# Undetectable Werner states using linear Bell inequalities

Ming-Xing Luo  1Information Security and National Computing Grid Laboratory,
Southwest Jiaotong University, Chengdu 610031, China,
2CSNMT, International Cooperation Research Center of China, Chengdu 610031,
China,
3Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

###### Abstract

Bell inequality serves as an important method to detect quantum entanglement,
a problem which is generally known to be NP-hard. Our goal in this work is to
detect Werner states using linear Bell inequality. Surprisingly, we show that
Werner states of almost all generalized multipartite Greenberger-Horne-
Zeilinger (GHZ) states cannot be detected by homogeneous linear Bell
inequalities with dichotomic inputs and outputs of each local observer. The
main idea is to estimate the largest violations of Werner states in the case
of general linear Bell inequalities. The presented method is then applied to
Werner states of all pure states to show a similar undetectable result.
Moreover, we provide an accessible method to determine undetectable Werner
states for general linear Bell inequality including sub-correlations. The
numeric algorithm shows that there are nonzero measures of Werner states with
small number of particles that cannot be detected by general linear Bell
inequalities.

## Introduction

Einstein-Podolski-Rosen (EPR) believed that local hidden variable (LHV) model
is a possible underlying explanation of physical reality EPR1 . Bell realized
that the simple assumptions behind any local hidden variable theory lead to
non-trivial restrictions on the strength of correlations and provide an
elegant way of verifying the validity of EPR’s belief Bell1 . The idea has
been described clearly as the violation of special inequalities which are
derived from LHV models. Bell theory Bell1  has become an utmost importance
tool for various applications in quantum information Bert  going beyond its
original motivations. Special Bell inequalities provide evidences for proof of
unconditional security for quantum key distribution Ek ; MY ; Acin1 ; Acin2 .
They are also useful for experimentally detecting entanglement without priori
hypothesis on the behavior of the experiment Cabel ; Brunn ; Kimb ; Piron ,
accomplishing multipartite interactive proof Bra ; Ben ; Kemp , estimating
dimensions of Hilbert space Brun2 ; Wehn , for others see reviews BCMD ; BCPS
.

The violations of Bell inequalities provide interesting ways of quantifying
inconsistence of quantum correlations with local classical description.
Unfortunately, for a given quantum state the violation primarily depends on
specially constructed Bell inequality. It turns out to be a daunting task to
build Bell inequality for general quantum states except for special cases.
Clauser-Horne-Shimony-Holt (CHSH) inequality as the first nontrivial example
CHSH1  can be applied for testing the inconsistence of generic bipartite
entangled pure states Gisin1 ; GP1 ; PR1  or special mixed states BMR  in the
case of dichotomic inputs and outputs of each local observer. Similar result
holds for multipartite entangled pure states LF ; YCZ  using Hardy’s
inequality Hard  which provides a state-dependent proof of the quantum
contextuality. However, Werner’s nonseparable mixed states
$\rho=v|\Phi\rangle\langle\Phi|+\frac{1-v}{4}\mathbbm{1}$ serves as a
counterexample to Gisin’s Theorem Gisin1  in the case of mixed states using
CHSH inequality. As mixtures of entangled pure state $|\Phi\rangle$ with white
noise, Werner states have been used to describe the state of each molecule
consisting of active nuclear spins in NMR Brau  and may be useful for single-
qubit teleportation Pop ; Benn . Werner states violate CHSH inequality for
$v>\frac{1}{\sqrt{2}}$ while they are separable if and only if
$v\leq\frac{1}{3}$. Moreover, Werner states admit a LHV model in the case of
projective measurements for $v\leq\frac{1}{2}$ Werner  or general measurements
for $v\leq\frac{5}{12}$ Bar . The range of $v$ for which a LHV model can be
constructed for Werner states has been further extended to $v\leq 0.66$ Pop1 ;
Bar ; AG . It is unknown whether Werner states admit a LHV model for $0.66\leq
v\leq\frac{1}{\sqrt{2}}$, or they violate a special Bell inequality.

Inspired by this open problem, in this letter we investigate the undetectable
problem of general Werner states. Although each entangled state can be
separated from all separable states using a linear functional from Hahn-Banach
theorem, however, Werner’s example demonstrated that the separating mapping
may be unavailable for some mixed entanglement in terms of LHV models. We
focus on LHV model with dichotomic inputs and outputs of each local observer.
For homogeneous linear Bell inequalities consisting of full correlations, we
prove that Werner states of almost all generalized multipartite Greenberger-
Horne-Zeilinger (GHZ) states cannot be completely detected. The main method is
to compare the critical visibility in terms of Bell inequality and the
critical parameter in terms of full separability. Different from the quantum
bound of Bell inequality using singular value decomposition of coefficient
matrix EKB , we estimate general quantum and classical bounds for Werner
states of all generalized multipartite GHZ states. Our result mainly relies on
direct estimation methods using Bell operators derived from Bell inequalities,
which are also different from operator space theory Ts ; JPP . Similar result
holds for Werner states of entangled pure state. Some generalized GHZ states
GHZ1  do not violate any Bell inequality involving two dichotomic observables
per local party ZBL , however, the result does not hold for sub-correlations
CHSH1  which are considered by us. For mixed states, the only result of the
undetectability holds for Werner state of the maximally entangled Einstein-
Podolsky-Rosen (EPR) state ZB1 ; EPR1 . We present a general undetectability
of homogeneous linear Bell inequalities consisting of dichotomic inputs and
outputs. For a general linear Bell inequality, we provide an accessible method
to test the undetectablity of Werner states of generalized multipartite GHZ
states or arbitrary entangled pure state. Based on numeric evaluations, there
exists nonzero measure of undetectable Werner states. These results are
related to the old mathematical problem of estimating Grothendieck-type
constants Ts ; AG , which are only useful for bipartite Bell inequalities.

## Preliminaries

In what follows, we focus on the scenario of dichotomic inputs and outputs of
each local observer. A general $m$-qubit state is represented by density
operator $\rho$ on Hilbert space $\mathbb{H}=\mathbb{H}^{\otimes m}_{2}$,
where $\mathbb{H}_{2}$ denotes Hilbert space of single qubit states. $\rho$ is
fully separable if there exist a normalized probability distribution
$\\{p_{i},i=1,\cdots,n\\}$ and density operators
$\\{\rho_{i}^{j},i=1,\cdots,n;j=1,\cdots,m\\}$ which are mixed states of the
$i$-th subsystem, such that
$\rho=\sum_{i=1}^{n}p_{i}\otimes_{j=1}^{m}\rho_{i}^{j}$. Otherwise, $\rho$ is
not fully separable. Without loss of generality, for each $j$ all states
$\rho_{i}^{j}(j=1,\cdots,m)$ may be assumed to be rank-1 projections, i.e.,
ensembles of pure states. Mathematically, all fully separable states are
convex polytopes of pure product states. The convexity implies the
separability of fully separable states and each entangled state from the
famous Hahn-Banach theorem.

Assume that the $j$-th space-like separated observer has two inputs labeled by
$x_{j}$ (which equals to 0 or 1 for simplicity), where the corresponding
outcomes are labeled by $a_{j}=\pm 1$. The key of LHV model is the probability
distribution of $a_{j}$ given $x_{j}$, that is, the conditional probability
$P(a_{1},\cdots,a_{m}|x_{1},\cdots,x_{m})$.

Definition 1. A LHV model Bell1 ; Pop1  is described by a conditional
probability

$\displaystyle
P(a_{1},\cdots,a_{m}|x_{1},\cdots,x_{m})=\int_{\Omega}d\mu(\lambda)\prod_{j=1}^{m}p_{\lambda}(a_{j}|x_{j},\lambda)$
(1)

for all dichotomic variables $a_{i}$s and $x_{i}$s, where
$(\Omega,\mu(\lambda))$ is a probability space of hidden variable $\lambda$
with normalization condition $\int_{\Omega}d\mu(\lambda)=1$.
$\\{p_{\lambda}(a_{j}=\pm 1|x_{j},\lambda)\\}$ ($j=1,\cdots,m$) are $m$
probability distributions depending on hidden variable $\lambda$.

Definition 2. A quantum model is described by a conditional probability

$\displaystyle P(a_{1},\cdots,a_{m}|x_{1},\cdots,x_{m})={\rm
Tr}(\otimes_{j=1}^{m}{\bf A}^{a_{j}}_{x_{j}}\rho)$ (2)

for all dichotomic variables $a_{i}$s and $x_{i}$s, where $\rho$ is a density
operator on Hilbert space $\mathbb{H}^{\otimes n}_{2}$. For each $i$, $\\{{\bf
A}^{a_{i}}_{x_{i}},a_{i}=\pm 1,x_{i}=0,1\\}$ consists of noncommuting
Hermitian operators with eigenvalues no more than $1$, which represent quantum
measurements on the $i$-th observer’s system, i.e., $\|{\bf
A}^{a_{i}}_{x_{i}}\|\leq 1$ for all $a_{i}$s and $x_{i}$s, $[{\bf
A}^{a_{i}}_{x_{i}=0},{\bf A}^{a_{i}}_{x_{i}=1}]\not=0$ for each $a_{i}$, and
$\sum_{a_{i}=\pm 1}{\bf A}^{a_{i}}_{x_{i}}=\mathbbm{1}_{2}$ (identity operator
on Hilbert space $\mathbb{H}_{2}$) for each $x_{i}$.

Quantum state $\rho$ used in Definition 2 is interpreted as shared quantum
resource. When $\rho$ is fully separable, Quantum model defined in Definition
2 is a special case of LHV defined in Definition 1 with discrete distribution
of $\lambda$, which may be obtained by measuring the subsystems with rank-1
projectors. Generally, a LHV model is represented by the following local
hidden state Ts :

Definition 3 A Local Hidden State (LHS) model is described by a conditional
probability

$\displaystyle P(a_{1},\cdots,a_{m}|x_{1},\cdots,x_{m})={\rm
Tr}(\otimes_{i=1}^{m}{\bf A}^{a_{i}}_{x_{i}}\rho)$ (3)

for all dichotomic variables $a_{i}$s and $x_{i}$s, where $\rho$ is a density
operator on Hilbert space $\mathbb{H}^{\otimes n}_{2}$, $\\{{\bf
A}^{a_{i}}_{x_{i}}\\}(i=1,\cdots,m)$ consists of commuting measurement
operators with eigenvalues no more than $1$, which represent projective
(classical) measurements on the $i$-th observer’s system, i.e., $\|{\bf
A}^{a_{i}}_{x_{i}}\|\leq 1$ for all $a_{i}$s and $x_{i}$s, $[{\bf
A}^{a_{i}}_{x_{i}=0},{\bf A}^{a_{i}}_{x_{i}=1}]=0$ for each $a_{i}$, and
$\sum_{a_{i}=\pm 1}{\bf A}^{a_{i}}_{x_{i}}=\mathbbm{1}_{2}$ for each $x_{i}$.

## Results

From Definition 1, define $\langle A_{x_{1}}\cdots
A_{x_{m}}\rangle=\sum_{a_{1},\cdots,a_{m}=-1,1}(-1)^{\sum_{i=1}^{m}(a_{i}+1)/2}\frac{1}{2^{m}}P(a_{1},\cdots,a_{m}|x_{1},\cdots,x_{m})$
as the expectation of outputs $a_{i}$s derived from $m$-partite correlations
$P(a_{1},\cdots,a_{m}|x_{1},\cdots,x_{m})$ with inputs $x_{i}$s, where we
assume the uniform probability distribution for each input $x_{i}$. Let
$\langle B\rangle=\sum_{x_{1},\cdots,x_{m}}\alpha_{x_{1}\cdots x_{m}}\langle
A_{x_{1}}\cdots A_{x_{m}}\rangle$ be linear superposition of some expectations
$\langle A_{x_{1}}\cdots A_{x_{m}}\rangle$. The corresponding Bell
inequalities are given by $|\langle B\rangle|_{c}\leq c_{1}$ and $|\langle
B\rangle|_{q}\leq c_{2}$, where $c_{1}$ denotes classical bound over all
conditional probabilities in Eq.(1) in terms of LHV model in Definition 1
while $c_{2}$ denotes quantum bound over all conditional probabilities in
Eq.(2) in terms of quantum model in Definition 2. The corresponding Bell
operator is given by ${\cal B}=\sum_{x_{1},\cdots,x_{m}}\alpha_{x_{1}\cdots
x_{m}}\sum_{a_{1},\cdots,a_{m}=-1,1}(-1)^{\sum_{i=1}^{m}(a_{i}+1)/2}{\bf
A}^{a_{1}}_{x_{1}}\otimes\cdots\otimes{\bf A}^{a_{m}}_{x_{m}}$. From
Definitions 2 and 3, we have $|\langle B\rangle|_{c}=\sup_{{\bf
A}^{a_{1}}_{x_{1}},\cdots,{\bf A}^{a_{m}}_{x_{m}},\rho}|{\rm Tr}({\cal
B}\rho)|:=\|{\cal B}\|_{c}$ over all commuting Hermitian operators ${\bf
A}^{a_{i}}_{x_{i}}$s and density operators $\rho$ on Hilbert space
$\mathbb{H}_{2}^{m}$, while $|\langle B\rangle|_{q}=\sup_{{\bf
A}^{a_{1}}_{x_{1}},\cdots,{\bf A}^{a_{m}}_{x_{m}},\rho}|{\rm Tr}({\cal
B}\rho)|:=\|{\cal B}\|_{q}$ over all general Hermitian operators ${\bf
A}^{a_{i}}_{x_{i}}$s and density operators $\rho$ on Hilbert space
$\mathbb{H}_{2}^{m}$.

Given an $m$-partite generalized GHZ state GHZ1
$|\Psi\rangle=\cos\theta|\vec{0}\rangle+\sin\theta|\vec{1}\rangle$ with
$\theta\in(0,\frac{\pi}{2})$, the corresponding Werner state Werner  is
defined as mixed state of $|\Psi\rangle\langle\Psi|$ and noise state
$\frac{1}{2^{m}}\mathbbm{1}$ with the following form:

$\displaystyle\rho_{v}=\frac{1-v}{2^{m}}\mathbbm{1}+v|\Psi\rangle\langle\Psi|,$
(4)

where $\mathbbm{1}$ denotes the identity operator on Hilbert space
$\mathbb{H}^{\otimes m}_{2}$, $|\vec{0}\rangle=|0\rangle^{\otimes
m},|\vec{1}\rangle=|1\rangle^{\otimes m}$ and $v\in[0,1]$. The weight $v$ may
operationally represent the observed interferometric contrast in experiment
Werner . Our first result is a typical undetectable feature of Werner states.
Formally, we prove the following theorem.

Theorem 1. The following results hold:

  * (i)

For almost all generalized GHZ states, the corresponding Werner states defined
in Eq.(4) cannot be completely detected by homogeneous linear Bell
inequalities;

  * (ii)

For most of generalized GHZ states, the corresponding Werner states defined in
Eq.(4) cannot be completely detected by a general linear Bell inequality if
the corresponding Bell operator ${\cal B}$ satisfies

$\displaystyle\|{\cal B}\|_{c}\geq\gamma_{i}\|{\cal B}_{i}\|_{c}$ (5)

for $i=1,\cdots,m-1$, where all positive constants $\gamma_{i}$s satisfy

$\displaystyle\sum_{i=1}^{m-1}\frac{1}{\gamma_{i}}<\sqrt{12}(2^{m}-1).$ (6)

Here, ${\cal B}_{i}$ denotes partial Bell operator of ${\cal B}$ including the
correlation of the $i$-th party but excluding all correlations of the $j$-th
parties, $j=1,\cdots,i-1$;

  * (iii)

For most of generalized GHZ states with $3\leq m\leq 6$, the corresponding
Werner states defined in Eq.(4) cannot be completely detected by general
linear Bell inequalities.

