# Nonlocality of All Quantum Networks

Ming-Xing Luo  Information Security and National Computing Grid Laboratory,
Southwest Jiaotong University, Chengdu 610031, China

###### Abstract

The multipartite correlations derived from local measurements on some
composite quantum systems are inconsistent with those reproduced classically.
This inconsistency is known as quantum nonlocality and shows a milestone in
the foundations of quantum theory. Still, it is NP hard to decide a nonlocal
quantum state. We investigate an extended question: how to characterize the
nonlocal properties of quantum states that are distributed and measured in
networks. We first prove the generic tripartite nonlocality of chain-shaped
quantum networks using semiquantum nonlocal games. We then introduce a new
approach to prove the generic activated nonlocality as a result of
entanglement swapping for all bipartite entangled states. The result is
further applied to show the multipartite nonlocality and activated nonlocality
for all nontrivial quantum networks consisting of any entangled states. Our
results provide the nonlocality witnesses and quantum superiorities of all
connected quantum networks or nontrivial hybrid networks in contrast to
classical networks.

###### pacs:

03.67.Lx, 76.60.-k, 89.80.+h

##  I Introduction

The joint probability distribution of the outcomes of local measurements
performed on spatially separated quantum systems sometimes exhibit
correlations that cannot be explained classically in terms of the information
shared beforehand. Such nonlocal correlations are revealed by the violation of
special inequalities [1-3]. Formally, Bell’s theorem [3] states that the
predictions of quantum mechanics are inconsistent with classical causal
relations those originate from a classical common local hidden variable (LHV).
The profound property holds for general quantum entangled systems [4-12] whose
joint state cannot be written in a mixture of states in product forms. These
typical correlations are found of great interest in information processing GH
; Masa ; PAMB , communication Ek ; SG ; MY ; AGM ; ABG , quantum theory and
potential applications BPA ; BCMD ; BCPS .

Quantum entanglement is a valuable resource for various tasks including
quantum key distribution, randomness extraction, and quantum communication
HHH0 ; CSS . Interestingly, different from classical states entangled states
can be swapped ZZHE ; SPG , where a local measurement can create an
entanglement for two parties who have no prior shared entanglement as shown in
Figure 1. The remarkable feature is the foundation of distributing quantum
entanglement in long distance TRO ; TBZ ; AB ; YC  and constructing large-
scale quantum networks Kimb ; Ritt ; BDCZ ; DLCZ ; ATL ; ZPD . Nonetheless, it
still remains an open problem to characterize these quantum behaviors, even if
for the simplest quantum network consisting of two bipartite entangled states.

Figure 1: (Color online) Schematic Bell testing of quantum entanglement
swapping using a generalized Bell inequality SQ . (a) Tripartite Bell testing
of two bipartite entangled states $\rho_{AB}\otimes\rho_{CD}$.
$\tau^{a_{1}},\tau^{a_{2}},\tau^{a_{3}}$ are input states of Alice, Bob, and
Charlie, respectively. $x_{1},x_{2},x_{3}$ are measurement outcomes of three
parties. (b) Classical hidden state model for testing the locality of two
shared sources $\lambda_{1},\lambda_{2}$ or separable states
$\rho_{AB}=\sum_{i}p_{i}\varrho^{(i)}_{A}\otimes\varrho^{(i)}_{B}$ and
$\rho_{CD}=\sum_{i}q_{i}\varrho^{(i)}_{C}\otimes\varrho^{(i)}_{D}$, where
$\\{p_{i}\\}$ and $\\{q_{i}\\}$ are probability distributions and
$\varrho^{(i)}_{A(B,C,D)}$ are density operators of system $A$ ($B,C,D$). The
probabilities of $\tau^{a_{i}}$s are uniform distributions.

In comparison to single entanglement, there are independent sources in a
general quantum network for distributing hidden states to space-like separated
parties in terms of the locally causal model Bell ; Cire ; PR ; ABH . The
standard Bell testing Bell ; CHSH  is useless for verifying these entangled
states in a distributive model. Recently, nonlinear Bell-type inequalities are
proposed for verifying the non-trilocality CAS  or non-bilocality BGP ; GMT
of correlations originating from the standard entanglement swapping. It is
then extended for star-shaped networks ACS  or small-sized networks LS ; CH .
Another procedure is iteratively expanding a given network into the desired
network RBB . Different from these methods, a polynomial-time algorithm is
proposed for constructing explicit nonlinear Bell-type inequalities for
general networks Luo . Their inequalities are useful for proving the generic
non-multilocality of quantum networks consisting of all bipartite entangled
pure states EPR1  and generalized Greenberger-Horne-Zeilinger (GHZ) states GHZ
. Certain quantum experiments are performed for verifying the nonlocalities CL
; BG ; TWE ; SBB ; CASB . Unfortunately, there is no result to feature all
quantum networks.

