# Network configuration theory for all networks

Ming-Xing Luo  The School of Information Science and Technology, Southwest
Jiaotong University, Chengdu 610031, China

###### Abstract

Entangled quantum networks provide great flexibilities and scalabilities for
quantum information processing or quantum Internet. Most of results are
focused on the nonlocalities of quantum networks. Our goal in this work is to
explore new characterizations of any networks with theory-independent
configurations. We firstly prove the configuration inequality for any network
using the fractional independent set of the associated graph. These
inequalities can be built with polynomial-time complexity. The new result
allows featuring correlations of any classical network depending only on its
network topology. Similar inequalities hold for all entangled quantum networks
with any local measurements. This shows an inherent feature of quantum
networks under local unitary operations. It is then applied for verifying
almost all multipartite entangled pure states with linear complexity, and
witnessing quantum network topology without assumption of inputs. The
configuration theory is further extended for any no-signalling networks. These
results may be interesting in entanglement theory, quantum information
processing, and quantum networks.

Bell inequality provides an important method to experimentally verify the
nonlocality of entangled states EPR ; Bell ; CHSH . It has been improved both
in theory and experiment to reveal the inherent features of quantum theory BCP
; HHH . As one special case quantum nonlocality is explored in network
scenarios BGP ; BRG ; Fritz ; TALR . The quantum network may consist of
independent entangled systems. This implies new limits on quantum correlations
beyond single-source network because each entanglement is shared by partial
observers. Another interest difference is from the local joint operations on
independent systems, which may result in distinctive quantum features BBC  or
activate new nonlocal correlations Activ ; Fritz . These quantum resources
allow distributed experimental settings for information processing in quantum
networks Kimble ; net1 ; net2 .

Although these improvements, however, the inherent or key features of quantum
nonlocality in networks is very limited Gisin ; TALR . Even for classical
networks, it is unknown how to characterize all the classical correlations
available in networks consisting of independent variables. The independence
assumptions of sources rule out the correlation polytope used for a single
common source BCP , but consist of non-convex correlation set. This imposes
special difficulties in exploring the facet inequalities. It may be resolved
for special networks by using Bell-type nonlinear inequalities [16-23] or
quantum game Luo2019 .

Beyond quantum nonlocality verified with Bell inequalities, another method is
to explore features of quantum correlations in networks. These limits depend
only on the specific network configurations Henson . In contrast to the
standard Bell scenario, the violations of the so-called configuration
inequalities, do not witness the nonlocality of quantum networks, but its
network topology under local operations. One example is tripartite network
with different configurations shown in Fig.1. An algorithmic method is to
explore information-theoretic correlations by using semidefinite programs CMG
; Alex ; ANDC ; Tava3 . The drawback is of limits on small-size networks.
Another way is from Bell-type inequalities Marc2019a . Different from quantum
nonlocality without involving any inputs Fritz ; Marc2019b , the network
topology imposes fundamental limitations on achievable (classical, quantum or
no-signalling) correlations CB ; GPC ; Marc2019a . So far, these results hold
for special quantum networks consisting of bipartite entangled resources
Marc2019a ; GPC ; Alex . A natural problem is how to characterize general
quantum networks. This is also related to another basic problem of witnessing
entanglement HHH .

Figure 1: (Color online) Schematically tripartite networks consisting of
single source, two independent sources or three independent sources.

Our goal in this work is to characterize correlations derived from any
networks. We firstly identify all the achievable correlations for chain
networks consisting of bipartite sources by constructing new nonlinear
inequalities. This can be used to verify triangle network beyond the recent
result Marc2019b ; Alex . It is further extended as a general method for all
networks consisting of any sources by exploring generalized Finner
inequalities Finner . Similar configuration inequalities hold for any quantum
networks under any local projections beyond special networks CMG ; Alex ; ANDC
; Tava3 ; Marc2019b . This can be used to resolve the compatibility problem of
quantum network topology in a device-independent manner. A specific network
topology can be verified by using the observed correlations. It further
motivates another remarkable application of verifying almost all multipartite
entangled pure states (except for one set with zero measure) with linear
complexity. Interestingly, the proposed configuration theory provides theory-
independent limits of correlations in any network scenarios independent of the
underlying physical model (for any possible no-signaling theory) PR ; BLM ;
Barrett  even for any noisy sources. It answers affirmatively a recent
conjecture Marc2019a ; GPC ; Alex .

General classical networks. Consider an $n$-partite network ${\cal N}$
consisting of independent sources $\lambda_{1},\cdots,\lambda_{m}$. Each party
$\textsf{A}_{j}$ shares some variables
$\Lambda_{j}\subseteq\\{\lambda_{1},\cdots,\lambda_{m}\\}$. $\Lambda$ is
arbitrary and could exist prior to the measurement choices. The measurement
outcome $a_{j}$ of $\textsf{A}_{j}$ depends only on its own resources
$\Lambda_{j}$, $j=1,\cdots,n$. The generalized locally causal model Bell ;
Luo2018  suggests a joint probability of the measurement outcomes as

$\displaystyle P({\bf
a})=\int_{\times_{j=1}^{n}\Omega_{j}}\prod_{j=1}^{n}p(a_{j}|\Lambda_{j})d\mu_{1}(\lambda_{1})\cdots
d\mu_{m}(\lambda_{m}),$ (1)

where ${\bf a}=(a_{1},\cdots{},a_{n})$, $a_{j}\in\\{0,\cdots,k_{j}\\}$, and
$(\Omega_{j},\Sigma_{j},\mu_{j})$ denotes the measure space of $\lambda_{j}$,
$\mu_{j}(\lambda_{j})$ is the measure of $\lambda_{j}$ with the normalization
condition $\int_{\Omega_{j}}d\mu_{j}(\lambda_{j})=1$, and
$p(a_{j}|\Lambda_{j})$ is the probability of $a_{j}$ conditional on
$\Lambda_{j}$ and satisfies $\sum_{a_{j}}p(a_{j}|\Lambda_{j})=1$ for any
$x_{j}$ and $\Lambda_{i}$. Different from the nonlocality [16-23], the
configuration of given network can be verified using the observed correlations
without assumption of inputs Henson ; CMG ; Alex ; ANDC ; Tava3 ; CB ; GPC .
These provide fundamental limitations on correlations achievable in a given
network regardless of specific sources. The only requirement is the no-
signaling principle PR .

A toy model is from tripartite networks consisting of single source GHZ ; Dur
, two bipartite sources BBC  or three bipartite sources Marc2019b , as shown
in Fig.1. It turns out the Finner inequality Finner ; Marc2019a  given by

$\displaystyle P(a,b,c)\leq\sqrt{p(a)p(b)p(c)}$ (2)

provides a configuration inequality for three bipartite sources. It is used
for verifying the GHZ-type source Marc2019a , but useless for other networks
(Appendix A). Other methods may be useful for special networks Henson ; CMG ;
Alex ; ANDC ; Tava3 ; CB ; GPC  or sources Marc2019a . There is few result to
distinguish these networks NWR ; Kraft ; Luo2021 ; BGP ; Marc2019b . A
triangle network Marc2019a  yields to the chain network BBC  regardless of one
source. This intrigues a new configuration inequality for the chain network as
(Appendix B)

$\displaystyle P(a,b,c)\leq
p(a)^{\frac{m-k}{m}}p(b)^{\frac{k}{m}}p(c)^{\frac{m-k}{m}}$ (3)

with $m\geq 2$ and $k\leq m-1$. It reduces to the inequality (2) for $m=2$ and
$k=1$. Interestingly, the inequality (3) is efficient for verifying the W-type
source of $P_{w}=p[001]+\frac{1-p}{2}([010]+[100])$ which cannot be verified
by the inequality (2), where $p_{xyz}[xyz]$ denotes the probability for
getting the outcomes $x,y,z$. Another example is the triangle network
consisting of three sources (Appendix C).

The inequality (2) shows new limits of correlations beyond the inequality (2).
A natural problem is how to feature general networks. Our goal in what follows
is to explore configuration inequalities for all networks with polynomial-time
complexity. The main idea is inspired by generalized fractional independent
set of graphs Schrijver . For a given $n$-partite network ${\cal N}$, there is
an associated undirected graph ${\cal G}=(V,E)$, where the vertex $i$ in $V$
denotes the party $\textsf{A}_{i}$, the edge $e_{ij}$ in $E$ connected with
the vertexes $i$ and $j$ denotes one bipartite source shared by
$\textsf{A}_{i}$ and $\textsf{A}_{j}$, and the $s$-hyper edge $e_{i_{1}\cdots
i_{s}}$ in $E$ connected with the vertexes ${i_{1}},\cdots,{i_{s}}$ denotes
one $s$-partite source shared by
$\textsf{A}_{i_{1}},\cdots,\textsf{A}_{i_{s}}$.

###### Definition 1

A fractional independent set of a graph ${\cal G}$ associated with the network
${\cal N}$ is a non-negative vector of ${\bf s}=(s_{1},\cdots,s_{n})$
satisfying

$\displaystyle s_{j}$ $\displaystyle\geq$ $\displaystyle 0,$
$\displaystyle\sum_{j:i\to j}s_{j}$ $\displaystyle\leq$ $\displaystyle
1,\forall i$ (4)

In the inequality (4) a nonnegative weight is setting for each node in ${\cal
G}$ such that for each source $j$ the summation of the weights of nodes
connected to it is at most 1. For any graph with bipartite sources, the
inequality (4) is known as edge inequality Schrijver . This allows an easy way
to characterize all solutions by using its extreme points. It is clique
inequality Schrijver  for a general graph with hyper edges, that is,
multipartite sources in the corresponding network. It is NP-complete hard to
find fractional clique cover numbers for any graph Schrijver . We provide two
algorithms with polynomial-time complexity for any finite networks (Appendix
D).

###### Theorem 1

(Informal) Let s be a fractional independent set of graph ${\cal G}$
associated with the network ${\cal N}$. Any correlation $P$ achievable in
${\cal N}$ satisfies

$\displaystyle P({\bf a})\leq\prod_{j=1}^{n}P(a_{j})^{s_{j}}$ (5)

for any outcome ${\bf a}$.

The Finner inequalities Marc2019a ; Finner  provide the configuration limits
for general networks consisting of bipartite sources. These kinds of
restrictions are optimal for specific networks such as triangle network or
cyclic networks Marc2019a ; Schrijver . Theorem 1 implies new configuration
limits for all the networks. One example is an $n$-partite chain network
${\cal N}$, where each pair of $\textsf{A}_{j}$ and $\textsf{A}_{j+1}$ shares
one bipartite source, $j=1,\cdots,n-1$. One fractional independent set is
given by ${\bf s}$ with $s_{i}=\frac{m-k}{m}$ for odd $i$’s and
$s_{j}=\frac{k}{m}$ for even $j$’s. The joint distribution $P({\bf a})$
satisfies the configuration inequality as

$\displaystyle P({\bf a})\leq\prod_{\textrm{odd
}j}P(a_{j})^{\frac{m-k}{m}}\prod_{\textrm{even}\,j}P(a_{j})^{\frac{k}{m}}$ (6)

for any $k,m$ with $1\leq k<m$ and $m\geq 2$. Another is $n$-partite star
networks, where each pair of $\textsf{A}_{j}$ and $\textsf{A}_{n}$ shares one
bipartite source, $j=1,\cdots,n-1$. The joint distribution $P({\bf a})$
satisfies

$\displaystyle P({\bf
a})\leq\prod_{j=1}^{n-1}P(a_{j})^{\frac{m-k}{m}}P(a_{n})^{\frac{k}{m}}$ (7)

where $s_{j}=\frac{m-k}{m}$ for $j=1,\cdots,n-1$ and $s_{n}=\frac{k}{m}$.
Similar results hold for other networks consisting of multipartite sources.
This provides a computationally efficient method for testing any network
configurations beyond recent results CMG ; Alex ; ANDC ; Tava3 . The proof is
based on the generalized Finner inequality Finner  (Appendix E), which is
itself interesting in applications.

Configuration limits of all quantum networks. Consider a general quantum
network consisting of any multipartite states. From Born rule, the joint
distribution after the local measurements performed on the total state $\rho$
is given by

$\displaystyle P_{q}({\bf a})={\rm tr}(M_{a_{1}}\otimes\cdots\otimes
M_{a_{n}}\rho)$ (8)

where $M_{a_{j}}$ are positive semidefinite operators satisfying
$\sum_{a_{j}}M_{a_{j}}=\mathbbm{1}$ with the identity operator $\mathbbm{1}$.
The quantum distribution is consistent with its from hidden variable model
with single source Bell . It also holds for quantum networks consisting of EPR
states EPR ; Marc2019a . A natural problem is for exploring general quantum
networks Henson ; CMG ; Alex ; ANDC ; Tava3 ; CB ; GPC ; 137 ; 138 ; 139 ; 140
. Our goal here is to present an efficient method for featuring all quantum
networks.

###### Theorem 2

(Quantum generalized Finner inequality) Let s be a fractional independent set
of graph ${\cal G}$ associated with quantum network ${\cal N}_{q}$. Any
correlation $P_{q}$ achievable under local projection measurements on ${\cal
N}_{q}$ satisfies

$\displaystyle P_{q}({\bf a})\leq\prod_{j=1}^{n}P_{q}(a_{j})^{s_{j}}$ (9)

for any outcome ${\bf a}$.

Theorem 2 means generalized Finner inequality (5) holds for quantum networks
with the same topology of classical network (Appendix F). Compared with the
recent result Marc2019a , the inequality (9) holds for all quantum networks
consisting of any entangled quantum states such as EPR state EPR ,
multipartite entangled states GHZ ; Dur  or noisy states Werner . Remarkably,
the generalized Finner inequality (5) holds for local joint measurements which
can change network configurations such as entanglement swapping experiment BBC
. It means Finner inequality depends on the inherent configuration under local
unitary operations. This provides a trivial feature of quantum networks
different from single entanglement NWR ; Kraft ; Luo2021 .

###### Corollary 1

Let s be a positive number set satisfying $\sum_{i=1}^{n}s_{i}\leq 1$. Any
correlation $P_{q}$ achievable in all $n$-partite quantum networks satisfies

$\displaystyle P_{q}({\bf a})\leq\prod_{j=1}^{n}P_{q}(a_{j})^{s_{j}}$ (10)

for any outcome ${\bf a}$.

As a special case of Theorem 2, the inequality (10) provides the first limit
of correlations from single-source networks without inputs. Any $n$-partite
quantum network ${\cal N}_{q}$ can be changed into a single-source network by
local entangled operations. All the correlations achievable under local joint
measurements on multiple-source networks are special cases of single-source
networks, and then satisfy both the inequalities (9) and (10). One example is
the chain network consisting of generalized EPR states EPR  with the
configuration inequality (10) (Appendix G). The other is the star networks
consisting of generalized EPR states EPR  with the configuration inequalities
(7) and (10) (Appendix H). For $n$-partite cyclic networks, Theorem 2 reduces
recent result Marc2019a  for an odd $n$ and new limits for an even $n$.
Generally, Theorem 2 provides a general method for solving the configuration
compatibility problem (Appendix I).

Witnessing Multipartite Entanglement. For an $n$-partite entangled pure state
$|\Phi\rangle_{A_{1}\cdots A_{n}}$ on Hilbert space
$\mathbb{H}_{A_{1}}\otimes\cdots\otimes\mathbb{H}_{A_{n}}$, lots of methods
such as Bell inequalities Bell ; CHSH ; BCP  and entanglement witness HHH  are
useful for verifying its entanglement. However, it is difficult to verify any
multipartite entanglement. Our goal here is to address this problem using
configuration inequalities in Theorem 2. Especially, for any $n$-partite
entangled pure state $|\Phi\rangle$, the local joint distribution
$P(a_{i},a_{i+1})$ can be obtained by performing local projection
measurements, $i=1,\cdots,n-1$. This provides an efficient method to verify
almost all entangled pure states without their density matrices.

###### Corollary 2

Almost all entangled pure states can be witnessed with linear complexity.

Corollary 2 means any multipartite entangled pure states can be verified with
linear complexity except for one set with zero measure. The main idea is from
the configuration inequality (3) (Appendix J). Take a three-qubit state
$|\phi\rangle_{A_{1}A_{2}A_{3}}$ as an example. If it is biseparable over the
bipartition $A_{1}$ and $\\{A_{2},A_{3}\\}$, it may be regarded as an
equivalent chain network ${\cal N}_{q}$ consisting of $A_{1}$,
$\\{A_{2},A_{3}\\}$ and an auxiliary party $C$. The output $c$ of $C$ is fixed
as $1$. We get the joint distribution given by
$P(a_{1},c=1,a_{2},a_{3})=P(a_{1},a_{2},a_{3})$ which satisfies the inequality
(3). This inspires an efficient method for witnessing almost all multipartite
entangled pure states $|\Phi\rangle_{A_{1}\cdots{}A_{n}}$ by using
distributions $P(a_{i},a_{i+1})$, $i=1,\cdots,n-1$. It is linear complexity
because of $n-1$ numbers of $2$-particle testing. One example is $n$-partite
GHZ state GHZ  using one distribution $P(a_{1},a_{2})$ from the symmetry. For
an $n$-partite W state Dur  of
$|W\rangle_{A_{1}\cdots{}A_{n}}=\sum_{i=1}^{n}\alpha_{i}|1\rangle_{A_{i}}\otimes_{j\not=i}|0\rangle_{A_{j}}$,
it can be verified by using the inequality (3) for
$\prod_{i=1}^{n}\alpha_{i}^{2}\not=0$ with large $m$.

Configuration limits of no-signalling networks. Classical networks or quantum
networks may be regarded as particular instances of general networks with no-
signalling sources PR . In this extension, the only condition is the no-
signaling principle requiring local operations on local systems cannot
instantaneously relay information to other systems. This shows the theory-
independent restrictions on space-like separated observations on any network.
Consider a no-signalling network ${\cal N}_{ns}$ with independent sources.
Each source can distribute no-signalling boxes to any parties who are
connected. These no-signalling boxes are then regarded as resources to
generate correlated outputs. Each party can locally wire the shared boxes in a
general way to determine an output PR . This model shows a general theory for
featuring most of well-known models Bell ; PR . A natural problem is to
feature general constraints of no-signaling correlations. So far, various
methods CB ; Fritz ; Tava3 ; Henson ; GPC ; 137 ; ANDC ; 138 ; 139 ; 140  have
been explored for special networks. Our consideration is to provide
configuration inequalities for any no-signalling networks. Note all theories
in the no-signalling framework share some axiomatic features such as the
convexity of state spaces, distinguishability of local measurement and
commutitity of local operations (Appendix K) BLM ; Barrett . With these
axioms, we prove the present Finner inequalities hold for any no-signalling
networks.

###### Theorem 3

(No-signalling Finner inequality) Let $s$ be a fractional independent set of
graph ${\cal G}$ associated with the no-signalling network ${\cal N}_{ns}$.
Any distribution $P$ achievable in ${\cal N}_{ns}$ satisfies

$\displaystyle P_{ns}({\bf a})\leq\prod_{j=1}^{n}P_{ns}(a_{j})^{s_{j}}$ (11)

for any outcome ${\bf a}$.

Theorem 3 shows the generalized Finner inequality holds for no-signalling
networks, that is, the inequality (5) is no-signalling. This implies a method
to verify distributions generated by local measurements on any no-signalling
networks without inputs. Different from previous approaches CB ; Fritz ; Tava3
; Henson ; GPC ; 137 ; ANDC ; 138 ; 139 ; 140 , the configuration inequality
(11) can be built for any finite networks within polynomial-time complexity.
This answered a recent conjecture for the Finner inequality Marc2019a .