Theorem 1 generalizes the undetectability of Werner state of the maximally
entangled GHZ state Werner ; ZB1 . In particular, the generic undetectability
holds for homogeneous linear Bell inequalities. Here, homogeneous means that
all terms of $m$-partite Bell inequality include full correlations of $m$
observers. Almost means for almost all $\theta\in(0,\frac{\pi}{2})$
(approximate full measure) Werner states defined in Eq.(4) cannot be
completely detected by any homogeneous linear Bell inequality, i.e., the
critical visibility $v$ in terms of the violation of some linear Bell
inequality is larger than the critical parameter $v$ in terms of the full
separability. Hence, there are Werner states that are entangled but cannot be
detected by Bell testing. The main idea is to estimate upper bounds of all
homogeneous linear Bell inequalities (Appendices A, B and C). CHSH inequality
CHSH1  and Mermin inequality Merm  are nontrivial examples that cannot
completely detect almost all Werner states defined in Eq.(4). A general linear
Bell inequality may have sub-correlations involving less than $m$ observers
and has no tight upper bound in terms of LHV model. The conditions presented
in Eqs.(5) and (6) provide an accessible way to determine whether a given
Werner state is undetectable or not. Some examples are shown in Appendix H.

Given an $m$-partite entangled pure state
$|\Phi\rangle=\sum_{i_{1},\cdots,i_{m}=0,1}\alpha_{\vec{i}}|\vec{i}\rangle$
with $m\geq 2$, Werner states Werner  of $|\Phi\rangle$ are defined by

$\displaystyle\rho_{v}=\frac{1-v}{2^{m}}\mathbbm{1}+v|\Phi\rangle\langle\Phi|,$
(7)

where $\alpha_{\vec{i}}$ are complex coefficients satisfying
$\sum_{\vec{i}}|\alpha_{\vec{i}}|^{2}=1$, $\vec{i}$ denote $m$-bit index
vectors $i_{1}\cdots i_{m}$, and $v\in[0,1]$. $\rho_{v}$ are density matrices
for all $v\in[0,1]$. Similar results hold for generalized Werner states.

Theorem 2. The following results hold:

  * (i)

For almost all pure states, the corresponding Werner states defined in Eq.(7)
cannot be completely detected by homogeneous linear Bell inequalities;

  * (ii)

There is a nonzero measure of entangled pure states whose Werner states
defined in Eq.(7) cannot be completely detected by a general linear Bell
inequality, if the corresponding Bell operator ${\cal B}$ satisfies

$\displaystyle\|{\cal B}\|_{c}\geq\gamma_{i}\|{\cal B}_{i}\|_{c}$ (8)

for $i=1,\cdots,m-1$, where positive constants $\gamma_{i}$ satisfy

$\displaystyle\sum_{i=1}^{m-1}\frac{1}{\gamma_{i}}<\frac{1}{\sqrt{3}}{\rm
Poly}(m)-1.$ (9)

Here, ${\cal B}_{i}$ denotes partial Bell operator including the correlation
of the $i$-th party but excluding all correlations of the $j$-th parties
($j=1,\cdots,i-1$), and ${\rm Poly}(\cdot)$ denotes any polynomial function;

  * (iii)

There is a nonzero measure of $m$-partite pure states with $3\leq m\leq 6$
with the property that the corresponding Werner states defined in Eq.(7)
cannot be completely detected by general linear Bell inequalities.

Theorem 2 generalizes the undetectability of Werner state of bipartite states
Pop1 ; Bar ; AG  and special multipartite states Werner ; ZB1 ; BFFH . Similar
to Theorem 1, there are many instances of undetectable Werner states of
general pure states in the case of homogeneous linear Bell inequalities such
as CHSH inequality CHSH1  and Mermin inequality Merm . Different from
generalized GHZ states, there is no tight upper bound of the critical
parameter $v$ for each pure state $|\Phi\rangle$ in terms of the full
separability (Appendices D, F and G). We cannot prove the typical
undetectability for Werner states of general pure states in the case of
general linear Bell inequalities. Although Grothendieck-type constants may be
used for bipartite, it is unclear how to extend the result to multipartite
states Ts ; AG . Our conditions given in Eqs.(8) and (9) may be useful for
determining whether a general Werner state is detectable or not for given
linear Bell inequalities.

## Discussion

Werner states as special mixed states can be geometrically described as a
curve of states in density operator space, which is from a fully separable
mixed state to an entangled pure state Werner . There always exists a critical
point separating fully separable states and entangled states on the curve.
These critical points generally have different mixing parameters $v$ that
depend on the given pure states. The relative critical visibility as the ratio
of quantum bound and classical bound of Bell inequalities characterizes the
delectability of these inequalities. Although all pure states admit a
universal violation YCZ , homogeneous Bell inequalities can only provide
bounded violations that are incompatible with infinitely small critical
parameter $v$ in terms of the full separability. We conjecture that this
incompatibility holds for general linear Bell inequalities even though there
are unbounded violations JPP . The possible reason is that almost all critical
points require exponential violations Deng1  (Appendices E, F) going beyond
polynomial violations JPP . Unfortunately, our algorithmic proof (Appendix F)
can only provide evidences for small $m$ due to high computational complexity.
It is an open problem to find new methods for estimating unbounded violations.
We further conjecture similar results for multipartite high-dimensional
states.

In summary, we have shown that Gisin’s theorem Gisin1  does not hold for most
Werner states in the case of homogeneous Bell inequalities consisting of full
correlations, where each observer can choose two dichotomic observables. For a
general linear Bell inequality without tight classical bound, we presented
accessible conditions to determine the detectability in the case of general
Werner states. Our conditions involve only classical bounds of $m-1$
restricted Bell operators. Moreover, with these conditions we numerically
proved the undetectable Werner states for small $m$. Of course, the presented
incompatibility may be resolved for different assumptions such as multiple
observables per local observer KC , which are also interesting problems. Our
results may shed a new light on entanglement testing, quantum supremacy or
quantum communication SG .

## Acknowledgements

We thank the helpful discussions of Luming Duan, Yaoyun Shi, Yuan Su, Xiubo
Chen and Yixian Yang. Authors thank for interesting suggestions by Joseph
Bowles. This work was supported by the National Natural Science Foundation of
China (Nos.61772437,61702427), Sichuan Youth Science and Technique Foundation
(No.2017JQ0048), Fundamental Research Funds for the Central Universities
(No.2682014CX095), Chuying Fellowship, and EU ICT COST CryptoAction
(No.IC1306).

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## Appendix A: Ideal bound of the critical visibility in the case of linear
Bell inequality

We firstly introduce some notation. Let $\vec{i}$ denote an $m$-bit index
vector $i_{1}\cdots i_{m}$, i.e, the concatenation of $m$ bits
$i_{1},\cdots,i_{m}$. Denote $\vec{i}_{c}=\vec{1}\oplus\vec{i}$ as binary
complement of $\vec{i}$, where $\vec{1}$ represents $m$-bit of $1\cdots 1$ and
addition is mod 2 per bit.

Given an $m$-partite generalized GHZ state GHZ
$|\Psi\rangle=\cos\theta|\vec{0}\,\rangle+\sin\theta|\vec{1}\,\rangle$ with
$\theta\in(0,\frac{\pi}{2})$, Werner state Wer  is a mixed state of pure state
$|\Psi\rangle\langle\Psi|$ and noise state (maximally mixed state)
$\frac{1}{2^{m}}$ with the following form:

$\displaystyle\rho_{v}=\frac{1-v}{2^{m}}\mathbbm{1}+v|\Psi\rangle\langle\Psi|,$
(A1)

where $\mathbbm{1}$ denotes the identity operator on Hilbert space
$\mathbb{H}^{\otimes m}_{2}$, $\mathbb{H}_{2}$ denotes Hilbert space of single
qubit states, and $v\in[0,1]$. It easily proves that $\rho_{v}$ are density
matrices (positive semidefinite) for all $v\in[0,1]$.

Define $\langle A_{x_{1}}\cdots
A_{x_{m}}\rangle=\sum_{a_{1},\cdots,a_{m}=-1,1}(-1)^{\sum_{i=1}^{m}(a_{i}+1)/2}P(a_{1},\cdots,a_{m}|x_{1},\cdots,x_{m})$
as the expectation of $m$-partite measurements $A_{x_{1}}\cdots A_{x_{m}}$
with inputs $x_{1},\cdots,x_{m}$, where we assume the uniform probability
distribution for each input $x_{i}$. Denote $\langle
B\rangle=\sum_{x_{1},\cdots,x_{m}}\alpha_{x_{1}\cdots x_{m}}\langle
A_{x_{1}}\cdots A_{x_{m}}\rangle$. The corresponding Bell inequalities are
given by $|\langle B\rangle|_{c}\leq c_{1}$ and $|\langle B\rangle|_{q}\leq
c_{2}$, where $c_{1}$ denotes classical bound over all conditional
probabilities shown in Eq.(1) in terms of LHV model in Definition 1 while
$c_{2}$ denotes quantum bound over all conditional probabilities given in
Eq.(2) in terms of quantum model in Definition 2. The corresponding Bell
operator is given by ${\cal B}=\sum_{x_{1},\cdots,x_{m}}\alpha_{x_{1}\cdots
x_{m}}\sum_{a_{1},\cdots,a_{m}=-1,1}(-1)^{\sum_{i=1}^{m}(a_{i}+1)/2}{\bf
A}^{a_{1}}_{x_{1}}\otimes\cdots\otimes{\bf A}^{a_{m}}_{x_{m}}$. From
Definitions 2 and 3, we have $|\langle B\rangle|_{c}=\sup_{{\bf
A}^{a_{1}}_{x_{1}},\cdots,{\bf A}^{a_{m}}_{x_{m}},\rho}|{\rm Tr}({\cal
B}\rho)|:=\|{\cal B}\|_{c}$ over all commuting Hermitian operators ${\bf
A}^{a_{i}}_{x_{i}}$s and density operator $\rho$ on Hilbert space
$\mathbb{H}_{2}^{m}$, while $|\langle B\rangle|_{q}=\sup_{{\bf
A}^{a_{1}}_{x_{1}},\cdots,{\bf A}^{a_{m}}_{x_{m}},\rho}|{\rm Tr}({\cal
B}\rho)|:=\|{\cal B}\|_{q}$ over all possible Hermitian operators ${\bf
A}^{a_{i}}_{x_{i}}$s and density operator $\rho$ on Hilbert space
$\mathbb{H}_{2}^{m}$.

The proof of Theorem 1 is completed by the following four sections. This
section contributes the lower bound of $v$ for which $\rho_{v}$ violates a
linear Bell inequality.

Given a Bell operator ${\cal B}$ derived from a linear $m$-partite Bell
inequality, assume that

$\displaystyle\|{\cal B}\|_{c}\leq c_{1}$ (A2)

in terms of LHS model, and

$\displaystyle\|{\cal B}\|_{q}\leq c_{2}$ (A3)

in terms of quantum model, where $\vec{\bf A}^{c}$ and $\vec{\bf A}^{q}$
denote $({\bf A}^{c}_{1,i_{1}},\cdots,{\bf A}^{c}_{1,i_{m}})$ and $({\bf
A}^{q}_{1,i_{1}},\cdots,{\bf A}^{q}_{1,i_{m}})$ respectively, ${\bf
A}^{c}_{j,i_{j}}$ denote commuting Hermitian operators of the $j$-th party,
and ${\bf A}^{q}_{j,i_{j}}$ denote general Hermitian operators of the $j$-th
party. To detect quantum entanglement $\rho_{v}$, $c_{i}$s satisfy
$c_{2}>c_{1}$.

Lemma 1. The critical visibility $v^{*}$ for which $\rho_{v}$ defined in
Eq.(A1) violates linear Bell inequality shown in Eq.(A2) is given by

$\displaystyle v^{*}\geq\frac{2^{m}c_{1}-c_{1}}{2^{m}c_{2}-c_{1}}.$ (A4)

Proof. Assume that Werner state $\rho_{v}$ defined in Eq.(A1) violates linear
Bell inequality shown in Eq.(A2). There exist general Hermitian measurements
$\vec{\bf A}^{q}$ and density operator $\rho$ on Hilbert space
$\mathbb{H}_{2}^{m}$ such that

$\displaystyle{\rm Tr}({\cal B}_{\vec{\bf A}^{q}}\rho)=$
$\displaystyle\frac{1-v}{2^{m}}{\rm Tr}({\cal B}_{\vec{\bf
A}^{q}}\mathbbm{1})+v{\rm Tr}({\cal B}_{\vec{\bf
A}^{q}}|\Psi\rangle\langle\Psi|)$ $\displaystyle>$ $\displaystyle c_{1},$ (A5)

where we assume ${\rm Tr}({\cal B}_{\vec{\bf A}^{q}}\rho)>0$; Otherwise, we
replace ${\cal B}_{\vec{\bf A}^{q}}$ with $-{\cal B}_{\vec{\bf A}^{q}}$, where
${\cal B}_{\vec{\bf A}^{q}}$ is the Bell operator derived from general
Hermitian measurement operators $\vec{\bf A}^{q}$. Since ${\rm Tr}({\cal
B}_{\vec{\bf A}^{q}}\mathbbm{1})\leq c_{1}$ ($\frac{1}{2^{m}}\mathbbm{1}$ is
the maximally mixed state which is fully separable) and ${\rm Tr}({\cal
B}_{\vec{\bf A}^{q}}|\Psi\rangle\langle\Psi|)\leq c_{2}$ from Eqs.(A2) and
(A3), Eq.(A5) implies that

$\displaystyle v>$ $\displaystyle\frac{2^{m}c_{1}-{\rm Tr}({\cal B}_{\vec{\bf
A}^{q}}\mathbbm{1})}{2^{m}{\rm Tr}({\cal B}_{\vec{\bf
A}^{q}}|\Psi\rangle\langle\Psi|)-{\rm Tr}({\cal B}_{\vec{\bf
A}^{q}}\mathbbm{1})}$ $\displaystyle\geq$ $\displaystyle\frac{2^{m}c_{1}-{\rm
Tr}({\cal B}_{\vec{\bf A}^{q}}\mathbbm{1})}{2^{m}c_{2}-{\rm Tr}({\cal
B}_{\vec{\bf A}^{q}}\mathbbm{1})}$ $\displaystyle\geq$
$\displaystyle\frac{2^{m}c_{1}-c_{1}}{2^{m}c_{2}-c_{1}},$ (A6)

where Eq.(A6) is from the inequality
$\frac{a-x}{b-x}\geq\frac{a-x^{\prime}}{b-x^{\prime}}$ for $x,x^{\prime}$
satisfying $0\leq x\leq x^{\prime}\leq\min\\{a,b\\}$. $\Box$

## Appendix B: Relative critical visibility in the case of homogeneous linear
Bell inequalities

This section contributes the relative critical visibility, i.e., the ratio of
the quantum upper bound and the classical upper bound of homogeneous linear
Bell inequalities. In what follows, we focus on the case of dichotomic inputs
and outputs for each observer. Two measurement operators ${\bf A}_{i,x_{i}=0}$
and ${\bf A}_{i,x_{i}=1}$ are used for the $i$-th observer, $i=1,\cdots,m$.
Assume that each measurement operator ${\bf A}_{i,x_{i}}$ has $\pm 1$
eigenvalues. $\langle{\bf A}_{i,x_{i}}\rangle$ denotes the expectation of
outputs of the $i$-th observer. With these denotations, from Definitions 2 and
3 a general homogeneous linear Bell inequality may be given by

$\displaystyle\sup_{\vec{\bf A}^{c(q)},\rho}|{\rm
Tr}(\sum_{i_{1},\cdots,i_{m}=0,1}\alpha_{\vec{i}}{\bf
A}^{c(q)}_{1,i_{1}}\otimes\cdots\otimes{\bf A}^{c(q)}_{m,i_{m}}\rho)|\leq$
$\displaystyle c_{1}(c_{2}),$ (B1)

where supremum is over all measurement operators $\vec{\bf A}^{c(q)}$ and
density operator $\rho$ on Hilbert space $\mathbb{H}_{2}^{m}$ Tsi1 ; Tsi2 ,
$c_{1}$ and $c_{2}$ denote the respective upper bound in the case of commuting
Hermitian operators ${\bf A}^{c}_{i,j}$s and general Hermitian operators ${\bf
A}^{q}_{i,j}$s, and $\alpha_{\vec{i}}$ are real coefficients. Here, general
Hermitian operators include quantum observeables and commuting Hermitian
operators. The corresponding Bell operator is given by

$\displaystyle{\cal B}=\sum_{i_{1},\cdots,i_{m}}\alpha_{\vec{i}}{\bf
A}_{1,i_{1}}\otimes\cdots\otimes{\bf A}_{m,i_{m}},$ (B2)

which is a homogeneous linear Hermitian operator. In this paper, homogeneous
means that all joint measurement operators ${\bf
A}^{c(q)}_{1,i_{1}}\otimes\cdots\otimes{\bf A}^{c(q)}_{m,i_{m}}$ of ${\cal B}$
include exactly $m$ measurement operators ${\bf
A}^{c(q)}_{1,i_{1}},\cdots,{\bf A}^{c(q)}_{m,i_{m}}$, or equivalently, the
corresponding Bell inequality includes all full correlations.

