$-partitionable, the entries of $\rho$
satisfy

$\frac{1}{2}|\textrm{Tr}[W_{P}\cdot\rho]|\leq\textrm{Tr}[W_{3-\text{prod}}\cdot\rho],$
(16)

where $W_{3-\text{prod}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with $a_{1}=2$, $a_{2}=a_{3}=a_{5}=a_{9}=1$ and
$a_{4}=a_{6}=a_{7}=a_{10}=a_{11}=a_{13}=1/2$. The witness in [38] can detect
$2$ ($3$ and $4$)-partite entanglement when $p>0.592$ ($p>0.729$ and
$p>0.873$) while our theorem can detect $2$ ($3$ and $4$)-partite entanglement
when $p>0.2$ ($p>0.304$ and $p>0.529$). The results compared with the best
result so far in [17] are illustrated in Fig. 3. The details can be found in
Appendix III.

Figure 3:  The illustration shows the results obtained by using our criteria
and Theorem 2 of Ref. [17]((I) in the figure). to detect the entanglement
depth.

Example 2. Now we consider the family of 4-qutrit quantum states,

$\rho(p,q)=p|W\rangle\langle W|+q\sigma^{\otimes 4}|W\rangle\langle
W|\sigma^{\otimes 4}+\frac{1-p-q}{3^{4}}\mathbb{I},$

where
$|W\rangle=\frac{1}{2\sqrt{2}}(\sum\limits_{i=1}^{2}|i000\rangle+|0i00\rangle+|00i0\rangle+|000i\rangle)$,
and $\sigma|0\rangle=|1\rangle$, $\sigma|1\rangle=|2\rangle$,
$\sigma|2\rangle=|0\rangle$. When $p>14(1-q)/95$ and $q>14(1-p)/95$, the two
inequalities (23) in Appendix III are violated and $\rho$ is genuinely
$3$-partite entangled. The results are compared with the ones in [17], which
are the best results so far in FIG. 4.

Figure 4:  The red line consists of the line $q=\frac{14}{95}(1-p)$,
$p\in[0,\frac{14}{109}]$ and the line $p=\frac{14}{95}(1-q)$,
$q\in[0,\frac{14}{109}]$. The area enclosed by the red line, $p$ axis, line
$q=1-p$ and the $q$ axis corresponds to the quantum states containing genuine
$3$-partite entanglement detected by our Theorem II. The blue line consists of
the line $q=\frac{16}{97}(1-p)$, $p\in[0,\frac{16}{113}]$ and the line
$p=\frac{16}{97}(1-q)$, $q\in[0,\frac{16}{113}]$. The area enclosed by the
blue line, $p$ axis, line $q=1-p$ and the $q$ axis corresponds to the quantum
states containing genuine $3$-partite entanglement detected by the Theorem 2
of Ref.[17].

For the $W$ states mixed with white noise in Example 1 and Example 2, our
inequalities perform also better compared with the results in [17].

Example 3. Consider the $5$-qubit mixed states,

$\rho_{CL_{5}^{s}}=(1-t)\frac{\mathbb{I}}{2^{5}}+t|CL_{5}^{s}\rangle\langle
CL_{5}^{s}|,$

where
$|CL_{5}^{s}\rangle=\frac{1}{2}(|00000\rangle+|01111\rangle+|10011\rangle+|11100\rangle)$.

The results are illustrated in Fig. 5, see also the details given in Appendix
III. The results in [23] show that $\rho_{CL_{5}^{s}}$ is not
$5$-partitionable ($4$\- or $3$-partitionable) when $t>{7}/{13}$ while the
ones by our criteria are $t>{1}/{9}$ ($t>{5}/{29}$ or $t>{9}/{25}$), and
$\rho_{CL_{5}^{s}}$ is $2$-partionable or genuine multipartite entangled when
$t>{9}/{13}$, while $t>{13}/{29}$ by our method.

Figure 5:  Illustration of the results obtained by using our criteria in
detecting the entanglement depth and intactness of $\rho_{CL_{5}^{s}}$.

Example 4. Consider the $6$-qubit mixed states,

$\rho_{GHZ_{6}}=(1-t)\frac{\mathbb{I}}{2^{6}}+t|GHZ_{6}\rangle\langle
GHZ_{6}|,$

where $|GHZ_{6}\rangle=\frac{1}{\sqrt{2}}(|000000\rangle+|111111\rangle)$.

In this case, the witness in [38] can detect $2$-partitional entanglement only
when $t>{32}/{63}$, the genuine entanglment only when $p>{31}/{47}$, $2$
($3$,$4$,$5$ and $6$)-partite entanglement when $t>0.366$
($t>0.448$,$p>0.522$,$0.556$ and $p>0.649$). The witness in [23] can only
detect the genuine mutipartite entanglement of $\rho_{GHZ_{6}}$ when
$p>{31}/{47}$. While our criteria can detect the genuine entanglement when
$p>{31}/{63}$, and 2(3,4, and 5)-partite entanglement when $p>{1}/{33}$
($p>{5}/{37}$, $p>{15}/{47}$, and $p>{25}/{57}$). The results are compared
with the ones in [17], which are the best known results so far, see Fig. 6 in
page 8 and further details are given in Appendix III. Our algorithm can detect
more entangled regions compared with the previous ones.

Figure 6:  Illustration of the results obtained by using our criteria in
detecting the $k$-partitionability and $k$-producibility of $\rho_{GHZ_{6}}$.
In the figure, (I) showcases the results derived from Theorem 3 of Ref. [17],
(IV) presents the results from Theorem 1 of Ref. [17], (II) demonstrates the
results based on Equation A25 of Ref. [38], and (III) illustrates the results
based on Equation A31 of Ref. [38].

Our criteria based on the hyperplane can detect more $k$-partionable,
$k$-producible and $k$-stretchable states compared with the previous
analytical results, and can be implemented experimentally by measuring local
observables. Take $6$-qubit GHZ state mixed with white noise, $W^{P}$ can be
realized by using $32$ local observables. To detect the $k$-nonseperable
states, the observable $W_{k-\text{part}}$ can be realized using $4$
$(34,14,14,4)$ local observables for $k=2$ $(3,4,5,6)$, respectively. To
detect the $k$-partite states, the observable $W_{k-\text{prod}}$ can be
realized by using $4(12,32,16,4)$ local observables for $k=2$ $(3,4,5,6)$,
respectively.

##  IV Conclusion

Based on the property that all states with the same entanglement structre
constitute a convex set, we have designed an algorithm by using SDP and the
gradient descent method to appromixmate the convex set from the inside of it.
The gradient descent method optimizes the initial points for the SDP, thus the
probability to obtain the global optimum is increased. And we have also
constructed the criteria from the outer polytopes of the convex set to
characterize the multipartite entanglement structure. These criteria can be
implemented through local observables and the number of the ones grows
linearly with the system size. The criteria can be optimized further by
optimizing the selection of the off-diagonal elements and diagnoal elements of
the density matrices. The algorithms and the criteria can characterize the
multipartite entanglement structure better compared with existing criteria.