Our goal in this work is to prove the nonlocality of all quantum networks
consisting of any entangled states, which is a weak problem of verifying all
quantum networks. We specially investigate the existence of generalized Bell
testing for verifying multipartite correlations generated by local
measurements in each nontrivial quantum network in terms of the generalized
locally causal model PR ; Cire . Our approach depends primarily on the recent
semiquantum nonlocal game SQ  that permits to verify single entanglement. We
firstly prove that tripartite correlations of the quantum network shown in
Figure 1 violate generalized Bell inequalities for all bipartite entangled
states. The generic non-multilocality is different from the nonlocality of
single entanglement using CHSH inequality CHSH ; GP ; PR ; LF  or Hardy
inequality Hardy ; YCZ . Furthermore, the nonlocality of two independent
parties without initially sharing entanglement can be activated by a local
measurement of the other party. It is of another remarkable feature of
entangled states SPG  and provides a way for detecting a single entanglement.
These results are then applied to the multipartite nonlocality of all
connected quantum networks consisting of any entangled states, and the
activated nonlocality for any subnetwork consisting of independent parties.
Finally, we show the multipartite nonlocality of nontrivial hybrid networks
consisting of entangled states and classical resources. Remarkably, the result
holds for any non-classical network and provides a quantum superiority of
hybrid Internet over all classical networks.

##  II Results

Multilocality structure of a network. Consider a network consisting of
classical systems that are shared by $n$ parties, $\texttt{P}_{1}$,
$\texttt{P}_{2}$, $\cdots$, $\texttt{P}_{n}$. In experiment, each party
$\texttt{P}_{i}$ perform $m$ local measurements on their respective subsystems
with $k$ possible outcomes, $x_{i}=1,\cdots,k$, where $k>2$ for general
systems Luo1 . Denote $a_{i}=1,\cdots,m$ as the measurement chosen by
$\texttt{P}_{i}$. The joint conditional probability distribution $P({\bf
x}|{\bf a})$ of all measurement outcomes with ${\bf x}=x_{1}\cdots x_{n}$ and
${\bf a}=a_{1}\cdots a_{n}$ is local whenever it can be explained as the
result of classically correlated datas, represented by hidden variables
$\lambda_{1},\cdots,\lambda_{s}$, that is,

$\displaystyle P_{local}({\bf x}|{\bf a})=\int_{\Omega}d\mu(\lambda_{1})\cdots
d\mu(\lambda_{s})\prod_{j=1}^{n}p(x_{j}|a_{j},\Lambda_{j}),$ (1)

where $\Lambda_{j}$ contains some variables $\lambda_{i}$s,
$\mu_{i}(\lambda_{i})$ denotes the measure of $\lambda_{i}$ with the
normalization condition $\int_{\Omega_{i}}d\mu_{i}(\lambda_{i})=1$ and
$(\Omega_{i},\Sigma_{i},\mu_{i})$ is the measure space of $\lambda_{i}$,
$i=1,2,\cdots,s$. Every local distribution (1) satisfies special constraint
known as Bell-type inequality Luo .

Local measurements on some composite quantum states $\rho_{1},\cdots,\rho_{k}$
in a distributive model or network lead to the violation of proper Bell
inequality that is then a signature of the non-multilocality BGP ; CH ; GMT ;
RBB ; Luo . Specifically, there exist joint conditional probability
distributions resulting from local quantum measurement outcomes of all
observers as