Our goal in this section is to estimate the relative critical visibility
$r_{cq}$ by taking use of Hermitian operator ${\cal B}$ as

$\displaystyle r_{cq}=\frac{\|{\cal B}\|_{q}}{\|{\cal B}\|_{c}},$ (B3)

where $\|{\cal B}\|_{c}:=\sup_{\vec{\bf A}^{c},\rho}|{\rm
Tr}(\sum_{i_{1},\cdots,i_{m}=0,1}\alpha_{\vec{i}}{\bf
A}^{c}_{1,i_{1}}\otimes\cdots\otimes{\bf A}^{c}_{m,i_{m}}\rho)|$ denotes the
norm of Bell operator ${\cal B}$ over commuting Hermitian operators ${\bf
A}^{c}_{j,i_{j}}$s and density operators $\rho$ while $\|{\cal
B}\|_{q}:=\sup_{\vec{\bf A}^{q},\rho}|{\rm
Tr}(\sum_{i_{1},\cdots,i_{m}=0,1}\alpha_{\vec{i}}{\bf
A}^{q}_{1,i_{1}}\otimes\cdots\otimes{\bf A}^{q}_{m,i_{m}}\rho)|$ denotes the
norm of Bell operator ${\cal B}$ over general Hermitian operators ${\bf
A}^{q}_{j,i_{j}}$s and density operators $\rho$. Here, $\|{\cal
B}\|_{c}\not=0$ for any nonzero Bell operators. The upper bound of $r_{cq}$
for homogeneous Bell operator ${\cal B}$ is shown in Lemma 2. Typical examples
are CHSH inequality CHSH  and Mermin inequality Mermin .

Lemma 2. For all homogeneous Bell operators of $m$-multipartite, the relative
critical visibility $r_{cq}$ defined in Eq.(B3) satisfies

$\displaystyle r_{cq}\leq\sqrt{3},$ (B4)

or $r_{cq}\leq\sqrt{\frac{5}{2}}$ when quantum observables are restricted to
anticommuting observables.

Proof. We first consider $m=2$ to explain the main idea. And then, we extend
the proof for $m>2$.

Case 1. $m=2$.

Let ${\bf A}_{1,0}$ and ${\bf A}_{1,1}$ be dichotomic measurement operators of
Alice, and ${\bf A}_{2,0}$ and ${\bf A}_{2,1}$ be dichotomic measurement
operators of Bob. ${\bf A}_{i,j}$ are Hermitian operators with eigenvalues
$\lambda$ satisfying $|\lambda|\leq 1$, i.e., $\|{\bf A}_{i,j}\|\leq 1$
($\|\cdot\|$ denotes the norm of operators in terms of quantum model). For
simplicity, we assume ${\bf A}_{i,j}^{2}=\mathbbm{1}_{2}$ (the identity
operator on single qubit state), i.e, the outcomes of observables are
associated to projective measurements Tsi1 ; Tsi2 .

Given a bipartite homogeneous linear Bell operator ${\cal B}$, we have

$\displaystyle{\cal B}=$ $\displaystyle\sum_{i,j=0}^{1}\alpha_{ij}{\bf
A}_{1,i}\otimes{\bf A}_{2,j}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{4}\alpha_{i}{\bf A}_{i}\otimes{\bf B}_{i},$ (B5)

where ${\bf A}_{1}={\bf A}_{2}={\bf A}_{1,0}$ and ${\bf A}_{3}={\bf
A}_{4}={\bf A}_{1,1}$, ${\bf B}_{1}={\bf B}_{3}=\frac{1}{2}({\bf A}_{2,0}+{\bf
A}_{2,1})$ and ${\bf B}_{2}={\bf B}_{4}=\frac{1}{2}({\bf A}_{2,0}-{\bf
A}_{2,1})$, $\alpha_{1}=\alpha_{00}+\alpha_{01}$,
$\alpha_{2}=\alpha_{00}-\alpha_{01}$, $\alpha_{3}=\alpha_{10}+\alpha_{11}$ and
$\alpha_{4}=\alpha_{10}-\alpha_{11}$. From Eq.(B5) we obtain

$\displaystyle{\cal B}^{2}=$
$\displaystyle\sum_{i,j=1}^{4}\alpha_{i}\alpha_{j}({\bf A}_{i}{\bf
A}_{j})\otimes({\bf B}_{i}{\bf B}_{j})$ $\displaystyle=$
$\displaystyle(\alpha_{1}{\bf A}_{1}+\alpha_{3}{\bf A}_{3})^{2}\otimes{\bf
B}_{1}^{2}+(\alpha_{2}{\bf A}_{2}+\alpha_{4}{\bf A}_{4})^{2}\otimes{\bf
B}_{2}^{2}$ $\displaystyle+\frac{1}{4}(\alpha_{1}{\bf A}_{1}+\alpha_{3}{\bf
A}_{3})(\alpha_{2}{\bf A}_{2}+\alpha_{4}{\bf A}_{4})\otimes[{\bf A}_{2,1},{\bf
A}_{2,0}]^{2}$ $\displaystyle+\frac{1}{4}(\alpha_{2}{\bf A}_{2}+\alpha_{4}{\bf
A}_{4})(\alpha_{1}{\bf A}_{1}+\alpha_{3}{\bf A}_{3})\otimes[{\bf A}_{2,0},{\bf
A}_{2,1}]^{2},$ (B6)

where $[{\bf X},{\bf Y}]={\bf X}{\bf Y}-{\bf Y}{\bf X}$.

In what follows, assume that $\alpha_{1}\geq 0$ and $\alpha_{2}\geq 0$;
Otherwise, by replacing $\alpha_{i}$ with $-\alpha_{i}$, the value of the
right side of Eq.(B6) is invariable. For commuting Hermitian operators ${\bf
A}^{c}_{i,j}$, ${\bf A}_{i}$ and ${\bf B}_{i}$ are commuting. Given a density
operator $\rho$ on Hilbert space $\mathbb{H}_{2}^{2}$, Eq.(B6) implies that

$\displaystyle{\rm Tr}({\cal B}^{2}\rho)={\rm Tr}((\alpha_{1}{\bf
A}_{1}+\alpha_{3}{\bf A}_{3})^{2}\otimes{\bf B}_{1}^{2}\rho)+{\rm
Tr}((\alpha_{2}{\bf A}_{2}+\alpha_{4}{\bf A}_{4})^{2}\otimes{\bf
B}_{2}^{2}\rho),$ (B7)

where Eq.(B7) is from the equalities $[{\bf B}_{i},{\bf B}_{j}]=0$ with
$i=1,3$ and $j=2,4$. Two different cases will be discussed as follows:

  * (i)

$\alpha_{3}\alpha_{4}\geq 0$. Eq.(B7) implies

$\displaystyle|{\rm Tr}({\cal B}^{2}\rho)|\leq$
$\displaystyle(\alpha_{1}+|\alpha_{3}|)^{2}\beta_{1}^{2}+(\alpha_{2}+|\alpha_{4}|)^{2}\beta_{2}^{2},$
(B8)

where Eq.(B8) is from the inequalities $\|{\bf A}_{i}\|\leq 1$
($i=1,\cdots,4$), $\beta^{2}_{1}={\rm Tr}({\bf
B}_{1}^{2}\rho_{B})=\frac{1}{2}+\frac{1}{4}{\rm Tr}({\bf A}^{c}_{2,0}{\bf
A}^{c}_{2,1}\rho_{B}+{\bf A}^{c}_{2,1}{\bf A}^{c}_{2,0}\rho_{B})$ and
$\beta_{2}^{2}={\rm Tr}({\bf B}_{2}^{2}\rho_{B})=\frac{1}{2}-\frac{1}{4}{\rm
Tr}({\bf A}^{c}_{2,0}{\bf A}^{c}_{2,1}\rho_{B}+{\bf A}^{c}_{2,1}{\bf
A}^{c}_{2,0}\rho_{B})$ with the reduced state $\rho_{B}={\rm Tr}_{A}(\rho)$ of
Bob. The equality of Eq.(B8) holds for separable density matrices
$\rho=\rho_{A}\otimes\rho_{B}$, and identity measurement operators ${\bf
A}_{1}={\bf A}_{2}=\mathbbm{1}_{2}$ and ${\bf
A}_{i}=\frac{\alpha_{i}}{|\alpha_{i}|}\mathbbm{1}_{2}$ ($i=3,4$) if
$\alpha_{i}\not=0$. Now, two subcases will be given as follows:

    * (a)

If $\alpha_{1}+|\alpha_{3}|\geq\alpha_{2}+|\alpha_{4}|$, Eq.(B8) implies that

$\displaystyle\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal
B}^{2}\rho)|^{\frac{1}{2}}=\alpha_{1}+|\alpha_{3}|,$ (B9)

where the supremum is achieved when $\rho=\frac{1}{4}\mathbbm{1}$ and ${\bf
A}^{c}_{2,0}={\bf A}^{c}_{2,1}=\mathbbm{1}_{2}$. From Schwartz inequality
$|{\rm Tr}({\cal B}\rho)|\leq\sqrt{|{\rm Tr}({\cal B}^{2}\rho)|}$ we have

$\displaystyle\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal B}\rho)|$
$\displaystyle\leq\sup_{\vec{\bf A}^{c},\rho}\sqrt{|{\rm Tr}({\cal
B}^{2}\rho)|}$ $\displaystyle=\alpha_{1}+|\alpha_{3}|,$ (B10)

where the equality holds for $\rho=\frac{1}{4}\mathbbm{1}$, ${\bf A}_{1}={\bf
A}_{2}={\bf A}^{c}_{2,0}={\bf A}^{c}_{2,1}=\mathbbm{1}_{2}$, ${\bf
A}_{3}=\frac{\alpha_{3}}{|\alpha_{3}|}\mathbbm{1}_{2}$ with
$\alpha_{3}\not=0$, and ${\bf
A}_{4}=\frac{\alpha_{4}}{|\alpha_{4}|}\mathbbm{1}_{2}$ with
$\alpha_{4}\not=0$.

    * (b)

If $\alpha_{1}+|\alpha_{3}|\leq\alpha_{2}+|\alpha_{4}|$, Eq.(B8) implies that

$\displaystyle\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal
B}\rho)|=\alpha_{2}+|\alpha_{4}|,$ (B11)

where the supremum is achieved when $\rho=\frac{1}{4}\mathbbm{1}$ and ${\bf
A}^{c}_{2,0}=-{\bf A}^{c}_{2,1}=\mathbbm{1}_{2}$. From Schwartz inequality we
have

$\displaystyle\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal B}\rho)|$
$\displaystyle\leq\sup_{\vec{\bf A}^{c},\rho}\sqrt{|{\rm Tr}({\cal
B}^{2}\rho)|}$ $\displaystyle=\alpha_{2}+|\alpha_{4}|,$ (B12)

where the equality holds for $\rho=\frac{1}{4}\mathbbm{1}$, ${\bf A}_{1}={\bf
A}_{2}={\bf A}^{c}_{2,0}=-{\bf A}^{c}_{2,1}=\mathbbm{1}_{2}$, ${\bf
A}_{3}=\frac{\alpha_{3}}{|\alpha_{3}|}\mathbbm{1}_{2}$ with
$\alpha_{3}\not=0$, and ${\bf
A}_{4}=\frac{\alpha_{4}}{|\alpha_{4}|}\mathbbm{1}_{2}$ with
$\alpha_{4}\not=0$.

From Eqs.(B10) and (B12), the norm of Bell operator ${\cal B}$ in the case of
commuting Hermitian measurements is given by

$\displaystyle\|{\cal B}\|_{c}=$ $\displaystyle\sup_{\vec{\bf
A}^{c},\rho}|{\rm Tr}({\cal B}\rho)|$ $\displaystyle=$
$\displaystyle\max\\{\alpha_{1}+|\alpha_{3}|,\alpha_{2}+|\alpha_{4}|\\}.$
(B13)

  * (ii)

$\alpha_{3}\alpha_{4}\leq 0$. Eq.(B7) implies that

$\displaystyle|{\rm Tr}({\cal B}^{2}\rho)|\leq$
$\displaystyle\max\\{(\alpha_{1}+|\alpha_{3}|)^{2}\beta_{1}^{2}+(\alpha_{2}-|\alpha_{4}|)^{2}\beta_{2}^{2},(\alpha_{1}-|\alpha_{3}|)^{2}\beta_{1}^{2}+(\alpha_{2}+|\alpha_{4}|)^{2}\beta_{2}^{2}\\},$
(B14) $\displaystyle\leq$
$\displaystyle\max\\{(\alpha_{1}+|\alpha_{3}|)^{2},(\alpha_{2}-|\alpha_{4}|)^{2},(\alpha_{1}-|\alpha_{3}|)^{2},(\alpha_{2}+|\alpha_{4}|)^{2}\\}$
(B15) $\displaystyle=$
$\displaystyle\max\\{(\alpha_{1}+|\alpha_{3}|)^{2},(\alpha_{2}+|\alpha_{4}|)^{2}\\},$
(B16)

where Eq.(B14) is from the inequalities $\|{\bf A}_{i}\|\leq 1$
($i=1,\cdots,4$), Eq.(B15) is from the inequalities
$(\alpha_{1}+|\alpha_{3}|)^{2}\beta_{1}^{2}+(\alpha_{2}-|\alpha_{4}|)^{2}\beta_{2}^{2}\leq\max\\{(\alpha_{1}+|\alpha_{3}|)^{2},(\alpha_{2}-|\alpha_{4}|)^{2}\\}$
and
$(\alpha_{1}-|\alpha_{3}|)^{2}\beta_{1}^{2}+(\alpha_{2}+|\alpha_{4}|)^{2}\beta_{2}^{2}\leq\max\\{(\alpha_{1}-|\alpha_{3}|)^{2},(\alpha_{2}-|\alpha_{4}|)^{2}\\}$
which may be proved similarly as those for Eqs.(B9)-(B12), and $\beta_{j}$ are
defined in Eq.(B8). The equality of Eq.(B14) holds for ${\bf A}_{1}={\bf
A}_{2}=\mathbbm{1}_{2}$, ${\bf
A}_{3}=\frac{\alpha_{3}}{|\alpha_{3}|}\mathbbm{1}_{2}$ ($\alpha_{3}\not=0$)
and ${\bf A}_{4}=-\frac{\alpha_{4}}{|\alpha_{4}|}\mathbbm{1}_{2}$
($\alpha_{4}\not=0$), or ${\bf A}_{1}={\bf A}_{2}=\mathbbm{1}_{2}$, ${\bf
A}_{3}=-\frac{\alpha_{3}}{|\alpha_{3}|}\mathbbm{1}_{2}$ and ${\bf
A}_{4}=\frac{\alpha_{4}}{|\alpha_{4}|}\mathbbm{1}_{2}$. The equality of
Eq.(B15) holds for the following cases respectively:

    * (a)

${\bf A}_{2,0}={\bf A}_{2,1}=\mathbbm{1}_{2}$ and
$(\alpha_{1}+|\alpha_{3}|)^{2}\geq(\alpha_{2}-|\alpha_{4}|)^{2}$;

    * (b)

${\bf A}_{2,0}=-{\bf A}_{2,1}=\mathbbm{1}_{2}$ and
$(\alpha_{1}+|\alpha_{3}|)^{2}\leq(\alpha_{2}-|\alpha_{4}|)^{2}$;

    * (c)

${\bf A}_{2,0}={\bf A}_{2,1}=\mathbbm{1}_{2}$ and
$(\alpha_{1}-|\alpha_{3}|)^{2}\geq(\alpha_{2}+|\alpha_{4}|)^{2}$;

    * (d)

${\bf A}_{2,0}=-{\bf A}_{2,1}=\mathbbm{1}_{2}$ and
$(\alpha_{1}-|\alpha_{3}|)^{2}\leq(\alpha_{2}+|\alpha_{4}|)^{2}$.