ACKNOWLEDGEMENTS This work is supported by the Fundamental Research Funds for
the Central Universities; the National Natural Science Foundation of China
(NSFC) under Grants 12071179, 12305007, 12371132, 12075159 and 12171044; Anhui
Provincial Natural Science Foundation (Grant No. 2308085QA29); the Alexander
von Humboldt Foundation; Natural Science Foundation of Shanghai (Grant No.
20ZR1426400); Shenzhen Institute for Quantum Science and Engineering, Southern
University of Science and Technology (Grant Nos. SIQSE202005); the Academician
Innovation Platform of Hainan Province.

##  V Appendix I

The proof of Proposition 1 is given as follows. Proof of Proposition 1

$\displaystyle\begin{aligned} &|\langle
m|\psi\rangle\langle\psi|n\rangle|=\sqrt{\langle
m|\psi\rangle\langle\psi|n\rangle\langle n|\psi\rangle\langle\psi|m\rangle}\\\
=&\sqrt{(\langle m_{X_{i}}|\langle
m_{\bar{X}_{i}}|)|\psi_{X_{i}}\rangle|\psi_{\bar{X}_{i}}\rangle}\times\sqrt{\langle\psi_{X_{i}}|\langle\psi_{\bar{X}_{i}}|(|n_{X_{i}}\rangle|n_{\bar{X}_{i}}\rangle)}\times\sqrt{(\langle
n_{X_{i}}|\langle
n_{\bar{X}_{i}}|)|\psi_{X_{i}}\rangle|\psi_{\bar{X}_{i}}\rangle}\times\sqrt{\langle\psi_{X_{i}}|\langle\psi_{\bar{X}_{i}}|(|m_{X_{i}}\rangle|m_{\bar{X}_{i}}\rangle)}\\\
=&\sqrt{\langle m_{X_{i}}|\psi_{X_{i}}\rangle\langle
m_{\bar{X}_{i}}|\psi_{\bar{X}_{i}}\rangle}\times\sqrt{\langle\psi_{X_{i}}|n_{X_{i}}\rangle\langle\psi_{\bar{X}_{i}}|n_{\bar{X}_{i}}\rangle}\times\sqrt{\langle
n_{X_{i}}|\psi_{X_{i}}\rangle\langle
n_{\bar{X}_{i}}|\psi_{\bar{X}_{i}}\rangle}\times\sqrt{\langle\psi_{X_{i}}|m_{X_{i}}\rangle\langle\psi_{\bar{X}_{i}}|m_{\bar{X}_{i}}\rangle}\\\
=&\sqrt{\langle m_{X_{i}}|\psi_{X_{i}}\rangle\langle
n_{\bar{X}_{i}}|\psi_{\bar{X}_{i}}\rangle}\times\sqrt{\langle\psi_{X_{i}}|m_{X_{i}}\rangle\langle\psi_{\bar{X}_{i}}|n_{\bar{X}_{i}}\rangle}\times\sqrt{\langle
n_{X_{i}}|\psi_{X_{i}}\rangle\langle
m_{\bar{X}_{i}}|\psi_{\bar{X}_{i}}\rangle}\times\sqrt{\langle\psi_{X_{i}}|n_{X_{i}}\rangle\langle\psi_{\bar{X}_{i}}|m_{\bar{X}_{i}}\rangle}\\\
=&\sqrt{(\langle m_{\bar{X}_{i}}|\langle
n_{X_{i}}|)|\psi\rangle\langle\psi|(|m_{\bar{X}_{i}}\rangle|n_{X_{i}}\rangle)}\times\sqrt{(\langle
m_{X_{i}}|\langle
n_{\bar{X}_{i}}|)|\psi\rangle\langle\psi|(|m_{X_{i}}\rangle|n_{\bar{X}_{i}}\rangle)}\\\
\leq&\frac{1}{2}[(\langle m_{\bar{X}_{i}}|\langle
n_{X_{i}}|)|\psi\rangle\langle\psi|(|m_{\bar{X}_{i}}\rangle|n_{X_{i}}\rangle)+(\langle
m_{X_{i}}|\langle
n_{\bar{X}_{i}}|)|\psi\rangle\langle\psi|(|m_{X_{i}}\rangle|n_{\bar{X}_{i}}\rangle)].\end{aligned}$

##  VI Appendix II

In the following part, we illustrate the optimization process for the
inequality based on outer polytopes. The left-hand side of the inequality
represents the sum of the absolute values of the non-diagonal elements in $P$,
while the right-hand side is the inner product of the vectors $\vec{\alpha}$
and $\vec{\beta}$. $\vec{\alpha}$ is derived from the coefficient matrix $M$
by taking the maximum value of per column. The entire optimization process can
be viewed as the optimization of a combination of inequalities.

The optimization strategy adheres to two fundamental principles: (1) Maximize
the number of all zero columns within matrix $M$. (2) Minimize the sum of the
elements comprising the vector $\vec{\alpha}$.

For each element $\rho_{m,n}$ within the set $P$, at least one corresponding
inequality can be formulated for every partition $\gamma_{i}$. We denote the
set of the inequalities with respect to $\rho_{m,n}$ and $\gamma_{i}$ as
$S_{m,n}^{\gamma_{i}}.$ We construct a candidate set from the inequalities in
$S_{m,n}^{\gamma_{i}}$ for all $\gamma_{i}\in\Gamma^{max}$ and given
$\rho_{mn}$, which satisfies two conditions: (1) containing one inequality for
every partition, (2) containing as few inequalities as possible. It is
acknowledged that there are multiple such sets that could fulfill these
conditions. Based on the principle of minimizing the number of diagonal
elements on the right-hand side of the inequalities, an optimal candidate set
$S_{m,n}$ is selected for each element $\rho_{m,n}$ from the set $P$.

Then the given partition $\gamma_{i}$ and each element $\rho_{m,n}$ and we
select one inequality from the set $S_{m,n}^{\gamma_{i}}\cap S_{m,n}$ and then
we add the left and the righ hand sides of all the inequalities respectively
for $m$ and $n,$ the inequality correspoing to the $i$-th row of $M$ is
attained. The selection princile is minimizing the summuation of the elements
that comprise the vector $\vec{\alpha}$, The detais of the selecting are as
follows.

There may exist a specific partition $\gamma$ such that for each part $X_{i}$
within $\gamma$, it holds that $|m_{X_{i}}\rangle=|n_{X_{i}}\rangle$. This
implies that for $\rho_{m,n}$, the corresponding inequality associated with
partition $\gamma$ is $|\rho_{m,n}|\leq(\rho_{m,m}+\rho_{n,n})/2$, which is
trivial from the positive definiteness of the density matrices. For each
partition $\gamma_{i}$ in $\Gamma^{\max}$, all trivial inequalities associated
with $\gamma_{i}$ can be determined. By summing the left and the right hand
sides of these trivial inequalities respectively, we can obtain the
corresponding trivial inequality for $\gamma_{i}$.