Robustness of network configurations. The present network configurations are
robust against general noises. Especially, Theorems 1-3 hold for any networks
consisting of noisy sources. The main reason is from the corresponding proofs
independent of specific sources (pure or noisy states in quantum scenarios for
example). This provides an efficient method to verify the correlations
available in noisy networks.

Figure 2: (Color online) (a) Achievable regions of $\theta$ and $v$ defined by
the inequality (2). (b) Achievable regions of $\theta$ and $v$ defined by the
inequality (34). Here, $m=1000$.

Example 1. Consider a noisy GHZ state GHZ ; Werner  as

$\displaystyle\rho_{ghz}=v|ghz\rangle\langle ghz|+\frac{1-v}{8}\mathbbm{1}$
(12)

where $|ghz\rangle=\cos\theta|000\rangle+\sin\theta|111\rangle$ with
$\theta\in(0,\frac{\pi}{2})$, and $v\in(0,1)$. From the inequality (3) with
large $m$, the visibility is given by $v^{*}=\frac{3-\sqrt{9-8\cos
2\theta^{2}}}{4\cos 2\theta^{2}}$ using the distribution $P(0,0,0)$ from local
projection measurement. The achievable regions of $\theta$ and $v$ with which
at least one distribution $P(a,b,c)$ violates the inequality (3) are shown in
Fig.2. The inequality (3) is useful for verifying noisy GHZ states for any
$\theta\in(0,\frac{\pi}{2})$ and $v>0.5$ beyond the inequality (2) for
$v>0.88$.

Example 2. Consider a noisy W state Dur ; Werner  as

$\displaystyle\rho_{W}=v|W\rangle\langle W|+\frac{1-v}{8}\mathbbm{1}$ (13)

where
$|w\rangle=\cos\theta\cos\gamma|001\rangle+\sin\theta\cos\gamma|010\rangle+\sin\gamma|100\rangle$
and $v\in(0,1)$. The joint distribution of $P(0,0,1)$ violates the inequality
(3) with large $m$ for any $v\in(0,1)$ while it is impossible for the
inequality (2).

Figure 3: (Color online) Achievable regions of $\theta$ and $v$. (a) Noisy
triangle network. (b) Noisy $4$-partite star-type network. The achievable
regions of $\theta$ and $v$ are obtained if at least one distribution
$P(a,b,c)$ violates the inequality (3) with $m=1000$.

Example 3. Consider a triangle network Marc2019b  consisting of three
generalized noisy EPR states EPR ; Werner  of

$\displaystyle\rho=v|EPR\rangle\langle{}EPR|+\frac{1-v}{4}\mathbbm{1}$ (14)

where $|EPR\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$ with
$\theta\in(0,\frac{\pi}{2})$, and $v\in(0,1)$. The inequality (3) can verify
all the correlations for $\theta\in(0,\frac{\pi}{2})$ and $v\geq 0.59$, as
shown in Fig.3(a). Similar result holds for $n$-partite star networks, where
one party $\textsf{A}_{i}$ is shared one noisy EPR state $\rho$ with
$\textsf{A}_{n}$, $i=1,\cdots,n-1$. The achievable regions of $\theta$ and $v$
are shown in Fig.3(b) with $n=4$.

Discussions. The inequality (3) provides configuration limits for any no-
signalling networks. An interesting feature is from large $m$. Especially, the
inequality (3) is changed into a linear form as

$\displaystyle P(a,b,c)\leq p(a)p(c)$ (15)

when $m\to\infty$ and $p(b)\not=0$. It may be regarded as the independence of
two parties in experiment. This provides the first facet inequality for any
tripartite chain networks. Similar results hold from Theorem 1 (Appendix L). A
natural problem is to explore other facet inequalities. This is related to the
completeness of configuration inequalities. Another problem is related to
verify generic entanglement such as many-body systems.

In summary, we provided an operational characterization of theory-independent
correlations from networks consisting of independent sources. The main idea is
inspired by the graph theory of generalized Finner inequalities. This allows
featuring correlations of all classical networks depending only on its network
topology. Similar results hold for all entangled quantum networks. This
implies a general method for witnessing almost all entangled pure states. It
is also applicable for testing entangled quantum network topology without
assumption of inputs. Similar result can be extended for no-signalling
networks. The present results should be interesting in entanglement theory,
quantum information processing, and quantum networks.

## Acknowledgements

We thank Marc-Olivier Renou, Alejandro Pozas-Kerstjens, Tavakoli Armin, Carlos
Palazuelos, Luming Duan, Donglin Deng, Shaoming Fei, and Salman Beigi. This
work was supported by the National Natural Science Foundation of China
(No.61772437), Fundamental Research Funds for the Central Universities
(No.2682014CX095), Chuying Fellowship, and EU ICT COST CryptoAction
(No.IC1306).

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##  Appendix A The Finner inequality (2)

In this section, we show the Finner inequality (2) for the triangle network
cannot be used for verifying other networks.

###  A.1 W-type distributions

Consider a generalized W state Dur  as

$\displaystyle|W\rangle=\alpha_{1}|001\rangle+\alpha_{2}|010\rangle+\alpha_{3}|100\rangle$
(16)

where $\alpha_{j}$ satisfies $\sum_{j=1}^{3}\alpha_{j}^{2}=1$. By performing
the projection measurement with basis $\\{|0\rangle,|1\rangle\\}$ on each
particle, the joint distribution is given by

$\displaystyle
P_{w}(a,b,c)=\alpha_{1}^{2}[001]+\alpha_{2}^{2}[010]+\alpha_{3}^{2}[100].$
(17)

It follows that

$\displaystyle p_{a}(0)=1-p_{3}^{2},p_{a}(1)=p_{3}^{2},$ $\displaystyle
p_{b}(0)=1-p_{2}^{2},p_{b}(1)=p_{2}^{2},$ $\displaystyle
p_{c}(0)=1-p_{1}^{2},p_{c}(1)=p_{1}^{2},$ (18)

where $p_{a}(x),p_{b}(y),p_{c}(y)$ denote the respective marginal distribution
of three parties for the outcomes $x,y,z$.

From Eqs.(17) and (18) we get

$\displaystyle P(0,0,1)^{2}-p_{a}(0)p_{b}(0)p_{c}(1)$ (19) $\displaystyle=$
$\displaystyle p_{1}^{4}-(1-p_{3}^{2})(1-p_{2}^{2})p_{1}^{2}$ $\displaystyle=$
$\displaystyle-p_{3}^{2}p_{2}^{2}p_{1}^{2}$ $\displaystyle\leq$ $\displaystyle
0$

for any $\alpha_{i}$’s. The distribution of $P(0,0,1)$ does not violate the
inequality (2).

Similarly, we can prove

$\displaystyle P(0,1,0)\leq\sqrt{p_{a}(0)p_{b}(1)p_{c}(0)}$ (20)
$\displaystyle P(1,0,0)\leq\sqrt{p_{a}(1)p_{b}(0)p_{c}(0)}$ (21)

for any $\alpha_{i}$’s. The Finner inequality (2) holds for all the
generalized W distributions given in Eq.(17).

###  A.2 Chain quantum networks

Consider an entanglement swapping network consisting of two generalized EPR
states $|\phi_{1}\rangle|\phi_{2}\rangle$ with
$|\phi_{i}\rangle=\cos\theta_{i}|00\rangle+\sin\theta_{i}|11\rangle$ and
$\theta_{i}\in(0,\frac{\pi}{2})$. Here, the parties $\textsf{A}_{1}$ and
$\textsf{A}_{2}$ share $|\phi_{1}\rangle$ while $\textsf{A}_{2}$ and
$\textsf{A}_{3}$ share $|\phi_{2}\rangle$. By performing the projection
measurement with the basis $\\{|0\rangle,|1\rangle\\}$ on each qubit, the
joint distribution is given by

$\displaystyle P(a,b,c)=p_{0}[000]+p_{1}[011]+p_{2}[120]+p_{3}[131]$ (22)

where the two-qubit states for $\textsf{A}_{2}$ are encoded into a four-
dimensional states with the basis
$|0\rangle:=|00\rangle,\cdots,|3\rangle=|11\rangle$,
$p_{0}=\cos^{2}\theta_{1}\cos^{2}\theta_{2}$,
$p_{1}=\cos^{2}\theta_{1}\sin^{2}\theta_{2}$,
$p_{2}=\sin^{2}\theta_{1}\cos^{2}\theta_{2}$ and
$p_{3}=\sin^{2}\theta_{1}\sin^{2}\theta_{2}$. It follows that

$\displaystyle p_{a}(0)=\cos^{2}\theta_{1},p_{a}(1)=\sin^{2}\theta_{1},$
$\displaystyle
p_{b}(0)=\cos^{2}\theta_{1}\cos^{2}\theta_{2},p_{b}(1)=\cos^{2}\theta_{1}\sin^{2}\theta_{2},$
$\displaystyle
p_{b}(2)=\sin^{2}\theta_{1}\cos^{2}\theta_{2},p_{b}(3)=\sin^{2}\theta_{1}\sin^{2}\theta_{2},$
$\displaystyle p_{c}(0)=\cos^{2}\theta_{2},p_{c}(1)=\sin^{2}\theta_{2}.$ (23)

For the distributions in Eqs.(22) and (23), from the forward evaluation we get

$\displaystyle P(a,b,c)=\sqrt{p(a)p(b)p(c)}$ (24)

for any $a,b,c\in\\{0,1\\}$. The Finner inequality (2) cannot verify any chain
networks even without noises.

###  A.3 Biseparable quantum networks

Consider a one-side entanglement network consisting of one-qubit state
$|\phi_{1}\rangle=\cos\theta_{1}|0\rangle+\sin\theta_{1}|1\rangle$ and one
generalized EPR state $|\phi_{2}\rangle$ with
$|\phi_{2}\rangle=\cos\theta_{2}|00\rangle+\sin\theta_{2}|11\rangle$,
$\theta_{1},\theta_{2}\in(0,\frac{\pi}{2})$. Here, the party $\textsf{A}_{1}$
has $|\phi_{1}\rangle$ while the parties $\textsf{A}_{2}$ and $\textsf{A}_{3}$
share $|\phi_{2}\rangle$. By performing the projection measurement on each
particle, the joint distribution is given by

$\displaystyle P(a,b,c)=p_{0}[000]+p_{1}[011]+p_{2}[100]+p_{3}[111]$ (25)

where $p_{0}=\cos^{2}\theta_{1}\cos^{2}\theta_{2}$,
$p_{1}=\cos^{2}\theta_{1}\sin^{2}\theta_{2}$,
$p_{2}=\sin^{2}\theta_{1}\cos^{2}\theta_{2}$, and
$p_{3}=\sin^{2}\theta_{1}\sin^{2}\theta_{2}$. It follows that

$\displaystyle p_{a}(0)=\cos^{2}\theta_{1},p_{a}(1)=\sin^{2}\theta_{1},$
$\displaystyle p_{b}(0)=\cos^{2}\theta_{2},p_{b}(1)=\sin^{2}\theta_{2},$
$\displaystyle p_{c}(0)=\cos^{2}\theta_{2},p_{c}(1)=\sin^{2}\theta_{2}.$ (26)

For the probability distributions in Eqs.(25) and (26), we can get

$\displaystyle P(a,b,c)\leq\sqrt{p(a)p(b)p(c)}$ (27)

for any $a,b,c\in\\{0,1\\}$. The Finner inequality (2) cannot verify any
biseparable quantum networks even without noises.

###  A.4 Fully separable quantum networks

Consider one fully separable network consisting of three single-qubit states
$\otimes_{i=1}^{3}|\phi_{i}\rangle$ with
$|\phi_{i}\rangle=\cos\theta_{i}|0\rangle+\sin\theta_{i}|1\rangle$,
$\theta_{i}\in(0,\frac{\pi}{2})$. Here, the party $\textsf{A}_{i}$ has
$|\phi_{i}\rangle$, $i=1,2,3$. By performing the projection measurement on
each qubit, the joint distribution is given by

$\displaystyle P(a,b,c)=\sum_{i,j,k=0}^{1}p_{ijk}[ijk]$ (28)

where $p_{ijk}$ is defined by
$p_{ijk}=\cos^{2-2i}\theta_{i}\sin^{2i}\theta_{i}\cos^{2-2j}\theta_{j}\sin^{2j}\theta_{j}\cos^{2-2k}\theta_{k}\sin^{2k}\theta_{k}$.
It follows that

$\displaystyle p_{a}(0)=\cos^{2}\theta_{1},p_{a}(1)=\sin^{2}\theta_{1},$
$\displaystyle p_{b}(0)=\cos^{2}\theta_{2},p_{b}(1)=\sin^{2}\theta_{2},$
$\displaystyle p_{c}(0)=\cos^{2}\theta_{3},p_{c}(1)=\sin^{2}\theta_{3}.$ (29)

For the probability distributions in Eqs.(28) and (29), we can get

$\displaystyle P(a,b,c)\leq\sqrt{p(a)p(b)p(c)}$ (30)

for any $a,b,c\in\\{0,1\\}$. The Finner inequality (2) cannot verify any fully
separable networks.

###  A.5 General cases

Consider a triangle network consisting of three independent sources
$\lambda_{1},\lambda_{2},\lambda_{3}$, where each pair of two parties share
one source. The joint probability distribution of $P(a,b,c)$ is defined by

$\displaystyle P(a,b,c)$ $\displaystyle=$
$\displaystyle\int_{\Omega}p(a|\lambda_{1},\lambda_{2})p(b|\lambda_{2},\lambda_{3})p(c|\lambda_{1},\lambda_{3})\mu_{1}(\lambda_{1})$
(31)
$\displaystyle\times\mu_{2}(\lambda_{2})\mu_{3}(\lambda_{3})d\lambda_{1}d\lambda_{2}d\lambda_{3}$

where $p(a|\lambda_{1},\lambda_{2})$ denotes the indicator function of the
first party with outcome $a$ conditional on the connected sources
$\lambda_{1}$ and $\lambda_{2}$, $p(b|\lambda_{2},\lambda_{3})$ and
$p(c|\lambda_{1},\lambda_{3})$ are defined similarly, $\mu_{i}(\lambda_{i})$
denotes the probability distribution of source $\lambda_{i}$, $i=1,2,3$.

Consider a chain network consisting of three parties $A_{1},A_{2},A_{3}$,
where the parties $A_{1}$ and $A_{3}$ are respectively connected the source
$\lambda_{1}$ and $\lambda_{2}$ while $A_{2}$ is connected with two sources.
Denote $a,b,c$ as the respective output of three parties. The joint
probability distribution of $P(a,b,c)$ is defined by

$\displaystyle P(a,b,c)$ $\displaystyle=$
$\displaystyle\int_{\Omega}p(a|\lambda_{1})p(b|\lambda_{1},\lambda_{2})p(c|\lambda_{2})$
(32)
$\displaystyle\times\mu_{1}(\lambda_{1})\mu_{2}(\lambda_{2})d\lambda_{1}d\lambda_{2}$

where $p(a|\lambda_{1})$ denotes the indicator function of the first party
with outcome $a$ conditional on the connected source $\lambda_{1}$,
$p(b|\lambda_{1},\lambda_{2})$ and $p(c|\lambda_{2})$ are defined similarly,
$\mu_{i}(\lambda_{i})$ denotes the probability distribution of source
$\lambda_{i}$, $i=1,2$. By regarding the source $\lambda_{1}$ in Eq.(31) as
constant, the joint distribution in Eq.(31) yields to its in Eq.(32). All the
correlations obtained from chain networks can be obtained from triangle
networks. This can be extended to general networks.

Proposition S1. Consider two $n$-partite networks ${\cal N}_{1}$ and ${\cal
N}_{2}$ consisting of some independent sources. Suppose ${\cal N}_{1}$ is a
sub-network of ${\cal N}_{2}$ (or ${\cal N}_{1}\subset{\cal N}_{2}$). All the
correlations $\\{P_{1}(a_{1}\cdots{}a_{n})|{\cal N}_{1}\\}$ available in
${\cal N}_{1}$ can be generated from ${\cal N}_{2}$, that is

$\displaystyle\\{P_{1}(a_{1}\cdots{}a_{n})|{\cal
N}_{1}\\}\subset\\{P_{2}(a_{1}\cdots{}a_{n})|{\cal N}_{2}\\}.$ (33)

From Proposition S1, all the correlations of
$\\{P_{1}(a_{1}\cdots{}a_{n})|{\cal N}_{1}\\}$ satisfy any configuration
inequality which holds for $\\{P_{2}(a_{1}\cdots{}a_{n})|{\cal N}_{2}\\}$.
Hence, the Finner inequality of triangle network cannot be used to verify the
correlations derived from chain networks.

##  Appendix B Proof of Inequality (3)

Consider an $n$-partite long-distance chain network consisting of $n$ parties
$\textsf{A}_{1},\cdots,\textsf{A}_{n}$. Each pair of two parties
$\textsf{A}_{j}$ and $\textsf{A}_{j+1}$ shares one source $\lambda_{j}$,
$j=1,\cdots,n-1$. Our goal in this section is to prove the generalized version
of the inequality (3) as

$\displaystyle P({\bf a})$ $\displaystyle\leq$
$\displaystyle\min\\{\prod_{\textrm{even}\,i}p(a_{i})^{\frac{m-k}{m}}\prod_{\textrm{odd}\,j}p(a_{j})^{\frac{k}{m}},$
(34)
$\displaystyle\prod_{\textrm{odd}\,i}p(a_{i})^{\frac{m-k}{m}}\prod_{\textrm{even}\,j}p(a_{j})^{\frac{k}{m}}\\}$

for any $m$ and $k$ with $m\geq 2$ and $k\leq m-1$, where ${\bf
a}=(a_{1},\cdots{},a_{n})$, and $p(a_{i})$ denotes the marginal distribution
of the outcome $a_{i}$ for the party $\textsf{A}_{i}$, $i=1,\cdots,n$.

