# Tight upper bound on the quantum value of Svetlichny operators
under local filtering and hidden genuine nonlocality

Lingyun Sun1 Li Xu1 Jing Wang1 Ming Li1 Shuqian Shen1 Lei Li1 Shao-Ming Fei2,3
1College of the Science, China University of Petroleum, 266580 Qingdao, China
2 School of Mathematical Sciences, Capital Normal University, 100048 Beijing,
China
3 Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

###### Abstract

Nonlocal quantum correlations among the quantum subsystems play essential
roles in quantum science. The violation of the Svetlichny inequality provides
sufficient conditions of genuine tripartite nonlocality. We provide tight
upper bounds on the maximal quantum value of the Svetlichny operators under
local filtering operations, and present a qualitative analytical analysis on
the hidden genuine nonlocality for three-qubit systems. We investigate in
detail two classes of three-qubit states whose hidden genuine nonlocalities
can be revealed by local filtering.

Suggested keywords

###### pacs:

03.67.Mn,03.65.Ud

##  I Introduction

As important physical resources, quantum correlations like entanglement play
fundamental roles in quantum information processing QE-2009-RevModPhys.81.865
; New_QE-2013-Front.Phys ; New_QE-2019-Front.Phys , with numerous applications
in quantum communication protocols with lower complexity communication-
complexity2002-PhysRevLett.89.197901 ; communication-
complexity2010-RevModPhys.82.665  and higher security
Security2001-PhysRevLett.87.117901 ; Security2006-PhysRevA.73.012314 . Two
systems $A$ and $B$ are entangled if the measurements on system $A$ does
affect the probabilities of the measurement outcomes from system $B$, and vise
versa. For tripartite systems, there exit correlations so called genuine
entanglement Quantum_entanglement2011-PhysRevLett.106.250404 . A tripartite
state may be genuine entangled even if any pair of the subsystems are
separable.

The stronger correlations than entanglement are nonlocal correlations. Two
systems $A$ and $B$ may be locally correlated even if they are entangled, as
long as the correlations of their measurement outcomes can be described by
classical correlation models of probability. For a bipartite $\rho$, let
$P(ab|XY)$ be the probability of measuring $X$ on subsystem $A$ with outcome
$a$ and $Y$ on subsystem $B$ with outcome $b$. If the probability correlation
$P(ab|XY)$ can be expressed in the form,
$P(ab|XY)=\sum\limits_{\lambda}{q_{\lambda}}P_{\lambda}(a|X)P_{\lambda}(b|Y)$,
where $\lambda$ is regarded as a shared local hidden variable,
$q_{\lambda}\geq 0$, and $\Sigma_{\lambda}q(\lambda)=1$, then the state $\rho$
is regarded as locally correlated, and admits a local hidden-variable (LHV)
model. The bipartite Bell nonlocality Bell-1964-Phys.1.195 ; Bell-
nonlocality2014-RevModPhys.86.419  can be witnessed by the violation of Bell
inequalities Bell-nonlocality2014-RevModPhys.86.419 .

Similar to the quantum entanglement, the quantum nonlocality becomes subtler
for multipartite and high dimensional systems Def-MNL-2013-PhysRevA.88.014102
; New_nonlocilty-FOP2012 ; New_nonlocilty-FOP2018 . Let $\rho$ be a tripartite
state. Performing local measurements $X$, $Y$ and $Z$ on the subsystems $A$,
$B$ and $C$ with outcomes $a$, $b$ and $c$, respectively, we say the state is
three local if the corresponding probability correlations $P(abc|XYZ)$ can be
written as

$P(abc|XYZ)=\sum\limits_{\lambda}{q_{\lambda}}P_{\lambda}(a|X)P_{\lambda}(b|Y)P_{\lambda}(c|Z),$
(1)

where $0\leq q_{\lambda}\leq 1$ and $\Sigma_{\lambda}q_{\lambda}=1$.
Otherwise, the state is called nonthree or full local. A nonthree local state
is said to be hybrid-nonlocal, admitting bi-LHV model, if

$P(abc|XYZ)=\sum\limits_{\lambda}{q_{\lambda}}P_{\lambda}(ab|XY)P_{\lambda}(c|Z)+\sum\limits_{\mu}{q_{\mu}}P_{\mu}(ac|XZ)P_{\mu}(b|Y)+\sum\limits_{\upsilon}{q_{\upsilon}}P_{\upsilon}(bc|YZ)P_{\upsilon}(a|X),$
(2)

where $0\leq q_{\lambda},q_{\mu},q_{\upsilon}\leq 1$, and
$\Sigma_{\lambda}q_{\lambda}+\Sigma_{\mu}q_{\mu}+\Sigma_{\upsilon}q_{\upsilon}=1$.