Similar to Eqs.(B10) and (B12), we have the same equality as that given in
Eq.(B13).

Now we estimate the norm of Bell operator ${\cal B}$ defined in Eq.(B5) in the
case of quantum model. For general Hermitian operators ${\bf A}_{i,j}$ and
density operator $\rho$, from Eq.(B6) and Schwartz inequality, we have

$\displaystyle|{\rm Tr}({\cal B}\rho)|\leq$ $\displaystyle|{\rm Tr}({\cal
B}^{2}\rho)|^{\frac{1}{2}}$ $\displaystyle\leq$ $\displaystyle|{\rm
Tr}((\alpha_{1}{\bf A}_{1}+\alpha_{3}{\bf A}_{3})^{2}\otimes{\bf
B}_{1}^{2}\rho)+{\rm Tr}((\alpha_{2}{\bf A}_{2}+\alpha_{4}{\bf
A}_{4})^{2}\otimes{\bf B}_{2}^{2}\rho)$ $\displaystyle+{\rm
Tr}((\alpha_{1}{\bf A}_{1}+\alpha_{3}{\bf A}_{3})(\alpha_{2}{\bf
A}_{2}+\alpha_{4}{\bf A}_{4})\rho_{A})$ $\displaystyle+{\rm
Tr}((\alpha_{2}{\bf A}_{2}+\alpha_{4}{\bf A}_{4})(\alpha_{1}{\bf
A}_{1}+\alpha_{3}{\bf A}_{3})\rho_{A})|^{\frac{1}{2}}$ (B17)
$\displaystyle\leq$
$\displaystyle\sqrt{(|\alpha_{1}|+|\alpha_{3}|)^{2}\beta_{1}^{2}+(|\alpha_{2}|+|\alpha_{4}|)^{2}\beta_{2}^{2}+2(|\alpha_{1}|+|\alpha_{3}|)(|\alpha_{2}|+|\alpha_{4}|)}$
(B18) $\displaystyle\leq$
$\displaystyle\max\\{\sqrt{(|\alpha_{1}|+|\alpha_{3}|)^{2}+2(|\alpha_{1}|+|\alpha_{3}|)(|\alpha_{2}|+|\alpha_{4}|)},$
$\displaystyle\qquad\sqrt{(|\alpha_{2}|+|\alpha_{4}|)^{2}+2(|\alpha_{1}|+|\alpha_{3}|)(|\alpha_{2}|+|\alpha_{4}|)}\\},$
(B19)

where Eq.(B17) is from the inequalities $\|[{\bf B}_{i},{\bf B}_{j}]\|\leq 2$
with $i=1,3$ and $j=2,4$, Eq.(B18) is from the inequalities $\|{\bf
A}_{i}\|\leq 1$ ($i=1,\cdots,4$); Eq.(B19) is from the fact that
$\beta_{i}^{2}$ defined in Eq(B8) have invariable upper bounds for commuting
or general Hermitian operators ${\bf B}_{i}$. Here, $\rho_{A}$ denotes the
reduced state of Alice. So, we have

$\displaystyle\|{\cal B}\|_{q}=$ $\displaystyle\sup_{\vec{\bf
A}^{q},\rho}|{\rm Tr}({\cal B}\rho)|$ $\displaystyle\leq$
$\displaystyle\max\\{\sqrt{(\alpha_{1}+|\alpha_{3}|)^{2}+2(\alpha_{1}+|\alpha_{3}|)(\alpha_{2}+|\alpha_{4}|)},$
$\displaystyle\qquad\sqrt{(\alpha_{2}+|\alpha_{4}|)^{2}+2(\alpha_{1}+|\alpha_{3}|)(\alpha_{2}+|\alpha_{4}|)}\\}$
$\displaystyle\leq$
$\displaystyle\max\\{\sqrt{3}(\alpha_{1}+|\alpha_{3}|),\sqrt{3}(\alpha_{2}+|\alpha_{4}|)\\},$
(B20)

where we have taken use of
$2(\alpha_{1}+|\alpha_{3}|)(\alpha_{2}+|\alpha_{4}|)\leq(\alpha_{1}+|\alpha_{3}|)^{2}+(\alpha_{2}+|\alpha_{4}|)^{2}$.

Eqs.(B3), (B13) and (B20) imply that the upper bound of $r_{cq}$ defined in
Eq.(B3) is given by

$\displaystyle r_{cq}\leq$
$\displaystyle\frac{\max\\{\sqrt{3}(\alpha_{1}+|\alpha_{3}|),\sqrt{3}(\alpha_{2}+|\alpha_{4}|)\\}}{\max\\{(\alpha_{1}+|\alpha_{3}|),(\alpha_{2}+|\alpha_{4}|)\\}}$
$\displaystyle=$ $\displaystyle\sqrt{3}.$ (B21)

Especially, for anticommuting operators (Pauli matrices) ${\bf A}_{i,j}$, we
have $\beta_{1}^{2}=\beta_{2}^{2}=\frac{1}{2}$. Eq.(B18) implies that

$\displaystyle\sup_{\vec{\bf A}^{q},\rho}|{\rm Tr}({\cal B}\rho)|\leq$
$\displaystyle\sqrt{\frac{1}{2}(\alpha_{1}+|\alpha_{3}|)^{2}+\frac{1}{2}(\alpha_{2}+|\alpha_{4}|)^{2}+2(\alpha_{1}+|\alpha_{3}|)(\alpha_{2}+|\alpha_{4}|)}$
$\displaystyle\leq$
$\displaystyle\max\\{\sqrt{\frac{5}{2}}(\alpha_{1}+|\alpha_{3}|),\sqrt{\frac{5}{2}}(\alpha_{2}+|\alpha_{4}|)\\}.$
(B22)

From Eqs.(B3), (B13) and (B22), we get an upper bound of the relative critical
visibility as

$\displaystyle r_{cq}\leq$
$\displaystyle\frac{\max\\{\sqrt{\frac{5}{2}}(\alpha_{1}+|\alpha_{3}|),\sqrt{\frac{5}{2}}(\alpha_{2}+|\alpha_{4}|)\\}}{\max\\{(\alpha_{1}+|\alpha_{3}|),(\alpha_{2}+|\alpha_{4}|)\\}}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{5}{2}}.$ (B23)

Case 2. $m>2$.

Consider an $m$-partite homogeneous linear Bell operator as

$\displaystyle{\cal B}_{m}=$ $\displaystyle\sum_{\vec{i}}\alpha_{\vec{i}}{\bf
A}_{1,i_{1}}\otimes\cdots\otimes{\bf A}_{m,i_{m}}$ $\displaystyle:=$
$\displaystyle\sum_{i=1}^{2^{m}}\alpha_{i}\tilde{\bf A}_{i}\otimes{\bf B}_{i}$
(B24)

where $\tilde{\bf A}_{i}:={\bf A}_{1,j_{1}}\otimes\cdots\otimes{\bf
A}_{m-1,j_{m-1}}$ in which $j_{1}\cdots j_{m-1}$ are the first $m-1$ bits
(from left to right) of the binary representation of integer $i$,
$\alpha_{i}=\alpha_{j_{1}\cdots j_{m-1}0}+\alpha_{j_{1}\cdots j_{m-1}1}$ for
odd integer $i$ with binary representation $j_{1}\cdots j_{m-1}0$,
$\alpha_{i}=\alpha_{j_{1}\cdots j_{m-1}0}-\alpha_{j_{1}\cdots j_{m-1}1}$ for
even integer $i$ with binary representation $j_{1}\cdots j_{m-1}1$, ${\bf
B}_{j}=\frac{1}{2}({\bf A}_{m,0}+{\bf A}_{m,1})$ for all odd integers $j\leq
2^{m}$, ${\bf B}_{j}=\frac{1}{2}({\bf A}_{m,0}-{\bf A}_{m,1})$ for all even
integers $j\leq 2^{m}$. From Eq.(B24) we have

$\displaystyle{\cal B}_{m}^{2}=$
$\displaystyle\sum_{i,j=1}^{2^{m}}\alpha_{i}\alpha_{j}(\tilde{\bf
A}_{i}\tilde{\bf A}_{j})\otimes({\bf B}_{i}{\bf B}_{j})$ $\displaystyle=$
$\displaystyle(\sum_{{\rm odd\,}i}\alpha_{i}\tilde{\bf A}_{i})^{2}\otimes{\bf
B}_{1}^{2}+(\sum_{{\rm even\,}i}\alpha_{i}\tilde{\bf A}_{i})^{2}\otimes{\bf
B}_{2}^{2}$ $\displaystyle+\frac{1}{4}(\sum_{{\rm odd\,}i}\alpha_{i}\tilde{\bf
A}_{i})(\sum_{{\rm even}\,i}\alpha_{i}\tilde{\bf A}_{i})\otimes[{\bf
A}_{m,1},{\bf A}_{m,0}]^{2}$ $\displaystyle+\frac{1}{4}(\sum_{{\rm
even\,}i}\alpha_{i}\tilde{\bf A}_{i})(\sum_{{\rm odd}\,i}\alpha_{i}\tilde{\bf
A}_{i})\otimes[{\bf A}_{m,0},{\bf A}_{m,1}]^{2}.$ (B25)

In what follows, we assume that $\alpha_{1}\geq 0$ and $\alpha_{2}\geq 0$;
Otherwise, by replacing $\alpha_{i}$ with $-\alpha_{i}$, the value of the
right side of Eq.(B25) is invariable. For commuting Hermitian operators ${\bf
A}_{i,j}$ ($\tilde{\bf A}_{i},{\bf B}_{i}$ are commuting) and density operator
$\rho$ on Hilbert space $\mathbb{H}_{2}^{m}$, we have $[\tilde{\bf
A}_{m,i},\tilde{\bf A}_{m,j}]=0$. From Eq.(B25) we get

$\displaystyle{\rm Tr}({\cal B}_{m}^{2}\rho)={\rm Tr}((\sum_{{\rm
odd\,}i}\alpha_{i}\tilde{\bf A}_{i})^{2}\otimes{\bf B}_{1}^{2}\rho)+{\rm
Tr}((\sum_{{\rm even\,}i}\alpha_{i}\tilde{\bf A}_{i})^{2}\otimes{\bf
B}_{2}^{2}\rho).$ (B26)

Two different cases will be discussed as follows:

  * (i)

$\alpha_{2i-1}\alpha_{2i}\geq 0$ for all $i=2,3,\cdots,2^{m-1}$. Eq.(B26)
implies

$\displaystyle|{\rm Tr}({\cal B}_{m}^{2}\rho)|\leq$
$\displaystyle\sqrt{(\sum_{{\rm
odd\,}i}|\alpha_{i}|)^{2}\beta_{1}^{2}+(\sum_{{\rm
even\,}i}|\alpha_{i}|)^{2}\beta_{2}^{2}},$ (B27)

where Eq.(B27) is from the inequalities $\|\tilde{\bf A}_{i}\|\leq 1$
($i=1,\cdots,2^{m}$), $\beta^{2}_{1}=|{\rm Tr}({\bf
B}_{1}^{2}\rho_{m})|=\frac{1}{2}+\frac{1}{4}({\rm Tr}({\bf A}_{m,0}{\bf
A}_{m,1}\rho_{m})+{\rm Tr}({\bf A}_{m,1}{\bf A}_{m,0}\rho_{m}))$ and
$\beta_{2}^{2}=|{\rm Tr}({\bf
B}_{2}^{2}\rho_{m})|=\frac{1}{2}-\frac{1}{4}({\rm Tr}({\bf A}_{m,0}{\bf
A}_{m,1}\rho_{m})+{\rm Tr}({\bf A}_{m,1}{\bf A}_{m,0}\rho_{m}))$, and
$\rho_{m}$ denotes the reduced state of the $m$-th party. The equality of
Eq.(B27) holds for $\rho=\otimes_{i=1}^{m}\rho_{i}$, $\tilde{\bf
A}_{1}=\tilde{\bf A}_{2}=\mathbbm{1}_{2}$ and $\tilde{\bf
A}_{j}=\frac{\alpha_{j}}{|\alpha_{j}|}\mathbbm{1}_{2}$ when $\alpha_{j}\not=0$
($j=3,\cdots,2^{m}$). Two subcases will be shown as:

    * (a)

If $\sum_{{\rm odd\,}i}|\alpha_{i}|\geq\sum_{{\rm even\,}i}|\alpha_{i}|$,
Eq.(B27) implies that

$\displaystyle\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal
B}^{2}\rho)|^{\frac{1}{2}}=\sum_{{\rm odd\,}i}|\alpha_{i}|,$ (B28)

where the supremum is achieved when $\rho=\frac{1}{2^{m}}\mathbbm{1}$ and
${\bf A}_{m,0}={\bf A}_{m,1}=\mathbbm{1}_{2}$. From Schwartz inequality $|{\rm
Tr}({\cal B}\rho)|\leq\sqrt{{\rm Tr}({\cal B}^{2}\rho)}$ we have

$\displaystyle\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal B}\rho)|$
$\displaystyle\leq\sup_{\vec{\bf A}^{c},\rho}\sqrt{{\rm Tr}({\cal
B}^{2}\rho)}$ $\displaystyle=\sum_{{\rm odd\,}i}|\alpha_{i}|,$ (B29)

where the equality holds for $\rho=\frac{1}{2^{m}}\mathbbm{1}$, $\tilde{\bf
A}_{1}=\tilde{\bf A}_{2}=\mathbbm{1}_{2}$, and $\tilde{\bf
A}_{i}=\frac{\alpha_{i}}{|\alpha_{i}|}\mathbbm{1}_{2}$ when $\alpha_{i}\not=0$
($i=3,\cdots,2^{m}$) and ${\bf A}_{m,0}={\bf A}_{m,1}=\mathbbm{1}_{2}$.

    * (b)

If $\sum_{{\rm odd\,}i}|\alpha_{i}|\leq\sum_{{\rm even\,}i}|\alpha_{i}|$,
Eq.(B27) implies that

$\displaystyle\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal
B}^{2}\rho)|^{\frac{1}{2}}=\sum_{{\rm even\,}i}|\alpha_{i}|,$ (B30)

where the supremum is achieved for $\rho=\frac{1}{2^{m}}\mathbbm{1}$ and ${\bf
A}_{m,0}=-{\bf A}_{m,1}=\mathbbm{1}_{2}$. From Schwartz inequality we have

$\displaystyle\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal B}\rho)|$
$\displaystyle\leq\sup_{\vec{\bf A}^{c},\rho}\sqrt{{\rm Tr}({\cal
B}^{2}\rho)}$ $\displaystyle=\sum_{{\rm even\,}i}|\alpha_{i}|,$ (B31)

where the equality holds for $\rho=\frac{1}{2^{m}}\mathbbm{1}$, $\tilde{\bf
A}_{1}=\tilde{\bf A}_{2}={\bf A}_{2,0}=-{\bf A}_{2,1}=\mathbbm{1}_{2}$, and
$\tilde{\bf A}_{i}=\frac{\alpha_{i}}{|\alpha_{i}|}\mathbbm{1}_{2}$ when
$\alpha_{i}\not=0$ ($i=3,\cdots,2^{m}$).