Proof of the inequality (34). We firstly prove the following inequality

$\displaystyle P({\bf a})$ $\displaystyle\leq$
$\displaystyle\prod_{\textrm{even}\,i}p(a_{i})^{\frac{m-k}{m}}\prod_{\textrm{odd}\,j}p(a_{j})^{\frac{k}{m}}$
(35)

for any $m\geq 2$. In fact, let
$(\Omega_{j},q_{j}(\lambda_{j}),\mu(\lambda_{j}))$ be the measurable space of
the source $\lambda_{j}$, where $q_{j}(\lambda_{j})$ denotes the probability
distribution of $\lambda_{j}$, and $\mu(\lambda_{j})$ denotes the measure of
$\lambda_{j}$, $j=1,\cdots,n-1$. From Eq.(1) we have

$\displaystyle P({\bf a})$ $\displaystyle=$
$\displaystyle\int_{\Omega_{1}\times\cdots\times\Omega_{n}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!p(a_{1}|\lambda_{1})\prod_{j=2}^{n-1}p(a_{j}|\lambda_{j-1},\lambda_{j})p(a_{n}|\lambda_{n-1})$
(38) $\displaystyle\times
q_{1}(\lambda_{1})\cdots{}q_{n-1}(\lambda_{n-1})d\mu(\lambda_{1})\cdots{}d\mu(\lambda_{n-1})$
$\displaystyle=$
$\displaystyle\int_{\Omega_{1}\times\cdots\times\Omega_{n}}\prod_{\textrm{even}\,i}p(a_{i}|\lambda_{i-1},\lambda_{i})q_{i-1}(\lambda_{i-1})^{\frac{m-k}{m}}$
$\displaystyle\times
q_{i}(\lambda_{i})^{\frac{m-k}{m}}\prod_{\textrm{odd}\,i}p(a_{j}|\lambda_{j-1},\lambda_{j})q_{j-1}(\lambda_{j-1})^{\frac{k}{m}}$
$\displaystyle\times{}q_{j}(\lambda_{j})^{\frac{k}{m}}d\mu(\lambda_{1})\cdots{}d\mu(\lambda_{n-1})$
$\displaystyle\leq$
$\displaystyle\prod_{\textrm{even}\,i}(\int_{\Omega_{i-1}\times\Omega_{i}}p(a_{i}|\lambda_{i-1},\lambda_{i})^{\frac{m-k}{m}}q_{i-1}(\lambda_{i-1})$
$\displaystyle\times{}q_{i}(\lambda_{i})d\mu(\lambda_{1})\cdots{}d\mu(\lambda_{n-1}))^{\frac{m-k}{m}}$
$\displaystyle\times\prod_{\textrm{odd}\,j}(\int_{\Omega_{j-1}\times\Omega_{j}}p(a_{j}|\lambda_{j-1},\lambda_{j})^{\frac{k}{m}}q_{j-1}(\lambda_{j-1})$
$\displaystyle\times{}q_{i}(\lambda_{i})d\mu(\lambda_{1})\cdots{}d\mu(\lambda_{n-1}))^{\frac{k}{m}}$
$\displaystyle=$
$\displaystyle\prod_{\textrm{even}\,i}(\int_{\Omega_{i-1}\times\Omega_{i}}p(a_{i}|\lambda_{i-1},\lambda_{i})q_{i-1}(\lambda_{i-1})$
$\displaystyle\times{}q_{i}(\lambda_{i})d\mu(\lambda_{1})\cdots{}d\mu(\lambda_{n-1}))^{\frac{m-k}{m}}$
$\displaystyle\times\prod_{\textrm{odd}\,j}(\int_{\Omega_{j-1}\times\Omega_{j}}p(a_{j}|\lambda_{j-1},\lambda_{j})q_{j-1}(\lambda_{j-1})$
$\displaystyle\times{}q_{i}(\lambda_{i})d\mu(\lambda_{1})\cdots{}d\mu(\lambda_{n-1}))^{\frac{k}{m}}$
$\displaystyle=$
$\displaystyle\prod_{\textrm{even}\,i}p(a_{i})^{\frac{m-1}{m}}\prod_{\textrm{odd}\,j}p(a_{j})^{\frac{k}{m}}$
(39)

In Eq.(38), $\lambda_{0}$ and $\lambda_{n}$ are two constants, and $m\geq 2$.
The inequality (38) follows from the Hölder inequality:

$\displaystyle\int{}fg{}dx\leq(\int{}f{}^{p}dx)^{\frac{1}{p}}(\int{}g{}^{q}dx)^{\frac{1}{q}}$
(40)

with positive constants $p$ and $q$ satisfying $\frac{1}{p}+\frac{1}{q}=1$.
Eq.(38) follows from the equalities:
$p(a_{j}|\lambda_{j-1},\lambda_{j})^{s_{j}}=p(a_{j}|\lambda_{j-1},\lambda_{j})\in\\{0,1\\}$
with positive constant $s_{j}$, where $p(a_{j}|\lambda_{j-1},\lambda_{j})$ is
indicator function, that is, $p(a_{j}|\lambda_{j-1},\lambda_{j})\in\\{0,1\\}$,
$j=1,\cdots,n$. Eq.(39) follows from the equalities:

$\displaystyle p(a_{j})$ $\displaystyle=$
$\displaystyle\int{}_{\Omega_{j-1}\times\Omega_{j}}p(a_{j}|\lambda_{j-1},\lambda_{j})q_{j-1}(\lambda_{j-1})$
(41)
$\displaystyle\times{}q_{j}(\lambda_{j})d\mu(\lambda_{j-1})d\mu(\lambda_{j})$

for $j=1,\cdots,n$.

Similarly, we can prove

$\displaystyle P({\bf a})$ $\displaystyle\leq$
$\displaystyle\prod_{\textrm{odd}\,i}p(a_{i})^{\frac{m-k}{m}}\prod_{\textrm{even}\,j}p(a_{j})^{\frac{k}{m}}$
(42)

for any $m\geq 2$ and $k\leq m-1$. This completes the proof.

##  Appendix C Verifying triangle network

In this section, we show the inequality (3) can be used to verify the
correlations available in triangle networks. Consider a triangle network
${\cal N}$ consisting of three independent sources with the probability
distributions

$\displaystyle P(x_{i},y_{i})=p_{i}[00]+q_{i}[11],i=1,2,3.$ (43)

Here, the party $\textsf{A}_{1}$ shares two different sources $x_{1}$ and
$y_{3}$, the party $\textsf{A}_{2}$ shares two different sources $y_{1}$ and
$x_{2}$, and the party $\textsf{A}_{3}$ shares two different sources $y_{2}$
and $x_{3}$. Let $a=x_{1}y_{3}$, $b=y_{1}x_{2}$ and $c=y_{2}x_{3}$. If each
party performs independently sampling on each source, the joint distribution
is given

$\displaystyle P(a,b,c)$ $\displaystyle=$ $\displaystyle
p_{1}p_{2}p_{3}[000]+p_{1}q_{2}p_{3}[012]+q_{1}p_{2}p_{3}[120]$ (44)
$\displaystyle+q_{1}q_{2}p_{3}[132]+p_{1}p_{2}q_{3}[201]+p_{1}q_{2}q_{3}[213]$
$\displaystyle+q_{1}p_{2}q_{3}[321]+q_{1}q_{2}q_{3}[333]$

where two bit series $xy$ are encoded into the four states with the basis
$\\{0,1,2,3\\}$. It follows that

$\displaystyle p_{a}(0)=p_{1}p_{3},p_{a}(1)=q_{1}p_{3},$ $\displaystyle
p_{a}(2)=p_{1}q_{3},p_{a}(3)=q_{1}q_{3},$ $\displaystyle
p_{b}(0)=p_{1}p_{2},p_{b}(1)=p_{1}q_{2},$ $\displaystyle
p_{b}(2)=q_{1}p_{2},p_{b}(3)=q_{1}q_{2},$ $\displaystyle
p_{c}(0)=p_{2}p_{3},p_{c}(1)=p_{2}q_{3},$ $\displaystyle
p_{c}(2)=q_{2}p_{3},p_{c}(3)=q_{2}q_{3}.$ (45)

From Eqs.(44) and (45) we obtain

$\displaystyle v_{000}$ $\displaystyle:=$ $\displaystyle
P(0,0,0)-p_{a}(0)^{\frac{m-1}{m}}p_{b}(0)^{\frac{1}{m}}p_{c}(0)^{\frac{m-1}{m}}$
(46) $\displaystyle=$ $\displaystyle
p_{1}p_{2}p_{3}-p_{1}p_{2}p_{3}^{\frac{2m-2}{m}}$ $\displaystyle>$
$\displaystyle 0$

for $m>2$ and $k=1$ and $p_{1},p_{2},p_{3}\in(0,1)$. The probability
distribution of $P(0,0,0)$ violates the inequality (3) for any
$p_{1},p_{2},p_{3}\in(0,1)$ and large $m$. However, it does not violate the
inequality (2).

##  Appendix D Fractional independent sets

Given an $n$-partite network ${\cal N}$ consisting of parties
$\textsf{A}_{1},\cdots,\textsf{A}_{n}$ who shares any independent sources
$\lambda_{1},\cdots,\lambda_{m}$ (classical variables or quantum states for
example). There is an undirected graph ${\cal G}=(V,E)$ associated with ${\cal
N}$, where the vertex $v_{i}$ in $V$ denotes one party $\textsf{A}_{i}$, the
edge $e_{ij}$ in $E$ connected with the vertexes $v_{i}$ and $v_{j}$ denotes
one bipartite source shared by the parties $\textsf{A}_{i}$ and
$\textsf{A}_{j}$, and the $s$-hyper edge $e_{i_{1}\cdots i_{s}}$ in $E$
connected with the vertexes $v_{i_{1}},\cdots,v_{i_{s}}$ denotes one
$s$-partite source shared by the parties
$\textsf{A}_{i_{1}},\cdots,\textsf{A}_{i_{s}}$. It is general difficult to
find optimal fractional independent sets for the graph ${\cal G}$ associated
with the given network ${\cal N}$. Here, we provide two efficient algorithms
for solving this problem regardless of the optimal solutions. The first one is
an application of bipartite graphs. The second is more adapt to network
configurations. Both algorithms can provide fractional independent sets for
each network even if they may be not optimal.

  *   * Input:

An undirected graph ${\cal G}=(V,E)$.

  * Output:

A fractional independent set $\textbf{s}=(s_{1},\cdots,s_{n})$

  * •

Set $x_{j}=\frac{1}{m_{j}}$ for each $m_{j}$-hyper edge $e_{j}\in E$;

  * •

Set $s_{i}=\min\\{s_{j}|j\to i\\}$;

  * •

Set $\textbf{s}=(s_{1},\cdots,s_{n})$.

Algorithm 1 Fractional independent set Figure 4: (Color online) 5-partite
network. $\textsf{A}_{1},\textsf{A}_{2}$ and $\textsf{A}_{3}$ share a triangle
network consisting of three bipartite sources. $\textsf{A}_{3},\textsf{A}_{4}$
and $\textsf{A}_{5}$ share a tripartite source corresponding to the $3$-hyper
edge in the graph.

For given an $n$-partite network consisting of $m$ sources, the total time
complexity is $O(mnN)$, where $N$ denotes the number of edges. It follows
Algorithm 1 has polynomial time-complexity in terms of sources and edges.

Example S1. Take the network shown in Fig.4 as example. There are five parties
$\textsf{A}_{1},\cdots,\textsf{A}_{5}$. $\textsf{A}_{1},\textsf{A}_{2}$ and
$\textsf{A}_{3}$ share a triangle network consisting of three bipartite
sources. $\textsf{A}_{3},\textsf{A}_{4}$ and $\textsf{A}_{5}$ share a
tripartite source associated with a hyper edge in graph. Here, we firstly set
$\frac{1}{2}$ for each edge while $\frac{1}{3}$ for the hyper edge. And then,
we update the covering number of $\textsf{A}_{3}$ as
$s_{3}=\min\\{\frac{1}{2},\frac{1}{3}\\}=\frac{1}{3}$, $s_{i}=\frac{1}{2}$ for
$\textsf{A}_{i}$ with $i=1,2$, and $s_{j}=\frac{1}{3}$ for $\textsf{A}_{j}$
with $j=4,5$. Finally, the fractional independent set
$\textbf{s}=(s_{1},\cdots,s_{6})$ is given by

$\displaystyle\textbf{s}=(\frac{1}{2},\frac{1}{2},\frac{1}{3},\frac{1}{3},\frac{1}{3})$
(47)

as shown in Fig.4(a).

  *   * Input

An undirected graph ${\cal G}=(V,E)$.

  * Output

A fractional independent set $\textbf{s}=(s_{1},\cdots,s_{n})$

  * •

Decompose ${\cal G}$ into a combination of different subgraphs ${\cal G}_{j}$,
that is, ${\cal G}=\cup_{j=2}^{t}{\cal G}_{j}$, where ${\cal G}_{2}$ denotes
the bipartite graph, and ${\cal G}_{j}$ denotes the $j$-hyper edge associated
with $j$-partite source;

  * •

Set a fractional independent set $\textbf{s}_{j}$ for the subgraph ${\cal
G}_{j}$;

  * •

Update $s_{i}$ for the $i$-th node as
$s_{i}=\min\\{s_{k}\in\textbf{s}_{j}|k\to i\\}$;

  * •

Set $\textbf{s}=(s_{1},\cdots,s_{n})$.

Algorithm 2 Fractional independent set

The main difference between Algorithms 1 and 2 is the covering set for each
subgraph. In Algorithm 2, one can choose different fractional independent sets
depending on the network configurations.

Example S2. For Algorithm 2, set $\frac{1}{2}$ for each edge of
$\textsf{A}_{1},\textsf{A}_{2}$ and $\textsf{A}_{3}$ in Fig.4 while
$\\{\frac{1}{2},\frac{1}{4},\frac{1}{4}\\}$ for three nodes of the hyper edge.
And then, we update the covering number as $s_{i}=\frac{1}{2}$ for $i=1,2,3$,
and $s_{j}=\frac{1}{4}$ for $j=4,5$. Finally, a fractional independent set
$\textbf{s}=(s_{1},\cdots,s_{6})$ is given by

$\displaystyle\textbf{s}=(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{4},\frac{1}{4})$
(48)

which is different from the fractional independent set given in Eq.(47).

For given an $n$-partite network consisting of $m$ sources, the total time
complexity of Algorithm 2 is $O(mLN)$, where $N$ denotes the number of edges
and $L$ denotes the maximal time complexity for setting fractional independent
set $\textbf{s}_{j}$. Note $L$ is upper bounded by some constant depending on
all the sub-graphs. This is because ${\cal G}_{j}\subset{\cal G}$. It follows
that Algorithm 2 has polynomial time-complexity in terms of sources and edges.

A more involved method depends on the connectedness of nodes. A general
principle is to set large fractional index for those nodes with large
connected edges. This may be completed by using the independent set Luo2018 .
The total time is polynomial in terms of sources and edges.

Figure 5: (Color online) An $n$-partite star network. The party
$\textsf{A}_{n}$ shares one independent bipartite source with
$\textsf{A}_{j}$, $j=1,\cdots,n-1$. Here, $m\geq 2$.

Example S3 (Star Network). Consider an $n$-partite star network, as shown in
Fig.5. Here, one party $\textsf{A}_{n}$ has $n-1$ connected edges while there
is one connected edge for each one $\textsf{A}_{j}$ of other parties,
$j=1,\cdots,n-1$. Hence, we can get $s_{n}=\frac{m-k}{m}$ for $\textsf{A}_{n}$
and $s_{j}=\frac{k}{m}$ for $\textsf{A}_{j}$, $j=1,\cdots,n-1$. The fractional
independent set $\textbf{s}=(s_{1},\cdots,s_{n-1},s_{n})$ is given by

$\displaystyle\textbf{s}=(\frac{k}{m},\cdots,\frac{k}{m},\frac{m-k}{m})$ (49)

with $m\geq 2$ and $k\leq m-1$, as shown in Fig.5(a). Another method is to get
the fractional independent set s as

$\displaystyle\textbf{s}=(\frac{m-k}{m},\cdots,\frac{m-k}{m},\frac{k}{m})$
(50)

as shown in Fig.5(b). Both fractional independent sets satisfy Definition 1.

Figure 6: (Color online) An $n$-partite chain network. Each pair of two
parties $\textsf{A}_{j}$ and $\textsf{A}_{j+1}$ shares one bipartite source,
$j=1,\cdots,n-1$. Here, $m\geq 2$ and $n$ is an even integer.

Example S4 (Chain network). Consider an $n$-partite chain network. Here, each
pair of two parties $\textsf{A}_{j}$ and $\textsf{A}_{j+1}$ shares one source,
$j=1,\cdots,n-1$. The set of
$\\{\textsf{A}_{2j}|j=1,\cdots,\lceil\frac{n}{2}\rceil\\}$ is an independent
set Luo2018 , that is, any pair of them have not shared sources. Hence, we can
get $s_{2j}=\frac{m-k}{m}$ for $\textsf{A}_{2j}$, and $s_{{2j+1}}=\frac{k}{m}$
for $\textsf{A}_{2j+1}$, $j=1,\cdots,\lceil\frac{n}{2}\rceil$, where
$\lceil{x}\rceil$ denotes the minimal integer no less than $x$, $m\geq 2$ and
$k\leq m-1$. The fractional independent set of
$\textbf{s}=(s_{1},s_{2},s_{3},\cdots,s_{{n}})$ is given by

$\displaystyle\textbf{s}=(\frac{k}{m},\frac{m-k}{m},\frac{k}{m},\cdots,\frac{m-k}{m})$
(51)

for an even integer $n$, as shown in Fig.6(a), and

$\displaystyle\textbf{s}=(\frac{m-k}{m},\frac{k}{m},\frac{m-k}{m},\cdots,\frac{k}{m},\frac{m-k}{m})$
(52)

for an odd integer $n$, as shown in Fig.6(b). Another method is to set
$s_{2j}=\frac{k}{m}$ for $\textsf{A}_{2j}$, and $s_{{2j+1}}=\frac{m-k}{m}$ for
$\textsf{A}_{2j+1}$, $j=1,\cdots,\lceil\frac{n}{2}\rceil$. The fractional
independent set of s is given by

$\displaystyle\textbf{s}=(\frac{m-k}{m},\frac{k}{m},\frac{m-k}{m},\cdots,\frac{k}{m})$
(53)

for an even integer $n$, as shown in Fig.6(b), and

$\displaystyle\textbf{s}=(\frac{m-k}{m},\frac{k}{m},\frac{m-k}{m},\cdots,\frac{k}{m},\frac{k}{m})$
(54)

for an integer odd $n$, where $2k\leq m$. Both fractional independent sets
satisfy Definition 1.

Figure 7: (Color online) An $n$-partite cyclic network. Each pair of two
parties $\textsf{A}_{j}$ and $\textsf{A}_{j+1}$ (or $\textsf{A}_{1}$ and
$\textsf{A}_{n}$ ) shares one bipartite source, $j=1,\cdots,n-1$. Here, $m\geq
2$ and $n$ is even integer.

Example S5 (Cyclic network). Consider an $n$-partite cyclic network, shown as
Fig.7. Here, each pair of two parties $\textsf{A}_{j}$ and $\textsf{A}_{j+1}$
(or $\textsf{A}_{1}$ and $\textsf{A}_{n}$ ) shares one bipartite source,
$j=1,\cdots,n-1$. Similar to $n$ chain network shown in Fig.6, we get the
fractional independent sets given in Eqs.(51)-(54).

Figure 8: (Color online) (a) An $n$-partite complete network. Each pair of two
parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$ shares one bipartite source,
$1\leq i\not=j\leq n$. (b) A $4$-partite complete network. Each three parties
$\textsf{A}_{i},\textsf{A}_{j},\textsf{A}_{k}$ shares one tripartite source,
$1\leq i\not=j\not=k\leq 4$. Here, $m\geq 2$.

Example S6 (Complete network). Consider an $n$-partite complete network
(associated with a complete hyper graph), shown in Fig.8, where each $m$
parties $\textsf{A}_{i_{1}},\cdots,\textsf{A}_{i_{m}}$ shares one source,
$1\leq i_{1},\cdots,i_{m}\leq n$. We set $s_{i}=\frac{1}{m}$ for each party
$\textsf{A}_{i}$. The fractional independent set of
$\textbf{s}=(s_{1},s_{2},\cdots,s_{{n}})$ is given by

$\displaystyle\textbf{s}=(\frac{1}{m},\cdots,\frac{1}{m})$ (55)

which satisfies Definition 1. For the special case of $m=2$, it reduces to the
Finner inequality Finner , as shown in Fig.8(a). Another example of $m=3$ is
shown in Fig.8(b).