$\displaystyle P_{Q}({\bf x}|{\bf
a})=\textrm{Tr}[(\otimes_{i=1}^{n}{M}^{x_{i}}_{a_{i}})(\otimes_{j=1}^{k}\rho_{j})],$
(2)

which cannot be reproduced by any local model (1), where
$\\{{M}^{x_{i}}_{a_{i}},\forall x_{i}\\}$ are positive operators describing
locally implementable quantum measurements by the observer $\mathtt{P}_{i}$
and satisfy $\sum_{x_{i}}M^{x_{i}}_{a_{i}}=\mathbbm{1}$ for each $a_{i}$, and
$\mathbbm{1}$ is the identity operator. The joint quantum system
$\rho_{1}\otimes\cdots\otimes\rho_{k}$ is multipartite nonlocal and reduces to
Bell’s nonlocal for $k=1$ Bell . A stronger nonlocality is possible in hidden
state scenario with quantum inputs SQ . $n$ random sources
$\\{\tau^{a_{1}},\forall a_{1}\\},\cdots,\\{\tau^{a_{n}},\forall a_{n}\\}$ are
assumed for all observers shown in Figure 1. It follows a new joint
conditional probability distribution as

$\displaystyle{P}_{Q}({\bf x}|{\bf a})\\!\\!\\!\\!\\!\\!$
$\displaystyle={P}_{Q}({\bf x}|\tau^{a_{1}},\cdots,\tau^{a_{n}})$ (3)
$\displaystyle=\textrm{Tr}[(\otimes_{i=1}^{n}{M}^{x_{i}}_{a_{i}})((\otimes_{j=1}^{k}\rho_{j})\otimes(\otimes_{s=1}^{n}\tau^{a_{s}}))].$

Similar to verifying single entanglement Bell ; CHSH , how to decide the
nonlocality of a quantum network is also a fundamental question. Nevertheless,
identifying the nonlocality of a general quantum network remains an extremely
difficult problem [40-47].

Here, we introduce a new framework for exploring the nonlocality of all
quantum networks. Given a quantum network consisting of a joint quantum system
$\rho=\rho_{1}\otimes\cdots\otimes\rho_{k}$ shared by $n$ observers, the main
idea is to create quantum subnetworks for special observers (Alice and Charlie
shown in Figure 1). If some nonlocal correlations can be observed in these
subnetworks, they are provided by the quantum state $\rho$, which is then
multipartite nonlocal. Within the new scenario, we investigate the activation
phenomena for these subnetworks consisting of all independent observers
without prior sharing entanglement. It is of the multipartite nonlocality in a
network scenario.

Definition 1. A quantum network ${\cal N}_{q}$ is multipartite nonlocal if a
set of observables existing for all observers such that multipartite quantum
correlations from local measurements are inconsistent with these from the
generalized local realism.

Definition 2. A quantum network ${\cal N}_{q}$ consisting of $n$ observers is
$k$-partite activated nonlocal if for any $s$ observers
$\texttt{p}_{i_{1}},\cdots,\texttt{p}_{i_{s}}$ with $2\leq s\leq k$, there is
a set of observables for all observers such that a local measurement of $n-s$
observers $\texttt{p}_{j}$s with
$j\in\\{1,\cdots,n\\}\backslash\\{i_{1},\cdots,i_{s}\\}$ creates a $s$-partite
nonlocal subnetwork.

Nonlocality of all $\Lambda$-shaped quantum networks. Consider a
$\Lambda$-shaped quantum network ${\cal N}_{q}$ consisting of three observers
Alice, Bob and Charlie shown in Figure 1. The nontrivial feature of ${\cal
N}_{q}$ is entanglement swapping ZZHE ; BGP . Different from the standard
nonlocality GP ; PR ; Gisin1 ; Pop  detected by linear Bell inequalities Bell
; CHSH , the tripartite nonlocality of ${\cal N}_{q}$ can be verified using
nonlinear Bell-type inequalities for all bipartite entangled pure states and
special entangled mixed states GMT ; Luo . Our goal here is to prove the
tripartite nonlocality and bipartite activated nonlocality of ${\cal N}_{q}$
for all bipartite entangled states. Specifically, assume that ${\cal N}_{q}$
consists of two bipartite entangled states $\rho_{AB}$ and $\rho_{CD}$, where
Alice has particle $A$, Bob has particles $B$ and $C$ while Charlie has
particle $D$. The joint state of ${\cal N}_{q}$ reading
$\rho=\rho_{AB}\otimes\rho_{CD}$ is activated nonlocal for all entangled pure
states GMT ; Luo . Our first result is to generalize the result to all
bipartite entangled states. Formally, we prove the following result

Theorem 1. Assume that a $\Lambda$-shaped quantum network ${\cal N}_{q}$
consists of any bipartite entangled states shared by three observers. Then the
following results hold: (i) ${\cal N}_{q}$ is tripartite nonlocal; (ii) ${\cal
N}_{q}$ is bipartite activated nonlocal.