From Eqs.(B29) and (B31) we obtain

$\displaystyle\|{\cal B}\|_{c}=$ $\displaystyle\sup_{\vec{\bf
A}^{c},\rho}|{\rm Tr}({\cal B}\rho)|$ $\displaystyle=$
$\displaystyle\max\\{\sum_{{\rm odd\,}i}|\alpha_{i}|,\sum_{{\rm
even\,}i}|\alpha_{i}|\\}.$ (B32)

  * (ii)

$\alpha_{2i-1}\alpha_{2i}\geq 0$ for $i\in{\cal I}_{1}$ and
$\alpha_{2j-1}\alpha_{2j}\leq 0$ for $j\in{\cal I}_{2}$, where ${\cal I}_{1}$
and ${\cal I}_{2}$ are partitions of the index set $\\{1,\cdots,2^{m}\\}$.
Eq.(B27) implies

$\displaystyle{\rm Tr}({\cal B}^{2}\rho)\leq$ $\displaystyle\max\\{(\sum_{{\rm
odd\,}i,\atop{i\in{\cal I}_{1}}}|\alpha_{i}|\pm\sum_{{\rm
odd\,}j,\atop{j\in{\cal I}_{2}}}|\alpha_{j}|)^{2}\beta_{1}^{2}+(\sum_{{\rm
even\,}k,\atop{k\in{\cal I}_{1}}}|\alpha_{k}|\mp\sum_{{\rm
even\,}s,\atop{s\in{\cal I}_{2}}}|\alpha_{s}|)^{2}\beta_{2}^{2}\\}$ (B33)
$\displaystyle\leq$ $\displaystyle\max\\{(\sum_{{\rm odd\,}i,\atop{i\in{\cal
I}_{1}}}|\alpha_{i}|\pm\sum_{{\rm odd\,}j,\atop{j\in{\cal
I}_{2}}}|\alpha_{j}|)^{2},(\sum_{{\rm even\,}k,\atop{k\in{\cal
I}_{1}}}|\alpha_{k}|\mp\sum_{{\rm even\,}s,\atop{s\in{\cal
I}_{2}}}|\alpha_{s}|)^{2}\\}$ (B34) $\displaystyle=$
$\displaystyle\max\\{(\sum_{{\rm odd\,}i}|\alpha_{i}|)^{2},(\sum_{{\rm
even\,}j}|\alpha_{j}|)^{2}\\},$ (B35)

where Eq.(B33) is from the inequalities $\|\tilde{\bf A}_{l}\|\leq 1$ for all
indexes $l$; Eq.(B34) is from the equality $\beta_{1}^{2}+\beta_{2}^{2}=1$ by
taking use of similar proofs of Eqs.(B9)-(B12); and $\beta_{i}$ are defined in
Eq.(B27). Here, the equality of Eq.(B34) holds for ${\bf A}_{m,0}={\bf
A}_{m,1}=\mathbbm{1}_{2}$ when the right side of Eq.(B33) equals to
$(\sum_{{\rm odd\,}i,i\in{\cal I}_{1}}|\alpha_{i}|\pm\sum_{{\rm
odd\,}j,j\in{\cal I}_{2}}|\alpha_{j}|)^{2}$, or ${\bf A}_{m,0}=-{\bf
A}_{m,1}=\mathbbm{1}_{2}$ when the right side of Eq.(B33) equals to
$(\sum_{{\rm even\,}k,{k\in{\cal I}_{1}}}|\alpha_{k}|\mp\sum_{{\rm
even\,}s,{s\in{\cal I}_{2}}}|\alpha_{s}|)^{2}$. The equality of Eq.(B33) holds
for $\tilde{\bf A}_{2i-1}=\tilde{\bf A}_{2i}=\mathbbm{1}_{2}$ with $i\in{\cal
I}_{1}$, and $\tilde{\bf
A}_{2j-1}=\pm\frac{\alpha_{2j-1}}{|\alpha_{2j-1}|}\mathbbm{1}_{2}$
($\alpha_{2j-1}\not=0$) and $\tilde{\bf
A}_{2j}=\mp\frac{\alpha_{2j}}{|\alpha_{2j}|}\mathbbm{1}_{2}$
($\alpha_{2j}\not=0$) with $j\in{\cal I}_{2}$. Using Schwartz inequality,
similar to Eq.(B29) or (B31) we get the same equality as that given in
Eq.(B32).

Now, we estimate the norm of ${\cal B}$ in terms of quantum model. For general
Hermitian operators ${\bf A}_{i,j}$, from Schwartz inequality and Eq.(B25), we
have

$\displaystyle|{\rm Tr}({\cal B}_{m}\rho)|\leq$ $\displaystyle{\rm Tr}({\cal
B}_{m}^{2}\rho)^{\frac{1}{2}}$ $\displaystyle\leq$ $\displaystyle|{\rm
Tr}((\sum_{{\rm odd\,\,}i}\alpha_{i}\tilde{\bf A}_{i})^{2}\otimes{\bf
B}_{1}^{2}\rho)+{\rm Tr}((\sum_{{\rm even\,\,}j}\alpha_{j}\tilde{\bf
A}_{j})^{2}\otimes{\bf B}_{2}^{2}\rho)+{\rm Tr}((\sum_{{\rm
odd\,\,}i}\alpha_{i}\tilde{\bf A}_{i})(\sum_{{\rm
even\,\,}j}\alpha_{j}\tilde{\bf A}_{j})\rho)$ $\displaystyle+{\rm
Tr}((\sum_{{\rm even\,\,}j}\alpha_{j}\tilde{\bf A}_{j})(\sum_{{\rm
odd\,\,}i}\alpha_{i}\tilde{\bf A}_{i})\rho)|^{\frac{1}{2}}$ (B36)
$\displaystyle\leq$ $\displaystyle\sqrt{(\sum_{{\rm
odd\,\,}i}|\alpha_{i}|)^{2}\beta_{1}^{2}+(\sum_{{\rm
even\,\,}i}|\alpha_{i}|)^{2}\beta_{2}^{2}+2\sum_{{\rm
odd\,\,}i}|\alpha_{i}|\sum_{{\rm even\,\,}j}|\alpha_{j}|}$ (B37)
$\displaystyle\leq$ $\displaystyle\max\\{\sqrt{(\sum_{{\rm
odd\,\,}i}|\alpha_{i}|)^{2}+2\sum_{{\rm odd\,\,}i}|\alpha_{i}|\sum_{{\rm
even\,\,}j}|\alpha_{j}|},\sqrt{(\sum_{{\rm
even\,\,}i}|\alpha_{i}|)^{2}+2\sum_{{\rm odd\,\,}i}|\alpha_{i}|\sum_{{\rm
even\,\,}j}|\alpha_{j}|}\\},$ (B38)

where Eq.(B36) is from the inequalities $\|[\tilde{\bf A}_{i},\tilde{\bf
A}_{j}]\|\leq 2$ for all $i,j$; Eq.(B37) is from the inequalities $\|{\bf
A}_{i,j}\|\leq 1$ for all $i,j$; and Eq.(B38) is from the fact $\beta_{i}^{2}$
have the same upper bounds for commuting or general Hermitian operators ${\bf
A}_{i,j}$, $i=0,1$. Hence, we obtain

$\displaystyle\|{\cal B}\|_{q}=$ $\displaystyle\sup_{\vec{\bf
A}^{q},\rho}|{\rm Tr}({\cal B}_{m}\rho)|$ $\displaystyle\leq$
$\displaystyle\max\\{\sqrt{(\sum_{{\rm odd\,}i}|\alpha_{i}|)^{2}+2\sum_{{\rm
odd\,}i}|\alpha_{i}|\sum_{{\rm even\,}j}|\alpha_{j}|},\sqrt{(\sum_{{\rm
even\,}i}|\alpha_{i}|)^{2}+2\sum_{{\rm odd\,}i}|\alpha_{i}|\sum_{{\rm
even\,}j}|\alpha_{j}|}\\}$ $\displaystyle=$
$\displaystyle\max\\{\sqrt{3}\sum_{{\rm
odd\,}i}|\alpha_{i}|,\sqrt{3}\sum_{{\rm even\,}j}|\alpha_{j}|\\},$ (B39)

where we have taken use of the inequality $2\sum_{{\rm
odd\,}i}|\alpha_{i}|\sum_{{\rm even\,}j}|\alpha_{j}|\leq(\sum_{{\rm
odd\,}i}|\alpha_{i}|)^{2}+(\sum_{{\rm even\,}j}|\alpha_{j}|)^{2}$.

From Eqs.(B3), (B32), and (B39), an upper bound of $r_{cq}$ defined in Eq.(B3)
is given by

$\displaystyle r_{cq}\leq$ $\displaystyle\frac{\max\\{\sqrt{3}\sum_{{\rm
odd\,}i}|\alpha_{i}|,\sqrt{3}\sum_{{\rm
even\,}j}|\alpha_{j}|\\}}{\max\\{\sum_{{\rm odd\,}i}|\alpha_{i}|,\sum_{{\rm
even\,}j}|\alpha_{j}|\\}}$ $\displaystyle\leq$ $\displaystyle\sqrt{3}.$ (B40)

In particular, for anticommuting operators (Pauli matrices) ${\bf A}_{i,j}$,
we have $\beta_{1}^{2}=\beta_{2}^{2}=\frac{1}{2}$. Eq.(B33) implies that

$\displaystyle\|{\cal B}\|_{q}\leq$
$\displaystyle\max\\{\sqrt{\frac{1}{2}(\sum_{{\rm
odd\,\,}i}|\alpha_{i}|)^{2}+2\sum_{{\rm odd\,\,}i}|\alpha_{i}|\sum_{{\rm
even\,\,}j}|\alpha_{j}|},\sqrt{\frac{1}{2}(\sum_{{\rm
even\,\,}i}|\alpha_{i}|)^{2}+2\sum_{{\rm odd\,\,}i}|\alpha_{i}|\sum_{{\rm
even\,\,}j}|\alpha_{j}|}\\}$ $\displaystyle\leq$
$\displaystyle\max\\{\sqrt{\frac{5}{2}}\sum_{{\rm
odd\,\,}i}|\alpha_{i}|,\sqrt{\frac{5}{2}}\sum_{{\rm
even\,\,}i}|\alpha_{i}|\\}.$ (B41)

Hence, from Eqs.(B3), (B32), and (B41), the upper bound of $r_{cq}$ is given
by

$\displaystyle r_{cq}\leq\sqrt{\frac{5}{2}}.$ (B40)

### Appendix C. Relative critical visibility in the case of general linear
Bell inequalities

Given a general linear Bell operator ${\cal
B}=\sum_{i_{1},\cdots,i_{m}}\alpha_{\vec{i}}\hat{\bf
A}_{1,i_{1}}\otimes\cdots\otimes\hat{\bf A}_{m,i_{m}}$, different from
homogeneous Bell operator shown in Eq.(B2), $\hat{\bf A}_{j,i_{j}}$ may be
dichotomic operators ${\bf A}_{j,i_{j}}$ or excluded. Although one may take
use of the identity operator to represent the case of absent observer, we
cannot obtain tight upper bound of $\|{\cal B}\|_{c}$ for all operators ${\cal
B}$ using similar method of Appendix B. Here, we provide an accessible method
to deal with each general linear Bell operator. A useful decomposition of
${\cal B}$ is given by

$\displaystyle{\cal B}=\sum_{j=1}^{m}{\cal B}_{j}$ (C1)

where ${\cal B}_{j}$ are partial Bell operators including measurement
operators of the $j$-th party but excluding all operators of $i$-parties for
$i=1,\cdots,j-1$, i.e, ${\cal
B}_{j}=\sum_{i_{j},\cdots,i_{m}=0,1}\beta_{i_{j}\cdots i_{m}}{\bf
A}_{j,i_{j}}\hat{\bf A}_{j+1,i_{j+1}}\cdots\hat{\bf A}_{m,i_{m}}$,
$\beta_{i_{j}\cdots i_{m}}$ are real coefficients. The main result of this
section is the following Lemma.

Lemma 3. For ${\cal B}$ presented in Eq.(C1), $r_{cq}$ defined in Eq.(B3)
satisfies

$\displaystyle r_{cq}\leq\sqrt{3}\sum_{i=1}^{m-1}\frac{1}{\gamma_{i}}+1$ (C2)

if $\|{\cal B}\|_{c}\geq\gamma_{i}\|{\cal B}_{i}\|_{c}$, $i=1,\cdots,m-1$,
where Bell operators ${\cal B}_{i}$ are defined in Eq.(C1) and $\gamma_{i}$
are positive constants.

Proof. From the decomposition shown in Eq.(C1), ${\cal B}_{i}$ may be
rewritten into

$\displaystyle{\cal B}_{i}=\sum_{j=1}^{2^{m+1-i}}\beta_{i,j}\hat{\bf
A}_{j}\otimes{\bf B}_{j},$ (C3)

where ${\bf B}_{j}=\frac{1}{2}({\bf A}_{m+1-i,0}+{\bf A}_{m+1-i,1})$ for all
odd integers $j$ satisfying $j\leq 2^{m+1-i}$ and ${\bf
B}_{j}=\frac{1}{2}({\bf A}_{m+1-i,0}-{\bf A}_{m+1-i,1})$ for all even integers
$j$ satisfying $j\leq 2^{m+1-k}$, $\beta_{i,j}$ are proper coefficients
depending on $\beta_{k_{1}\cdots k_{m+1-i}}$, and $\hat{\bf A}_{j}$ denote the
operators of $\hat{\bf A}_{1,s_{1}}\otimes\cdots\otimes\hat{\bf
A}_{m+2-i,s_{m+2-i}}$ that are Hermitian and satisfy $\|\hat{\bf A}_{j}\|\leq
1$ and $\hat{\bf A}^{2}_{j}=\mathbbm{1}$.

Similar to Eqs.(B24)-(B35), from straight forward computations we obtain

$\displaystyle\|{\cal B}_{k}\|_{c}=$ $\displaystyle\sup_{\vec{\bf
A}^{c},\rho}{\rm Tr}({\cal B}_{k}\rho)$ $\displaystyle=$
$\displaystyle\max\\{\sum_{{\rm odd\,}i\atop{i\leq
N_{k}}}|\beta_{k,i}|,\sum_{{\rm even\,\,}j\atop{j\leq
N_{k}}}|\beta_{k,j}|\\},$ (C4)

where $N_{k}=2^{m+1-k}$, $k=1,2,\cdots,m-1$. Moreover, similar to
Eqs.(B36)-(B39), we have

$\displaystyle\|{\cal B}_{k}\|_{c}\leq\max\\{\sqrt{3}\sum_{{\rm
odd\,\,}i\atop{i\leq N_{k}}}\beta_{k,i},\sqrt{3}\sum_{{\rm
even\,\,}j\atop{j\leq N_{k}}}\beta_{k,j}\\},$ (C5)

where $k=1,2,\cdots,m-1$. Note that ${\cal B}_{m}$ is a linear operator
including ${\bf A}_{m,0},{\bf A}_{m,1}$. It follows that

$\displaystyle\frac{\sup_{{\bf A}^{q}_{m,i_{m}},\rho_{m}}{\rm Tr}({\cal
B}_{m}\rho_{m})}{\sup_{{\bf A}^{c}_{m,i_{m}},\rho_{m}}{\rm Tr}({\cal
B}_{m}\rho_{m})}=1,$ (C6)

where $\rho_{m}$ is the reduced density operator of the $m$-th party.