The right-hand side of the trivial inequalities for all partitions in
$\Gamma^{max}$ can also be expressed as the inner product of a matrix
$M^{\prime}$ with the vector $\vec{\beta}$. Subsequently, by taking the
maximum value across the columns of $M^{\prime}$, we obtain the vector
$\vec{\eta}$. The maximum values in all columns of matrix $M$, which
correspond to all trivial inequalities, are determined by $\vec{\eta}$. The
vector $\vec{\eta}$ can be considered as the zeroth row of the matrix $M$.
Then, when constructing a certain row of the matrix $M$, if the value in the
zeroth row of a certain column is not equal to zero, and the value in that row
and column of $M$ is smaller than the value in the zeroth row, it may be
beneficial to adjust the selection of inequalities such that the value in that
row and column approaches the value in the zeroth row. This approach aligns
well with our guiding principle for selection. This process involves exact
matching, for which we have utilized the Dancing Links algorithm [48]. An
illustration of this approach will be provided in the forthcoming details of
Example 1. All of the aforementioned constitutes our comprehensive
optimization strategies and techniques.

##  VII Appendix III

In the following section, we give the details to detect the entanglement
structure of the examples using the criteria based on the outerplane. To
represent the partitions for short in the examples, we give the definitions of
the type of the partitions firstly. The type of the partition $\gamma$ (the
integer partition of $N$) is denoted as
$\hat{\gamma}=\\{|X_{1}|,\cdots,|X_{|\gamma|}|\\}$, where $X_{i}$ is the part
of $\gamma$. Disregarding the sequence of elements, the type of the partition
$\gamma$ is uniquely identified. $\hat{\gamma}$ is the refinement of
$\hat{\xi}$ if $\gamma$ is the refinement of $\xi$. as depicted in Fig. 7,
Young diagrams provide a graphical representation for the types of partitions.
If $\hat{\gamma}$ is the refinement of $\hat{\xi}$, then for each partition
$\gamma_{1}$ in $\hat{\gamma}$, there exists a partition $\xi_{1}$ in
$\hat{\xi}$ such that $\gamma_{1}\preceq\xi_{1}$.

Figure 7:  The types of all partitions of the set $\\{1,2,3,4\\}$ are
represented by Young diagrams, along with their poset relationships.

For example, as depicted in Fig. 7, when we consider the set of partition
types for $\Gamma_{2-\text{part}}$ of $\\{1,2,3,4\\}$, we find that it
consists of $\\{\\{1,1,1,1\\},\\{2,1,1\\},\\{2,2\\},\\{3,1\\}\\}$. Among
these, the maximal elements within the set of partition types are
$\\{\\{2,2\\},\\{3,1\\}\\}$. Consequently, $\Gamma_{2-\text{part}}^{\max}$
encompasses all partitions whose types are either $\\{2,2\\}$ or $\\{3,1\\}$,
reflecting the partitions that achieve the greatest refinement within the
poset defined by $\Gamma_{2-\text{part}}$ .

Then we illustrated the details of the Examples in E of Section III.

The details of Example 1:

We select the set of non-diagonal elements
$P=\\{\rho_{2,3},\rho_{2,5},\rho_{2,9},\rho_{3,5},\rho_{3,9},\rho_{5,9}\\}$.

The sets of maximal elements of all $k$-stretchable partitions of
$\\{1,2,3,4\\}$ are

$\displaystyle\Gamma_{(-1)-\text{str}}^{\max}$ $\displaystyle=$
$\displaystyle\\{12|3|4,13|2|4,14|2|3,23|1|4,24|1|3,34|1|2\\},$
$\displaystyle\Gamma_{0-\text{str}}^{\max}$ $\displaystyle=$
$\displaystyle\\{12|34,13|24,14|23\\},$
$\displaystyle\Gamma_{1-\text{str}}^{\max}$ $\displaystyle=$
$\displaystyle\Gamma_{0-\text{str}}^{\max}\bigcup\\{123|4,124|3,134|2,234|1\\}.$

If $\rho$ is $3$-partitionable or $(-1)$-stretchable, the trivial inequality
corresponding to $12|3|4$ is $|\rho_{5,9}|\leq(\rho_{5,5}+\rho_{9,9})/2$.
Additionally, the trivial inequality corresponding to $12|3|4$ is
$|\rho_{2,3}|\leq(\rho_{2,2}+\rho_{3,3})/2$. As a result, the vector
$\vec{\eta}=[\eta_{1},\eta_{2},...,\eta_{16}]$ can be defined with
$\eta_{2}=\eta_{3}=\eta_{5}=\eta_{9}=1/2$. When constructing the inequality
corresponding to $1|2|34$, if we select
$|\rho_{5,9}|\leq(\rho_{1,1}+\rho_{13,13})/2$, the coefficient of $\rho_{1,1}$
in the final inequality becomes $2.5$. However, if we select
$|\rho_{5,9}|\leq(\rho_{5,5}+\rho_{9,9})/2$, it appears to add a trivial
inequality, but the coefficient of $\rho_{1,1}$ in the final inequality
becomes $2$, with no change in the coefficients of other terms. This is the
optimization method for adjusting inequality selection introduced in Appendix
II.

Therefore, if $\rho$ is $3$-partitionable or $(-1)$-stretchable, the partition
$12|3|4$ and $34|1|2$ correspond to the following inequality:

$\displaystyle\begin{aligned} \sum_{\rho_{m,n}\in
P}|\rho_{m,n}|\leq&2\rho_{1,1}+\frac{1}{2}(\rho_{6,6}+\rho_{7,7}+\rho_{10,10}+\rho_{11,11}\\\
&+\rho_{2,2}+\rho_{3,3}+\rho_{5,5}+\rho_{9,9}),\end{aligned}$ (17)

and the partition $13|2|4$ and $24|1|3$ correspond to the following
inequality:

$\displaystyle\begin{aligned} \sum_{\rho_{m,n}\in
P}|\rho_{m,n}|\leq&2\rho_{1,1}+\frac{1}{2}(\rho_{4,4}+\rho_{7,7}+\rho_{10,10}+\rho_{13,13}\\\
&+\rho_{2,2}+\rho_{3,3}+\rho_{5,5}+\rho_{9,9}),\end{aligned}$ (18)

and the partition $14|2|3$ and $23|1|4$ correspond to the following
inequality:

$\displaystyle\begin{aligned} \sum_{\rho_{m,n}\in
P}|\rho_{m,n}|\leq&2\rho_{1,1}+\frac{1}{2}(\rho_{4,4}+\rho_{6,6}+\rho_{11,11}+\rho_{13,13}\\\
&+\rho_{2,2}+\rho_{3,3}+\rho_{5,5}+\rho_{9,9}).\end{aligned}$ (19)

Thus one of the criteria is

$\displaystyle\begin{aligned} \sum_{\rho_{m,n}\in
P}|\rho_{m,n}|\leq&2\rho_{1,1}+\frac{1}{2}(\rho_{4,4}+\rho_{6,6}+\rho_{7,7}+\rho_{10,10}+\rho_{11,11}\\\
&+\rho_{13,13}+\rho_{2,2}+\rho_{3,3}+\rho_{5,9}+\rho_{9,9}).\end{aligned}$
(20)

If $\rho$ is $2$-producible or $0$-stretchable, the partition $12|34$
corresponds to the inequality (17), and the partition $13|24$ corresponds to
the inequality (18), and the partition $14|23$ corresponds to the inequality
(19). Thus one of the criteria is the inequality (20).