Different from previous Bell inequalities for verifying special
$\Lambda$-shaped quantum networks ZZHE ; BGP ; GMT ; Luo , Theorem A shows
that a generalized Bell-type inequality exists for a given $\Lambda$-shaped
quantum network consisting of any bipartite entangled states. Generally, this
Bell-type inequality is state-dependent. Furthermore, one measurement of Bob
can create an entanglement between Alice and Charlie who have no prior shared
entanglement. The interesting feature of bipartite entangled states is going
beyond classical resources SPG  and key to build large-scale quantum networks
Kimb ; Ritt ; ZPD . The proof of the tripartite nonlocality stated in Theorem
A is a straight forward application of the semiquantum nonlocal game for each
entanglement SQ . For the bipartite activated nonlocality, our proof will be
completed for a reduced quantum network consisting of qubit-based entangled
states. The main idea is that Alice and Bob are allowed to firstly perform
local projections on high-dimensional systems and local distilling of
entangled mixed states HHH ; SI . These assumptions are reasonable because
local operations and classical communication (LOCC) or local operations and
shared randomness (LOSR) cannot create entanglement between two observers
initially sharing no entanglement HHH ; SQ . Additionally, the proof of
Theorem 1 also provided an interesting by-product that universal Bell
inequality exists for detecting a single entanglement by using local
projection and entanglement distilling SQ .

Figure 2: (Color online) A schematic connected quantum network. Quantum
resources consist of 4 bipartite entangled states, one tripartite entanglement
and one 4-partite entanglement that are shared by 10 observers
$\texttt{P}_{1},\cdots,\texttt{P}_{10}$. $\texttt{P}_{i}$ and $\texttt{P}_{9}$
share one bipartite entanglement for $i=2,4,5$. $\texttt{P}_{2}$ and
$\texttt{P}_{7}$ share one bipartite entanglement.
$\texttt{P}_{3},\texttt{P}_{7}$ and $\texttt{P}_{8}$ share one tripartite
entanglement. $\texttt{P}_{1},\texttt{P}_{6},\texttt{P}_{7}$ and
$\texttt{P}_{10}$ share one 4-partite entanglement. Each ball denotes one
physical particle.

Nonlocality of all connected quantum networks. The $\Lambda$-shaped quantum
network can be generalized to chain-shaped quantum networks RBB ; Luo  or
star-shaped quantum networks with several observers ACS ; RBB ; Luo . Here, we
further consider a general quantum network ${\cal N}_{q}$ with the
connectivity in its schematic graph scenario. Our goal is to prove the
multipartite nonlocality and activated locality of these networks.
Specifically, assume that ${\cal N}_{q}$ is connected network if for any two
observers $\texttt{P}_{k}$ and $\texttt{P}_{s}$ there is a set of observers
$\texttt{P}_{i_{1}},\cdots,\texttt{P}_{i_{t}}$ such that any adjacent two
observers of
$\texttt{P}_{k},\texttt{P}_{i_{1}},\cdots,\texttt{P}_{i_{t}},\texttt{P}_{s}$
share at least one entanglement. Otherwise, ${\cal N}_{q}$ is disconnected.
The definition can be also defined as the connectivity of its equivalent graph
BLW , where each $s$-partite entanglement is schematically represented by a
complete graph with $s$ distinct vertices, and each node represents an
observer shown in Figure 2. Our applications in the following are all
connected quantum networks. Similar to Theorem 1, we can prove the
multipartite nonlocality and activated nonlocality for generic connected
quantum networks. It is formally stated as the following result

Theorem 2. Assume that a connected quantum network ${\cal N}_{q}$ consists of
any multipartite entangled states shared by $n$ observers. Then the following
results hold: (i) ${\cal N}_{q}$ is multipartite nonlocal; (ii) ${\cal N}_{q}$
is $k$-partite activated nonlocal for any $2\leq k\leq n-1$.