From Eqs.(C4) and (C6) we have $\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal
B}\rho)|>0$ for any nonzero linear operator ${\cal B}$. In fact, suppose
$\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal B}\rho)|=0$ for all commuting
operators ${\bf A}_{i,j}$, ${\cal B}$ must be the zero operator. The reason is
that $\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal B}\rho)|=0$ implies $|{\rm
Tr}({\cal B}\rho)|=0$ for all commuting operators ${\bf A}_{i,j}$ and density
operator $\rho$. Hence, From Eqs.(C4)-(C6) we have

$\displaystyle r_{cq}=$ $\displaystyle\frac{\sup_{\vec{\bf A}^{q},\rho}|{\rm
Tr}({\cal B}\rho)|}{\sup_{\vec{\bf A}^{c},\rho}|{\rm Tr}({\cal B}\rho)|}$
$\displaystyle\leq$ $\displaystyle\frac{\sum_{k=1}^{m}\sup_{\vec{\bf
A}^{q},\hat{\rho}_{k}}|{\rm Tr}({\cal B}_{k}\hat{\rho}_{k})|}{\sup_{\vec{\bf
A}^{c},\rho}|{\rm Tr}({\cal B}\rho)|}$ (C7) $\displaystyle\leq$
$\displaystyle\sum_{k=1}^{m}\frac{\sup_{\vec{\bf A}^{q},\hat{\rho}_{k}}|{\rm
Tr}({\cal B}_{k}\hat{\rho}_{k})|}{\gamma_{k}\sup_{\vec{\bf
A}^{c},\hat{\rho}_{k}}|{\rm Tr}({\cal B}_{k}\hat{\rho}_{k})|}$ (C8)
$\displaystyle\leq$
$\displaystyle\sqrt{3}\sum_{i=1}^{m-1}\frac{1}{\gamma_{i}}+1,$ (C9)

where $\hat{\rho}_{k}$ denote the reduced density operators on the subspace
determined by ${\cal B}_{k}$, Eq.(C7) is from the triangle inequality
$\sup\\{x+y\\}\leq\sup\\{x\\}+\sup\\{y\\}$, Eq.(C8) is from the assumptions of
$\|{\cal B}\|_{c}\geq\gamma_{i}\|{\cal B}_{i}\|_{c}$ and $\gamma_{i}>0$, and
Eq.(C9) is from Eqs.(C4)-(C6).

### Appendix D. Proof of Theorem 1

Given an $m$-partite generalized GHZ state, the corresponding Werner state
$\rho_{v}$ is defined in Eq.(A1). We assume $\theta\in(0,\frac{\pi}{2})$. From
the equivalent theorem Deng , $\rho_{v}$ is fully separable if and only if
$v\leq\frac{\alpha}{2^{m}+\alpha}$ with $\alpha=\frac{2}{\sin 2\theta}$. It
follows that the critical parameter $v^{*}$ for which Werner states $\rho_{v}$
with $v\leq v^{*}$ are fully separable is given by

$\displaystyle v^{*}=\frac{1}{2^{m-1}\sin 2\theta+1}.$ (D1)

In what follows, we prove that the critical visibility $v^{\prime}$ for which
Werner state violates some linear Bell inequality is strictly larger than
$v^{*}$ presented in Eq.(D1). Assume that an arbitrary $m$-partite Bell
inequality is given by

$\displaystyle\|{\cal B}\|_{c(q)}\leq c_{1}(c_{2}),$ (D2)

where $c_{1}$ and $c_{2}$ denote the respective upper bound of the norm of
Bell operator ${\cal B}$ in the case of commuting Hermitian measurement
operators and general Hermitian measurement operators of all observers. From
Lemma 1, the critical visibility $v^{\prime}$ for which Werner state defined
in Eq.(A1) violates Bell inequality of Eq.(D2) with $c_{1}$ is given by

$\displaystyle v^{\prime}\geq\frac{2^{m}-1}{2^{m}r_{cq}-1},$ (D3)

where $r_{cq}$ is defined in Eq.(B3).

Case 1. Homogeneous linear Bell inequalities.

From Lemma 2 and Eq.(D3), we get
$v^{\prime}\geq\frac{2^{m}-1}{2^{m}\sqrt{3}-1}$. When $\sin
2\theta>\frac{2\sqrt{3}-2}{2^{m}-1}$ we have
$v^{\prime}\geq\frac{2^{m}-1}{2^{m}\sqrt{3}-1}>\frac{1}{2^{m-1}\sin
2\theta+1}=v^{*}$ from Eq.(D1). So, Werner states shown in Eq.(A1) cannot be
completely detected by homogeneous linear Bell inequalities when $\theta$
satisfies

$\displaystyle\frac{1}{2}\arcsin(\frac{2\sqrt{3}-2}{2^{m}-1})<\theta<\pi-\frac{1}{2}\arcsin(\frac{2\sqrt{3}-2}{2^{m}-1}).$
(D4)

Numerical bounds of $\theta$ presented in Eq.(D4) are shown in Table I. It
follows that for almost all generalized GHZ states, the corresponding Werner
states cannot be completely detected by homogeneous linear Bell inequalities.

Table 1: Undetectable Werner states of $m$-partite generalized GHZ states defined in Eq.(A1) in the case of homogeneous linear Bell inequalities. $\theta_{u}$ denotes the upper bound of $\theta$. $\theta_{l}$ denotes the lower bound of $\theta$. $\theta_{u}$ and $\theta_{l}$ are given in Eq.(D4). $r$ is defined by $r=\frac{1}{\pi}(\theta_{u}-\theta_{l})$, which denotes the measure of generalized GHZ state (in the set of all generalized GHZ states) whose Werner states cannot be completely detected. $m$ | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---
${\theta_{u}}/{\pi}$ | 0.9189 | 0.9665 | 0.9844 | 0.9925 | 0.9963
${\theta_{l}}/{\pi}$ | 0.0811 | 0.0335 | 0.0156 | 0.0075 | 0.0037
$r$ | 83.77% | 83.29% | 96.89% | 98.5% | 99.26%

Case 2. General linear Bell inequalities.

Denote $\gamma=\sqrt{3}\sum_{i=1}^{m-1}\gamma_{i}^{-1}+1$. From Lemma 3 and
Eq.(D3), we obtain $v^{\prime}\geq\frac{2^{m}-1}{2^{m}\gamma-1}$ when $\|{\cal
B}\|_{c}\geq\gamma_{i}\|{\cal B}_{i}|_{c}$, $i=1,\cdots,m-1$. From Eq.(D1),
when $\sin
2\theta>\frac{2\sqrt{3}}{2^{m}-1}\sum_{i=1}^{m-1}\frac{1}{\gamma_{i}}$ we have
$v^{\prime}\geq\frac{2^{m}-1}{2^{m}\gamma-1}>\frac{1}{2^{m-1}\sin
2\theta+1}=v^{*}$. Hence, there are Werner states defined in Eq.(A1) that
cannot be completely detected by a given linear Bell inequality shown in
Eq.(D2) when $\theta$ satisfies

$\displaystyle\frac{1}{2}\arcsin(\frac{2\sqrt{3}}{2^{m}-1}\sum_{i=1}^{m-1}\frac{1}{\gamma_{i}})<\theta<\pi-\frac{1}{2}\arcsin(\frac{2\sqrt{3}}{2^{m}-1}\sum_{i=1}^{m-1}\frac{1}{\gamma_{i}}).$
(D5)

It implies that Werner states of generalized GHZ states with large $m$ cannot
be completely detected by linear Bell inequalities with all $\gamma_{i}$s
satisfying $\gamma_{i}=\frac{1}{{\rm Poly}(m)}$, i.e, polynomial functions of
$m$.

In order to complete the proof, we need to estimate the bounds of
$\gamma_{i}$. Unfortunately, we cannot get explicit bounds for all
$\gamma_{i}$s. Here, we take use of numeric methods. For a general linear Bell
operator ${\cal B}=\sum_{i=1}^{m-1}{\cal B}_{i}$ given in Eq.(C1), the
classical bound is always achieved from extremal points (i.e., ${\bf
A}_{i,j}=\pm\mathbbm{1}_{2}$) when $\|{\bf A}_{i,j}\|\leq 1$ Tsi1 ; Tsi2 . In
this case, all correlations $\langle\hat{\bf A}_{1,i_{1}}\cdots\hat{\bf
A}_{m,i_{m}}\rangle$ of $\langle{\cal B}\rangle$ equal to
$\prod_{j=1}^{m}\langle\hat{\bf A}_{j,i_{j}}\rangle$. From Eq.(C1), for each
Bell operator ${\cal B}$, $\langle{\cal B}\rangle$ may be vectorized as
$\vec{\alpha}\cdot\vec{A}$ and $\langle{\cal B}_{j}\rangle$ may be vectorized
as $\vec{\alpha}_{j}\cdot\vec{A}_{j}$, where $\vec{\alpha}$ denotes
$3^{m}-1$-dimensional vector consisting of all coefficients
$\beta_{i_{1}\cdots i_{m}}$ of ${\cal B}$, $\vec{A}$ denotes
$3^{m}-1$-dimensional vector consisting of all correlations
$\prod_{j=1}^{m}\langle\hat{\bf A}_{j,i_{j}}\rangle$ of $\langle{\cal
B}\rangle$, $\vec{\alpha}_{j}$ denotes $\ell_{j}$-dimensional vector
consisting of all coefficients $\beta_{i_{j}\cdots i_{m}}$ of ${\cal B}_{j}$,
$\vec{A}_{j}$ denotes $\ell_{j}$-dimensional vector consisting of all
correlations $\prod_{s=j}^{m}\langle{\bf A}_{s,i_{s}}\rangle$ of $\langle{\cal
B}_{j}\rangle$, and $\ell_{j}=2\times 3^{m-j}$, $j=1,\cdots,m$. From Eqs.(C9)
and (D5), it is sufficient to get the tight bound of the relative critical
visibility $r_{cq}$ by estimating $\min_{\cal B}\gamma_{j}$ from $\min_{\cal
B}\frac{\|{\cal B}\|_{c}}{\|{\cal B}_{j}\|_{c}}$. Here, we only need to
consider all coefficients of ${\cal B}$ on $3^{m}-1$-dimensional unit
hypersphere. Denote $M=[\vec{\bf A}]$ as a $4^{m}\times(3^{m}-1)$ matrix and
$M_{j}=[\vec{\bf A}_{j}]$ as a $4^{m+1-j}\times\ell_{j}$ matrix, where
$\langle{\bf A}_{j,i_{j}}\rangle,\cdots,\langle{\bf
A}_{m,i_{m}}\rangle\in\\{\pm 1\\}$, $j=1,\cdots,m$. We present Algorithm 1.

Input: $m$, $N$

Output: $\min_{\cal B}{\gamma}_{1},\cdots,\min_{\cal B}{\gamma}_{m}$

  * ${\Xi}$ is an $N$-partition of $3^{m}-1$-dimensional (column) sphere;

  * For $i=1:m$

  * $X_{i}={\Xi}(:,L_{i-1}:L_{i})$;

  * For $j=1:N$

  * $\vec{x}_{ij}=X_{i}(:,j)$;

  * $\|\hat{\cal B}_{j}\|_{c}=\max|M\vec{x}_{ij}|$;

  * $\|\hat{\cal B}_{ij}\|_{c}=\max|M_{i}\vec{x}_{ij}(L_{i-1}:L_{i})|$;

  * $\gamma_{ij}=\frac{\|\hat{\cal B}_{j}\|_{c}}{\|\hat{\cal B}_{ij}\|_{c}}$;

  * ${\gamma_{i}}=\min_{j}\\{\gamma_{ij}\\}$;

Algorithm 1 Evaluating $\min_{\cal B}\gamma_{i}$ in the case of $m$-partite
Bell inequalities

In Algorithm 1, $N$ denotes the total number of unit vectors, where each unit
vector can determine a general $m$-partite Bell operator. ${\Xi}$ is a
$(3^{m}-1)\times L_{m}$ matrix, where each column vector is a
$3^{m}-1$-dimensional real unit vector, $L_{i}=\sum_{j=1}^{i-1}\ell_{j}$ and
$L_{-1}=0$. $N$-partition of unit sphere means that there are $N$ points on
unit sphere such that the minimum distance of any two points has the lower
bound $O(N^{1/m})$. $X_{i}={\Xi}(:,L_{i-1}:L_{i})$ denotes the submatrix of
${\Xi}$, which includes all components from the $L_{i-1}$-th column to the
$L_{i}$-th column, $i=1,\cdots,m$. $\vec{x}_{ij}$ denotes the $j$-th column of
the matrix $X_{i}$. ${\cal B}_{j}$ and ${\cal C}_{ij}$ are the corresponding
Bell operators derived from coefficients $\vec{x}_{ij}$. $\|\hat{\cal
B}_{j}\|_{c}$ and $\|\hat{\cal B}_{ij}\|_{c}$ are computed as the maximal
bound of extremal points. Note that $\frac{\|\hat{\cal
B}_{j}\|_{c}}{\|\hat{\cal B}_{ij}\|_{c}}$ is continue function in terms of
variables $\vec{\alpha}_{j}$. Algorithm 1 provides a useful approximate
algorithm to obtain lower bound $\gamma_{i}$. The computation complexity is
$O(mN^{2}3^{m}2^{5m+m^{2}}\prod_{i=1}^{m}\ell_{i})$ that is exponential in
$m$. Numeric evaluations of small $m$ are shown in Table II. From Table II,
all $\gamma_{i}$s satisfy $\gamma_{i}\geq 1$. Numeric bounds of $\theta$ given
by Eq.(D5) for $\gamma_{1}=\cdots=\gamma_{m}=1$ are shown in Table III.
Although there are Werner states that cannot be detected for $m=2$ ZB ,
however, from Table III there is no example. The reason is that the quantum
bound shown in Eq.(B40) is not tight.

Table 2: The lower bounds of $\gamma_{i}$s in the case of $m$-partite Bell inequalities. All lower bounds are larger than 1. We conjecture that all lower bounds of $\gamma_{i}$ for each general Bell inequality are larger than $\frac{1}{{\rm Poly}(m)}$ for some polynomial function ${\rm Poly}(m)$. $m$ | $\gamma_{1}$ | $\gamma_{2}$ | $\gamma_{3}$ | $\gamma_{4}$ | $\gamma_{5}$ | $\gamma_{6}$
---|---|---|---|---|---|---
$2$ | 1.00 | 1.01 |  |  |  |
$3$ | 1.00 | 1.10 | 1.72 |  |  |
$4$ | 1.00 | 1.10 | 1.80 | 3.10 |  |
$5$ | 1.00 | 1.20 | 1.90 | 3.40 | 6.70 |
$6$ | 1.00 | 1.23 | 2.00 | 3.95 | 6.98 | 12.00
Table 3: Numeric bounds of $\theta$ given in Eq.(D5). Here, we assume $\gamma_{1}=\cdots=\gamma_{m}=1$. $\theta_{u}$ denotes the upper bound of $\theta$ given in Eq.(D5). $\theta_{l}$ denotes the lower bound of $\theta$. $r$ is defined by $r=\frac{1}{\pi}(\theta_{u}-\theta_{l})$, which denotes the measure of generalized GHZ states (in the set of all generalized GHZ states) whose Werner states cannot be completely detected. $m$ | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---
${\theta_{l}}/{\pi}$ | - | 0.2272 | 0.1218 | 0.0738 | 0.0443
${\theta_{u}}/{\pi}$ | - | 0.7728 | 0.8782 | 0.9262 | 0.9557
$r$ | - | 54.56% | 75.64% | 85.24% | 91.14%

## Appendix E. Critical parameters of Werner states

In this section, we take use of the notations of index vector defined in
Appendix A. Given an $m$-partite pure state
$|\Phi\rangle=\sum_{\vec{i}}\alpha_{\vec{i}\,}|\vec{i}\,\rangle$, the
corresponding Werner state is defined by

$\displaystyle\rho_{v}=\frac{1-v}{2^{m}}\mathbbm{1}+v|\Phi\rangle\langle\Phi|,$
(E1)

where $\mathbbm{1}$ denotes the identity operator on Hilbert space
$\mathbb{H}^{\otimes m}_{2}$, $\alpha_{\vec{i}\,}$ are complex coefficients
satisfying $\sum_{\vec{i}\,}|\alpha_{\vec{i}\,}|^{2}=1$ and $v\in[0,1]$. It
easily proves that $\rho_{v}$ are density matrices for all $v\in[0,1]$.