If $\rho$ is $3$-producible, $2$-partitionable, or $1$-stretchable, one of the
criteria is

$\displaystyle\begin{aligned} \sum_{\rho_{m,n}\in
P}|\rho_{m,n}|\leq&2\rho_{1,1}+\frac{1}{2}(\rho_{4,4}+\rho_{6,6}+\rho_{7,7}+\rho_{10,10}+\rho_{11,11}\\\
&+\rho_{13,13})+(\rho_{2,2}+\rho_{3,3}+\rho_{5,9}+\rho_{9,9}).\end{aligned}$
(21)

The observable $W_{P}$ and $W_{k-\text{prod}}$ in the left and right hand
sides of the inequalies in this example can be expressed as

$\displaystyle\begin{aligned} &W_{P}=\sum_{1\leq i<j\leq n}[\bigotimes_{k\neq
i,j}Z^{+}_{k}]\otimes H_{i,j},\\\
&W_{2-\text{prod}}=2\Delta_{1}+\frac{1}{2}(\Delta_{2}+\Delta_{3})\\\
&W_{3-\text{prod}}=2\Delta_{1}+\Delta_{2}+\frac{1}{2}\Delta_{3}\end{aligned}$
(22)

with

$\displaystyle\begin{aligned}
&Z_{i}^{+}=\frac{(\mathbb{I}_{2}+\sigma^{(i)}_{z})}{2},Z_{k}^{-}=\frac{(\mathbb{I}_{2}-\sigma^{(i)}_{z})}{2},\\\
&H_{i}^{+}=|0\rangle_{i}\langle 1|,H_{i}^{-}=|1\rangle_{i}\langle 0|\\\
&H_{i,j}=H_{i}^{+}\otimes H_{j}^{-}+H_{i}^{-}\otimes H_{j}^{+}\\\
&\Delta_{1}=\bigotimes\limits_{i=1}^{4}Z_{k}^{+}\\\
&\Delta_{2}=\sum\limits_{i=1}^{4}Z_{i}^{-}\otimes[\bigotimes\limits_{j=1,j\neq
i}^{4}Z_{j}^{+}]\\\ &\Delta_{3}=\sum\limits_{1\leq i_{1}<i_{2}\leq
4}Z_{i_{1}}^{+}\otimes Z_{i_{2}}^{+}\otimes[\bigotimes\limits_{j\neq
i_{1},i_{2}}Z_{j}^{-}]\end{aligned}$

and $\sigma_{z}^{(i)}$ being the pauli matrice of the $i$-th system.

The details of Example 2.

We select $P_{1}=\\{$ $\rho_{2,3}$, $\rho_{2,4}$, $\rho_{2,7}$, $\rho_{2,10}$,
$\rho_{2,19}$, $\rho_{2,28}$, $\rho_{2,55}$, $\rho_{3,4}$, $\rho_{3,7}$,
$\rho_{3,10}$, $\rho_{3,19}$, $\rho_{3,28}$, $\rho_{3,55}$, $\rho_{4,7}$,
$\rho_{4,10}$, $\rho_{4,19}$, $\rho_{4,28}$, $\rho_{4,55}$, $\rho_{7,10}$,
$\rho_{7,19}$, $\rho_{7,28}$, $\rho_{7,55}$, $\rho_{10,19}$, $\rho_{10,28}$,
$\rho_{10,55}$, $\rho_{19,28}$, $\rho_{19,55}$, $\rho_{28,55}$ $\\}$, and
$P_{2}=\\{$ $\rho_{14,32}$, $\rho_{14,38}$, $\rho_{14,40}$, $\rho_{14,42}$,
$\rho_{14,44}$, $\rho_{14,50}$, $\rho_{14,68}$, $\rho_{32,38}$,
$\rho_{32,40}$, $\rho_{32,42}$, $\rho_{32,44}$, $\rho_{32,50}$,
$\rho_{32,68}$, $\rho_{38,40}$, $\rho_{38,42}$, $\rho_{38,44}$,
$\rho_{38,50}$, $\rho_{38,68}$, $\rho_{40,42}$, $\rho_{40,44}$,
$\rho_{40,50}$, $\rho_{40,68}$, $\rho_{42,44}$, $\rho_{42,50}$,
$\rho_{42,68}$, $\rho_{44,50}$, $\rho_{44,68}$, $\rho_{50,68}$ $\\}$.

If $\rho$ is $2$-producible, then

$\displaystyle{\sum\limits_{\rho_{m,n}\in
P_{i}}|\rho_{m,n}|}\leq\textrm{Tr}[W_{2-\text{prod}}^{P_{i}}\cdot\rho],$ (23)

where $W_{2-\text{prod}}^{P_{i}}=\textrm{diag}\\{a^{i}_{1},\cdots\\}$
$(i=1,2)$ is the diagonal matrix with $a^{1}_{1}=8,$
$a^{1}_{2}=a^{1}_{3}=a^{1}_{4}=a^{1}_{7}=a^{1}_{10}=a^{1}_{19}=a^{1}_{28}=a^{1}_{55}=1$
and
$a^{1}_{5}=a^{1}_{6}=a^{1}_{8}=a^{1}_{9}=a^{1}_{11}=a^{1}_{12}=a^{1}_{13}=a^{1}_{16}=a^{1}_{20}=a^{1}_{21}=a^{1}_{22}=a^{1}_{25}=a^{1}_{29}=a^{1}_{30}=a^{1}_{31}=a^{1}_{34}=a^{1}_{37}=a^{1}_{46}=a^{1}_{56}=a^{1}_{57}=a^{1}_{58}=a^{1}_{61}=a^{1}_{64}=a^{1}_{73}=\frac{1}{2}.$
and $a^{2}_{41}=8,$
$a^{2}_{14}=a^{2}_{32}=a^{2}_{38}=a^{2}_{40}=a^{2}_{42}=a^{2}_{44}=a^{2}_{50}=a^{2}_{68}=1$
and
$a^{2}_{5}=a^{2}_{11}=a^{2}_{13}=a^{2}_{15}=a^{2}_{17}=a^{2}_{23}=a^{2}_{28}=a^{2}_{31}=a^{2}_{33}=a^{2}_{35}=a^{2}_{37}=a^{2}_{39}=a^{2}_{43}=a^{2}_{45}=a^{2}_{47}=a^{2}_{49}=a^{2}_{51}=a^{2}_{53}=a^{2}_{59}=a^{2}_{65}=a^{2}_{67}=a^{2}_{69}=a^{2}_{71}=a^{2}_{77}=\frac{1}{2}.$

The observable $W_{P_{1}}$, $W_{P_{2}}$, $W_{2-\text{prod}}^{P_{1}}$ and
$W_{2-\text{prod}}^{P_{2}}$ in the left and right hand sides of the inequalies
in this example can be expressed as