In contrast to the genuine multipartite nonlocality of special quantum
networks SS ; CGP  detected by Svetlichny inequalities Sve  or multipartite
nonlocality detected by nonlinear Bell-type inequalities BGP ; GMT ; Luo ,
Theorem 2 provides the existence of generalized Bell-type inequalities for
verifying each connected quantum network. The result is further extended to
any subnetwork with the activated nonlocality. The surprising part is similar
to the standard entanglement swapping that permits some observers in ${\cal
N}_{q}$ help independent observers to construct nonlocal quantum correlations
going beyond classical scenario SPG . Theorem 2 implies that this interesting
feature is generic for all connected quantum networks consisting of any
entangled states going beyond chain-shaped or star-shaped quantum networks ACS
; RBB ; Luo . It is special important for constructing general quantum
networks. Similar to Theorem 1, the first proof of Theorem 2 is an application
of the semiquantum nonlocal game SQ ; SI . To prove the activated nonlocality,
an iterative algorithm will be proposed to reduce a general quantum network
into a hybrid network consisting of several chain-shaped and star-shaped
quantum subnetworks. The projection method and entanglement distilling are
then used to complete the proof HHH ; SI .

Figure 3: (Color online) A schematic hybrid network. There are five
subnetworks ${\cal N}_{1},\cdots,{\cal N}_{5}$ consisting of 12 observers
$\texttt{P}_{1},\cdots,\texttt{P}_{12}$. $\texttt{P}_{i}$ and
$\texttt{P}_{i+1}$ share one bipartite entanglement for $i=1,10,11$.
$\texttt{P}_{2},\texttt{P}_{3}$ and $\texttt{P}_{4}$ share one tripartite
entanglement. $\texttt{P}_{6},\cdots,\texttt{P}_{9}$ share one 4-partite
entanglement. $\texttt{P}_{4},\texttt{P}_{5}$ and $\texttt{P}_{6}$ share one
tripartite fully separable state. $\texttt{P}_{8}$ and $\texttt{P}_{10}$ share
one bipartite separable state. ${\cal N}_{1},{\cal N}_{2}$ and ${\cal N}_{3}$
are connected quantum subnetworks while ${\cal N}_{4}$ and ${\cal N}_{5}$ are
classical subnetworks. Here, each fully separable state can be equivalently
described by a local hidden variable in terms of LHV model.

Quantum superiority of all nontrivial hybrid networks. Theorem 2 provides a
strong multipartite nonlocality in terms of the activated nonlocality for all
connected quantum networks. An available network can consist of hybrid
resources including quantum entangled states and classical resources (or fully
separable states) shown in Figure 3. A natural problem states: is it possible
to characterize these hybrid networks? So far, no related result has been
obtained for this problem. Note that these networks are generally disconnected
quantum networks and then cannot provide the activated nonlocality for all
subnetworks. Although the multipartite correlations from local measurements of
hybrid networks do not satisfy the activated nonlocality given in Definition
2, our goal here is to explore the global quantum nonlocality obtained from
these networks. Specifically, is it possible to define a generalized Bell-type
inequality that can witness a hybrid network in contrast with classical
networks? Different from Theorem 2 we prove weak multipartite nonlocality or
global quantum supremacy for hybrid networks as follows:

Theorem 3. Assume that a nontrivial hybrid network ${\cal N}_{cq}$ consist of
any multipartite entangled states and fully separable states. Then ${\cal
N}_{cq}$ is multipartite nonlocal.

A nontrivial hybrid network in Theorem 3 denotes any hybrid network consisting
of at least one multipartite entangled state. Different from the entanglement
detection Bell  or the verification of quantum networks CAS ; BGP ; GMT ; Luo
, Theorem 3 provides the first generic result of nontrivial hybrid networks.
The following equivalent interpretation of Theorem 3 is more interesting. The
global nonlocality of ${\cal N}_{cq}$ can be used to acquire quantum
superiority over all classical networks in terms of the collaborative
semiquantum game SQ ; SI , where without communicating with each other all
players maximize a negative payoff using local measurement outcomes.
Especially, from Theorem 3 for all players there is a general payoff $\wp$
depending on multipartite correlations $p(\textbf{x}|\textbf{a})$ satisfying a
strictly inequality $\wp_{cq}>\wp_{c}$, where $\wp_{cq}$ and $\wp_{c}$ are
maximal payoffs using multipartite correlations of ${\cal N}_{cq}$ and all
classical networks, respectively. It suggests an interesting way to design
distributive tasks for any hybrid network Kimb . More importantly, it provides
a quantum superiority of each nontrivial hybrid network over all classical
networks.