##  Appendix E Proof of Theorem 1

Suppose $(\Omega_{i},{\cal A}_{i},\mu_{i}),i\in I_{n}=\\{1,\cdots,n\\}$ are
measure spaces. Let $(\Omega_{S},{\cal A}_{S},\mu_{S})$ be the product space
of $(\Omega_{i},{\cal A}_{i},\mu_{i})$ with $i\in S$, that is,
$\Omega_{S}=\times_{i\in S}\Omega_{i},{\cal A}_{S}=\otimes_{i\in S}{\cal
A}_{i}$ and $\mu_{S}=\otimes_{i\in S}\mu_{i}$ for any set $S\subset I_{n}$.
Let $L^{p}(\Omega_{S},{\cal A}_{S},\mu_{S})$ be the space consisting of
equivalence classes of measurable functions $f:\Omega_{S}\to\mathbb{R}$ such
that $\int_{\Omega_{S}}|f|^{p}d\mu<\infty$. Theorem 1 is formalized as

Theorem 1’. Let $m\in\mathbb{N}$, $M=\\{1,\cdots,m\\}$,
$\emptyset\not=S_{j}\subseteq{}I_{n}$, $M_{j}=\\{j\in M:i\in S_{j}\\}$ for
$i\in I_{n}$ such that $\sum_{j\in M_{i}}s_{ij}\leq 1$ for all $i\in I_{n}$,
and $0<s_{ij}<1$. Suppose $f_{j}\in L^{\frac{1}{q_{j}}}(\Omega_{S_{j}},{\cal
A}_{S_{j}},\mu_{S_{j}})$ with $q_{j}=\min_{j}\\{s_{t},t\in S_{j}\\}$ and
$|f_{j}|\leq 1$, $j\in M$. We have $\prod_{j}|f_{j}|\in
L^{1}(\Omega_{I_{n}},{\cal A}_{S_{n}},\mu_{S_{n}})$ and

$\displaystyle\int\prod_{j\in M}|f_{j}|d\mu_{I_{n}}\leq\prod_{j\in
M}(\int|f_{j}|^{\frac{1}{q_{j}}})^{q_{j}}$ (56)

Moreover, if $f_{j}\in\\{0,\pm 1\\}$, that is, $f_{j}$ being characteristic
function, we get

$\displaystyle\int\prod_{j\in M}|f_{j}|d\mu_{I_{n}}\leq\prod_{j\in
M}(\int|f_{j}|^{\frac{1}{q_{j}}})^{q_{j}}$ (57)

where $q_{j}=\min_{j}\\{s_{t},t\in S_{j}\\}$.

Proof. The proof is inspired by the Fubini Theorem Finner  and completed by
induction on $n$. For $n=1$, from the generalized Hölder inequality Finner  it
follows that

$\displaystyle\int\prod_{j\in M}|f_{j}|d\mu_{I_{n}}\leq\prod_{j\in
M}(\int|f_{j}|^{\frac{1}{p_{j}}})^{p_{j}}$ (58)

with $\sum_{j}p_{j}=1$ and $p_{j}\in(0,1)$. Combined with the norm inequality
of

$\displaystyle(\int|f_{j}|^{\frac{1}{s_{j}}}d\mu_{S_{j}})^{s_{j}}\leq(\int|f_{j}|^{\frac{1}{q_{j}}}d\mu_{S_{j}})^{q_{j}}$
(59)

with $q_{j}\leq s_{j}$, it follows the inequality (56).

Now, assume the inequality (56) holds for $n-1$ with $n\geq 2$. Let $i\in
I_{n}$, $J_{i}=I_{n}\backslash\\{i\\}$, $\emptyset\not=S_{i}\subseteq{}I_{n}$,
and $M_{i}=\\{i\in M:j\in S_{i}\\}$.

From the Fubini Theorem Finner  we get

$\displaystyle\int_{\Omega_{I_{n}}}\prod_{j\in M}|f_{j}|d\mu_{I_{n}}$
$\displaystyle=$
$\displaystyle\int_{\Omega_{J_{i}}}\int_{\Omega_{i}}\prod_{j\in
M}|f_{j}|d\mu_{i}d\mu_{J_{i}}$ (60) $\displaystyle=$
$\displaystyle\\!\\!\\!\int_{\Omega_{J_{i}}}\prod_{j\in
M\backslash{}M_{i}}\\!\\!\\!|f_{j}|(\int_{\Omega_{i}}\prod_{j\in
M_{i}}|f_{j}|d\mu_{i})d\mu_{J_{i}}$ $\displaystyle\leq$
$\displaystyle\int_{\Omega_{J_{i}}}\prod_{j\in
M\backslash{}M_{i}}|f_{j}|d\mu_{J_{i}}$
$\displaystyle\times(\int_{\Omega_{i}}\prod_{j\in
M_{i}}|f_{j}|^{\frac{1}{s_{ij}}}d\mu_{i})^{s_{ij}}$

which follows from the Hölder inequality. By decomposing $j\in M_{i}$ as $j\in
M_{i}:S_{j}=\\{i\\}$ and $t\in M_{i}:S_{t}\not=\\{i\\}$, it follows that

$\displaystyle\int_{\Omega_{I_{n}}}\prod_{j\in M}|f_{j}|d\mu_{I_{n}}$
$\displaystyle\leq$ $\displaystyle\prod_{j\in
M_{i}:\atop{S_{j}=\\{i\\}}}(\int_{\Omega_{i}}|f_{j}|^{\frac{1}{s_{ij}}}d\mu_{i})^{s_{ij}}$
$\displaystyle\times\int_{\Omega_{J_{i}}}\prod_{j\in
M\backslash{}M_{i}}|f_{j}|\prod_{t\in M_{i}:\atop{S_{t}\not=\\{i\\}}}$
$\displaystyle\times(\int_{\Omega_{i}}|f_{k}|^{\frac{1}{s_{it}}}d\mu_{i})^{s_{it}}d\mu_{J_{i}}$
$\displaystyle\leq$ $\displaystyle\prod_{j\in
M_{i}:\atop{S_{j}=\\{i\\}}}(\int_{\Omega_{i}}|f_{j}|^{\frac{1}{q_{j}}}d\mu_{i})^{q_{j}}$
$\displaystyle\times\int_{\Omega_{J_{i}}}\prod_{j\in
M\backslash{}M_{i}}|f_{j}|$ $\displaystyle\times\prod_{t\in
M_{i}:\atop{S_{t}\not=\\{i\\}}}(\int_{\Omega_{i}}|f_{k}|^{\frac{1}{q_{t}}}d\mu_{i})^{q_{t}}d\mu_{J_{i}}$

from the norm inequality (59). From the assumption of $n-1$, it follows from
the inequality (LABEL:E7)

$\displaystyle\int_{\Omega_{I_{n}}}\prod_{j\in M}|f_{j}|d\mu_{I_{n}}$
$\displaystyle\leq$ $\displaystyle\prod_{j\in
M_{i}:\atop{S_{j}=\\{i\\}}}(\int_{\Omega_{i}}|f_{j}|^{\frac{1}{q_{j}}}d\mu_{i})^{q_{j}}$
(62) $\displaystyle\times\prod_{j\in
M\backslash{}M_{i}}(\int_{\Omega_{S_{j}}}|f_{j}|^{\frac{1}{q_{j}}}d\mu_{S_{j}})^{q_{j}}$
$\displaystyle\times\\!\\!\\!\prod_{j\in
M_{i}:\atop{S_{j}\not=\\{i\\}}}(\int_{\Omega_{S_{j}\backslash\\{i\\}}}\\!\\!\\!\\!\\!\\!|f_{j}|^{\frac{1}{q_{j}}}d\mu_{i})^{q_{j}}d\mu_{S_{j}\backslash\\{i\\}}$
$\displaystyle=$ $\displaystyle\prod_{j\in
M}(\int_{\Omega_{S_{j}}}|f_{j}|^{\frac{1}{q_{j}}}d\mu_{S_{j}})^{q_{j}}$

This has completed the proof of the inequality (56). The inequality (57) is a
special case of the inequality (56). It completes the proof.

Different from the Finner inequality Finner , the necessary condition may be
not true because of the inequality $\sum_{j\in M_{i}}s_{ij}\leq 1$.

##  Appendix F Proof of Theorem 2

The generalized Finner inequality holds for all quantum networks. It can be
formalized as follows:

Theorem 2’(Quantum generalized Finner inequality). Let
$s=\\{s_{1},\cdots,s_{n}\\}$ be a fractional independent set of graph ${\cal
G}$ associated with quantum network ${\cal N}_{q}$. Let $f_{j}$ be any real
positive local post-processing (function) of the classical output (under the
local projections with computation basis) of the party $\textsf{A}_{j}$. Then,
any correlation $P$ achievable in ${\cal N}_{q}$ satisfies

$\displaystyle\mathbb{E}_{q}[\prod_{j=1}^{n}f_{j}]\leq\prod_{j}\|f_{j}\|_{\frac{1}{s_{j}}}$
(63)

where $\|f\|=\mathbb{E}[f^{1/s}]^{s}$ and the expectation
$\mathbb{E}(\cdot{})$ is with respect to $P$. In particular, if $f_{j}$ is the
indicator function of the output of $\mathsf{A}_{j}$ associated with the
outcome $a_{j}$ (under any projections with general orthogonal basis), we have

$\displaystyle
P_{q}(a_{1},\cdots,a_{n})\leq\prod_{j=1}^{n}P_{q}(a_{j})^{s_{j}}$ (64)

A well-known result for single source is given by

Lemma 1. Any correlation $P$ achievable in $n$-partite states can be generated
by $n$-partite variable.

Proof. Different from previous proof for general network consisting of
bipartite sources Marc2019a , we do not take use of special features such as
the convexity or extra point of fractional independent sets. Our goal is to
present a general method with less restriction. This is essential for
extensions in no-signalling model. In fact, for each $m$-partite state
$\rho_{j}$, we define a set of $\overline{s}_{j}:=\\{s_{i\to j}\\}$ for all
the parties $\textsf{A}_{i}$’s who share the state $\rho_{j}$, where $s_{i\to
j}\in[0,1]$ and satisfies $\sum_{i}s_{i\to j}\leq 1$. This set is further
regarded as a fractional independent set of one hyper-edge in the associated
graph of ${\cal N}_{q}$. Consider any set of ${\bf
s}=\cup_{j}\overline{s}_{j}=(s_{1},\cdots,s_{n})$ being a total fractional
independent set of the associated graph ${\cal G}$, where
$s_{j}=\min\\{s_{i\to j}\\}$.

Case 1. Proof of the inequality (63)

Similar to previous discussions Marc2019a , for $s_{j}=0$, we get

$\displaystyle\|f_{j}\|_{\frac{1}{s_{j}}}=\|f_{j}\|_{\infty}=\max_{a_{j}:P_{A_{j}}(a_{j})\not=0}f(a_{j})$
(65)

and

$\displaystyle\mathbb{E}_{q}[\prod_{i=1}^{n}f_{i}]\leq\mathbb{E}_{q}[\prod_{i\not=j}f_{i}]\|f_{j}\|_{\infty}$
(66)

If the generalized Finner inequality (63) holds by ignoring the $n$-th party
(for the joint distribution $P(a_{1},\cdots,a_{n})$), it also holds including
this party by putting $s_{j}=0$. Hence, we can restrict all the weights to be
$\textbf{s}=(s_{1},\cdots,s_{n})$ such that $s_{j}\in(0,1]$ for all $j$’s.

Moreover, for $s_{j}=1$, the $j$-th party cannot share any source with others.
Here, we get

$\displaystyle\mathbb{E}_{q}[\prod_{i=1}^{n}f_{i}]\leq\mathbb{E}_{q}[\prod_{i\not=j}f_{j}]\mathbb{E}_{q}[f_{j}]$
(67)

where $f_{j}$ satisfies
$\|f_{j}\|_{\frac{1}{s_{j}}}=\|f_{j}\|=\mathbb{E}_{q}[f_{j}]$. As a result,
all the parties with weights being equal to 1 can be ignored.

To sum up, assume $\textbf{s}=(s_{1},\cdots,s_{n})$ such that $s_{j}\in(0,1)$
for all $j$’s. In this case, it is sufficient to prove

$\displaystyle\mathbb{E}_{q}[\prod_{j=1}^{n}f_{j}]\leq\prod_{j}\|f_{j}\|_{\frac{1}{s_{j}}}$
(68)

Note $f_{j}$ is an arbitrary positive function on the output of the $j$-th
party $\textsf{A}_{j}$. Composing the projection measurement operators of
$\textsf{A}_{j}$ with this classical post-processing, assume $f_{j}$ is the
outcome of some quantum observable $X_{j}$ that the $j$-th party applies on
quantum systems in her hand.

Suppose $X_{j}$ is decomposed into the superposition of complete orthogonal
projection operators. In this case, we assume

$\displaystyle X_{j}=\sum_{a_{j}}f_{j}(a_{j})M_{a_{j}}$ (69)

where $\\{M_{a_{j}}=|a_{j}\rangle\langle a_{j}|\\}$ are orthogonal projection
operators satisfying
$M_{a_{j}=x_{1}}M_{a_{j}=x_{2}}=\delta_{x_{1}x_{2}}M_{a_{j}=x_{1}}$ with the
delta function $\delta_{x_{1}x_{2}}$, that is, $\delta_{x_{1}x_{2}}=1$ for
$x_{1}=x_{2}$ and $\delta_{x_{1}x_{2}}=0$ for $x_{1}\not=x_{2}$.

From the Born rule, we get

$\displaystyle\mathbb{E}_{q}[\prod_{j=1}^{n}f_{j}]={\rm
tr}[\otimes_{j}X_{j}\rho]$ (70)

where $\rho=\otimes_{i}\rho_{i}$ and $\rho_{i}$ are pure states. Otherwise,
consider the purification $|\Phi\rangle_{A_{1}\cdots A_{n};B}$ of $\rho$ with
auxiliary particle $B$. It is easy to check

$\displaystyle\mathbb{E}_{q}[\prod_{j=1}^{n}f_{j}]={\rm
tr}[\mathbbm{1}_{B}\otimes(\otimes_{j=1}^{n}X_{j})|\Phi\rangle\langle\Phi|]$
(71)

where $\mathbbm{1}_{B}$ denotes the identity operator on the particle $B$.

From Eqs.(70) and (71), we get

$\displaystyle\mathbb{E}_{q}[\prod_{j=1}^{n}f_{j}]$ $\displaystyle=$
$\displaystyle{\rm
tr}[\otimes_{j=1}^{n}\sum_{a_{j}}f_{j}(a_{j})|a_{j}\rangle\langle a_{j}|\rho]$
(72) $\displaystyle=$ $\displaystyle{\rm tr}[\sum_{{\bf
a}}f_{1}(a_{1})\cdots{}f_{n}(a_{n})\otimes_{j=1}^{n}|a_{j}\rangle\langle{}a_{j}|\rho]$
$\displaystyle=$ $\displaystyle\sum_{{\bf a}}\gamma_{{\bf
a}}f_{1}(a_{1})\cdots{}f_{n}(a_{n})$ $\displaystyle\leq$
$\displaystyle\mathbb{E}_{c}[f_{1}\cdots{}f_{n}]$ (73) $\displaystyle\leq$
$\displaystyle\prod_{j=1}^{n}\|f_{j}\|_{\frac{1}{s_{j}}}$ (74)

In Eq.(72), we have $\gamma_{{\bf a}}={\rm
tr}[\otimes_{j=1}^{n}|a_{j}\rangle\langle{}a_{j}|\rho]$. From Eq.(69), it
follows $\sum_{i_{1}\cdots i_{n}}\gamma_{i_{1}\cdots{}i_{n}}=1$. It means
$\\{\gamma_{i_{1}\cdots{}i_{n}}\\}$ can be regarded as a classical probability
distribution $\\{P_{c}({\bf a})\\}_{{\bf a}=i_{1}\cdots{}i_{n}}$ with
$P_{c}({\bf a})=\gamma_{{\bf a}}^{2}$ for any $a_{1},\cdots,a_{n}$. Note the
local projection measurement
$\\{\otimes_{j=1}^{n}|a_{j}\rangle\langle{}a_{j}|\\}$ does not change the
network configuration of ${\cal N}_{q}$. From Lemma 1, any correlation
achievable in single state can be generated by one classical variable.
Moreover, $f_{j}(a_{j})\geq 0$. These facts imply the correlation of
$\gamma_{{\bf a}}$ is achievable in classical network ${\cal N}$ with the same
network configuration of ${\cal N}_{q}$. This implies the inequality (73) from
Theorem 1’. $\mathbb{E}_{c}[f_{1}\cdots{}f_{n}]$ in the inequality (73)
denotes the classical expect of the positive function $f_{1}\cdots{}f_{n}$
associated with the classical probability distribution $\\{P_{c}({\bf a})\\}$.
The equality follows if all the orthogonal bases of $\\{|a_{j}\rangle\langle
a_{j}|,\forall j\\}$ are complete. This completes the proof of the inequality
(63).

Figure 9: (Color online) Schematic procedure for generating the distributions
in Eqs.(81)-(84).

Case 2. Proof of the inequality (64) with local projection measurements under
the computation basis.

In this case, we have

$\displaystyle P(a_{1},\cdots,a_{n})={\rm
tr}[\otimes_{j=1}^{n}|a_{j}\rangle\langle a_{j}|\rho],$ (75) $\displaystyle
P(a_{j})={\rm tr}[|a_{j}\rangle\langle
a_{j}|\otimes\mathbbm{1}_{\overline{j}}\rho],j=1,\cdots,n$ (76)

when $X_{j}=|a_{j}\rangle\langle a_{j}|$, where $\mathbbm{1}_{\overline{j}}$
denotes the identity operator performed on all the systems not owned by the
party $\textsf{A}_{j}$, $j=1,\cdots,n$. From the inequality (74), it follows
that

$\displaystyle
P_{q}(a_{1},\cdots,a_{n})\leq\prod_{j=1}^{n}P_{q}(a_{j})^{s_{j}}$ (77)

This has proved the inequality (64) for local projection measurements.

Case 3. Proof of the inequality (64) with local projection measurements.

In what follows, we prove the inequality (64) holds for any local joint
projection measurements $\\{|\phi_{a_{j}}\rangle=U_{A_{i}}|a_{j}\rangle\\}$
with the computation basis $\\{|a_{j}\rangle\\}$. For given an $n$-partite
quantum network ${\cal N}_{q}$ shared by parties
$\textsf{A}_{1},\cdots,\textsf{A}_{n}$, suppose the total state is given by
$\rho_{A_{1}\cdots A_{n}}$, where the party $\textsf{A}_{j}$ own the system
$A_{j}$, $j=1,\cdots,n$. From Born rule, the joint probability is given by

$\displaystyle P(a_{1},\cdots,a_{n})={\rm
tr}[\otimes_{j=1}^{n}|\phi_{a_{j}}\rangle\langle\phi_{a_{j}}|\rho]$ (78)

From Bayes’ Rule, it follows that

$\displaystyle
P(a_{1},\cdots,a_{n})=p(a_{1})p(a_{2}|a_{1})\cdots{}p(a_{n}|a_{1}\cdots{}a_{n-1})$
(79)

Hence, for given a distribution $\\{P(a_{1},\cdots,a_{n})\\}$, we get $n$
distributions given by

$\displaystyle P_{a_{j}}=\\{p(a_{j}|a_{1}\cdots{}a_{j-1})\\},j=1,\cdots,n$
(80)

where $a_{0}=1$. The main idea is to generate the distributions $P_{a_{j}}$’s
by performing local projection measurements on ${\cal N}_{q}$ assisted by some
additional states for each party. So, the network configuration is unchanged.
The proof is completed by induction on the Bayes’ Rule in Eq.(79), shown as
Fig.9.