$\displaystyle\begin{aligned} &W_{P_{1}}=\sum_{1\leq i<j\leq
4}[(\bigotimes_{k\neq
i,j}Z^{0}_{k})\otimes(H_{i,j}+J_{i,j}+\Delta_{i,j}^{1}+\Delta_{j,i}^{1})]\\\
&W_{P_{2}}=\sum_{1\leq i<j\leq 4}[(\bigotimes_{k\neq
i,j}Z^{1}_{k})\otimes(H_{i,j}+K_{i,j}+\Delta_{i,j}^{2}+\Delta_{j,i}^{2})]\\\
&W_{2-\text{prod}}^{P_{1}}=8\cdot(\bigotimes\limits_{i=1}^{4}Z_{i}^{0})+\frac{1}{2}\sum_{1\leq
i<j\leq 4}[(\sum_{a,b\in\\{1,2\\}}Z_{i}^{a}\otimes Z_{j}^{b})\\\
&\otimes(\bigotimes_{k\neq i,j}Z_{k}^{0})]+\sum_{1\leq i\leq
4}[(\bigotimes_{k\neq i}Z^{0}_{k})\otimes(Z^{1}_{i}+Z^{2}_{i})]\\\
&W_{2-\text{prod}}^{P_{2}}=8\cdot(\bigotimes\limits_{i=1}^{4}Z_{i}^{1})+\frac{1}{2}\sum_{1\leq
i<j\leq 4}[(\sum_{a,b\in\\{0,2\\}}Z_{i}^{a}\otimes Z_{j}^{b})\\\
&\otimes(\bigotimes_{k\neq i,j}Z_{k}^{1})]+\sum_{1\leq i\leq
4}[(\bigotimes_{k\neq i}Z^{1}_{k})\otimes(Z^{0}_{i}+Z^{2}_{i})]\end{aligned}$
(24)

with

$\displaystyle\begin{aligned} &Z_{i}^{0}=|0\rangle_{i}\langle
0|,Z_{i}^{1}=|1\rangle_{i}\langle 1|,Z_{i}^{2}=|2\rangle_{i}\langle 2|\\\
&H_{i}^{+}=|0\rangle_{i}\langle 1|,H_{i}^{-}=|1\rangle_{i}\langle 0|\\\
&J_{i}^{+}=|0\rangle_{i}\langle 2|,J_{i}^{-}=|2\rangle_{i}\langle 0|\\\
&K_{i}^{+}=|1\rangle_{i}\langle 2|,K_{i}^{-}=|2\rangle_{i}\langle 1|\\\
&H_{i,j}=H_{i}^{+}\otimes H_{j}^{-}+H_{i}^{-}\otimes H_{j}^{+}\\\
&J_{i,j}=J_{i}^{+}\otimes J_{j}^{-}+J_{i}^{-}\otimes J_{j}^{+}\\\
&K_{i,j}=K_{i}^{+}\otimes K_{j}^{-}+K_{i}^{-}\otimes K_{j}^{+}\\\
&\Delta_{i,j}^{1}=H_{i}^{+}\otimes J_{j}^{-}+H_{i}^{-}\otimes J_{j}^{+}\\\
&\Delta_{i,j}^{2}=H_{i}^{+}\otimes K_{j}^{-}+H_{i}^{-}\otimes K_{j}^{+}\\\
\end{aligned}$ (25)

The details of Example 3.

Unless otherwise specified, we select
$P=\\{\rho_{1,16},\rho_{1,20},\rho_{1,29},\rho_{16,20},\rho_{16,29},\rho_{20,29}\\}$.

If $\rho$ is $(-2)$-stretchable or $4$-partitionable,
$\Gamma_{4-\text{part}}^{\max}$ is a set of partitions having the type
$\\{2,1,1,1\\}$. By the greedy algorithm, one of the criteria is

$\displaystyle{\sum_{\rho_{m,n}\in
P}|\rho_{m,n}|}\leq\textrm{Tr}[W_{4-\text{part}}\cdot\rho],$ (26)

where $W_{4-\text{part}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with
$a_{2}=a_{3}=a_{4}=a_{5}=a_{8}=a_{9}=a_{12}=a_{13}=a_{14}=a_{15}=a_{17}=a_{18}=a_{19}=a_{21}=a_{24}=a_{25}=a_{28}=a_{30}=a_{31}=a_{32}=\frac{1}{2}$.

If $\rho$ is $(-1)$-stretchable or $2$-producible,
$\Gamma_{2-\text{prod}}^{\max}$ is a set of partitions having the type
$\\{2,2,1\\}$. We select $P_{2}=\\{\rho_{1,16}\\}$. By the codes in [44], one
of the criteria is

${|\rho_{1,16}|}\leq\textrm{Tr}[W_{2-\text{prod}}\cdot\rho],$ (27)

where $W_{2-\text{prod}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with $a_{4}=a_{6}=a_{7}=a_{10}=a_{11}=a_{13}=\frac{1}{2}$.

If $\rho$ is $0$-stretchable or $1$-stretchable or $3$-producible or
$3$-partitionable, one of the criteria is

${\sum_{\rho_{m,n}\in
P}|\rho_{m,n}|}\leq\textrm{Tr}[W_{3-\text{part}}\cdot\rho],$ (28)

where $W_{3-\text{part}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with $a_{4}=a_{13}=a_{17}=a_{32}=1$, and
$a_{1}=a_{2}=a_{3}=a_{5}=a_{6}=a_{7}=a_{8}=a_{9}=a_{10}=a_{11}=a_{12}=a_{14}=a_{15}=a_{16}=a_{18}=a_{19}=a_{20}=a_{21}=a_{22}=a_{23}=a_{24}=a_{25}=a_{26}=a_{27}=a_{28}=a_{29}=a_{30}=a_{31}=\frac{1}{2}$.

If $\rho$ is $2$-stretchable or $2$-partitionable or $4$-producible, one of
the criteria is

$\displaystyle{\sum_{\rho_{m,n}\in
P}|\rho_{m,n}|}\leq\textrm{Tr}[W_{4-\text{prod}}\cdot\rho],$ (29)

where $W_{4-\text{prod}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with
$a_{2}=a_{3}=a_{4}=a_{5}=a_{8}=a_{9}=a_{12}=a_{13}=a_{14}=a_{15}=a_{17}=a_{18}=a_{19}=a_{21}=a_{24}=a_{25}=a_{28}=a_{30}=a_{31}=a_{32}=1$,
and
$a_{1}=a_{6}=a_{7}=a_{10}=a_{11}=a_{16}=a_{20}=a_{22}=a_{23}=a_{26}=a_{27}=a_{29}=\frac{1}{2}$.