## Discussions

Different from previous multipartite nonlocality using nonlinear Bell-type
inequalities BGP ; CH ; GMT ; RBB ; Luo , Definition 1 provides the weakest
version to show a global nonlocality of quantum entangled states in network
scenario CAS , where one entanglement can imply the global nonlocality from
Theorem 3. Definition 2 can be described as an extension of the genuine
multipartite nonlocality of single entanglement Sve ; SS ; CGP  for a quantum
network consisting of multiple entangled states. In the strong statement, the
multipartite correlations $P({\bf x}|{\bf a})$ cannot be decomposed into a
convex combination of $s$-partite with $1\leq s\leq n-1$ and $n-s$ partite
correlations with respect to any fixed bipartite partition of $n$ observers,
i.e., $P({\bf x}|{\bf a})\not=\sum_{\lambda}p(\lambda)P({\bf x}_{s}|{\bf
a}_{k},\lambda)P({\bf x}_{n-s}|{\bf a}_{n-s},\lambda)$, where ${\bf
x}_{s}=x_{i_{1}}\cdots x_{i_{s}}$, ${\bf a}_{s}=a_{i_{1}}\cdots a_{i_{s}}$,
${\bf x}_{n-s}$ and ${\bf a}_{n-s}$ denote all the other variables of $x_{i}$s
and $a_{i}$s except for ${\bf x}_{s}$ and ${\bf a}_{s}$, respectively.
Theorems 1 and 2 imply that all connected quantum networks consisting of $n$
observers are genuine multipartite nonlocal, where $2\leq k\leq n-1$. This
result can be easily followed from the definition of connected quantum
networks.

A weak version of Definition 2 for a quantum network ${\cal N}_{q}$ is to
consider the activated nonlocality of special subnetworks. In this case, the
activated nonlocality does not hold for all subnetworks. Formally, ${\cal
N}_{q}$ is $k$-partite weak activated nonlocal if for some independent $k$
observers there exist local measurements of all the other $n-k$ observers such
that one of their outcomes creates a nonlocal subnetwork. The special case is
also useful for special goals. From new definition, the weak activated
nonlocality can be proved for some nontrivial hybrid networks with connected
subnetworks, where LOCC or LOSR cannot create entanglement between two parties
initially sharing no entanglement HHH ; SQ . Interestingly, recent results CHW
; BRL  provide a method to construct Bell-type inequalities for single
entanglement based on its entanglement witness. Unfortunately, there is no
explicit algorithm to construct the entanglement witness for all entangled
states. New explorations should be interesting for quantum many-body systems
Deng ; LGLS ; ACTC . Besides, it is unknown how to verify general entanglement
swapping without entanglement distilling HHH . We provide some further
discussions by classifying entangled mixed states SQ .

We showed that any $\Lambda$-shaped quantum network is tripartite nonlocal and
bipartite activated nonlocal, which provided an evidence of generic
entanglement swapping for all entangled states. As a nontrivial application,
we obtained a generalized result that the multipartite nonlocality holds for
all connected quantum network consisting of any multipartite entangled states.
The interesting part is the activated nonlocality of these quantum networks,
which is a natural extension of the genuine multipartite nonlocality of single
entanglement. As another by-product, each nontrivial hybrid network can
provide quantum superiority over all classical networks in terms of the
semiquantum nonlocal game, in which the joint question-answer probability
distributions of a nontrivial hybrid network cannot be described classically
with any shared randomness. These results show the usefulness of quantum
networks including hybrid Internet going beyond classical networks.

## Acknowledgements

We thank the helpful discussions of Luming Duan. This work is supported by the
National Natural Science Foundation of China (Grants No.61772437, 61702427),
Sichuan Youth Science and Technique Foundation (Grant No.2017JQ0048),
Fundamental Research Funds for the Central Universities (Grant No.
2682014CX095), and financial support from Chuying fellowship.

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Supplementary Information “Nonlocality of All Quantum Networks”
Ming-Xing Luo

Information Security and National Computing Grid Laboratory,

Southwest Jiaotong University, Chengdu 610031, China

## Appendix A: Proof of Theorem 1

Before we present the detailed proofs of Theorem 1 and the other results, we
introduce some necessary definitions inspired by the semiquantum nonlocal game
SQ  to rigorously state the main result. In what follows, all quantum systems
are finite dimensions (i.e., their Hilbert spaces, denoted by ${\cal{H}}$, are
finite dimensions) and index sets (denoted by ${\cal{S}},{\cal{T}},{\cal{X}}$,
and ${\cal{Y}}$) contain only a finite number of elements. The convex set of
probability distributions defined on an index set ${\cal{X}}$ is denoted by
${\cal{P}}({\cal{X}})$. The set of linear operators acting on a Hilbert space
${\cal{H}}$ is denoted by $\mathbb{L}({\cal{H}})$. The set of density matrices
(i.e., positive semidefinite, trace-one operators) is denoted by
$\mathbb{S}({\cal{H}})\subset\mathbb{L}({\cal{H}})$.