  * •

Firstly, we generate the distribution of $\\{p(a_{1})\\}$. From the assumption
of the local measurement, we get

$\displaystyle p(a_{1})$ $\displaystyle=$ $\displaystyle{\rm
tr}[|\phi_{a_{1}}\rangle\langle\phi_{a_{1}}|\rho_{A_{1}}]$ (81)
$\displaystyle=$ $\displaystyle{\rm tr}[|a_{1}\rangle\langle
a_{1}|\hat{\rho}_{A_{1}}]$

where $\hat{\rho}_{A_{1}}=U_{A_{1}}\rho_{A_{1}}U_{A_{1}}^{\dagger}$ and
$\rho_{A_{1}}$ is the reduced density matrix of the system $A_{1}$. After this
local measurement, the $n$-partite network ${\cal N}_{q}$ is changed into
$n-1$-partite new network ${\cal N}_{q}^{(n-1)}$, where each party of
$\textsf{A}_{2},\cdots,\textsf{A}_{n}$ may perform local unitary operation
only on its own particles which are entangled with the particles owned by the
party $\textsf{A}_{1}$ conditional on the outcome $a_{1}$. These operations do
not change the configuration of the reduced $n-1$-partite network after the
local measurement of $\textsf{A}_{1}$ being performed.

  * •

Secondly, the party $\textsf{A}_{2}$ performs the local measurement under the
basis $\\{|\phi_{a_{2}}\rangle\\}$, where its reduced density matrix is
denoted by $\rho_{A_{2}}$. Similar to Eq.(81), we get

$\displaystyle p(a_{2}|a_{1})$ $\displaystyle=$ $\displaystyle{\rm
tr}[|\phi_{a_{2}}\rangle\langle\phi_{a_{2}}|\rho_{A_{2}}]$ (82)
$\displaystyle=$ $\displaystyle{\rm tr}[|a_{2}\rangle\langle
a_{2}|\hat{\rho}_{A_{2}}]$

where $\hat{\rho}_{A_{2}}=U_{A_{2}}\rho_{A_{2}}U_{A_{2}}^{\dagger}$. After
this local measurement, the $n-1$-partite network ${\cal N}_{q}^{(n-1)}$ is
changed into an $n-2$-partite new network ${\cal N}_{q}^{(n-2)}$, where each
party of $\textsf{A}_{3},\cdots,\textsf{A}_{n}$ may perform local unitary
operation only on its own particles which are entangled with the particles
owned by the party $\textsf{A}_{2}$ conditional on the outcome $a_{2}$. These
operations do not change the configuration of the reduced $n-2$-partite
network after the local measurement of $\textsf{A}_{2}$ being performed.

  * •

This procedure can be iteratively performed. Take the party $\textsf{A}_{k}$
for example. After the party $\textsf{A}_{k-1}$ performs the local measurement
under the basis $\\{|\phi_{a_{k-1}}\rangle\\}$, the party $\textsf{A}_{k}$
performs the local measurement under the basis $\\{|\phi_{a_{2}}\rangle\\}$,
where its reduced density matrix is denoted by $\rho_{A_{k}}$. Similar to
Eq.(81), we get

$\displaystyle p(a_{k}|a_{1}\cdots{}a_{k-1})$ $\displaystyle=$
$\displaystyle{\rm tr}[|\phi_{a_{k}}\rangle\langle\phi_{a_{k}}|\rho_{A_{k}}]$
(83) $\displaystyle=$ $\displaystyle{\rm tr}[|a_{k}\rangle\langle
a_{k}|\hat{\rho}_{A_{k}}]$

where $\hat{\rho}_{A_{k}}=U_{A_{k}}\rho_{A_{k}}U_{A_{k}}^{\dagger}$. After
this local measurement, the $n-1$-partite network ${\cal N}_{q}^{(n-k+1)}$ is
changed into an $n-k$-partite new network ${\cal N}_{q}^{(n-k)}$, where each
party of $\textsf{A}_{k+1},\cdots,\textsf{A}_{n}$ may perform local unitary
operation on its own particles which are entangled with the particles owned by
$\textsf{A}_{k}$ conditional on the outcome $a_{k}$. These operations do not
change the configuration of the reduced $n-k$-partite network after the local
measurement of $\textsf{A}_{k}$ being performed.

  * •

The procedure is completed after the party $\textsf{A}_{n}$ performs the local
measurement under the basis $\\{|\phi_{a_{n}}\rangle\\}$ in order to get

$\displaystyle p(a_{n}|a_{1}\cdots{}a_{n-1})$ $\displaystyle=$
$\displaystyle{\rm tr}[|\phi_{a_{n}}\rangle\langle\phi_{a_{n}}|\rho_{A_{n}}]$
(84) $\displaystyle=$ $\displaystyle{\rm tr}[|a_{n}\rangle\langle
a_{n}|\hat{\rho}_{A_{n}}]$

where $\hat{\rho}_{A_{n}}=U_{A_{n}}\rho_{A_{n}}U_{A_{n}}^{\dagger}$.

Figure 10: (Color online) Schematic procedure for generating the distributions
in Eq.(85).

So far, we have generated the joint distribution of
$\\{P(a_{1},\cdots,a_{n})\\}$ by rebuilding $n$ distributions
$p(a_{1}),\cdots,p(a_{n}|a_{1}\cdots{}a_{n-1})$ given in Eqs.(81)-(84).

Now, suppose each party $\textsf{A}_{k}$ has an additional system
$A_{k}^{\prime}$ in the state $\hat{\rho}$ defined in Eq.(83). A new $n$-party
network ${\cal N}_{q}^{\prime}$ is defined by considering all the systems of
$A_{1}A_{1}^{\prime},\cdots,A_{n}A_{n}^{\prime}$, where each party
$\textsf{A}_{k}$ owns two systems of $A_{k}$ and $A_{k}^{\prime}$. The
configuration of ${\cal N}_{q}^{\prime}$ is same as its of the original
network ${\cal N}_{q}$ without additional systems. In what follows, we
construct new distributions of $q(a_{1}),\cdots,q(a_{n}|a_{1}\cdots{}a_{n-1})$
by performing local projection measurement on ${\cal N}_{q}^{\prime}$ and
post-processing of measurement outcomes such that

$\displaystyle
q(a_{k}|a_{1}\cdots{}a_{k-1})=p(a_{k}|a_{1}\cdots{}a_{k-1}),j=1,\cdots,n$ (85)

The rebuilding procedure is shown in Fig.10. The details are given as follows.

  * •

Firstly, we generate the distribution of $\\{q(a_{1})\\}$. The party
$\textsf{A}_{1}$ performs the local projection measurement under the
computation basis $\\{|a_{1}\rangle_{A_{1}^{\prime}}|i_{1}\rangle_{A_{1}}\\}$
on its own systems $A_{1}^{\prime}A_{1}$, where $A_{1}^{\prime}$ is an
axillary system in the state $\hat{\rho}_{A_{1}^{\prime}}$ defined in Eq.(81).
We get

$\displaystyle q(a_{1})$ $\displaystyle=$ $\displaystyle\sum_{i_{1}}{\rm
tr}[|a_{1}\rangle\langle a_{1}|_{A_{1}^{\prime}}\otimes|i_{1}\rangle\langle
i_{1}|_{A_{1}}(\hat{\rho}_{A_{1}^{\prime}}\otimes\rho_{A_{1}})]$ (86)
$\displaystyle=$ $\displaystyle{\rm tr}[|a_{1}\rangle\langle
a_{1}|\hat{\rho}_{A_{1}}]$ $\displaystyle=$ $\displaystyle p(a_{1})$

where $\rho_{A_{1}}$ is the reduced density matrix of the system $A_{1}$.
Here, the summation can be regarded as a linearly post-processing of
measurement outcomes. After this local measurement, the $n$-partite network
${\cal N}_{q}$ is changed into an $n-1$-partite network ${\cal N}_{n-1}$,
where each party of $\textsf{A}_{2},\cdots,\textsf{A}_{n}$ may perform local
unitary operation only on its own particles that are entangled with the
particles owned by the party $\textsf{A}_{1}$ conditional on the outcome
$a_{1}$. These operations do not change the configuration of the reduced
$n-1$-partite network after the local measurement of the party
$\textsf{A}_{1}$ being performed.

  * •

Secondly, the party $\textsf{A}_{2}$ performs the local measurement under the
basis $\\{|a_{2}\rangle_{A_{2}^{\prime}}|i_{1}\rangle_{A_{2}}\\}$ on own
systems $A_{2}^{\prime}$ and $A_{2}$, where $A_{2}^{\prime}$ is an axillary
system in the state $\hat{\rho}_{A_{2}^{\prime}}$ defined in Eq.(82). Similar
to Eq.(86), we get

$\displaystyle q(a_{2}|a_{1})$ $\displaystyle=$ $\displaystyle\sum_{i_{2}}{\rm
tr}[|a_{2}\rangle\langle a_{2}|_{A_{2}^{\prime}}\otimes|i_{2}\rangle\langle
i_{2}|_{A_{2}}(\hat{\rho}_{A_{2}^{\prime}}\otimes\rho_{A_{2}})]$ (87)
$\displaystyle=$ $\displaystyle{\rm tr}[|a_{2}\rangle\langle
a_{2}|\hat{\rho}_{A_{2}}]$ $\displaystyle=$ $\displaystyle p(a_{2}|a_{1})$

where $\rho_{A_{2}}$ is the reduced density matrix of $A_{2}$. After this
local measurement, the $n-1$-partite network ${\cal N}_{n-1}$ is changed into
$n-2$-partite new network ${\cal N}_{n-2}$, where each party of
$\textsf{A}_{3},\cdots,\textsf{A}_{n}$ may perform local unitary operation on
its own particles that are entangled with the particles owned by the party
$\textsf{A}_{2}$ conditional on the outcome $a_{2}$. These operations do not
change the configuration of the reduced $n-2$-partite network after the local
measurement of $\textsf{A}_{2}$ being performed.

  * •

This procedure is iteratively performed. After the party $\textsf{A}_{k-1}$
performs the local measurement, the party $\textsf{A}_{k}$ performs the local
measurement under the basis
$\\{|a_{k}\rangle_{A_{k}^{\prime}}|i_{k}\rangle_{A_{k}}\\}$ on the systems
$A_{k}^{\prime}$ and $A_{k}$, where $A_{k}^{\prime}$ is an auxiliary system in
the state $\hat{\rho}_{A_{k}^{\prime}}$ defined in Eq.(83). So, we get

$\displaystyle q(a_{k}|a_{1}\cdots{}a_{k-1})$ (88) $\displaystyle=$
$\displaystyle\sum_{i_{k}}{\rm tr}[|a_{k}\rangle\langle
a_{k}|_{A_{k}^{\prime}}\otimes|i_{k}\rangle\langle
i_{k}|_{A_{k}}(\hat{\rho}_{A_{k}^{\prime}}\otimes\rho_{A_{k}})]$
$\displaystyle=$ $\displaystyle{\rm tr}[|a_{k}\rangle\langle
a_{k}|\hat{\rho}_{A_{k}}]$ $\displaystyle=$ $\displaystyle
p(a_{k}|a_{1}\cdots{}a_{k-1})$

where $\rho_{A_{k}}$ is the reduced density matrix of $A_{k}$. After this
local measurement, the $n-k+1$-partite network ${\cal N}_{n-k+1}$ is changed
into an $n-k$-partite new network ${\cal N}_{n-k}$, where each party of
$\textsf{A}_{k+1},\cdots,\textsf{A}_{n}$ may perform local unitary operation
only on its own particles that are entangled with the particles owned by
$\textsf{A}_{k}$ conditional on the outcome $a_{k}$. These operations do not
change the configuration of the reduced $n-k$-partite network after the local
measurement of $\textsf{A}_{k}$ being performed. The procedure is completed
after the party $\textsf{A}_{n}$ performs the local measurement under the
basis $\\{|a_{n}\rangle_{A_{n}^{\prime}}|i_{n}\rangle_{A_{n}}\\}$ on the
systems $A_{n}^{\prime}$ and $A_{n}$.

From Eqs.(86)-(88), we can generate all the distributions
$\\{p(a_{k}|a_{1}\cdots{}a_{k-1})\\}$ for $k=1,\cdots,n$ by performing local
projection measurements under the computation basis on the extended network
${\cal N}^{\prime}_{q}$. From Eq.(79), it means the joint distribution
$\\{p(a_{1},\cdots{}a_{n})\\}$ is rebuilt by performing local projection
measurements under the computation basis on the extended network ${\cal
N}^{\prime}_{q}$ with the same configuration as ${\cal N}_{q}$. Hence, from
Case 2 stated above, the distribution of $\\{p(a_{1},\cdots{}a_{n})\\}$ should
satisfy the inequality (77). This has completed the proof. $\Box$

Figure 11: (Color online) Tripartite star quantum network. The upper sub-
figure is used to generate the joint distribution $P(a_{1},a_{2},a_{3})$ by
performing local joint projection measurements. The lower sub-figure is used
to rebuild $P(a_{1},a_{2},a_{3})$ by performing local projection measurements
with the same measurement order in the upper sub-figure.

Examples S7. Consider a tripartite star quantum network shown in Fig.11, where
each pair of two adjacent parties $\textsf{A}_{i}$ and $\textsf{A}_{i+1}$
shares one bipartite entanglement, $i=1,2$. The upper sub-figure is used to
generate the joint distribution $P(a_{1},a_{2},a_{3})$ from Eq.(79) using the
measurement order from $\textsf{A}_{1}$, $\textsf{A}_{2}$ to $\textsf{A}_{3}$.
$p(a_{1})$ and $p(a_{3}|a_{1}a_{2})$ are completed by local projection
measurements. $p(a_{2}|a_{1})$ is completed by the party $\textsf{A}_{2}$. One
method is to perform local joint projection measurement under the basis
$\\{U|k,s\rangle,\forall k,s\\}$ (such as Bell basis) with any unitary matrix
$U$. The other is firstly to perform the unitary operation $U$, and then
performs local projection measurements. The lower sub-figure is used to
rebuild $P(a_{1},a_{2},a_{3})$ from Eq.(85) with the same measurement order in
the upper sub-figure. $q(a_{1})=p(a_{1})$ is completed by local projection
measurements, where the local operations of $\textsf{A}_{1}$ do not change the
network configuration. $p(a_{2}|a_{1})$ is completed by the party
$\textsf{A}_{2}$. From Eq.(82), $\textsf{A}_{2}$ firstly adds one additional
particle $A_{2}^{\prime}$ in the state $\hat{\rho}_{A_{2}^{\prime}}$
corresponding to the reduced state of $\textsf{A}_{2}$ in the upper sub-
figure. And then, he performs local projection measurement and post-processing
from Eq.(87). Similarly, $\textsf{A}_{3}$ can get
$q(a_{3}|a_{1}a_{2})=p(a_{3}|a_{1}a_{2})$ without changing of network
configuration from Eq.(88). It means the quantum Finner inequalities hold for
correlations from star networks under any local joint measurements.

Another method is analytically to verify the inequality (34) for any given
local measurements such as Bell basis. Especially, consider a tripartite
network consisting of two generalized EPR states EPR  given by
$|\Phi\rangle_{12}\otimes|\Phi\rangle_{34}$ with
$|\Phi\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$,
$\theta\in(0,\frac{\pi}{2})$. The parties $\textsf{A}_{1}$ and
$\textsf{A}_{3}$ perform projection measurement under the computation basis
$\\{|0\rangle,|1\rangle\\}$. The party $\textsf{A}_{2}$ performs joint
measurement under the Bell basis
$\\{|0\rangle:=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle),|1\rangle:=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle),|2\rangle:=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle),|3\rangle:=\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle)\\}$.
The joint probability $P(a_{1},a_{2},a_{3})$ of the measurement outcome
$a_{1}a_{2}a_{3}$ is given by

$\displaystyle P(0,0,0)$ $\displaystyle=$ $\displaystyle
P(0,3,0)=\frac{1}{2}\cos^{4}\theta$ $\displaystyle P(0,1,1)$ $\displaystyle=$
$\displaystyle P(0,2,1)=P(1,1,0)=P(1,2,0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\cos^{2}\theta\sin^{2}\theta$ $\displaystyle
P(1,0,1)$ $\displaystyle=$ $\displaystyle P(1,3,1)=\frac{1}{2}\sin^{4}\theta$
(89)

From Eq.(89), it follows that

$\displaystyle
P(0,0,0)-(p_{a_{1}}(0)p_{a_{3}}(0))^{\frac{m-1}{m}}p_{a_{2}}(0)^{\frac{1}{m}}$
(90) $\displaystyle=$
$\displaystyle\frac{1}{2}\cos^{4}\theta-\frac{1}{2^{\frac{1}{m}}}\cos^{\frac{4m-4}{m}}\theta(\cos^{4}\theta+\sin^{4}\theta)^{\frac{1}{m}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\cos^{4}\theta-\frac{1}{2^{\frac{2}{m}}}\cos^{\frac{4m-4}{m}}\theta$
$\displaystyle\leq$ $\displaystyle 0$

for any $m\geq 2$ and $\theta\in(0,\frac{\pi}{2})$, where we have used the
inequality:
$\cos^{4}\theta+\sin^{4}\theta\geq\frac{1}{2}(\cos^{2}\theta+\sin^{2}\theta)^{2}=\frac{1}{2}$.
Similar results hold for $P(0,3,0),P(1,0,1)$ and $P(1,3,1)$.

Moreover, we have

$\displaystyle
P(0,1,1)-(p_{a_{1}}(0)p_{a_{3}}(1))^{\frac{m-1}{m}}p_{a_{2}}(1)^{\frac{1}{m}}$
(91) $\displaystyle=$
$\displaystyle\frac{1}{2}\cos^{2}\theta\sin^{2}\theta-\frac{1}{2^{\frac{1}{m}}}\cos^{2}\theta\sin^{2}\theta$
$\displaystyle\leq$ $\displaystyle 0$

for any $m\geq 2$ and $\theta\in(0,\frac{\pi}{2})$. Similar results hold for
$P(0,2,1),P(1,1,0)$ and $P(1,2,0)$. So, we have proved the distribution
$P(a_{1},a_{2},a_{3})$ satisfies the Finner inequality:
$P(a_{1},a_{2},a_{3})\leq(p({a_{1}})p(a_{3}))^{\frac{m-1}{m}}p(a_{2})^{\frac{1}{m}}$
for any $m\geq 2$. Similar proof holds for the inequality of
$P(a_{1},a_{2},a_{3})\leq(p({a_{1}})p(a_{3}))^{\frac{1}{m}}p(a_{2})^{\frac{m-1}{m}}$.
It follows that the distribution derived from local joint projection
measurement on star network satisfies the Finner inequality (34). For general
measurements one can verify the inequality (2) by numeric evaluations.

Figure 12: (Color online) Triangle quantum network. The upper sub-figure is
used to generate the joint distribution $P(a_{1},a_{2},a_{3})$ by performing
local joint projection measurements. The lower sub-figure is used to rebuild
$P(a_{1},a_{2},a_{3})$ by performing local projection measurements with the
same measurement order in the upper sub-figure.