If the probability correlation can not be written in form (2), the state is
called genuine tripartite nonlocal. The genuine tripartite nonlocality of a
state can be detected by the Svetlichny inequality (SI)
Svetlichny1987-PhysRevD.35.3066 . The violation of SI is a sufficient
condition for the genuine tripartite nonlocality. However, generally it is not
easy to verify such violations. In LM2017-PhysRevA.96.042323 , the authors
presented a tight upper bound for the maximal quantum value of the Svetlichny
operator.

For bipartite case, it has been shown that the nonlocality of certain quantum
states can be revealed by using local filters before performing a standard
Bell test, known as genuine hidden nonlocality
localfilter-2013-PhysRevLett.111.160402 . In Max-entanglement-
PhysRevA.68.012103 , Verstraete et. al demonstrated that the optimal local
filtering operations can maximize certain entanglement measures. Moreover,
quantum properties such as Bell nonlocality and steerability
Steering2007-PhysRevLett.98.140402  of specific quantum states can be revealed
by local filtering. In lf-CHSH-PhysRevLett.74.2619 ; PhysLettA.210.151 , the
authors considered two-qubit states which do not violate CHSH inequality
before, but do violate after performing local filtering operations. The
maximal violation of the CHSH inequality and the lower bound of the maximal
violation of V$\acute{e}$rtesi inequality under local filtering operations
were computed analytically in li2017maximal . In Hidden-
steerability2019-PhysRevA.99.030101 , Pramanik et al. showed that there exist
initially unsteerable bipartite states which show steerability after local
filtering.

For a tripartite state $\rho$, under local filtering transformations one gets,

${\rho}^{\prime}=\frac{1}{N}\left(F_{A}\otimes F_{B}\otimes
F_{C}\right)\rho\left(F_{A}\otimes F_{B}\otimes F_{C}\right)^{\dagger},$ (3)

where $N={\rm tr}\left[\left(F_{A}\otimes F_{B}\otimes
F_{C}\right)\rho\left(F_{A}\otimes F_{B}\otimes F_{C}\right)^{\dagger}\right]$
is a normalization factor, $F_{A}$, $F_{B}$ and $F_{C}$ are positive operators
acting on the local subsystems, respectively. In
AON2020-PhysRevLett.124.050401 , Tendick et al. discussed the relation between
entanglement and nonlocality in the hidden nonlocality scenario, and presented
a fully biseparable three-qubit bound entangled state with a local model for
the most general measurements. By using the ${\rm\acute{S}}$liwa’s inequality
and an iterative sequence of semidefinite programs it is shown that the local
model breaks down when suitable local filters are applied, which demonstrates
the activation of nonlocality in bound entanglement, as well as that genuine
hidden nonlocality does not imply entanglement distillability.

In this paper, we first study the maximal quantum value of the Svetlichny
operators after local filtering operations for any three-qubit system. A tight
upper bound for the maximal value of the Svetlichny operators after local
filtering is obtained. Then we take the color noised Greenberger-Horne-
Zeilinger (GHZ)-class states as examples to illustrate how local filter
operations work in nonlocality improvement. We show that the hidden genuine
nonlocalities can be revealed by local filtering for these classes of three-
qubit states.

##  II Tight upper bound on the value of Svetlichny operator under local
filtering

The Svetlichny operator in Svetlichny inequality
Svetlichny1987-PhysRevD.35.3066  reads

$\mathcal{S}=A\otimes[(B+B^{\prime})\otimes C+(B-B^{\prime})\otimes
C^{\prime}]+A^{\prime}\otimes[(B-B^{\prime})\otimes C-(B+B^{\prime})\otimes
C^{\prime}],$ (4)

where $A,A^{\prime},B,B^{\prime},C$ and $C^{\prime}$ denote the local
observables of the form
$G=\vec{g}\cdot\vec{\sigma}=\Sigma_{k=1}^{3}g_{k}\sigma_{k}$,
$G\\!\in\\!\\{A,A^{\prime},B,B^{\prime},C,C^{\prime}\\}$ and
$\vec{g}\\!\in\\!\\{\vec{a},\vec{a}\,^{\prime},\vec{b},\vec{b}\,^{\prime},\vec{c},\vec{c}\,^{\prime}\\}$,
respectively. $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ with
$\sigma_{i}$, $i=1,2,3$, the standard Pauli matrices. $\vec{g}$ is a three-
dimensional real unit vector.