The observables $W_{P}$ and $W_{k-\text{prod}},$ $W_{k-\text{part}}$ and
$W_{k-\text{str}}$ are expressed as follows

$\displaystyle\begin{aligned}
&W_{P}=Z_{1}^{+}\otimes(\bigotimes\limits_{i=2}^{5}H_{i}^{+})+Z_{2}^{+}\otimes
Z_{3}^{+}\otimes H_{1}^{+}\otimes H_{4}^{+}\otimes H_{5}^{+}\\\
&+Z_{4}^{+}\otimes Z_{5}^{+}\otimes H_{1}^{+}\otimes H_{2}^{+}\otimes
H_{3}^{+}+Z_{2}^{-}\otimes Z_{3}^{-}\otimes H_{1}^{+}\\\ &\otimes
H_{4}^{-}\otimes H_{5}^{-}+Z_{4}^{-}\otimes Z_{5}^{-}\otimes H_{1}^{+}\otimes
H_{2}^{-}\otimes H_{3}^{-}\\\ &+Z_{1}^{-}\otimes H_{2}^{+}\otimes
H_{3}^{+}\otimes H_{4}^{-}\otimes H_{5}^{-}\\\
&W_{P_{2}}=Z_{1}^{+}\otimes(\bigotimes\limits_{i=2}^{5}H_{i}^{+})\\\
&W_{4-\text{part}}=\frac{\mathbb{I}_{32}-\Delta_{1}}{2}\\\
&W_{2-\text{prod}}=\frac{1}{2}Z_{1}^{+}\otimes\sum_{2\leq i<j\leq
5}[Z_{i}^{+}\otimes Z_{j}^{+}\otimes(\bigotimes_{\begin{subarray}{c}2\leq
k\leq 5,\\\ k\neq i,j\end{subarray}}Z_{k}^{-})]\\\
&W_{3-\text{part}}=\frac{\mathbb{I}}{2}+\frac{\Delta_{2}}{2}\\\
&W_{4-\text{prod}}=\mathbb{I}-\frac{\Delta_{1}}{2}\end{aligned}$ (30)

with

$\displaystyle\begin{aligned}
&\Delta_{1}=(Z_{1}^{-}+Z_{1}^{+})\otimes[\sum_{\begin{subarray}{c}2\leq i\leq
3,\\\ 4\leq j\leq 5\end{subarray}}Z_{i}^{+}\otimes
Z_{j}^{+}\otimes(\bigotimes_{\begin{subarray}{c}2\leq k\leq 5,\\\ k\neq
i,j\end{subarray}}Z_{k}^{-})]\\\
&+\bigotimes_{i=1}^{5}Z_{i}^{-}+Z_{1}^{-}\otimes(\bigotimes_{i=2}^{5}Z_{i}^{+})+Z_{1}^{+}\otimes(Z_{2}^{-}\otimes
Z_{3}^{-}\otimes Z_{4}^{+}\\\ &\otimes Z_{5}^{+}+Z_{2}^{+}\otimes
Z_{3}^{+}\otimes Z_{4}^{-}\otimes Z_{5}^{-})\\\
&\Delta_{2}=Z_{1}^{-}\otimes(Z_{2}^{-}\otimes Z_{3}^{-}\otimes
Z_{4}^{+}\otimes Z_{5}^{+}+Z_{2}^{+}\otimes Z_{3}^{+}\otimes Z_{4}^{-}\\\
&\otimes
Z_{5}^{-}+Z_{1}^{+}\otimes(\bigotimes_{i=2}^{5}Z_{i}^{-}+\bigotimes_{i=2}^{5}Z_{i}^{+})\end{aligned}$
(31)

The details of Example 4.

Unless otherwise specified, we select $P=\\{\rho_{1,64}\\}$.

Different from other examples, we fix the right-hand side of the inequality
and seek a larger left-hand side, the details are listed as follows.

If $\rho$ is $1$-producible, one of the criteria is

$\displaystyle{|\rho_{1,64}|}\leq\textrm{Tr}[W_{1-\text{prod}}\cdot\rho],$
(32)

where $W_{1-\text{prod}}=\textrm{diag}\\{a_{1},\cdots,a_{64}\\}$ is the
diagonal matrix with $a_{2}=a_{63}=\frac{1}{2}$.

If $\rho$ is $5$-partitionable or $(-3)$-stretchable,
$\Gamma_{5-\text{part}}^{\max}$ is a set of partitions of type
$\\{2,1,1,1,1\\}$. Consider six inequalities derived from Proposition 1,
corresponding to the parts $\\{1\\}$, $\\{2\\}$, $\\{3\\}$, $\\{4\\}$,
$\\{5\\}$, and $\\{6\\}$, respectively. Given each partition in the current
set includes four parts, summing the corresponding four inequalities yields
the inequality for that partition. Thus the criterion is

$\displaystyle{4|\rho_{1,64}|}\leq\textrm{Tr}[W_{5-\text{part}}\cdot\rho],$
(33)

where $W_{5-\text{part}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with
$a_{2}=a_{3}=a_{5}=a_{9}=a_{17}=a_{32}=a_{33}=a_{48}=a_{56}=a_{60}=a_{62}=a_{63}=\frac{1}{2}$.

If $\rho$ is $4$-partitionable, $\Gamma_{4-\text{part}}^{\max}$ is a set of
partitions of type $\\{2,2,1,1\\}$ and $\\{3,1,1,1\\}$. Given each partition
in the current set includes two parts, with each part containing one
subsystem, the criterion is

$\displaystyle{2|\rho_{1,64}|}\leq\textrm{Tr}[W_{4-\text{part}}\cdot\rho],$
(34)

where $W_{4-\text{part}}=W_{5-\text{part}}$.

If $\rho$ is $(-2)$-stretchable, $\Gamma_{(-2)-\text{str}}^{\max}$ is a set of
partitions of type $\\{2,2,1,1\\}$. Therefore, the criterion is given by the
inequality (34).

If $\rho$ is $(-1)$-stretchable, $\Gamma_{(-1)-\text{str}}^{\max}$ is a set of
partitions of types $\\{2,2,2\\}$ and $\\{3,1,1,1\\}$. We consider 15
inequalities derived from Proposition 1, each corresponding to a part that
includes two subsystems. For each partition in the set of partitions of type
$\\{2,2,2\\}$, which includes three parts each containing two subsystems, the
inequality for that partition is obtained by summing the corresponding three
inequalities. Similarly, we consider six inequalities, each corresponding to a
part that contains one subsystem. For each partition in the set of partitions
of type $\\{3,1,1,1\\}$, which includes three parts each containing one
subsystem, the inequality for that partition is obtained by summing the
corresponding three inequalities. Therefore, the criterion is of the form,

$\displaystyle{3|\rho_{1,64}|}\leq\textrm{Tr}[W_{(-1)-\text{str}}\cdot\rho],$
(35)

where $W_{(-1)-\text{str}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with
$a_{2}=a_{3}=a_{4}=a_{5}=a_{6}=a_{7}=a_{9}=a_{10}=a_{11}=a_{13}=a_{16}=a_{17}=a_{18}=a_{19}=a_{21}=a_{24}=a_{25}=a_{28}=a_{30}=a_{31}=a_{32}=a_{33}=a_{34}=a_{35}=a_{37}=a_{40}=a_{41}=a_{44}=a_{46}=a_{47}=a_{48}=a_{49}=a_{52}=a_{54}=a_{55}=a_{56}=a_{58}=a_{60}=a_{59}=a_{61}=a_{62}=a_{63}=\frac{1}{2}$.