Examples S8. Consider a tripartite cyclic quantum network shown in Fig.12,
where each pair of two parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$ shares
one bipartite entanglement, $1\leq i\not=j\leq 3$. The upper sub-figure is
used to generate the joint distribution $P(a_{1},a_{2},a_{3})$ by performing
local joint projection measurements. From Eq.(79), the measurement is
completed by using the measurement order from $\textsf{A}_{1}$,
$\textsf{A}_{2}$ to $\textsf{A}_{3}$. One method is to perform local joint
projection measurement under the basis $\\{U_{i}|k,s\rangle,\forall k,s\\}$
(such as Bell basis) with any unitary matrix $U_{i}$. The other is firstly to
perform the unitary operation $U_{i}$, and then performs local projection
measurements. The lower sub-figure is used to rebuild $P(a_{1},a_{2},a_{3})$
with the same measurement order in the upper sub-figure. $p(a_{1})$ is
completed by the party $\textsf{A}_{1}$. From Eq.(81), $\textsf{A}_{1}$
firstly adds one additional particle $A_{1}^{\prime}$ in the state
$\hat{\rho}_{A_{1}^{\prime}}$ corresponding to the reduced state of
$\textsf{A}_{1}$ in the upper sub-figure. And then, he performs local
projection measurement and post-processing from Eq.(86).
$q(a_{2}|a_{1})=p(a_{2}|a_{1})$ is completed by the party $\textsf{A}_{2}$.
From Eq.(82), $\textsf{A}_{2}$ firstly adds one additional particle
$A_{2}^{\prime}$ in the state $\hat{\rho}_{A_{2}^{\prime}}$ corresponding to
the reduced state of $\textsf{A}_{2}$ in the upper sub-figure. And then, he
performs local projection measurement and post-processing from Eq.(87).
Similarly, $\textsf{A}_{3}$ can get $q(a_{3}|a_{1}a_{2})=p(a_{3}|a_{1}a_{2})$
without changing the network configuration. The quantum generalized Finner
inequality holds for correlations under any local joint measurements on
triangle networks.

Another method is analytically to verify the inequality (2) for any given
local measurements. Especially, consider a triangle network consisting of
three generalized EPR states EPR  given by
$|\Phi\rangle_{12}\otimes|\Phi\rangle_{34}|\Phi\rangle_{56}$, where
$|\Phi\rangle$ is defined in Example S7. Each party $\textsf{A}_{i}$ performs
joint measurement Marc2019a  under the basis
$\\{|0\rangle:=|00\rangle,|1\rangle:=\cos\gamma|01\rangle+\sin\gamma|10\rangle,|2\rangle:=\sin\gamma|01\rangle-\cos\gamma|10\rangle,|3\rangle:=|11\rangle\\}$
with $\gamma\in(0,\frac{\pi}{2})$. The joint probability
$P(a_{1},a_{2},a_{3})$ for the measurement outcome $a_{1}a_{2}a_{3}$ is given
by

$\displaystyle P(0,0,0)$ $\displaystyle=$ $\displaystyle\cos^{6}\theta$
$\displaystyle P(1,1,0)$ $\displaystyle=$ $\displaystyle
P(0,2,1)=\sin^{4}\gamma\cos^{4}\theta\sin^{2}\theta$ $\displaystyle P(1,2,0)$
$\displaystyle=$ $\displaystyle P(2,1,0)=P(0,2,2)=P(2,0,1)$ $\displaystyle=$
$\displaystyle P(0,1,1)=P(1,0,2)$ $\displaystyle=$
$\displaystyle\sin^{2}\gamma\cos^{2}\gamma\cos^{4}\theta\sin^{2}\theta$
$\displaystyle P(2,2,0)$ $\displaystyle=$ $\displaystyle
P(0,1,2)=P(1,0,1)=\cos^{4}\gamma\cos^{4}\theta\sin^{2}\theta$ $\displaystyle
P(1,3,1)$ $\displaystyle=$ $\displaystyle P(2,0,2)=P(3,1,2)=P(2,2,3)$
$\displaystyle=$ $\displaystyle\sin^{4}\gamma\cos^{2}\theta\sin^{4}\theta$
$\displaystyle P(2,3,1)$ $\displaystyle=$ $\displaystyle
P(2,1,3)=P(3,1,1)=P(1,3,2)$ $\displaystyle=$ $\displaystyle P(3,2,2)=P(1,2,3)$
$\displaystyle=$
$\displaystyle\sin^{2}\gamma\cos^{2}\gamma\cos^{2}\theta\sin^{4}\theta$
$\displaystyle P(3,2,1)$ $\displaystyle=$ $\displaystyle
P(2,3,2)=P(1,1,3)=\cos^{4}\gamma\cos^{2}\theta\sin^{4}\theta$ $\displaystyle
P(3,3,3)$ $\displaystyle=$ $\displaystyle\sin^{6}\theta$ (92)

From Eq.(92), it follows that $p_{a_{i}}(0)=\cos^{4}\theta$,
$p_{a_{i}}(1)=p_{a_{i}}(2)=\sin^{2}\theta\cos^{2}\theta$, and
$p_{a_{i}}(3)=\sin^{4}\theta$ (independent of $\gamma$) for $i=1,2,3$. So, we
get

$\displaystyle P(0,0,0)-\sqrt{p_{a_{1}}(0)p_{a_{2}}(0)p_{a_{3}}(0)}=0$ (93)

Similarly, we can prove the inequality (2) for the distribution
$P(a_{1},a_{2},a_{3})$. So, the distribution derived from local joint
projection measurements on triangle network satisfies the Finner inequality
(2). For general measurements, we can verify the inequality (2) by numeric
evaluations (code available for requirement).

Note 1. From Theorem 2’, the quantum generalized Finner inequality holds for
local joint measurement. In entanglement swapping experiment, such as
tripartite star network shown in Fig.11, the middle party can perform joint
measurement under Bell basis. This is equivalent to local projection
measurement after a two-particle entangling operation. The quantum generalized
Finner inequality holds for new networks even if this local joint operation
can change the network configuration. It means the quantum generalized Finner
inequality provides correlation invariance under local unitary operations.

Note 2. Theorem 2’ does not contradict to the recent result Marc2019a  in
which authors provide quantum correlations beyond classical correlations in a
triangle network consisting of three generalized EPR states EPR . The possible
reason is the quantum Finner inequalities do not imply all the facets of
correlations from the networks. Hence, an interesting problem is to explore
network nonlocality.

##  Appendix G $n$-partite chain quantum networks

Consider an $n$-partite chain network ${\cal N}_{q}$, as shown in Fig.6,
consisting of $n$ generalized EPR states EPR  of

$\displaystyle|\phi_{i}\rangle=\cos\theta_{i}|00\rangle+\sin\theta_{i}|11\rangle$
(94)

where the parties $\textsf{A}_{i}$ and $\textsf{A}_{i+1}$ share
$|\phi_{i}\rangle$, $i=1,\cdots,n-1$. After the projection measurement being
performed by each party $\textsf{A}_{1}$ and $\textsf{A}_{n}$, and Bell basis
$\\{|0\rangle:=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle),|1\rangle:=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle),|2\rangle:=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle),|3\rangle:=\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle)\\}$
for $\textsf{A}_{j}$ with $j=2,\cdots,n-1$, the joint distribution of the
outcome ${\bf 0}$ is given by

$\displaystyle P_{ch}({\bf
0})=\frac{\prod_{i=1}^{n}\cos^{2}\theta_{i}}{2^{n-2}}[{\bf 0}]$ (95)

It is easy to evaluate

$\displaystyle p_{a_{1}}(0)=\cos^{2}\theta_{1},$ $\displaystyle
p_{a_{n}}(0)=\cos^{2}\theta_{n},$ $\displaystyle
p_{a_{i}}(0)=\frac{1}{2}\cos^{2}\theta_{i}\cos^{2}\theta_{i+1},i=2,\cdots,n-1$
(96)

From Eqs.(95) and (96), it follows that

$\displaystyle v_{{\bf 0}}$ $\displaystyle:=$ $\displaystyle P_{ch}({\bf
0})-\prod_{i=1}^{n}p_{a_{i}}(0)^{s_{i}}$ (97) $\displaystyle=$
$\displaystyle\frac{1}{2^{n-2}}\prod_{i=1}^{n}\cos^{2}\theta_{i}-\frac{1}{2^{\sum_{i=2}^{n-1}s_{i}}}$
$\displaystyle\times\cos^{2s_{1}}\theta_{1}\cos^{2s_{n}}\theta_{n}\prod_{i=2}^{n-1}\cos^{2s_{i}+2s_{i+1}}\theta_{i}$
$\displaystyle\leq$ $\displaystyle 0$

for any $s_{i}$ and $\theta_{1},\cdots,\theta_{n}\in(0,\frac{\pi}{2})$, where
we have used the inequalities: $\sum_{i=2}^{n-1}s_{i}\leq 1$, $s_{1},s_{n}\leq
1$, and $s_{i}+s_{i+1}\leq 1$, $i=2,\cdots,n-1$. Similar results hold for
other local measurements. This implies the inequality (10) is the
configuration inequality for chain networks.

##  Appendix H $n$-partite star quantum networks

Consider an $n$-partite star network ${\cal N}_{q}$, as shown in Fig.5,
consisting of $n$ generalized EPR states EPR  given in Eq.(94), where the
parties $\textsf{A}_{i}$ and $\textsf{A}_{n}$ share $|\phi_{i}\rangle$,
$i=1,\cdots,n-1$. Under the projection measurement for each party
$\textsf{A}_{i}$ with $i=1,\cdots,n-1$, and joint measurement with the
$n-1$-partite Bell basis
$\\{|i\rangle:=\frac{1}{\sqrt{2}}(|i\rangle+|2^{n-1}-1-i\rangle),|2^{n-2}+i\rangle:=\frac{1}{\sqrt{2}}(|i\rangle+|2^{n-1}-1-i\rangle),i=0,\cdots,2^{n-2}\\}$,
the joint distribution of the outcome ${\bf 0}$ is given by

$\displaystyle P_{st}({\bf
0})=\frac{1}{2^{n-1}}\prod_{i=1}^{n}\cos^{2}\theta_{i}[{\bf 0}]$ (98)

It is easy to evaluate

$\displaystyle p_{a_{i}}(0)=\cos^{2}\theta_{i},i=1,\cdots,n-1$ $\displaystyle
p_{a_{n}}(0)=\frac{1}{2^{n-1}}\prod_{i=1}^{n-1}\cos^{2}\theta_{i}$ (99)

From Eqs.(98) and (99), it follows that

$\displaystyle v_{{\bf 0}}$ $\displaystyle:=$ $\displaystyle P_{st}({\bf
0})-\prod_{i=1}^{n}p_{a_{i}}(0)^{s_{i}}$ (100) $\displaystyle=$
$\displaystyle\frac{1}{2^{n-1}}\prod_{i=1}^{n}\cos^{2}\theta_{i}-\frac{1}{2^{ns_{n}-s_{n}}}\prod_{i=1}^{n-1}\cos^{2s_{i}}\theta_{i}\cos^{2s_{n}}\theta_{n}$
$\displaystyle\leq$ $\displaystyle 0$

for any $s_{i}$ and $\theta_{1},\cdots,\theta_{n}\in(0,\frac{\pi}{2})$, where
we have used the inequalities: $ns_{n}-s_{n}\leq n-1$, $s_{i}\leq 1$,
$i=1,\cdots,n$. Similar results hold for other measurements. It follows that
the inequality (10) is the configuration inequality for star networks.

##  Appendix I Verifying the configuration of quantum networks

In this section, we firstly present a general algorithm to verify the
configuration of quantum networks in the first subsection. And then, we prove
the correlations available in any networks consisting of bipartite entangled
pure states and GHZ states violate specific configuration inequality in the
second subsection. This provides a general method to construct different
networks with incompatible configuration inequalities. Moreover, we provide
some special networks including tripartite single-source networks, chain
networks and 4-party network in the last subsection.

###  I.1 Algorithm for verifying the configuration of quantum networks

Given an $n$-partite quantum network ${\cal N}_{q}$ consisting of any
multipartite entangled states, the configuration compatibility problem is
related to test the configuration of ${\cal N}_{q}$ by using observed
correlations.

The configuration compatibility problem. Given a correlation $P({\bf a})$, the
configuration compatibility problem is to determine whether $P({\bf a})$ is
compatible with the a given quantum network ${\cal N}_{q}$, that is, whether
$P({\bf a})$ can be generated by specific local measurements on ${\cal
N}_{q}$.

Theorem 2 is useful for solving the configuration compatibility problem by
testing the inequality (9) with the given correlation $P({\bf a})$. This may
be further applied to verify quantum networks. Unfortunately, it is impossible
to verify the specific configuration of general quantum networks with
polynomial-time complexity because there are exponential numbers of different
configurations for given a network. Our goal here is to provide a procedure to
complete this task regardless of time complexity. It may be useful for small-
size networks. Another application is for configuration compatibility problem.

  * Input:

An $n$-partite network ${\cal N}_{q}$.

  * Output:

Configuration of network.

  * •

Consider another $n$-partite network ${\cal N}_{q}^{\prime}$.

  * •

Find a fractional independent set $\textbf{s}=(s_{1},\cdots,s_{n})$ for ${\cal
N}_{q}^{\prime}$.

  * •

Obtain the configuration inequality from Theorem 2 for ${\cal
N}_{q}^{\prime}$.

  * •

Get the joint distribution $P({\bf a})$ for ${\cal N}_{q}$.

  * •

Verify whether $P({\bf a})$ violates the present configuration inequality or
not.

Algorithm 3 Verifying network configuration

This algorithm provides a general method to test the configuration of given
networks. It depends on the proposed configuration inequality in Theorem 2.
Take the network ${\cal N}_{q}^{\prime}$ in Fig.14(b) as an example. Different
from Case 1, all the parties $\textsf{A}_{k+1},\cdots,\textsf{A}_{n}$ in
Fig.14(b) can be regarded as one party B. In this case, from Eq.(49) and
Theorem 2 we get the following configuration inequality

$\displaystyle P({\bf
a})\leq\prod_{i=1}^{k}p(a_{i})^{\frac{1}{m}}p(a_{k+1},\cdots,a_{n})^{\frac{m-1}{m}}$
(101)

Figure 13: (Color online) (a) Triangle network consisting of three generalized
EPR states. (b) $4$-partite network consisting of two generalized GHZ states.

###  I.2 Any networks consisting of bipartite entangled pure states and GHZ
states

In this subsection, we prove for an $n$-partite network ${\cal N}_{q}$
consisting of any bipartite entangled pure states EPR  or generalized GHZ
states GHZ , its joint distribution can violate specific configuration
inequality associated with other networks.

Proposition S2. There exists an $n$-partite network ${\cal N}_{q}^{\prime}$
such that the joint distribution $P({\bf 0})$ achievable in ${\cal N}_{q}$
violates the configuration inequality of ${\cal N}_{q}^{\prime}$.

Proof of Corollary 3. Consider an $n$-partite quantum network ${\cal N}_{q}$
consisting of any bipartite entangled pure states EPR  or generalized GHZ
states GHZ  as: $|\Phi_{1}\rangle,\cdots,|\Phi_{m}\rangle$. Assume each party
$\textsf{A}_{j}$ owns some particles contained in entangled systems
$|\Phi_{j_{1}}\rangle,\cdots,|\Phi_{j_{s_{j}}}\rangle$. Note ${\cal N}_{q}$
may be disconnected, that is, consisting of some disconnected networks. All
the particles owned by one party can be regarded as one high-dimensional
state. For example, two qubits with basis $\\{|00\rangle,\cdots,|11\rangle\\}$
can be regarded as $4$-dimensional system with basis
$\\{|0\rangle,\cdots,|3\rangle\\}$. Here, each entanglement $|\Phi_{j}\rangle$
can be locally transformed into

$\displaystyle|\Phi_{j}\rangle=\sqrt{q_{j}}|0\rangle^{\otimes\ell_{j}}+\sum_{i\not=0}\alpha_{ij}|i\rangle^{\otimes\ell_{j}}$
(102)

For simplicity, assume $|\Phi_{j}\rangle$ is shared by $\ell_{j}$ parties.
Otherwise, all the particles in $|\Phi_{j}\rangle$ owned by one party are
regarded as one high-dimensional system. Each party performs projective
measurement under the computation basis $\\{|0\rangle,\cdots,|d-1\rangle\\}$
for a $d$-dimensional system. It is easy to evaluate the joint distribution of
$P({\bf 0})$ given by

$\displaystyle P({\bf 0})=\prod_{j=1}^{m}q_{j}$ (103)

In what follows, we need to find specific network ${\cal N}_{q}^{\prime}$ and
its fractional independent set ${\bf s}$. The main idea is inspired by the
$k$-independent set Luo2018 . Especially, for each network ${\cal N}_{q}$, we
find a $k$-independent set $\\{\textsf{A}_{1},\cdots,\textsf{A}_{k}\\}$ (for
example) with polynomial time complexity Luo2018 , where each party
$\textsf{A}_{i}$ ($1\leq i\leq k$) does not share any entanglement with other
party $\textsf{A}_{j}$ ($1\leq j\not=i\leq k$). Different from the result
Luo2018 , $k$ may be any integer, that is, $k\geq 1$.

From Eq.(102), for each party $\textsf{A}_{j}$ the probability for the outcome
$a_{j}=0$ is given by

$\displaystyle p_{a_{j}}(0)=\prod_{t=1}^{s_{j}}q_{j_{t}}$ (104)

where $q_{j_{t}}$ is derived from the local measurement on
$|0\rangle^{\otimes\ell_{j_{t}}}$ of $|\Phi_{j_{t}}\rangle$.

Define a new network ${\cal N}_{q}^{\prime}$ consisting of all the bipartite
entangled states, as shown in Fig.14. Especially, for $k\leq\frac{n}{2}$, that
is, $k\leq n-k$, ${\cal N}_{q}^{\prime}$ is a star network shown in Fig.14(a),
where all the parties $\textsf{A}_{k+1},\cdots,\textsf{A}_{n}$ are
independent. For $k>\frac{n}{2}$, that is, $k>n-k$, ${\cal N}_{q}^{\prime}$ is
another star network shown in Fig.14(b), where all the parties
$\textsf{A}_{1},\cdots,\textsf{A}_{k}$ are independent.

Figure 14: (Color online) A star network. The parties
$\textsf{A}_{k+1},\cdots,\textsf{A}_{n}$ are independent, and all the parties
$\textsf{A}_{1},\cdots,\textsf{A}_{k}$ are regarded as one party B. (a)
$k\leq\frac{n}{2}$. (b) $k>\frac{n}{2}$. The parties
$\textsf{A}_{1},\cdots,\textsf{A}_{k}$ are independent, and all the parties
$\textsf{A}_{k+1},\cdots,\textsf{A}_{n}$ are regarded as one party B.