The mean value of the Svetlichny operator for an arbitrary three-qubit state
$\rho$ admitting a bi-LHV model satisfies the following inequality
Svetlichny1987-PhysRevD.35.3066 ,

$|\langle\mathcal{S}\rangle_{\rho}|\leq 4,$ (5)

where $\langle\mathcal{S}\rangle_{\rho}$=tr$(\mathcal{S}\rho)$. A state
violating the inequality (5) is called genuine three-qubit nonlocal. It has
been shown that the maximal quantum value of the Svetlichny operator for
three-qubit systems is upper bounded LM2017-PhysRevA.96.042323 ,

$\mathcal{Q}(\mathcal{S})\equiv{\rm max}|\langle\mathcal{S}\rangle_{\rho}|\leq
4\lambda_{1},$ (6)

where $\lambda_{1}$ is the maximal singular value of the matrix
$M=(M_{j,ik})$, with
$M_{ijk}$=tr$[\rho(\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k})],i,j,k=1,2,3.$
The upper bound is tight if the degeneracy of $\lambda_{1}$ is more than 1,
and the two degenerate nine-dimensional singular vectors corresponding to
$\lambda_{1}$ take the form of
$\vec{a}\otimes\vec{c}-\vec{a}\,^{\prime}\otimes\vec{c}\,^{\prime}$ and
$\vec{a}\otimes\vec{c}\,^{\prime}+\vec{a}\,^{\prime}\otimes\vec{c}$.

Let $F_{A}=U\Sigma_{A}U^{\dagger}$, $F_{B}=V\Sigma_{B}V^{\dagger}$ and
$F_{C}=W\Sigma_{C}W^{\dagger}$ be the spectral decompositions of the filter
operators $F_{A}$, $F_{B}$ and $F_{C}$, respectively, where $U$, $V$ and $W$
are unitary operators. Set $\delta_{l}=\Sigma_{A}\sigma_{l}\Sigma_{A}$,
$\eta_{m}=\Sigma_{B}\sigma_{m}\Sigma_{B}$ and
$\gamma_{n}=\Sigma_{C}\sigma_{n}\Sigma_{C}$. Without loss of generality, we
assume that

$\Sigma_{A}=\begin{pmatrix}x&0\\\ 0&1\end{pmatrix}$,
$\Sigma_{B}=\begin{pmatrix}y&0\\\ 0&1\end{pmatrix}$ and
$\Sigma_{C}=\begin{pmatrix}z&0\\\ 0&1\end{pmatrix}$

with $x,y,z\geq 0$. Let $X=(X_{m,ln})$ be a matrix with entries given by

$X_{lmn}={\rm
tr}[\varrho(\delta_{l}\otimes\eta_{m}\otimes\gamma_{n})],~{}~{}~{}l,m,n=1,2,3,$
(7)

where $\varrho$ is any state that is locally unitary equivalent to $\rho$.

######  Theorem 1.

For the local filtered quantum state
${\rho}^{\prime}=\frac{1}{N}\left(F_{A}\otimes F_{B}\otimes
F_{C}\right)\rho\left(F_{A}\otimes F_{B}\otimes F_{C}\right)^{\dagger}$ of a
three-qubit $\rho$, the maximal quantum value of the Svetlichny operator
$\mathcal{S}$ defined in Eq. (4) satisfies

$\mathcal{Q}(\mathcal{S})^{\prime}={\rm
max}\left|\left\langle\mathcal{S}\right\rangle_{{\rho}^{\prime}}\right|\leq
4{\lambda_{1}}^{\prime},$ (8)

where $\left\langle\mathcal{S}\right\rangle_{{\rho}^{\prime}}={\rm
tr}(\mathcal{S}{\rho}^{\prime})$, ${\lambda_{1}}^{\prime}$ is the maximal
singular value of the matrix $X/N$, with $X$ defined in Eq. (7), taking over
all quantum states $\varrho$ which are locally unitary equivalent to $\rho$.
Equivalently, ${\lambda_{1}}^{\prime}$ is also the maximal singular value of
the matrix ${M}^{\prime}=({M_{j,ik}}^{\prime})$, with ${M_{ijk}}^{\prime}={\rm
tr}[{\rho}^{\prime}(\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k})]$,
$i,j,k=1,2,3.$

###### Proof.

The normalization factor $N$ has the following form,

$\displaystyle N$ $\displaystyle={\rm
tr}\left[(U\Sigma_{A}^{2}U^{\dagger}\otimes V\Sigma_{B}^{2}V^{\dagger}\otimes
W\Sigma_{C}^{2}W^{\dagger})\rho\right]$ $\displaystyle={\rm
tr}\left[(\Sigma_{A}^{2}\otimes\Sigma_{B}^{2}\otimes\Sigma_{C}^{2})(U^{\dagger}\otimes
V^{\dagger}\otimes W^{\dagger})\rho(U\otimes V\otimes W)\right]$
$\displaystyle={\rm
tr}\left[(\Sigma_{A}^{2}\otimes\Sigma_{B}^{2}\otimes\Sigma_{C}^{2})\varrho\right],$

where $\varrho=(U^{\dagger}\otimes V^{\dagger}\otimes
W^{\dagger})\rho(U\otimes V\otimes W)$. Since $\rho$ and $\varrho$ are local
unitary equivalent, they have the same value of the maximal violation of the
SI.