The proof of Theorem 2 is completed within the following three sections.
Appendices E and F contribute to the upper bound of the critical parameter
$v^{*}$ for which all Werner states $\rho_{v}$ defined in Eq.(E1) with $v\leq
v^{*}$ are fully separable. Appendix G provides the proof of Theorem 2. In
this section, we present a parameter-dependent upper bound of $v^{*}$ as
follows:

Lemma 4. The critical parameter $v^{*}$ for which Werner states defined in
Eq.(E1) with $v\leq v^{*}$ are fully separable satisfies

$\displaystyle v^{*}\leq$ $\displaystyle\min\\{\min_{\vec{i}\in{\cal
I},\vec{j}}\frac{1}{\sqrt{|4^{m}|\alpha_{\vec{j}}|^{2}|\alpha_{\vec{j}_{c}}|^{2}-4^{m}|\alpha_{\vec{i}}|^{2}|\alpha_{\vec{i}_{c}}|^{2}+2^{m}(|\alpha_{\vec{i}}|^{2}+|\alpha_{\vec{i}_{c}}|^{2})-1|}},1\\}$
(E2)

where ${\cal I}$ denotes the index set ${\cal
I}=\\{\vec{i}\,||\alpha_{\vec{i}}|^{2}+|\alpha_{\vec{i}_{c}}|^{2}\leq\frac{1}{2^{m-1}}\\}$
and the index vector $\vec{j}$ can be any $m$-bit vector.

Proof. The proof is to obtain the necessary condition of Werner states being
fully separable states. From Pitt , the necessary condition that $\rho_{v}$ in
Eq.(E1) is fully separable is given by

$\displaystyle\min_{\vec{i}}\sqrt{\langle\vec{i}|\rho_{v}|\vec{i}\,\rangle\langle\vec{i}_{c}|\rho_{v}|\vec{i}_{c}\rangle}\geq\max_{\vec{j}}|\langle\vec{j}|\rho|\vec{j}_{c}\rangle|.$
(E3)

In what follows, we estimate the upper bound of $v^{*}$ from Eq.(E3). From
Eq.(E1), we have

$\displaystyle\langle\vec{j}|\rho_{v}|\vec{j}_{c}\rangle^{2}-\langle\vec{i}|\rho_{v}|\vec{i}\,\rangle\langle\vec{i}_{c}|\rho_{v}|\vec{i}_{c}\rangle=$
$\displaystyle
v^{2}|\alpha_{\vec{j}}|^{2}|\alpha_{\vec{j}_{c}}|^{2}-(\frac{1-v}{2^{m}}+v|\alpha_{\vec{i}}|^{2})(\frac{1-v}{2^{m}}+v|\alpha_{\vec{i}_{c}}|^{2})$
$\displaystyle=$
$\displaystyle\frac{1}{4^{m}}[(4^{m}|\alpha_{\vec{j}}|^{2}|\alpha_{\vec{j}_{c}}|^{2}-(2^{m}|\alpha_{\vec{i}}|^{2}-1)(2^{m}|\alpha_{\vec{i}_{c}}|^{2}-1))v^{2}$
$\displaystyle-(2^{m}|\alpha_{\vec{i}}|^{2}+2^{m}|\alpha_{\vec{i}_{c}}|^{2}-2)v-1]$
$\displaystyle\leq$ $\displaystyle 0$ (E4)

for all $m$-bit index vectors $\vec{i},\vec{j}$.

Denote two functions $f(\vec{i},\vec{j}\,)$ and $g(\vec{i}\,)$ as

$\displaystyle f(\vec{i},\vec{j}\,)=$ $\displaystyle
4^{m}|\alpha_{\vec{j}}|^{2}|\alpha_{\vec{j}_{c}}|^{2}-4^{m}|\alpha_{\vec{i}}|^{2}|\alpha_{\vec{i}_{c}}|^{2}+2^{m}(|\alpha_{\vec{i}}|^{2}+|\alpha_{\vec{i}_{c}}|^{2})-1,$
(E5) $\displaystyle g(\vec{i}\,)=$ $\displaystyle
2^{m}|\alpha_{\vec{i}}|^{2}+2^{m}|\alpha_{\vec{i}_{c}}|^{2}-2.$ (E6)

From the normalization condition of
$\sum_{\vec{i}}(|\alpha_{\vec{i}}|^{2}+|\alpha_{\vec{i}_{c}}|^{2})=1$, we get
$|\alpha_{\vec{i}}|^{2}+|\alpha_{\vec{i}_{c}}|^{2}\leq\frac{1}{2^{m-1}}$ for
some index vector $\vec{i}$ using the pigeonhole principle because there are
$2^{m-1}$ index vectors $\vec{i}$. So, from Eq.(E6) we obtain

$\displaystyle\min_{\vec{i}}g(\vec{i}\,)=\min_{\vec{i}\in{\cal
I}}g(\vec{i}\,)\leq 0,$ (E7)

where ${\cal I}$ denotes the index set of ${\cal
I}=\\{\vec{i}\,|g(\vec{i}\,)\leq 0\\}$. The following proof is divided into
three cases.

Case 1. $f(\vec{i},\vec{j}\,)>0$ for some index vectors $\vec{i}\in{\cal I}$
and $\vec{j}$. Eq.(E4) implies that

$\displaystyle
v\leq\frac{g(\vec{i}\,)+\sqrt{\Delta(\vec{i},\vec{j}\,)}}{2f(\vec{i},\vec{j}\,)},$
(E8)

where $\Delta(\vec{i},\vec{j}\,)=g(\vec{i}\,)^{2}+4f(\vec{i},\vec{j}\,)$.
Since
$\sqrt{\Delta(\vec{i},\vec{j}\,)}\leq|g(\vec{i}\,)|+2\sqrt{f(\vec{i},\vec{j}\,)}$,
Eq.(E8) yields to

$\displaystyle
v\leq\sqrt{\frac{g(\vec{i}\,)+|g(\vec{i}\,)|}{2f(\vec{i},\vec{j}\,)}+\frac{1}{f(\vec{i},\vec{j}\,)}}.$
(E9)

Note that Eq.(E4) holds for any index vectors $\vec{i},\vec{j}$. From Eqs.(E7)
and (E9) we have

$\displaystyle v\leq$
$\displaystyle\min_{\vec{i},\vec{j}}\sqrt{\frac{g(\vec{i}\,)+|g(\vec{i}\,)|}{2f(\vec{i},\vec{j}\,)}+\frac{1}{f(\vec{i},\vec{j}\,)}}$
$\displaystyle\leq$ $\displaystyle\min_{\vec{i}\in{\cal
I},\vec{j}}\frac{1}{\sqrt{f(\vec{i},\vec{j}\,)}},$ (E10)

where ${\cal I}$ is defined in Eq.(E7).

Case 2. $f(\vec{i},\vec{j}\,)<0$ and $\Delta(\vec{i},\vec{j}\,)\geq 0$ for
some index vectors $\vec{i}\in{\cal I}$ and $\vec{j}$. Eq.(E4) implies that

$\displaystyle
v\leq\frac{g(\vec{i}\,)+\sqrt{\Delta(\vec{i},\vec{j}\,)}}{2f(\vec{i},\vec{j}\,)}$
(E11)

or

$\displaystyle
v\geq\frac{g(\vec{i}\,)-\sqrt{\Delta(\vec{i},\vec{j}\,)}}{2f(\vec{i},\vec{j}\,)},$
(E12)

where $\Delta(\vec{i},\vec{j}\,)$ is defined in Eq.(E8). Since
$|g(\vec{i}\,)|-2\sqrt{-f(\vec{i},\vec{j}\,)}\leq\sqrt{\Delta(\vec{i},\vec{j}\,)}\leq|g(\vec{i}\,)|$,
from Eq.(E11) we obtain

$\displaystyle v\leq$
$\displaystyle\frac{-g(\vec{i}\,)-\sqrt{\Delta(\vec{i},\vec{j}\,)}}{-2f(\vec{i},\vec{j}\,)}$
$\displaystyle\leq$
$\displaystyle\frac{-g(\vec{i}\,)+2\sqrt{-f(\vec{i},\vec{j}\,)}-|g(\vec{i}\,)|}{-2f(\vec{i},\vec{j}\,)}$
(E13)

for any index vectors $\vec{i},\vec{j}$ because Eq.(E4) holds for any index
vectors $\vec{i},\vec{j}$.

Note that Eq.(E13) implies

$\displaystyle v\leq$
$\displaystyle\min_{\vec{i},\vec{j}}\frac{-g(\vec{i}\,)+2\sqrt{-f(\vec{i},\vec{j}\,)}-|g(\vec{i}\,)|}{-2f(\vec{i},\vec{j}\,)}$
$\displaystyle\leq$ $\displaystyle\min_{\vec{i}\in{\cal
I},\vec{j}}\frac{1}{\sqrt{-f(\vec{i},\vec{j}\,)}}.$ (E14)

Moreover, from Eq.(E12) we get

$\displaystyle v\geq$
$\displaystyle\frac{g(\vec{i}\,)+|g(\vec{i}\,)|}{-2f(\vec{i},\vec{j}\,)}$
$\displaystyle\geq$
$\displaystyle\max_{\vec{i}}\frac{g(\vec{i}\,)+|g(\vec{i}\,)|}{-2f(\vec{i},\vec{j}\,)}$
$\displaystyle\geq$ $\displaystyle 0.$ (E15)

Case 3. $f(\vec{i},\vec{j}\,)<0$ and $\Delta(\vec{i},\vec{j}\,)<0$ for some
index vectors $\vec{i}\in{\cal I}$ and $\vec{j}$. For this case, Eq.(E4)
implies $v\in[0,1]$.

To sum up, since Eq.(E4) holds for all index vectors $\vec{i},\vec{j}$, from
Eqs.(E10) and (E14) we have

$\displaystyle v\leq\min\\{\min_{\vec{i}\in{\cal
I},\vec{j}}\frac{1}{\sqrt{|f(\vec{i},\vec{j}\,)|}},1\\}$ (E16)

Here, we do not consider the case of $f(\vec{i},\vec{j}\,)=0$ for some index
vectors $\vec{i}\in{\cal I}$ and $\vec{j}$, which yields to
$v\leq-1/g(\vec{i}\,)$. The reason is from $f(\vec{i},\vec{j}\,)\not=0$ for
almost all pure states. Of course, the minimal upper bound of $v^{*}$ derived
from Eq.(E4) is lower than the bound presented in Eq.(E16). Hence, we have
completed the proof.

## Appendix F. Special upper bound of critical parameters

In this section, we present a special upper bound of critical parameters which
is independent of coefficients of pure states. Note that the separability of
each pure is independent of global phases. We only need to consider all pure
states which consist of a unit hypersphere (embedded in a higher-dimensional
Euclidean space).

Lemma 5. There is a nonzero measure of pure states
$|\Phi\rangle=\sum_{i_{1},\cdots,i_{m}=0,1}\alpha_{\vec{i}\,}|\vec{i}\,\rangle$
satisfying

$\displaystyle\max_{\vec{i}\in{\cal
I},\vec{j}}\sqrt{|f(\vec{i},\vec{j}\,)|}>\frac{2^{m}{\rm Poly}(m)-1}{2^{m}-1}$
(F1)

for large $m$, where nonzero measure means the ratio of Lebesgue measures of
pure states satisfying Eq.(F1) and all pure states is nonzero,
$f(\vec{i},\vec{j}\,)$ is defined in Eq.(E5), ${\rm Poly}(m)$ denotes any
polynomial function of $m$, and ${\cal I}$ is defined in Eq.(E2).

Proof. Denote $\delta=\frac{2^{m}{\rm Poly}(m)-1}{2^{m}-1}$. Define two index
sets ${\cal J}_{1}$ and ${\cal J}_{2}$ as

$\displaystyle{\cal J}_{1}:=\\{(\vec{i},\vec{j}\,)|f(\vec{i},\vec{j}\,)\geq
0,\vec{i}\in{\cal I}\\},$ (F2) $\displaystyle{\cal
J}_{2}:=\\{(\vec{i},\vec{j}\,)|f(\vec{i},\vec{j}\,)\leq 0,\vec{i}\in{\cal
I}\\}.$ (F3)

Since $\max_{\vec{i}\in{\cal
I}}\\{|\alpha_{\vec{i}}|^{2}|\alpha_{\vec{i}_{c}}|^{2}-\frac{1}{2^{m}}(|\alpha_{\vec{i}}|^{2}+|\alpha_{\vec{i}_{c}}|^{2})\\}\leq
0$, we obtain

$\displaystyle\max_{(\vec{i},\vec{j}\,)\in{\cal
J}_{2}}|f(\vec{i},\vec{j}\,)|=$
$\displaystyle\max_{(\vec{i},\vec{j}\,)\in{\cal
J}_{1}}\\{1-4^{m}|\alpha_{\vec{j}}|^{2}|\alpha_{\vec{j}_{c}}|^{2}+4^{m}(|\alpha_{\vec{i}}|^{2}|\alpha_{\vec{i}_{c}}|^{2}-\frac{1}{2^{m}}(|\alpha_{\vec{i}}|^{2}+|\alpha_{\vec{i}_{c}}|^{2})\\}$
$\displaystyle\leq$ $\displaystyle\max_{(\vec{i},\vec{j}\,)\in{\cal
J}_{2}}\\{1-4^{m}|\alpha_{\vec{j}}|^{2}|\alpha_{\vec{j}_{c}}|^{2}\\}$
$\displaystyle\leq$ $\displaystyle 1.$ (F4)

Eq.(F4) means that there is no index pair $(\vec{i},\vec{j}\,)\in{\cal J}_{2}$
satisfying the inequality shown in Eq.(F1) because $\delta>1$ from its
definition.

Now, assume that

$\displaystyle\max_{(\vec{i},\vec{j}\,)\in{\cal
J}_{1}}|f(\vec{i},\vec{j}\,)|=\max_{(\vec{i},\vec{j}\,)\in{\cal
J}_{1}}f(\vec{i},\vec{j}\,)>\delta^{2}.$ (F5)

Since $\max_{\vec{i}\in{\cal
I}}(|\alpha_{\vec{i}}|^{2}|\alpha_{\vec{i}_{c}}|^{2}-\frac{1}{2^{m}}(|\alpha_{\vec{i}}|^{2}+|\alpha_{\vec{i}_{c}}|^{2}))\leq
0$, Eq.(F5) holds when the following inequality is satisfied

$\displaystyle\max_{\vec{j}}|\alpha_{\vec{j}}|^{2}|\alpha_{\vec{j}_{c}}|^{2}>\frac{({\rm
Poly}(m)+1)^{2}}{4^{m}}>\frac{\delta^{2}}{4^{m}}.$ (F6)

Define $\mathbb{S}$ as the set of all normalized $m$-partite pure states and
$\mathbb{S}_{m}$ as the set of all normalized $m$-partite pure states
satisfying Eq.(F6), i.e.,

$\displaystyle\mathbb{S}_{m}:=\\{|\Phi\rangle|\max_{\vec{j}}|\alpha_{\vec{j}}|^{2}|\alpha_{\vec{j}_{c}}|^{2}>\frac{({\rm
Poly}(m)+1)^{2}}{4^{m}},\sum_{\vec{j}}|\alpha_{\vec{j}}|^{2}=1\\}.$ (F7)

Note that $(\vec{i},\vec{j}\,)\in{\cal J}_{1}$ when Eq.(F7) is satisfied. It
means that all states in $\mathbb{S}_{m}$ satisfy Eq.(F1). Since
$\max_{\vec{j}}|\alpha_{\vec{j}}|^{2}|\alpha_{\vec{j}_{c}}|^{2}\in(0,\frac{1}{4}]$,
one may informally conclude that the most of pure states $|\Phi\rangle$
satisfy Eq.(F1) for large integer $m$, where the upper bound $\frac{1}{4}$ is
achieved for the maximally entangled $m$-partite GHZ state.