From Eq.(49), define ${\bf s}$ as follows

$\displaystyle{\bf s}$ $\displaystyle=$
$\displaystyle({\footnotesize{\underbrace{\frac{1}{m},\cdots,\frac{1}{m}}\atop
k},{\underbrace{\frac{m-1}{m},\cdots,\frac{m-1}{m}}\atop{n-k}}}),k\leq\frac{n}{2}$
(105) $\displaystyle{\bf s}$ $\displaystyle=$
$\displaystyle({\footnotesize{\underbrace{\frac{m-1}{m},\cdots,\frac{m-1}{m}}\atop
k},{\underbrace{\frac{1}{m},\cdots,\frac{1}{m}}\atop n-k}}),k>\frac{n}{2}$
(106)

Consider the network in Fig.14(a). For each bipartite source shared by two
parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$ in
$\\{\textsf{A}_{1},\cdots,\textsf{A}_{k}\\}$, we have $s_{i}+s_{j}\leq 1$.
Moreover, for each bipartite source shared by two parties
$\textsf{A}_{i},\textsf{A}_{j}$ with
$\textsf{A}_{i}\in\\{\textsf{A}_{1},\cdots,\textsf{A}_{k}\\}$ and
$\textsf{A}_{j}\in\\{\textsf{A}_{k+1},\cdots,\textsf{A}_{n}\\}$, we have
$s_{i}+s_{j}=1$. It means ${\bf s}$ in Eq.(105) is the fractional independent
set of the network in Fig.14(a). Similarly, we can prove ${\bf s}$ in Eq.(106)
is the fractional independent set of the network in Fig.14(b). In
applications, ${\cal N}_{q}^{\prime}$ may be consisted of multipartite
entangled states.

From Theorem 2, the correlation $P({\bf a})$ derived from local measurements
on the network ${\cal N}_{q}^{\prime}$ satisfies the configuration inequality

$\displaystyle P({\bf
a})\leq\prod_{i=1}^{k}p(a_{i})^{\frac{1}{m}}\prod_{j=k+1}^{n}p(a_{j})^{\frac{m-1}{m}},\mbox{
if }k\leq\frac{n}{2},$ (107) $\displaystyle P({\bf
a})\leq\prod_{i=1}^{k}p(a_{i})^{\frac{m-1}{m}}\prod_{j=k+1}^{n}p(a_{j})^{\frac{1}{m}},\mbox{
if }k>\frac{n}{2}.$ (108)

Define four sets as follows:

  * •

${S}_{1}$ consists of all the bipartite entangled states shared by one party
in the independent set $\\{\textsf{A}_{1},\cdots,\textsf{A}_{k}\\}$ and one
party not in the independent set.

  * •

${S}_{2}$ consists of all the bipartite entangled states shared by two parties
not in the independent set.

  * •

$S_{3}$ denotes all the multipartite entangled states shared by one party in
the the independent set $\\{\textsf{A}_{1},\cdots,\textsf{A}_{k}\\}$ and other
parties not in the independent set.

  * •

$S_{4}$ denotes all the multipartite entangled states shared by all parties
not in the independent set.

It is easy to prove $S_{1},S_{2},S_{3}$ and $S_{4}$ consist of a four-partite
partition of all entangled states $|\Phi_{1}\rangle,\cdots,|\Phi_{m}\rangle$.

In what follows, we prove the joint distribution $P({\bf 0})$ in Eq.(103)
violates the inequality (107) or (108). For $k\leq\frac{n}{2}$, from Eqs.(103)
and (104) we have

$\displaystyle v_{{\bf 0}}$ $\displaystyle:=$ $\displaystyle P({\bf
0})-\prod_{i=1}^{k}p_{a_{i}}(0)^{\frac{1}{m}}\prod_{j=k+1}^{n}p_{a_{j}}(0)^{\frac{m-1}{m}}$
$\displaystyle=$
$\displaystyle\prod_{j=1}^{m}q_{j}-\prod_{i=1}^{k}(\prod_{t=1}^{s_{i}}q_{i_{t}})^{\frac{1}{m}}\prod_{j=k+1}^{n}(\prod_{t=1}^{s_{j}}q_{j_{t}})^{\frac{m-1}{m}}$
$\displaystyle=$ $\displaystyle\prod_{j=1}^{m}q_{j}-\prod_{i\in
S_{1}}q_{i}\prod_{i\in S_{2}}q_{i}^{\frac{2m-2}{m}}$
$\displaystyle\times\prod_{j\in{}S_{3}}q_{j}^{1+\frac{\ell_{j}-2}{m}}\prod_{j\in{}S_{4}}q_{j}^{\frac{\ell_{j}(m-1)}{m}}$
$\displaystyle=$ $\displaystyle\prod_{j=1}^{m}q_{j}(1-\prod_{i\in
S_{2}}q_{i}^{\frac{m-2}{m}}$
$\displaystyle\times\prod_{j\in{}S_{3}}q_{j}^{\frac{\ell_{j}-2}{m}}\prod_{j\in{}S_{4}}q_{j}^{\frac{\ell_{j}m-m-\ell_{j}}{m}})$
$\displaystyle>$ $\displaystyle 0$ (110)

In Eq.(I.2), for any entangled state $|\Phi_{i}\rangle\in S_{1}$ shared by two
parties, we have $q_{i}=q_{i}^{\frac{1}{m}}q_{i}^{\frac{m-1}{m}}$, where one
weight is $\frac{1}{m}$ and other is $1-\frac{1}{m}$. For any entangled state
$|\Phi_{i}\rangle\in S_{2}$ shared by two parties, we have
$q_{i}^{\frac{2m-2}{m}}=q_{i}^{\frac{m-1}{m}}q_{i}^{\frac{m-1}{m}}$, where
both weights are $1-\frac{1}{m}$. For any entangled state $|\Phi_{j}\rangle\in
S_{3}$ shared by $\ell_{j}$ parties, we get
$q_{j}^{1+\frac{\ell_{j}-2}{m}}=q_{j}^{\frac{m-1}{m}}q_{j}^{\frac{1}{m}\times(\ell_{j}-1)}$,
where one weight is $1-\frac{1}{m}$ and all the others are $\frac{1}{m}$. For
any entangled state $|\Phi_{j}\rangle\in S_{4}$ shared by $\ell_{j}$ parties,
it follows
$q_{j}^{\frac{\ell_{j}(m-1)}{m}}=q_{j}^{\frac{m-1}{m}\times\ell_{j}}$, where
all the weights are $1-\frac{1}{m}$. The inequality (110) follows from the
inequalities: $\prod_{i\in S_{2}}q_{i}^{\frac{m-2}{m}}\leq 1$,
$\prod_{j\in{}S_{3}}q_{j}^{\frac{\ell_{j}-2}{m}}<1$,
$\prod_{j\in{}S_{4}}q_{j}^{\frac{\ell_{j}m-m-\ell_{j}}{m}}<1$ for large $m\geq
3$. From the inequality (110), it follows the joint distribution $P({\bf 0})$
in Eq.(103) violates the inequality (107).

Similarly, we can prove the joint distribution $P({\bf 0})$ in Eq.(103)
violates the inequality (108) for $k\leq\frac{n}{2}$. This completes the
proof. $\Box$

###  I.3 Some examples

Example S9 (Tripartite single-source network). Consider a tripartite network
shown in Fig.1. For the given correlation of $P(a,b,c)$ from local
measurements on single-source network such as generalized GHZ state GHZ  or W
state Dur . The goal in this subsection is to verify $P(a,b,c)$ violates some
configuration inequalities compatible with the tripartite networks consisting
of two sources or three sources, that is, chain networks or triangle networks
consisting of three generalized EPR states EPR .

Note for a single-source network, from Corollary 1 the correlation should
satisfy

$\displaystyle P({\bf a})\leq\prod_{i=1}^{3}p(a_{i})^{s_{i}}$ (111)

with $\sum_{i=1}^{3}s_{i}\leq 1$.

Firstly, consider a chain network ${\cal N}_{q}$ consisting of two generalized
EPR states (112), where the parties $\textsf{A}_{1}$ and $\textsf{A}_{2}$
share $|\phi_{1}\rangle$ while the parties $\textsf{A}_{2}$ and
$\textsf{A}_{3}$ share $|\phi_{1}\rangle$. From Appendix G, the inequality
(111) is compatible with the chain network.

In what follows, consider a triangle network ${\cal N}_{q}$, as shown in
Fig.13(a), consisting of three generalized EPR states EPR  of

$\displaystyle|\phi_{i}\rangle=\cos\theta_{i}|00\rangle+\sin\theta_{i}|11\rangle$
(112)

where each party shares two qubits from two EPR states, and
$\theta_{i}\in(0,\frac{\pi}{2})$. After the projection measurement under the
computation basis
$\\{|0\rangle:=|00\rangle,|1\rangle:=|01\rangle,|2\rangle:=|10\rangle,|3\rangle:=|11\rangle\\}$,
the joint distribution is given by Eq.(44) with $p_{i}=\cos^{2}\theta_{i}$ and
$q_{i}=\sin^{2}\theta_{i}$, $i=1,2,3$. From Eqs.(44) and (45), it follows that

$\displaystyle v_{000}$ $\displaystyle:=$ $\displaystyle
P_{tri}(0,0,0)-p_{a}(0)^{s_{1}}p_{b}(0)^{s_{2}}p_{c}(0)^{s_{3}}$ (113)
$\displaystyle=$ $\displaystyle
p_{1}p_{2}p_{3}-p_{1}^{s_{1}+s_{2}}p_{2}^{s_{2}+s_{3}}p_{3}^{s_{1}+s_{3}}$
$\displaystyle<$ $\displaystyle 0$

for any $s_{i}$ with $s_{i}\not=0$, and
$\theta_{1},\theta_{2},\theta_{3}\in(0,\frac{\pi}{2})$, where $s_{i}+s_{j}\leq
1$. The distribution $P_{tri}(0,0,0)$ satisfies the inequality (111).

Similar result holds for general two-qubit joint measurement. Take the Bell
basis
$\\{|0\rangle:=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle),|1\rangle:=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle),|2\rangle:=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle),|3\rangle:=\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle)\\}$
as an example. The joint distribution is given

$\displaystyle P_{tri}({\bf
0})=\frac{1}{8}(\prod_{i=1}^{3}\cos\theta_{i}+\prod_{i=1}^{3}\sin\theta_{i})^{2}[000]$
(114)

It is easy to evaluate

$\displaystyle p_{a_{1}}(0)=\frac{1}{2}\cos^{2}\theta_{1}\cos^{2}\theta_{3},$
(115) $\displaystyle
p_{a_{2}}(0)=\frac{1}{2}\cos^{2}\theta_{1}\cos^{2}\theta_{2},$ (116)
$\displaystyle p_{a_{3}}(0)=\frac{1}{2}\cos^{2}\theta_{2}\cos^{2}\theta_{3}.$
(117)

From Eqs.(114) and (117), it follows that

$\displaystyle v_{000}$ $\displaystyle:=$ $\displaystyle
P_{tri}(0,0,0)-\prod_{i=1}^{3}p_{a_{i}}(0)^{s_{i}}$ (118) $\displaystyle=$
$\displaystyle\frac{1}{8}(\prod_{i=1}^{3}\cos\theta_{i}+\prod_{i=1}^{3}\sin\theta_{i})^{2}$
$\displaystyle-\frac{1}{4}\cos^{2s_{1}+2s_{2}}\theta_{1}\cos^{2s_{2}+2s_{3}}\theta_{2}\cos^{2s_{1}+2s_{3}}\theta_{3}$
$\displaystyle<$
$\displaystyle\frac{1}{8}(\prod_{i=1}^{3}\cos\theta_{i}+\prod_{i=1}^{3}\sin\theta_{i})^{2}$
$\displaystyle-\frac{1}{4}\cos^{2}\theta_{1}\cos^{2}\theta_{2}\cos^{2}\theta_{3}$
$\displaystyle<$
$\displaystyle\frac{1}{8}\prod_{i=1}^{3}\sin^{2}\theta_{i}+\frac{1}{4}\prod_{i=1}^{3}\sin\theta_{i}\cos\theta_{i}-\frac{1}{8}\prod_{i=1}^{3}\cos^{2}\theta_{i}$
$\displaystyle<$ $\displaystyle 0$

for any $s_{i}$ with $s_{i}\not=0$, and
$\theta_{1},\theta_{2},\theta_{3}\in(\frac{\pi}{4},\frac{\pi}{2})$. The
distribution $P_{tri}(0,0,0)$ satisfies the inequality (111). Similar results
hold for other distributions.

To sum up, the inequality (111) should be satisfied by all networks.

Example S10 (Chain network). Consider a tripartite network shown in Fig.1. For
the given correlation $P(a,b,c)$ from local measurements on the chain network
consisting of two sources. The goal in this subsection is to verify $P(a,b,c)$
violates some configuration inequalities compatible with the tripartite
network consisting of one source or three sources, that is, generalized GHZ
states GHZ  or W states Dur , or triangle network consisting of three
generalized EPR states EPR . Actually, this can be completed by using the
inequality (3).

It is easy to verify correlations from local measurements on the generalized
GHZ states $|ghz\rangle=\cos\theta|000\rangle+\sin\theta|111\rangle$ with
$\theta\in(0,\frac{\pi}{2})$ violates the inequality (3). Consider a
generalized W state Dur  given by

$\displaystyle|w\rangle$ $\displaystyle=$
$\displaystyle\cos\theta\cos\gamma|001\rangle+\sin\theta\cos\gamma|010\rangle$
(119) $\displaystyle+\sin\gamma|100\rangle$

where $\theta,\gamma\in(0,\frac{\pi}{2})$. By performing the projection
measurement under the basis $\\{|0\rangle,|1\rangle\\}$ for each party, the
joint distribution is given by

$\displaystyle P_{w}(a,b,c)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta\cos^{2}\gamma[001]+\sin^{2}\theta\cos^{2}\gamma[010]$
(120) $\displaystyle+\sin^{2}\gamma[100]$

It follows that

$\displaystyle p_{a}(0)$ $\displaystyle=$
$\displaystyle\cos^{2}\gamma,p_{a}(1)=\sin^{2}\gamma$ $\displaystyle p_{b}(0)$
$\displaystyle=$ $\displaystyle 1-\sin^{2}\theta\cos^{2}\gamma$ $\displaystyle
p_{b}(1)$ $\displaystyle=$ $\displaystyle\sin^{2}\theta\cos^{2}\gamma$
$\displaystyle p_{c}(0)$ $\displaystyle=$ $\displaystyle
1-\cos^{2}\theta\cos^{2}\gamma$ $\displaystyle p_{a}(1)$ $\displaystyle=$
$\displaystyle\cos^{2}\theta\cos^{2}\gamma$ (121)

From Eqs.(120) and (121), we get

$\displaystyle v_{001}$ $\displaystyle:=$ $\displaystyle
P_{w}(0,0,1)-p_{a}(0)^{\frac{m-1}{m}}p_{b}(0)^{\frac{1}{m}}p_{c}(1)^{\frac{m-1}{m}}$
(122) $\displaystyle=$
$\displaystyle\cos^{2}\theta\cos^{2}\gamma-\cos^{\frac{2m-2}{m}}\gamma(1-\sin^{2}\theta\cos^{2}\gamma)^{\frac{1}{m}}$
$\displaystyle\times\cos^{\frac{2m-2}{m}}\theta\cos^{\frac{2m-2}{m}}\gamma$
$\displaystyle>$
$\displaystyle\cos^{2}\theta\cos^{2}\gamma-\cos^{\frac{2m-2}{m}}\theta\cos^{\frac{4m-4}{m}}\gamma$
$\displaystyle>$ $\displaystyle 0$ (123)

if $\theta$ and $\gamma$ satisfy the following inequality

$\displaystyle|\cos\theta|<|\cos\gamma|^{m-2}$ (124)

where the inequality (123) follows the inequality of
$1-\sin^{2}\theta\cos^{2}\gamma<1$ for $\theta,\gamma\in(0,\frac{\pi}{2})$.
The distribution $P_{w}(0,0,1)$ in Eq.(120) violates the configuration
inequality (107) if the inequality (124) holds.

Moreover, from Eqs.(120) and (121) we get

$\displaystyle v_{010}$ $\displaystyle:=$ $\displaystyle
P_{w}(0,1,0)-p_{a}(0)^{\frac{m-1}{m}}p_{b}(1)^{\frac{1}{m}}p_{c}(0)^{\frac{m-1}{m}}$
(125) $\displaystyle=$
$\displaystyle\sin^{2}\theta\cos^{2}\gamma-\cos^{2}\gamma\sin^{\frac{2}{m}}\theta$
$\displaystyle\times(1-\cos^{2}\theta\cos^{2}\gamma)^{\frac{m-1}{m}}$
$\displaystyle>$ $\displaystyle 0$

if $\theta$ and $\gamma$ satisfy the following inequality

$\displaystyle\sin^{2}\theta-\sin^{2}\theta\cos^{2}\theta\cos^{2}\gamma<1$
(126)

So, the distribution $P_{w}(0,1,0)$ in Eq.(120) violates configuration
inequality (3) if the inequality (126) holds.

Similarly, we get

$\displaystyle v_{100}$ $\displaystyle:=$ $\displaystyle
P_{w}(1,0,0)-p_{a}(1)^{\frac{m-1}{m}}p_{b}(0)^{\frac{1}{m}}p_{c}(0)^{\frac{m-1}{m}}$
(127) $\displaystyle=$
$\displaystyle\sin^{2}\gamma-\sin^{\frac{2m-2}{m}}\gamma(1-\sin^{2}\theta\cos^{2}\gamma)^{\frac{1}{m}}$
$\displaystyle\times(1-\cos^{2}\theta\cos^{2}\gamma)^{\frac{m-1}{m}}$
$\displaystyle>$ $\displaystyle 0$ (128)

if $\theta$ and $\gamma$ satisfy the following inequality

$\displaystyle\sin^{2}\gamma-\sin^{2}\gamma\cos^{2}\gamma\cos^{2}\theta<1$
(129)

The distribution $P_{w}(1,0,0)$ in Eq.(120) violates the configuration
inequality (3) if the inequality (129) holds.

Figure 15: (Color online) Violations of W states (a) According to the
inequality (2). (b) According to the inequality (34). Here, $m=1000$ and
$v_{max}$ denotes the maximal violation over all the distributions $p(a,b,c)$.

From the inequalities (124), (126) and (129), the distribution of $P_{w}$ in
Eq.(120) violates the inequality (3) for $\theta,\gamma\in(0,\frac{\pi}{2})$.
The maximal violation of

$\displaystyle
v_{\max}=\max_{a,b,c}\\{P_{q}(a,b,c)-p(a)^{\frac{m-1}{m}}p(b)^{\frac{1}{m}}p(c)^{\frac{m-1}{m}}\\}$
(130)

is shown in Fig.15(b). The inequality (3) can be used for verifying all the
generalized W states. However, from Fig.15(a), the inequality (2) is useless
for verifying the correlation of $P_{w}$ in Eq.(120).

In what follows, consider a triangle network ${\cal N}_{q}$, as shown in
Fig.13(a), consisting of three generalized EPR states EPR  in Eq.(112), where
each party shares two qubits from two EPR states. After the projection
measurement under the basis
$\\{|0\rangle:=|00\rangle,|1\rangle:=|00\rangle,|2\rangle:=|10\rangle,|3\rangle:=|11\rangle\\}$,
the correlation is given by Eq.(44) with $p_{i}=\cos^{2}\theta_{i}$ and
$q_{i}=\sin^{2}\theta_{i}$, $i=1,2,3$. From Eqs.(44) and (45), it follows that

$\displaystyle v_{000}$ $\displaystyle:=$ $\displaystyle
P_{tri}(0,0,0)-p_{a}(0)^{\frac{m-1}{m}}p_{b}(0)^{\frac{1}{m}}p_{c}(0)^{\frac{m-1}{m}}$
(131) $\displaystyle=$ $\displaystyle
p_{1}p_{2}p_{3}-p_{1}p_{2}p_{3}^{\frac{2m-2}{m}}$ $\displaystyle>$
$\displaystyle 0$

for $m>2$ and $\theta_{1},\theta_{2},\theta_{3}\in(0,\frac{\pi}{2})$. The
distribution of $P_{tri}(0,0,0)$ violates the inequality (3). Similar results
hold for other correlations. The maximal violation of $v_{\max}$ in Eq.(130)
is shown in Fig.16(a). The inequality (3) is useful for verifying all the
correlations derived from ${\cal N}_{3}$ in Fig.13(a).