From the double cover relationship SO(3)-1995-PhysRevA.52.4396 ;
SU(3)-LM-2014-PhysRevA.89.062325  between the special unitary group $SU(2)$
and the special orthogonal group $SO(3)$,
$U\sigma_{{}_{i}}U^{\dagger}=\sum_{j=1}^{3}O_{ij}\sigma_{j}$, where $U$ is any
given unitary operator and the matrix $O$ with entries $O_{ij}$ belongs to
$SO(3)$, we have

$\displaystyle{M_{ijk}}^{\prime}$ $\displaystyle={\rm
tr}[{\rho}^{\prime}(\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k})]$
$\displaystyle=\frac{1}{N}{\rm tr}\left[\left(F_{A}\otimes F_{B}\otimes
F_{C}\right)\rho\left(F_{A}^{\dagger}\otimes F_{B}^{\dagger}\otimes
F_{C}^{\dagger}\right)(\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k})\right]$
$\displaystyle=\frac{1}{N}{\rm
tr}\left[\rho(U\Sigma_{A}U^{\dagger}\sigma_{i}U\Sigma_{A}U^{\dagger}\otimes
V\Sigma_{B}V^{\dagger}\sigma_{j}V\Sigma_{B}V^{\dagger}\otimes
W\Sigma_{C}W^{\dagger}\sigma_{k}W\Sigma_{C}W^{\dagger})\right]$
$\displaystyle=\frac{1}{N}\sum_{l,m,n}{\rm tr}\left[(U^{\dagger}\otimes
V^{\dagger}\otimes W^{\dagger})\rho(U\otimes V\otimes
W)(\Sigma_{A}O_{il}^{A}\sigma_{l}\Sigma_{A}\otimes\Sigma_{B}O_{jm}^{B}\sigma_{m}\Sigma_{B}\otimes\Sigma_{C}O_{kn}^{C}\sigma_{n}\Sigma_{C})\right]$
$\displaystyle=\frac{1}{N}\sum_{l,m,n}O_{il}^{A}O_{jm}^{B}O_{kn}^{C}{\rm
tr}\left[\varrho(\Sigma_{A}\sigma_{l}\Sigma_{A}\otimes\Sigma_{B}\sigma_{m}\Sigma_{B}\otimes\Sigma_{C}\sigma_{n}\Sigma_{C})\right]$
$\displaystyle=\frac{1}{N}\sum_{l,m,n}O_{il}^{A}O_{jm}^{B}O_{kn}^{C}{\rm
tr}\left[\varrho(\delta_{l}\otimes\eta_{m}\otimes\gamma_{n})\right]$
$\displaystyle=\frac{1}{N}\left[O_{A}X\left(O_{B}^{T}\otimes
O_{C}^{T}\right)\right]_{ijk}.$ (9)

Therefore, we have ${M}^{\prime}=\left[O_{A}X\left(O_{B}^{T}\otimes
O_{C}^{T}\right)\right]/N$, and

$\left({M}^{\prime}\right)^{\dagger}{M}^{\prime}=\frac{1}{N^{2}}\left(O_{B}\otimes
O_{C}\right)X^{\dagger}O_{A}^{\dagger}O_{A}X\left(O_{B}\otimes
O_{C}\right)^{\dagger}=\frac{1}{N^{2}}\left(O_{B}\otimes
O_{C}\right)X^{\dagger}X\left(O_{B}\otimes O_{C}\right)^{\dagger}.$ (10)

By noticing the orthogonality of the operator $O_{B}\otimes O_{C}$, one
obtains that ${M}^{\prime}\left({M}^{\prime}\right)^{\dagger}$ has the same
eigenvalues as $X^{\dagger}X/N^{2}$. Hence, $M^{\prime}$ has the same singular
values as $X/N$. Let $\vec{v}$ be a nine-dimensional singular vector of the
matrix $X/N$. Then $(O_{B}\otimes O_{C})\vec{v}$ is a nine-dimensional
singular vector of the matrix $M^{\prime}$. ∎

##  III Tightness of the upper bound and hidden genuine nonlocality

As applications of the Theorem 1, we consider the activation of the hidden
genuine nonlocality of three-qubit systems. We present two classes of three-
qubit states which admit bi-LHV model before local filtering, but display
genuine nonlocality after local filtering.