If $\rho$ is $3$-partitionable, $\Gamma_{3-\text{part}}^{\max}$ is a set of
partitions of types $\\{2,2,2\\}$, $\\{3,2,1\\}$ and $\\{4,1,1\\}$. For each
partition in the set of partitions of type $\\{2,2,2\\}$, we also consider 15
inequalities as before. For each partition in the set of partitions of type
$\\{4,1,1\\}$, we further consider six inequalities, each corresponding to a
part that, as in the previous cases, contains one subsystem. Employing the
codes in [44], for each partition in the set of partitions of type
$\\{3,2,1\\}$, we sum the six inequalities twice. Therefore, we have

$\displaystyle{3|\rho_{1,64}|}\leq\textrm{Tr}[W_{3-\text{part}}\cdot\rho],$
(36)

where $W_{3-\text{part}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with
$a_{2}=a_{3}=a_{5}=a_{9}=a_{17}=a_{32}=a_{33}=a_{48}=a_{56}=a_{60}=a_{62}=a_{63}=1$
and
$a_{4}=a_{6}=a_{7}=a_{10}=a_{11}=a_{13}=a_{16}=a_{18}=a_{19}=a_{21}=a_{24}=a_{25}=a_{28}=a_{30}=a_{31}=a_{34}=a_{35}=a_{37}=a_{40}=a_{41}=a_{44}=a_{46}=a_{47}=a_{49}=a_{52}=a_{54}=a_{55}=a_{58}=a_{59}=a_{61}=\frac{1}{2}$.

If $\rho$ is $0$-stretchable, $\Gamma_{0-\text{str}}^{\max}$ is a set of
partitions of types $\\{2,2,2\\}$ and $\\{3,2,1\\}$. The criteria is given by
the inequality (36).

The inequalities (33, 34, 35, 36) are all derived using the approach of fixing
the right-hand side of the inequality and seeking a larger left-hand side.

If $\rho$ is $2$-producible, $\Gamma_{2-\text{prod}}^{\max}$ is a set of
partitions of type $\\{2,2,2\\}$. By the codes in [44], one of the criteria is

$\displaystyle{|\rho_{1,64}|}\leq\textrm{Tr}[W_{2-\text{prod}}\cdot\rho],$
(37)

where $W_{2-\text{prod}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with
$a_{4}=a_{7}=a_{11}=a_{19}=a_{30}=a_{35}=a_{46}=a_{54}=a_{58}=a_{61}=\frac{1}{2}$.

If $\rho$ is $3$-producible, $\Gamma_{3-\text{prod}}^{\max}$ is a set of
partitions of types $\\{2,2,2\\}$ and $\\{3,3\\}$. By the codes in [44], one
of the criteria is

$\displaystyle{|\rho_{1,64}|}\leq\textrm{Tr}[W_{3-\text{prod}}\cdot\rho],$
(38)

where $W_{3-\text{prod}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with
$a_{4}=a_{7}=a_{8}=a_{11}=a_{12}=a_{14}=a_{15}=a_{19}=a_{20}=a_{22}=a_{23}=a_{26}=a_{27}=a_{29}=a_{30}=a_{35}=a_{36}=a_{38}=a_{39}=a_{42}=a_{43}=a_{45}=a_{46}=a_{50}=a_{51}=a_{53}=a_{54}=a_{57}=a_{58}=a_{61}=\frac{1}{2}$

If $\rho$ is $1$-stretchable, $\Gamma_{1-\text{str}}^{\max}$ is a set of
partitions of types $\\{2,2,2\\}$, $\\{3,3\\}$ and $\\{4,1,1\\}$.By the codes
in [44], one of the criteria is

$\displaystyle{|\rho_{1,64}|}\leq\textrm{Tr}[W_{1-\text{str}}\cdot\rho],$ (39)

where $W_{1-\text{str}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with
$a_{2}=a_{3}=a_{4}=a_{5}=a_{7}=a_{8}=a_{9}=a_{11}=a_{12}=a_{14}=a_{15}=a_{17}=a_{19}=a_{20}=a_{22}=a_{23}=a_{26}=a_{27}=a_{29}=a_{30}=a_{35}=a_{36}=a_{38}=a_{39}=a_{42}=a_{43}=a_{45}=a_{46}=a_{48}=a_{50}=a_{51}=a_{53}=a_{54}=a_{56}=a_{57}=a_{58}=a_{60}=a_{61}=a_{62}=a_{63}=\frac{1}{2}$.

If $\rho$ is $4$-producible or $2$-stretchable,
$\Gamma_{4-\text{prod}}^{\max}$ is a set of partitions of types $\\{4,2\\}$
and $\\{3,3\\}$, and the criterion is given by

$\displaystyle{|\rho_{1,64}|}\leq\textrm{Tr}[W_{4-\text{prod}}\cdot\rho],$
(40)

where $W_{4-\text{prod}}^{P}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with
$a_{4}=a_{6}=a_{7}=a_{8}=a_{10}=a_{11}=a_{12}=a_{13}=a_{14}=a_{15}=a_{16}=a_{18}=a_{19}=a_{20}=a_{21}=a_{22}=a_{23}=a_{24}=a_{25}=a_{26}=a_{27}=a_{28}=a_{29}=a_{30}=a_{31}=a_{34}=a_{35}=a_{36}=a_{37}=a_{38}=a_{39}=a_{40}=a_{41}=a_{42}=a_{43}=a_{44}=a_{45}=a_{46}=a_{47}=a_{49}=a_{50}=a_{51}=a_{52}=a_{53}=a_{54}=a_{55}=a_{57}=a_{58}=a_{59}=a_{61}=\frac{1}{2}$.

If $\rho$ is $5$-producible or $3$-stretchable,
$\Gamma_{4-\text{prod}}^{\max}$ is a set of partitions of types $\\{5,1\\}$,
$\\{4,2\\}$ and $\\{3,3\\}$, and the criterion is

$\displaystyle{|\rho_{1,64}|}\leq\textrm{Tr}[W_{5-\text{prod}}\cdot\rho],$
(41)

where $W_{5-\text{prod}}=\textrm{diag}\\{a_{1},\cdots\\}$ is the diagonal
matrix with
$a_{2}=a_{3}=a_{4}=a_{5}=a_{6}=a_{7}=a_{8}=a_{9}=a_{10}=a_{11}=a_{12}=a_{13}=a_{14}=a_{15}=a_{16}=a_{17}=a_{18}=a_{19}=a_{20}=a_{21}=a_{22}=a_{23}=a_{24}=a_{25}=a_{26}=a_{27}=a_{28}=a_{29}=a_{30}=a_{31}=a_{32}=a_{33}=a_{34}=a_{35}=a_{36}=a_{37}=a_{38}=a_{39}=a_{40}=a_{41}=a_{42}=a_{43}=a_{44}=a_{45}=a_{46}=a_{47}=a_{48}=a_{49}=a_{50}=a_{51}=a_{52}=a_{53}=a_{54}=a_{55}=a_{56}=a_{57}=a_{58}=a_{59}=a_{60}=a_{61}=a_{62}=a_{63}=\frac{1}{2}$.