Figure 16: (Color online) Violations of networks (a) Triangle network
consisting of three generalized EPR states. Here,
$\theta_{1}=\theta_{2}=\theta_{3}$. (b) 4-partite network consisting of two
generalized GHZ states. Here, $\theta_{1}=\theta_{2}$. $v_{\max}$ is defined
in Eq.(130).

To sum up, the inequality (3) is useful for verifying the chain network
consisting of two sources.

Example S11 (4-partite network). Consider a 4-partite network ${\cal N}_{4}$,
there are lots of different configurations which may consist of single
sources, two sources, three sources or four sources. Here, we consider one
network topology, as shown in Fig.13(b), consisting of two generalized GHZ
states GHZ : $|ghz_{1}\rangle_{123}|ghz_{2}\rangle_{456}$, where
$|ghz_{i}\rangle$ is given by

$\displaystyle|ghz_{i}\rangle=\cos\theta_{i}|000\rangle+\sin\theta_{i}|111\rangle$
(132)

for $\theta_{i}\in(0,\frac{\pi}{2})$, $i=1,2$. Here, two parties
$\textsf{A}_{1}$ and $\textsf{A}_{4}$ hold respectively one qubit 1 and 6,
while $\textsf{A}_{2}$ and $\textsf{A}_{3}$ hold respectively two qubits
$(2,4)$ and $(3,5)$. From Theorem 2, a new configuration inequality is given
by

$\displaystyle
p(a_{1},\cdots,a_{4})\leq(p(a_{1})p(a_{2}))^{\frac{m-2}{m}}(p(a_{3})p(a_{4}))^{\frac{1}{m}}$
(133)

for $m\geq 3$. This inequality is useful for verifying the distributions
derived from ${\cal N}_{4}$.

Especially, $\textsf{A}_{1}$ and $\textsf{A}_{4}$ perform local projection
measurements under the basis $\\{|0\rangle,|1\rangle\\}$. $\textsf{A}_{2}$ and
$\textsf{A}_{3}$ perform local projection measurements under the basis
$\\{|0\rangle:=|00\rangle,|1\rangle:=|01\rangle,|2\rangle:=|10\rangle,|3\rangle:=|11\rangle\\}$.
The joint distribution is given by

$\displaystyle P(a_{1},\cdots,a_{4})$ $\displaystyle=$
$\displaystyle\cos^{2}\theta_{1}\cos^{2}\theta_{2}[0000]$ (134)
$\displaystyle+\cos^{2}\theta_{1}\sin^{2}\theta_{2}[0111]$
$\displaystyle+\sin^{2}\theta_{1}\cos^{2}\theta_{2}[1220]$
$\displaystyle+\sin^{2}\theta_{1}\sin^{2}\theta_{2}[1331]$

It follows that

$\displaystyle
p_{a_{1}}(0)=\cos^{2}\theta_{1},p_{a_{1}}(1)=\sin^{2}\theta_{1}$
$\displaystyle
p_{a_{2}}(0)=p_{a_{3}}(0)=\cos^{2}\theta_{1}\cos^{2}\theta_{2},$
$\displaystyle p_{a_{2}}(1)=p_{a_{3}}(1)=\cos^{2}\theta_{1}\sin^{2}\theta_{2}$
$\displaystyle
p_{a_{2}}(2)=p_{a_{3}}(2)=\sin^{2}\theta_{1}\cos^{2}\theta_{2},$
$\displaystyle
p_{a_{2}}(3)=p_{a_{3}}(3)=\sin^{2}\theta_{1}\sin^{2}\theta_{2},$
$\displaystyle
p_{a_{4}}(0)=\cos^{2}\theta_{2},p_{a_{4}}(1)=\sin^{2}\theta_{2}$ (135)

From Eqs.(134) and (135), we get

$\displaystyle v_{0000}$ $\displaystyle:=$ $\displaystyle
P(0,0,0,0)-(p_{a_{1}}(0)p_{a_{2}}(0))^{\frac{m-2}{m}}(p_{a_{3}}(0)p_{a_{4}}(0))^{\frac{1}{m}}$
(136) $\displaystyle=$
$\displaystyle\cos^{2}\theta_{1}\cos^{2}\theta_{2}-\cos^{\frac{6m-12}{m}}\theta_{1}\cos^{\frac{6m-12}{m}}\theta_{2}$
$\displaystyle>$ $\displaystyle 0$

for $m\geq 3$ and $\theta_{i}\in(0,\frac{\pi}{2})$, $i=1,2$. The distribution
$P(0,0,0,0)$ violates the inequality (133). The maximal violation of

$\displaystyle v_{\max}$ $\displaystyle=$
$\displaystyle\max_{a_{1},\cdots,a_{4}}\\{P(a_{1},a_{2},a_{3},a_{4})-(p(a_{1})p(a_{2}))^{\frac{m-2}{m}}$
(137) $\displaystyle\times(p(a_{3})p(a_{4}))^{\frac{1}{m}}\\}$

is shown in Fig.16(b). The inequality (133) is useful for verifying all the
correlations derived from ${\cal N}_{4}$ in Fig.13(b).

##  Appendix J Witnessing any multipartite entangled pure states

In this section, we firstly prove Corollary 2 in Subsec.J.1 which implies an
efficient method for witnessing almost all multipartite entangled pure states.
Moreover, we provide one example for witnessing generalized tripartite
entangled pure states in Subsec.J.2. We present a general method for testing
the $k$-separablity of multipartite pure states in Subsec.J.3.

###  J.1 Proof of Corollary 2

Consider an $n$-partite state $|\Phi\rangle_{A_{1}\cdots{}A_{n}}$ on Hilbert
space $\otimes_{j=1}^{n}\mathbb{H}_{j}$.

Figure 17: (Color online) (a) An equivalent tripartite chain network. Each
pair of $A_{1}$ and $C$ (or $C$ and $A_{2}$ shares an auxiliary source
$\lambda_{1}$ (or $\lambda_{2}$). (b) An equivalent $k$-partite chain network.
Each pair of $S_{i}$ and $C$ share one auxiliary source $\lambda_{i}$,
$i=1,\cdots,k$. The output $c$ of $C$ is fixed as $1$ with unit probability.

Assume $|\Phi\rangle_{A_{1}\cdots{}A_{n}}$ is separable. There is an integer
$i$ such that $A_{i}$ and $A_{i+1}$ are separable, that is,

$\displaystyle\rho_{A_{i}A_{i+1}}=\rho_{A_{i}}\rho_{A_{i+1}}$ (138)

where $\rho_{A_{i}A_{i+1}}={\rm
tr}_{A_{j},j\not=i,i+1}(|\Phi\rangle\langle\Phi|)$, $\rho_{A_{i}}={\rm
tr}_{A_{j},j\not=i}(|\Phi\rangle\langle\Phi|)$ and $\rho_{A_{i+1}}={\rm
tr}_{A_{j},j\not={i+1}}(|\Phi\rangle\langle\Phi|)$ denote the reduced density
matrices. Under the local projection under the computation basis
$\\{|j\rangle,\forall j\\}$, the joint distribution $P(a_{i},a_{i+1})$
satisfies

$\displaystyle P(a_{i},a_{i+1})=p(a_{i})p(a_{i+1})$ (139)

With this fact, it only needs to consider the reduced distribution
$P(a_{i},a_{i+1})$. Take $A_{1}$ and $A_{2}$ as an example. If they are
separable, $A_{1}$, $A_{2}$ and auxiliary particle $C$ may be regarded as an
equivalent tripartite chain network ${\cal N}_{q}$, as shown in Fig.17(a),
where each pair of $A_{1}$ and $C$ (or $C$ and $A_{2}$ shares an auxiliary
source $\lambda_{1}$ (or $\lambda_{2}$). The output $c$ of $C$ is fixed as $1$
with unit probability. From Eq.(1), the joint distribution satisfies

$\displaystyle P(a_{1},c=1,a_{2})=P(a_{1},a_{2})$ (140)

where $a_{i}$ is independent of the shared auxiliary source $\lambda_{i}$,
$i=1,2$. Note $A_{1}$ and $A_{2}$ are separable. From Theorem 2, the joint
distribution $P(a_{1},c=1,a_{2})$ satisfies the configuration inequality

$\displaystyle P(a_{1},c=1,a_{2})$ $\displaystyle\leq$ $\displaystyle
p(a_{1})^{\frac{m-1}{m}}p(c=1)^{\frac{1}{m}}p(a_{2})^{\frac{m-1}{m}}$ (141)
$\displaystyle=$ $\displaystyle
p(a_{1})^{\frac{m-1}{m}}p(a_{2})^{\frac{m-1}{m}}$

for any $m\geq 2$.

Fact. The inequality (141) holds if and only if the following inequality holds

$\displaystyle P(a_{1},a_{2})\leq p(a_{1})p(a_{2})$ (142)

This can be proved by using the equality of
$\lim_{m\to\infty}p(a_{1})^{\frac{m-1}{m}}p(c=1)^{\frac{1}{m}}p(a_{2})^{\frac{m-1}{m}}=p(a_{1})p(a_{2})$.
Eq.(139) consists of one facet of correlations for bipartite entangled pure
states. So, $a_{i}$ and $a_{i+1}$ are dependent ($A_{i}$ and $A_{i+1}$ are
entangled in $|\Phi\rangle_{A_{1}\cdots{}A_{n}}$) if and only if the reduced
probability distribution $P(a_{1},a_{2})$ is not independent for some $a_{i}$
and $a_{2}$, that is, $P(a_{1},a_{2})$ violates the inequality (142) for some
$a_{i}$ and $a_{2}$.

Now, we provide an efficient method to witness any multipartite entanglement
as follows.

  * Input:

An $n$-partite entangled state $|\Phi\rangle_{A_{1}\cdots{}A_{n}}$.

  * Output:

$n$-partite entanglement.

  * •

For $i$ from $1$ to $n-1$.

  * •

Obtain the probability distribution $P(a_{i},a_{i+1})$ under the local
measurement basis of $\\{|s\rangle_{A_{i}}\\}$ and
$\\{|t\rangle_{A_{i+1}}\\}$.

  * •

Test the configuration inequality (142).

  * •

Return: $|\Phi\rangle$ is $n$-partite entanglement if all the distributions
$P(a_{i},a_{i+1})$’s violate the inequality (140) for some $a_{i}$ and
$a_{i+1}$.

Algorithm 4 Witnessing multipartite entanglement

The proof is completed by verifying the inequality (142) for all the reduced
probability distributions $P(a_{i},a_{i+1})$, $i=1,\cdots,n-1$, that is,
testing $n-1$ inequalities for all the distributions $P(a_{i},a_{i+1})$’s.
Consider a joint probability distribution $P(a_{i},a_{i+1})$. If
$P(a_{i},a_{i+1})$ violates the inequality (142) for some specific values of
$a_{i}$ and $a_{i+1}$, the particles $A_{i}$ and $A_{i+1}$ are entangled.
Otherwise, all the probability distributions of $P(a_{i},a_{i+1})$ do not
violate the inequality (142). Note that

$\displaystyle\sum_{a_{i},a_{i+1}}P(a_{i},a_{i+1})=\sum_{a_{i},a_{i+1}}p(a_{1})p(a_{2})=1$
(143)

It follows Eq.(139) holds for all the values of $a_{i}$ and $a_{i+1}$. These
equalities may hold for some entangled states (fo example,
$|\Phi\rangle=\frac{1}{2}|00\rangle+|01\rangle+|10\rangle-|11\rangle$).
Fortunately, these equalities has only defined a zero-measure of subset in all
the values of $a_{i}$ and $a_{i+1}$.

Note each distribution $P(a_{i},a_{i+1})$ can be obtained with constant
complexity under the local projection measurements. Moreover, for each
probability distribution $P(a_{i},a_{i+1})$, the test of the configuration
inequality (142) has constant complexity. It follows the total time of
Algorithm 4 is linear complexity because only $n-1$ number of two-partite
inequalities will be tested. This completes the proof.

###  J.2 Witnessing a general three-qubit state

Consider a general three-qubit state $|\Phi\rangle_{ABC}$ with the following
normal form Acin  of

$\displaystyle|\Phi\rangle_{ABC}$ $\displaystyle=$
$\displaystyle\alpha_{0}|000\rangle+\alpha_{1}e^{i\phi}|100\rangle+\alpha_{2}|101\rangle$
(144) $\displaystyle+\alpha_{3}|110\rangle+\alpha_{4}|111\rangle$

where $\alpha_{i}$ satisfy $\sum_{i}\alpha_{i}^{2}=1$. Its entanglement can be
verified by using the bipartitions $\\{A\\}$ and $\\{B,C\\}$, $\\{B\\}$ and
$\\{A,C\\}$, and $\\{C\\}$ and $\\{A,B\\}$ by using the inequality (3).

After the projection measurement under the basis $\\{|0\rangle,|1\rangle\\}$,
the joint distribution is given by

$\displaystyle P(a,b,c)$ $\displaystyle=$
$\displaystyle\alpha_{0}^{2}[000]+\alpha_{1}^{2}[100]+\alpha_{2}^{2}[101]$
(145) $\displaystyle+\alpha_{3}^{2}[110]+\alpha_{4}^{2}[111]$

It follows that

$\displaystyle p_{a}(0)=\alpha_{0}^{2},p_{a}(1)=1-\alpha_{0}^{2},$
$\displaystyle
p_{b}(0)=\alpha_{3}^{2}+\alpha_{4}^{2},p_{b}(1)=1-\alpha_{3}^{2}-\alpha_{4}^{2},$
$\displaystyle
p_{c}(0)=\alpha_{2}^{2}+\alpha_{4}^{2},p_{c}(1)=1-\alpha_{2}^{2}-\alpha_{4}^{2},$
$\displaystyle
P_{ab}(0,0)=\alpha_{0}^{2},P_{ab}(1,0)=\alpha_{1}^{2}+\alpha_{2}^{2},$
$\displaystyle
P_{ab}(11)=\alpha_{3}^{2}+\alpha_{4}^{2},P_{bc}(0,0)=\alpha_{0}^{2}+\alpha_{1}^{2},$
$\displaystyle P_{bc}(0,1)=\alpha_{2}^{2},P_{bc}(1,0)=\alpha_{3}^{2},$
$\displaystyle P_{bc}(1,1)=\alpha_{4}^{2},P_{ac}(0,0)=\alpha_{0}^{2},$
$\displaystyle
P_{ac}(1,0)=\alpha_{1}^{2}+\alpha_{3}^{2},P_{ac}(11)=\alpha_{2}^{2}+\alpha_{4}^{2}$
(146)

Firstly, if $|\Phi\rangle_{ABC}$ is separable in terms of the bipartition
$\\{A\\}$ and $\\{B,C\\}$, the reduced distribution $\\{P(a,b)\\}$ satisfies
the inequality (142). From Eqs.(145) and (146), it follows that

$\displaystyle v_{00}$ $\displaystyle:=$ $\displaystyle
P_{ab}(0,0)-p_{a}(0)p_{b}(0)$ (147) $\displaystyle=$
$\displaystyle\alpha_{0}^{2}-\alpha_{0}^{2}(\alpha_{3}^{2}+\alpha_{4}^{2})$
$\displaystyle>$ $\displaystyle 0$

if $\alpha_{0}^{2}$ and $\alpha_{3}^{2}+\alpha_{4}^{2}\not=0$, which holds for
$\alpha_{0},\alpha_{3},\alpha_{4}\in(0,1)$ and
$\alpha_{1},\alpha_{2}\in[0,1)$. The distribution $P_{ab}(0,0)$ in Eq.(145)
violates the inequality (142) for $\alpha_{0},\alpha_{3},\alpha_{4}\in(0,1)$
and $\alpha_{1},\alpha_{2}\in[0,1)$.

Secondly, if $|\Phi\rangle_{ABC}$ is separable in terms of the bipartition
$\\{B\\}$ and $\\{A,C\\}$, the reduced distribution $\\{P(a,c)\\}$ satisfies
the inequality (142). From Eqs.(145) and (146), it follows that

$\displaystyle v_{00}$ $\displaystyle:=$ $\displaystyle
P_{ac}(0,0)-p_{a}(0)p_{c}(0)$ (148) $\displaystyle=$
$\displaystyle\alpha_{0}^{2}-\alpha_{0}^{2}(\alpha_{2}^{2}+\alpha_{4}^{2})$
$\displaystyle>$ $\displaystyle 0$

if $\alpha_{0}^{2}$ and $\alpha_{2}^{2}+\alpha_{4}^{2}\not=0$, which holds for
$\alpha_{0},\alpha_{2},\alpha_{4}\in(0,1)$ and
$\alpha_{1},\alpha_{3}\in[0,1)$. The distribution $P_{ac}(0,0)$ in Eq.(145)
violates the inequality (142) for $\alpha_{0},\alpha_{2},\alpha_{4}\in(0,1)$
and $\alpha_{1},\alpha_{3}\in[0,1)$.

Thirdly, if $|\Phi\rangle_{ABC}$ is separable in terms of the bipartition
$\\{A\\}$ and $\\{B,C\\}$, the reduced distribution $\\{P(b,c)\\}$ satisfies
the inequality (142). From Eqs.(145) and (146), it follows that

$\displaystyle v_{00}$ $\displaystyle:=$ $\displaystyle
P_{bc}(0,0)-p_{b}(0)p_{c}(0)$ (149) $\displaystyle=$
$\displaystyle\alpha_{0}^{2}+\alpha_{1}^{2}-(\alpha_{3}^{2}+\alpha_{4}^{2})(\alpha_{2}^{2}+\alpha_{4}^{2})$
$\displaystyle v_{01}$ $\displaystyle:=$ $\displaystyle
P_{bc}(0,1)-p_{b}(0)p_{c}(1)$ (150) $\displaystyle=$
$\displaystyle\alpha_{2}^{2}-(\alpha_{3}^{2}+\alpha_{4}^{2})(1-\alpha_{2}^{2}-\alpha_{4}^{2})$
$\displaystyle v_{10}$ $\displaystyle:=$ $\displaystyle
P_{bc}(1,0)-p_{b}(1)p_{c}(0)$ (151) $\displaystyle=$
$\displaystyle\alpha_{3}^{2}-(1-\alpha_{3}^{2}-\alpha_{4}^{2})(\alpha_{2}^{2}+\alpha_{4}^{2})$
$\displaystyle v_{11}$ $\displaystyle:=$ $\displaystyle
P_{bc}(1,1)-p_{b}(1)p_{c}(1)$ (152) $\displaystyle=$
$\displaystyle\alpha_{4}^{2}-(1-\alpha_{3}^{2}-\alpha_{4}^{2})(1-\alpha_{2}^{2}-\alpha_{4}^{2})$

Note $\sum_{b,c}P(b,c)=\sum_{b,c}p(b)p(c)$. It follows $v_{ij}>0$ for some