Let us begin with the two-qubit isotropic states

${\chi}_{iso}(p)=p|\phi\rangle\langle\phi|+(1-p)\frac{I_{4}}{4},$ (11)

where $|\phi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ is the
maximally entangled state, $I_{4}$ denotes the $4\times 4$ identity matrix,
and $0\leq p\leq 1$. The state ${\chi}_{iso}(p)$ is fully local for $0\leq
p\leq 0.4167$ and the related local model can be generalized from isotropic
states to the following states LHV-PhysRevLett.99.040403 ,

$\hat{\chi}(p,\theta)=p|\psi_{s}\rangle\langle\psi_{s}|+(1-p)\frac{I_{4}}{4},$
(12)

where $|\psi_{s}\rangle={\rm cos}\,\theta|00\rangle+{\rm
sin}\,\theta|11\rangle$, and $0\leq\theta\leq\pi/4$.

Consider the following states,

$\chi(p,\theta)=p|\psi_{s}\rangle\langle\psi_{s}|+(1-p)|0\rangle\langle
0|\otimes\frac{I_{2}}{2},$ (13)

where $I_{2}$ denotes the $2\times 2$ identity matrix, $0\leq\theta\leq\pi/4$,
and $0\leq p\leq 1$. Following the protocol presented in
localfilter-2013-PhysRevLett.111.160402 , we have that $\chi(p,\pi/4)$ admits
an LHV model as $\hat{\chi}(p,\pi/4)$. Namely, $\chi(p,\theta)$, with
$0\leq\theta\leq\pi/4$, admits an LHV model for $0\leq p\leq 0.4167$.

Any bipartite local states can be converted to multipartite states with a bi-
local model construction-PhysRevLett.115.030404 . Following the construction
given in construction-PhysRevLett.115.030404 , we can transform the states
$\chi(p,\theta)$ into the mixture of the colored noise and the three-qubit
GHZ-class states,

$\rho_{\chi}(p,\theta)=p|\Psi_{s}\rangle\langle\Psi_{s}|+(1-p)|00\rangle\langle
00|\otimes\frac{I_{2}}{2},$ (14)

where $|\Psi_{s}\rangle={\rm cos}\,\theta|000\rangle+{\rm
sin}\,\theta|111\rangle$. Analogously, the state $\rho_{\chi}(p,\theta)$
admits an bi-LHV model for $0\leq p\leq 0.4167$. In the following, we set
$\theta=\pi/8$ and consider the activation of the hidden genuine nonlocality
of $\rho_{\chi}(p,\pi/8)$ under local filtering.

Firstly, based on the genuine multipartite concurrence of three-qubit _X_
states X.GME-2012-PhysRevA.86.062303 , one can show that
$\rho_{\chi}(p,\pi/8)$ is genuine multipartite entangled for $0<p\leq 1$.
Secondly, the quantum state $\rho_{\chi}(p,\pi/8)$ attains the upper bound on
the mean values of the SI operators, but never violates the SI, which can be
seen from the matrix $M$ of $\rho_{\chi}(p,\pi/8)$ defined in (6),

$M=\left(\begin{array}[]{ccccccccc}\displaystyle\frac{\sqrt{2}p}{2}&0&0&0&\displaystyle-\frac{\sqrt{2}p}{2}&0&0&0&0\\\
0&\displaystyle-\frac{\sqrt{2}p}{2}&0&\displaystyle-\frac{\sqrt{2}p}{2}&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&\displaystyle\frac{\sqrt{2}p}{2}\end{array}\right).$ (15)