The observables $W_{P}$ and $W_{k-\text{prod}},$ $W_{k-\text{part}}$ and
$W_{k-\text{str}}$ are expressed as follows

$\displaystyle\begin{aligned}
&W_{P}=\bigotimes_{i=1}^{n}H_{i}^{+}+\bigotimes_{i=1}^{n}H_{i}^{-}\\\
&W_{1-\text{prod}}=\frac{1}{2}(Z_{1}^{+}Z_{2}^{+}Z_{3}^{+}Z_{4}^{+}Z_{5}^{+}Z_{6}^{-}+Z_{1}^{-}Z_{2}^{-}Z_{3}^{-}Z_{4}^{-}Z_{5}^{-}Z_{6}^{+})\\\
&W_{2-\text{prod}}=\frac{1}{2}(\Delta_{1}+X\Delta_{1}X)\\\
&W_{3-\text{prod}}=\frac{1}{2}(\Delta_{1}+X\Delta_{1}X+\Delta_{2}+X\Delta_{2}X+\Delta_{3}+X\Delta_{3}X)\\\
&W_{4-\text{prod}}=\frac{\mathbb{I}_{64}}{2}-\frac{1}{2}\bigotimes_{k=1}^{6}Z_{k}^{+}-\frac{1}{2}\bigotimes_{k=1}^{6}Z_{k}^{-}-\\\
&\quad\quad\frac{1}{2}\sum_{k=1}^{6}Z_{k}^{-}\otimes(\bigotimes_{j\neq
k}Z_{j}^{+})-\frac{1}{2}\sum_{k=1}^{6}Z_{k}^{+}\otimes(\bigotimes_{j\neq
k}Z_{j}^{-})\end{aligned}$ (42)

$\displaystyle\begin{aligned}
&W_{5-\text{prod}}=\frac{\mathbb{I}_{64}}{2}-\frac{1}{2}\bigotimes_{k=1}^{6}Z_{k}^{+}-\frac{1}{2}\bigotimes_{k=1}^{6}Z_{k}^{-}\\\
&W_{1-\text{str}}=\frac{\mathbb{I}_{64}}{2}-\frac{1}{2}(\Delta_{4}+X\Delta_{4}X+\otimes_{i=2}^{6}Z_{i}^{+}+\otimes_{i=2}^{6}Z_{i}^{-})\end{aligned}$

$\displaystyle\begin{aligned}
&W_{-1-\text{str}}=\frac{\mathbb{I}_{64}}{2}-\frac{1}{2}(\Delta_{2}+X\Delta_{2}X+\Delta_{3}+X\Delta_{3}X+\\\
&\quad\quad\otimes_{i=1}^{6}Z_{i}^{+}+\otimes_{i=1}^{6}Z_{i}^{-})\\\
&W_{3-\text{part}}=\frac{\mathbb{I}_{64}}{2}+\frac{1}{2}(\Delta_{5}+X\Delta_{5}X-\Delta_{2}-X\Delta_{2}X-\Delta_{3}\\\
&\quad\quad-X\Delta_{3}X-\otimes_{i=1}^{6}Z_{i}^{+}-\otimes_{i=1}^{6}Z_{i}^{-})\\\
\end{aligned}$ $\displaystyle\begin{aligned}
&W_{4-\text{part}}=W_{5-\text{part}}^{P}=\sum_{k=1}^{6}Z_{k}^{-}\otimes(\bigotimes\limits_{j=1,j\neq
k}^{6}Z_{j}^{+})\\\
&\quad\quad+\sum_{k=1}^{6}Z_{k}^{+}\otimes(\bigotimes\limits_{j=1,j\neq
k}^{6}Z_{j}^{-})\end{aligned}$ (43)

with

$\displaystyle\begin{aligned} &X=\otimes_{i=1}^{6}\sigma_{x}^{(i)}\\\
&\Delta_{1}=Z_{1}^{+}Z_{2}^{+}(Z_{3}^{+}Z_{4}^{+}Z_{5}^{-}Z_{6}^{-}+Z_{3}^{+}Z_{4}^{-}Z_{5}^{-}Z_{6}^{+}+Z_{3}^{-}Z_{4}^{+}Z_{5}^{-}Z_{6}^{+})\\\
&\quad\quad+(Z_{1}^{+}Z_{2}^{-}+Z_{1}^{-}Z_{2}^{+})Z_{3}^{+}Z_{4}^{+}Z_{5}^{-}Z_{6}^{+}\\\
&\Delta_{2}=Z_{1}^{+}Z_{2}^{+}(Z_{3}^{+}Z_{4}^{+}Z_{5}^{+}Z_{6}^{-}+Z_{3}^{+}Z_{4}^{+}Z_{5}^{-}Z_{6}^{+}+Z_{3}^{+}Z_{4}^{-}Z_{5}^{+}Z_{6}^{+}\\\
&\quad\quad+Z_{3}^{-}Z_{4}^{+}Z_{5}^{+}Z_{6}^{+})\\\
&\Delta_{3}=Z_{1}^{+}Z_{2}^{-}(Z_{3}^{+}Z_{4}^{+}Z_{5}^{-}Z_{6}^{-}+Z_{3}^{+}Z_{4}^{-}Z_{5}^{-}Z_{6}^{+}+Z_{3}^{-}Z_{4}^{+}Z_{5}^{-}Z_{6}^{+}\\\
&\quad\quad+Z_{3}^{-}Z_{4}^{-}Z_{5}^{+}Z_{6}^{+}+Z_{3}^{-}Z_{4}^{+}Z_{5}^{+}Z_{6}^{-}+Z_{3}^{+}Z_{4}^{-}Z_{5}^{+}Z_{6}^{-})\\\
&\Delta_{4}=(Z_{1}^{+}Z_{2}^{-}+Z_{1}^{-}Z_{2}^{+})(Z_{3}^{+}Z_{4}^{+}Z_{5}^{+}Z_{6}^{-}+Z_{3}^{+}Z_{4}^{-}Z_{5}^{+}Z_{6}^{+}\\\
&\quad\quad+Z_{3}^{-}Z_{4}^{+}Z_{5}^{+}Z_{6}^{+})+Z_{1}^{+}Z_{2}^{+}(Z_{3}^{+}Z_{4}^{-}Z_{5}^{+}Z_{6}^{-}\\\
&\quad\quad+Z_{3}^{-}Z_{4}^{+}Z_{5}^{+}Z_{6}^{-}+Z_{3}^{-}Z_{4}^{-}Z_{5}^{+}Z_{6}^{+}Z_{3}^{-}Z_{4}^{-}Z_{5}^{-}Z_{6}^{-})\\\
&\Delta_{5}=\sum\limits_{i=1}^{6}Z_{i}^{-}\otimes(\bigotimes\limits_{j=1,j\neq
i}^{6}Z_{j}^{+})\end{aligned}$ (44)

and $\sigma_{x}^{(i)}$ being the pauli matrice in the $i$-th subsystem.

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