We may formally prove the result as follows. Denote
$\overline{\mathbb{S}}_{m}:=\\{|\Phi\rangle|\max_{\vec{j}}|\alpha_{\vec{j}}|^{2}|\alpha_{\vec{j}_{c}}|^{2}\leq
4^{-m}({\rm Poly}(m)+1)^{2},\sum_{\vec{j}}|\alpha_{\vec{j}}|^{2}=1\\}$, i.e.,
all pure states that do not satisfy Eq.(F1). By representing each coefficient
$\alpha_{\vec{j}}$ by
$\alpha_{\vec{j}}=r_{\vec{j\,}}e^{\sqrt{-1}\theta_{\vec{j\,}}}$,
$\overline{\mathbb{S}}_{m}$ and $\overline{\mathbb{S}}$ are decomposed into
product spaces of
$\overline{\mathbb{S}}_{m}=\mathbb{S}^{r}_{m}\times[0,2\pi]^{\times 2^{m}}$
and $\mathbb{S}=\mathbb{S}^{r}\times[0,2\pi]^{\times 2^{m}}$, where
$\overline{\mathbb{S}}^{r}_{m}:=\\{|\Phi\rangle|\max_{\vec{j}}r_{\vec{j}}^{2}r_{\vec{j}_{c}}^{2}\leq
4^{-m}({\rm Poly}(m)+1)^{2},\sum_{\vec{j}}r_{\vec{j}}^{2}=1,r_{\vec{j}}\geq
0\\}$,
$\mathbb{S}^{r}_{m}:=\\{|\Phi\rangle|\sum_{\vec{j}}r_{\vec{j}}^{2}=1,r_{\vec{j}}\geq
0\\}$, and $[0,2\pi]^{\times 2^{m}}$ denotes $2^{m}$-dimensional phase space
of $(\theta_{\vec{0}},\cdots,\theta_{\vec{1}})$. Hence, we only need to
consider subspaces $\mathbb{S}^{r}$ and $\overline{\mathbb{S}}^{r}_{m}$ which
are independent of the phase space. Specially, we have

$\displaystyle\frac{M(\mathbb{S}_{m})}{M(\mathbb{S})}$
$\displaystyle=1-\frac{M(\overline{\mathbb{S}}_{m})}{M(\mathbb{S})}$
$\displaystyle=1-\frac{M(\overline{\mathbb{S}}^{r}_{m})}{M(\mathbb{S}^{r})},$
(F8)

where $M(\cdot)$ denotes Lebesgue measure on a $2^{m}$-dimensional real
hypersphere (embedded in a $2^{m}+1$-dimensional Euclidean space).

Denote $c=2^{-m}({\rm Poly}(m)+1)$, for all states in
$\overline{\mathbb{S}}^{r}_{m}$, we have $r_{1}r_{2^{m}}\leq c$. Denote
$\mathbb{S}_{+}^{2^{m}}$ as the subset of $2^{m}$-dimensional unit hypersphere
$\mathbb{S}^{2^{m}}$ with all positive components. Note that the measure of
$2^{m}$-dimensional hypersphere $\mathbb{S}^{2^{m}}$ with radius $x$ is given
by $M(\mathbb{S}^{2^{m}})=\frac{2\pi^{2^{m-1}}x^{2^{m}}}{\Gamma(2^{m-1})}$,
where Gamma function $\Gamma(k)=k!$. From the symmetry of hypersphere we get

$\displaystyle M(\mathbb{S}^{r})=$ $\displaystyle M(\mathbb{S}_{+}^{2^{m}})$
$\displaystyle=$ $\displaystyle\frac{1}{2^{2^{m}}}M(\mathbb{S}^{2^{m}})$
$\displaystyle=$
$\displaystyle\frac{2\pi^{2^{m-1}}}{2^{2^{m}}\Gamma(2^{m-1})}.$ (F9)

For set $\overline{\mathbb{S}}_{m}^{r}$ we have

$\displaystyle M(\overline{\mathbb{S}}_{m}^{r})\leq$
$\displaystyle\int_{\vec{r}\in S_{+}^{2^{m}}|r_{1}r_{2^{m}}\leq c}dx$
$\displaystyle=$
$\displaystyle\frac{2\pi^{2^{m-1}-1}}{2^{2^{m}}\Gamma(2^{m-1}-1)}\int_{r_{1}r_{2^{m}}\leq
c}(1-r_{1}^{2}-r_{2^{m}}^{2})^{2^{m-1}-1}dr_{1}dr_{2^{m}}.$ (F10)

Let $r_{1}=r\cos\theta,r_{2^{m}}=r\sin\theta$, $r\in[0,c]$ and
$\theta\in[0,\frac{\pi}{2}]$. From Eq.(F10) we get

$\displaystyle M(\overline{\mathbb{S}}_{m}^{r})\leq$
$\displaystyle\frac{2\pi^{2^{m-1}-1}}{2^{2^{m}}\Gamma(2^{m-1}-1)}\int_{r\in[0,c],\theta\in[0,\frac{\pi}{2}]|r\sin
2\theta\leq 2c}r(1-r^{2})^{2^{m-1}-1}drd\theta$ $\displaystyle=$
$\displaystyle\frac{2\pi^{2^{m-1}-1}}{2^{2^{m}}\Gamma(2^{m-1}-1)}\int_{r\in[0,c]}\int_{\theta\in[0,\frac{1}{2}\arcsin\frac{2c}{r}]}r(1-r^{2})^{2^{m-1}-1}drd\theta$
$\displaystyle\leq$
$\displaystyle\frac{\pi^{2^{m-1}}}{2^{2^{m}}\Gamma(2^{m-1}-1)}\int_{r\in[0,c]}r(1-r^{2})^{2^{m-1}-1}dr$
$\displaystyle\leq$
$\displaystyle\frac{2\pi^{2^{m-1}}}{2^{2^{m}}\Gamma(2^{m-1})}-\frac{\pi^{2^{m-1}}}{2^{2^{m}}\Gamma(2^{m-1})}(1-c^{2})^{2^{m-1}}.$
(F11)

Consequently, Eqs.(F8)-(F11) implies that

$\displaystyle\frac{M(\mathbb{S}_{m})}{M(\mathbb{S})}$
$\displaystyle\geq(1-c^{2})^{2^{m-1}}.$ (F12)

It means that there is a nonzero measure of pure states satisfying Eq.(F1).

## Appendix G. Proof of Theorem 2

Given an $m$-partite pure state
$|\Phi\rangle=\sum_{\vec{i}}\alpha_{\vec{i}}|\vec{i}\,\rangle$, the
corresponding Werner state is defined in Eq.(E1). In what follows, we prove
that the critical visibility $v^{\prime}$, for which $\rho_{v}$ with
$v>v^{\prime}$ violates some linear Bell inequality, is strictly larger than
$v^{*}$ given in Eq.(F1).

Assume an $m$-partite Bell inequality is given by

$\displaystyle\|{\cal B}\|\leq c_{1}(c_{2}),$ (G1)

where $c_{1}$ and $c_{2}$ denote the respective classical bound (in terms of
LHS model) and quantum bound (in terms of quantum model). From Lemma 1, the
critical visibility $v^{\prime}$ of Werner state $\rho_{v}$ defined in Eq.(E1)
is given by

$\displaystyle v^{\prime}\geq\frac{2^{m}-1}{2^{m}r_{cq}-1}$ (G2)

for which Werner state violates linear Bell inequality presented in Eq.(G1)
with $c_{1}$, where $r_{cq}$ is defined in Eq.(B3). The following proof will
be divided into two cases.

Case 1. Homogeneous linear Bell inequalities. From Lemma 2 and Eq.(G2), we
have $v^{\prime}\geq\frac{2^{m}-1}{2^{m}\sqrt{3}-1}$. From Lemma 5, there is a
nonzero measure of pure states satisfying $\max_{\vec{i}\in{\cal
I},\vec{j}}\sqrt{f(\vec{i},\vec{j}\,)}>\frac{2^{m}\sqrt{3}-1}{2^{m}-1}$, where
$f(\vec{i},\vec{j}\,)$ is defined in Eq.(E5). From Lemmas 4 and 5, we get
$v^{*}<\frac{2^{m}-1}{2^{m}\sqrt{3}-1}\leq v^{\prime}$. Hence, there is a
nonzero measure of pure states whose Werner states cannot be completely
detected by homogenous linear Bell inequalities.

Case 2. General linear Bell inequalities. From Lemma 2 and Eq.(G2), we have
$v^{\prime}\geq\frac{2^{m}-1}{2^{m}(\sqrt{3}\sum_{i=1}^{m-1}\gamma_{i}^{-1}+1)-1}$
when $\|{\cal B}\|_{c}\geq\gamma_{i}\|{\cal B}_{i}\|_{c}$, $i=1,\cdots,m-1$.
Moreover, when all $\gamma_{i}$s satisfy
$\sum_{i=1}^{m-1}\frac{1}{\gamma_{i}}<\frac{1}{\sqrt{3}}{\rm Poly}(m)-1$ we
get

$\displaystyle\frac{2^{m}-1}{2^{m}(\sqrt{3}\sum_{i=1}^{m-1}\frac{1}{\gamma_{i}}+1)-1}>\frac{2^{m}-1}{2^{m}{\rm
Poly}(m)-1}>v^{*}$ (G3)

from Lemmas 4 and 5. It means that there is a nonzero measure of pure states
whose Werner states cannot be completely detected by a general linear Bell
inequality if the corresponding Bell operators ${\cal B}$ and ${\cal B}_{i}$
satisfy $\|{\cal B}\|_{c}\geq\gamma_{i}\|{\cal B}_{i}\|_{c}$ for
$i=1,\cdots,m-1$, where $\gamma_{i}$ satisfy
$\sum_{i=1}^{m-1}\frac{1}{\gamma_{i}}<\frac{1}{\sqrt{3}}{\rm Poly}(m)-1$.
Furthermore, for $m\leq 6$, from Table II there is a nonzero measure of pure
states whose Werner states cannot be detected by general linear Bell
inequalities.

## Appendix H. Examples

Example 1. CHSH inequality CHSH , Mermin inequality Mermin  and Svetlichny
inequality Sv ; BGP ; CWKO  are homogeneous Bell inequalities that cannot
detect Werner states of almost all generalized GHZ states or Werner states of
most general pure states.

Example 2. CH inequality CH  depending on probabilities is given by

$\displaystyle
r_{1}+r^{\prime}_{0}+r_{0}r^{\prime}_{1}-r_{1}r^{\prime}_{1}-r_{1}r^{\prime}_{0}-r_{0}r^{\prime}_{0}\leq
1$ (H1)

for $r_{i},r^{\prime}_{i}\in[0,1]$. It is equivalent to the following operator
inequality

$\displaystyle\langle\textbf{A}_{1}+\textbf{B}_{0}+\textbf{A}_{0}\otimes\textbf{B}_{1}-\textbf{A}_{1}\otimes\textbf{B}_{1}-\textbf{A}_{1}\otimes\textbf{B}_{0}-\textbf{A}_{0}\otimes\textbf{B}_{0}\rangle_{c}\leq
4$ (H2)

for $\|\textbf{A}_{i}\|,\|\textbf{B}_{i}\|\leq 1$. Moreover, we easily obtain
that $\gamma_{1}=\frac{4}{3}$ from
$\langle\textbf{A}_{1}+\textbf{B}_{0}+\textbf{A}_{0}\otimes\textbf{B}_{1}-\textbf{A}_{1}\otimes\textbf{B}_{1}-\textbf{A}_{1}\otimes\textbf{B}_{0}-\textbf{A}_{0}\otimes\textbf{B}_{0}\rangle_{c}\leq
3$ and $\gamma_{2}=4$ from $\langle\textbf{B}_{0}\rangle\leq 1$. From Theorems
1 and 2, CH inequality cannot detect Werner states of the most generalized GHZ
states or Werner states of general pure states with nonzero measure.

Example 3. Scarani-Acin-Schenck-Aspelmeyer inequality

$\displaystyle\langle\textbf{A}_{0}\otimes\textbf{C}_{1}\otimes\textbf{D}_{0}+\textbf{A}_{0}\otimes\textbf{C}_{0}\otimes\textbf{D}_{1}+\textbf{A}_{1}\otimes\textbf{B}_{0}\otimes\textbf{C}_{0}\otimes\textbf{D}_{0}-\textbf{A}_{1}\otimes\textbf{B}_{0}\otimes\textbf{C}_{1}\otimes\textbf{D}_{1}\rangle_{c}\leq
2$ (H3)

has been used for detecting four-qubit cluster state SASA . Note that all
correlations include $A_{i}$. This inequality is equivalent to a homogeneous
Bell inequality as Example 1 using Lemma 3.

Example 4. Brunner-Sharam-Vertesi inequalities

$\displaystyle
I_{4}=[-1\,\,\,\,-1;-2\,\,\,\,0\,\,\,\,-2;\,\,\,\,-2\,\,\,\,1\,\,\,\,1\,\,\,\,-2]\leq
8$ (H4)

for $4$-partite entanglement and

$\displaystyle
I_{5}=[0\,\,\,\,0;-2\,\,\,\,0\,\,\,\,-1;0\,\,\,\,0\,\,\,\,0\,\,\,\,0;-4\,\,\,\,0\,\,\,\,2\,\,\,\,0\,\,\,\,1]\leq
15$ (H5)

for $5$-partite entanglement as general Bell inequalities including sub-
correlations have been used to test the structure of multipartite entanglement
BSV , where we have taken use of the symmetric notation BSV . For the
inequality presented in Eq.(H4), we can obtain $\gamma_{1}=\frac{1}{2}$,
$\gamma_{2}=\frac{4}{5}$, $\gamma_{3}=\frac{4}{3}$ and $\gamma_{4}=4$ assisted
by computer. If we take use of Bell inequality $|I_{4}|\leq 32$, we have
larger $\gamma_{i}$ as $\gamma_{1}=\frac{8}{5}$, $\gamma_{2}=\frac{16}{5}$,
$\gamma_{3}=\frac{16}{3}$ and $\gamma_{4}=16$, which satisfy the lower bounds
presented in Table II. From Theorems 1 and 2, there is a nonzero measure of
generalized GHZ states or general pure states whose Werner states cannot be
detected by this inequality. Moreover, for the inequality shown in Eq.(H5), we
obtain $\gamma_{1}=\frac{15}{29}$, $\gamma_{2}=\frac{5}{4}$ and
$\gamma_{3}=\frac{15}{4}$. Similarly, for Bell inequality $|I_{5}|\leq 45$ we
have larger $\gamma_{i}$ as $\gamma_{1}=\frac{45}{29}$,
$\gamma_{2}=\frac{15}{4}$ and $\gamma_{3}=\frac{45}{4}$ that satisfy the lower
bounds shown in Table II. Hence, from Theorems 1 and 2 there is a nonzero
measure of generalized GHZ states or general pure states whose Werner states
cannot be detected by this inequality.

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