The singular values of the matrix $M$ are $p$, $p$ and
$\displaystyle\frac{\sqrt{2}p}{2}$. Hence, $\lambda_{1}=p$. The upper bound of
the maximal mean value of the Svetlichny operator is
$\mathcal{Q}(\mathcal{S})={\rm
max}|\langle\mathcal{S}\rangle_{\rho_{\chi}(p,\pi/8)}|\leq 4\lambda_{1}=4p$.
In order to assure that the upper bound can be used to determine the violation
of the SI, one needs to prove that the bound is attained for the state
$\rho_{\chi}(p,\pi/8)$, which requires that two nine-dimensional singular
vectors have the forms
$\vec{a}\otimes\vec{c}-\vec{a}\,^{\prime}\otimes\vec{c}\,^{\prime}$ and
$\vec{a}\otimes\vec{c}\,^{\prime}+\vec{a}\,^{\prime}\otimes\vec{c}$ exist. We
select the two singular vectors corresponding to the degenerated $\lambda_{1}$
as
$\vec{v_{1}}=(1,0,0,0,-1,0,0,0,0)^{T}=(1,0,0)^{T}\otimes(1,0,0)^{T}-(0,-1,0)^{T}\otimes(0,-1,0)^{T}$
and
$\vec{v_{2}}=(0,-1,0,-1,0,0,0,0,0)^{T}=(1,0,0)^{T}\otimes(0,-1,0)^{T}+(0,-1,0)^{T}\otimes(1,0,0)^{T}$.
Setting $\vec{a}=(1,0,0)^{T}$, $\vec{a}\,^{\prime}=(0,-1,0)^{T}$,
$\vec{c}=(1,0,0)^{T}$ and $\vec{c}\,^{\prime}=(0,-1,0)^{T}$, and choosing
$\vec{b}$ and $\vec{b}\,^{\prime}$ to be suitable unit vectors of proper
measurement directions of $B$ and $B^{\prime}$ in Eq. (4), we can show that
the upper bound $4p$ is attained for $\rho_{\chi}(p,\pi/8)$. Nevertheless, the
violation of the SI never happens for the quantum state $\rho_{\chi}(p,\pi/8)$
as expected.

Now we consider the local filtering of $\rho_{\chi}(p,\pi/8)$. By direct
computation, we have the matrix $\tilde{M}=(\tilde{M}_{m,ln})=({\rm
tr}[\rho_{\chi}(p,\pi/8)(\delta_{l}\otimes\eta_{m}\otimes\gamma_{n})])$,
$l,m,n=1,2,3$,

$\tilde{M}=\left(\begin{array}[]{ccccccccc}\displaystyle\frac{\sqrt{2}pxyz}{2}&0&0&0&\displaystyle-\frac{\sqrt{2}pxyz}{2}&0&0&0&0\\\
0&\displaystyle-\frac{\sqrt{2}pxyz}{2}&0&\displaystyle-\frac{\sqrt{2}pxyz}{2}&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&D\\\ \end{array}\right),$ (16)

where
$D=\displaystyle-\frac{2-\sqrt{2}}{4}p-\frac{1-p}{2}x^{2}y^{2}+\frac{2+\sqrt{2}p}{4}x^{2}y^{2}z^{2}$.
The singular values of the matrix $\tilde{M}$ are $pxyz$, $pxyz$ and $D.$
Since $\rho_{\chi}(p,\pi/8)$ and $\varrho_{\chi}(p,\pi/8)$ are locally unitary
equivalent, we conclude that ${pxyz}/{N}$, ${pxyz}/{N}$ and ${D}/{N}$ are the
singular values of the matrix $X/N$, where $N={\rm
tr}[\rho_{\chi}(p,\pi/8)(\Sigma^{2}_{A}\otimes\Sigma^{2}_{B}\otimes\Sigma^{2}_{C})]=\displaystyle\frac{2-\sqrt{2}}{4}p+\frac{1-p}{2}x^{2}y^{2}+\frac{2+\sqrt{2}p}{4}x^{2}y^{2}z^{2}$,
which are also the singular values of the matrix $M^{\prime}$ in Theorem 1.
The maximal singular value $\lambda_{1}^{\prime}$ is ${pxyz}/{N}$ for given
$p$, with ${pxyz}/{N}>{D}/{N}$. Then the upper bound of the maximal value of
the Svetlichny operator is given by

$\mathcal{Q}(\mathcal{S})^{\prime}={\rm
max}|\langle\mathcal{S}\rangle_{\rho_{\chi}^{\prime}(p,\pi/8)}|\leq
4\lambda_{1}^{\prime}=\displaystyle\frac{4pxyz}{N}.$ (17)

The matrix $X/N$ also has the singular vectors $\vec{v_{1}}$ and $\vec{v_{2}}$
with respect to the singular value $\lambda_{1}^{\prime}$. According to
Theorem 1, the singular vectors of $M^{\prime}$ corresponding to
$\lambda_{1}^{\prime}$ are $(O_{B}\otimes
O_{C})\vec{v_{1}}=O_{B}\,\vec{a}\otimes
O_{C}\,\vec{c}-O_{B}\,\vec{a}\,^{\prime}\otimes O_{C}\,\vec{c}$ and
$(O_{B}\otimes O_{C})\vec{v_{2}}=O_{B}\,\vec{a}\otimes
O_{C}\,\vec{c}\,^{\prime}+O_{B}\,\vec{a}\,^{\prime}\otimes O_{C}\,\vec{c}$,
where $O_{B}$ and $O_{C}$ belong to $SU(3)$. The upper bound is saturated as
the singular vectors can be written in required decomposition forms. The SI is
violated if and only if $\lambda_{1}^{\prime}=\displaystyle\frac{pxyz}{N}>1$.
Maximizing $\lambda_{1}^{\prime}$ under the restriction
$\displaystyle\frac{pxyz}{N}>\frac{D}{N}$, we obtain that the quantum states
$\rho_{\chi}^{\prime}(p,\pi/8)$ violate the SI and are genuine three-qubit
nonlocal for $0.3697\leq p\leq 1$, although $\rho_{\chi}(p,\pi/8)$ is bi-
local, see Fig. 1.

Figure 1: The state $\rho_{\chi}(p,\pi/8)$ admits a bi-local hidden model for
$0\leq p\leq 0.4167$ and never violates SI for $0\leq p\leq 1$ as the upper
bound in Theorem 1 is saturated. The locally filtered state shows the genuine
nonlocality for $0.3697\leq p\leq 1$. The hidden genuine tripartite
nonlocality is revealed for $0.3697\leq p\leq 0.4167$.

Now we consider another class of three-qubit states. Consider the mixture of
the three-qubit GHZ states and the colored noise,

$\rho=p|{\rm GHZ}\rangle\langle{\rm
GHZ}|+\frac{1-p}{4}\widetilde{I_{0}}\otimes I_{4},$ (18)

where $|{\rm GHZ}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$,
$\widetilde{I_{0}}=\left(\begin{array}[]{cc}1&0\\\ 0&0\\\ \end{array}\right)$
and $0\leq p\leq 1$. The state $\rho$ is genuine multipartite entangled for
$0<p\leq 1$ by using the criterion given in X.GME-2012-PhysRevA.86.062303 .
The corresponding matrix $M$,

$M=\left(\begin{array}[]{ccccccccc}p&0&0&0&-p&0&0&0&0\\\
0&-p&0&-p&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0\\\ \end{array}\right),$ (19)

has singular values $\sqrt{2}p$, $\sqrt{2}p$ and 0, i.e.,
$\lambda_{1}=\sqrt{2}p$. The upper bound of the maximal value of the
Svetlichny operator satisfies
$\mathcal{Q}(\mathcal{S})=\rm{max}|\langle\mathcal{S}\rangle_{\rho}|\leq
4\lambda_{1}=4\sqrt{2}p$. This upper bound is saturated since two nine-
dimensional singular vectors of the forms,
$\vec{a}\otimes\vec{c}-\vec{a}\,^{\prime}\otimes\vec{c}\,^{\prime}$ and
$\vec{a}\otimes\vec{c}\,^{\prime}+\vec{a}\,^{\prime}\otimes\vec{c}$ can be
found in the following way. Take the singular vectors corresponding to
$\lambda_{1}$ to be $\vec{v_{1}}=(1,0,0,0,-1,0,0,0,0)^{T}$ and
$\vec{v_{2}}=(0,-1,0,-1,0,0,0,0,0)^{T}$, which have exactly the forms,
$(1,0,0)^{T}\otimes(1,0,0)^{T}-(0,-1,0)^{T}\otimes(0,-1,0)^{T}$ and
$(1,0,0)^{T}\otimes(0,-1,0)^{T}+(0,-1,0)^{T}\otimes(1,0,0)^{T}$, respectively.
By defining $\vec{a}=(1,0,0)^{T}$, $\vec{a}\,^{\prime}=(0,-1,0)^{T}$,
$\vec{c}=(1,0,0)^{T}$ and $\vec{c}\,^{\prime}=(0,-1,0)^{T}$, and selecting
suitable $\vec{b}$ and $\vec{b}\,^{\prime}$, the upper bound is attained.
Therefore, the state $\rho$ in Eq. (18) violates the SI if and only if
$0.707107<p\leq 1$.

The matrix $\tilde{M}=(\tilde{M}_{m,ln})=({\rm
tr}[\rho(\delta_{l}\otimes\eta_{m}\otimes\gamma_{n})])$, $l,m,n=1,2,3$, has
the form,

$\tilde{M}=\left(\begin{array}[]{ccccccccc}pxyz&0&0&0&-pxyz&0&0&0&0\\\
0&-pxyz&0&-pxyz&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&D\end{array}\right),$ (20)

where
$D=-\frac{1}{2}p+\frac{1-p}{4}x^{2}-\frac{1-p}{4}x^{2}y^{2}-\frac{1-p}{4}x^{2}z^{2}+\frac{1+p}{4}x^{2}y^{2}z^{2}.$
The singular values of $\tilde{M}$ are $\sqrt{2}pxyz,$ $\sqrt{2}pxyz$ and $D.$
Due to the local unitary equivalence between $\rho$ and $\varrho$, the
singular values of the matrix $X/N$ are ${\sqrt{2}pxyz}/{N}$,
${\sqrt{2}pxyz}/{N}$ and ${D}/{N}$, where $N={\rm
tr}[\rho(\Sigma^{2}_{A}\otimes\Sigma^{2}_{B}\otimes\Sigma^{2}_{C})]=\frac{1}{2}\,p+\frac{1-p}{4}x^{2}+\frac{1-p}{4}x^{2}y^{2}+\frac{1-p}{4}x^{2}z^{2}+\frac{1+p}{4}x^{2}y^{2}z^{2}.$
According to Theorem 1, these values are also the singular values of the
matrix $M^{\prime}$. We have $\lambda_{1}^{\prime}=\frac{\sqrt{2}pxyz}{N}$ for
$\frac{\sqrt{2}pxyz}{N}>\frac{D}{N}$. The matrix $X/N$ also has the same
singular vectors $\vec{v_{1}}$ and $\vec{v_{2}}$ with respect to
$\lambda_{1}^{\prime}$. The singular vectors of $M^{\prime}$ with respect to
$\lambda_{1}^{\prime}$ are $(O_{B}\otimes O_{C})\vec{v_{1}}$ and
$(O_{B}\otimes O_{C})\vec{v_{2}}$, namely, $O_{B}\,\vec{a}\otimes
O_{C}\,\vec{c}-O_{B}\,\vec{a}\,^{\prime}\otimes O_{C}\,\vec{c}\,^{\prime}$ and
$O_{B}\,\vec{a}\otimes
O_{C}\,\vec{c}\,^{\prime}+O_{B}\,\vec{a}\,^{\prime}\otimes O_{C}\,\vec{c},$
with $O_{B}$ and $O_{C}$ belonging to $SU(3)$. Hence the upper bound is also
saturated for the locally filtered state. Therefore, the state violates the SI
if and only if $\lambda_{1}^{\prime}=\frac{\sqrt{2}pxyz}{N}>1$. Then the upper
bound of the maximal value of the Svetlichny operator satisfies

$\mathcal{Q}(\mathcal{S})^{\prime}={\rm
max}|\langle\mathcal{S}\rangle_{\rho}|\leq
4\lambda_{1}^{\prime}=\displaystyle\frac{4\sqrt{2}pxyz}{N}.$ (21)

Based on the above analysis, the genuine nonlocality of the quantum state
$\rho^{\prime}$ is detected by the SI for $0.3334\leq p\leq 1$. Therefor, the
hidden genuine nonlocality of $\rho$ is revealed by local filtering operations
for $0.3334\leq p\leq 0.7071$, see Fig. 2.

Figure 2:  Denote $f(p)$ the maximal value of $\mathcal{Q}(\mathcal{S})$:
Max$|\langle\mathcal{S}\rangle_{\rho}|$ (dashed line)
max$|\langle\mathcal{S}\rangle_{\rho^{\prime}}|$ (solid line). As the upper
bound in Theorem 1 is saturated, for $0.3334\leq p\leq 0.7071$ the state
$\rho=p\,|{\rm GHZ}\rangle\langle{\rm
GHZ}|+\frac{1-p}{4}\widetilde{I_{0}}\otimes I_{4}$ does not violate the SI,
but its locally filtered state $\rho^{\prime}$ shows genuine nonlocality.

##  IV Conclusions and Discussions

We have presented a qualitative analytical analysis of the hidden genuine
nonlocality for three-qubit systems by providing a tight upper bound on the
maximal quantum value of the Svetlichny operators under local filtering
operations. The tightness of the upper bounds have been investigated through
detailed color noised quantum states. We have presented two classes of three-
qubit states whose hidden genuine nonlocalities can be revealed by local
filtering. Our results give rise to an operational method in investigating the
genuine nonlocality for three-qubit mixed states. Moreover, the method
presented in this paper can also be used in optimizing the maximal quantum
violations of other Bell-type inequalities for tripartite or multipartite
quantum systems under local filtering.

## Acknowledgment

Acknowledgments This work is supported by NSFC (11775306, 11701568, 11675113),
the Fundamental Research Funds for the Central Universities (18CX02035A,
18CX02023A, 19CX02050A), Beijing Municipal Commission of Education under Grant
No. KZ201810028042, Beijing Natural Science Foundation (Z190005), and Academy
for Multidisciplinary Studies, Capital Normal University.

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