A random source of states of a quantum system $A$ is represented by an
ensemble $\tau=(\\{p(s),\tau^{s}\\};s\in{\cal{S}}\\})$, where
$p\in{\cal{P}}({\cal{S}})$ and $\tau^{s}\in\mathbb{S}({\cal{H}}_{A})$, for all
$s$. Given an outcome set ${\cal{X}}=\\{x\\}$ and a quantum system $A$ on
Hilbert space ${\cal{H}}_{A}$, an ${\cal{X}}$-probability operator-valued
measure (${\cal X}$-POVM, for short) on $A$ is a family
$P=(P^{x};x\in{\cal{X}})$ of positive semidefinite operators
$P^{x}\in\mathbb{L}({\cal{H}}_{A})$, such that
$\sum_{x\in{\cal{X}}}P^{x}=\mathbbm{1}$. We denote by ${\cal{M}}(A;{\cal{X}})$
the convex set of all ${\cal{X}}$-POVMs on $A$. A POVM $P\in{\cal
M}(A;{\cal{X}})$ induces, via the relation $p(x)=\textrm{Tr}[P^{x}\varrho]$, a
linear function $P:\varrho\mapsto P\varrho$ from $\mathbb{S}({\cal{H}}_{A})$
to ${\cal P}({\cal{X}})$. POVMs in ${\cal M}(A;{\cal{X}})$ are used to
represent physical available measurements performed on a quantum system $A$
with outcomes in ${\cal{X}}$.

We firstly verify the following standard $\Lambda$-shaped quantum network
consisting of two entangled states. Here, we present detailed statements of
Theorem 1 as:

Theorem 1. For any network ${\cal{N}}_{q}$ consisting of three observers
Alice, Bob and Charlie, assume that Alice and Bob share one bipartite
entanglement $\rho_{AB}$ on the Hilbert space
${\cal{H}}_{A}\otimes{\cal{H}}_{B}$, and Bob and Charlie share one bipartite
entanglement $\rho_{CD}$ on the Hilbert space
${\cal{H}}_{C}\otimes{\cal{H}}_{D}$. Then $\rho_{AB}\otimes\rho_{CD}$ is
tripartite nonlocal. Moreover, for the subnetwork consisting of Alice and
Charlie, there are local observables for three observers such that one local
measurement of Bob can create a bipartite entanglement between Alice and Bob,
i.e., ${\cal{N}}_{q}$ is bipartite activated nonlocal.

Proof. Since $\rho_{AB}$ is bipartite entangled, there is a semiquantum game
$\mathbbm{G}_{sq}$ (see Figure S1(a)) SQ , constants
$\alpha_{a_{1}a_{2}}^{x_{1}x_{2}}$, auxiliary states
$\tau_{A_{0}}^{a_{1}}\in{\cal{H}}_{A_{0}}$ and ${\mathscr{X}}_{1}$-POVM
$\\{P^{x_{1}}\\}\in{\cal{M}}_{A_{0}A;{\mathscr{X}}_{1}}$ for Alice, auxiliary
states $\tau_{B_{0}}^{a_{2}}\in{\cal{H}}_{B_{0}}$ and ${\mathscr{X}}_{2}$-POVM
$\\{Q^{x_{2}}\\}\in{\cal M}_{BB_{0};{\mathscr{X}}_{2}}$ for Bob such that

$\displaystyle\sum_{a_{1},a_{2},x_{1},x_{2}}\alpha_{a_{1}a_{2}}^{x_{1}x_{2}}P(x_{1},x_{2}|a_{1},a_{2})>c_{1}$
(A1)

where $P(x_{1},x_{2}|a_{1},a_{2})$ are joint conditional probability
distributions computed as

$\displaystyle
P(x_{1},x_{2}|a_{1},a_{2})=\textrm{Tr}[(P^{x_{1}}_{A_{0}A}\otimes
Q^{x_{2}}_{BB_{0}})(\tau^{a_{1}}_{A_{0}}\otimes\rho_{AB}\otimes\tau^{a_{2}}_{B_{0}})],$
(A2)

$c_{1}$ denotes the maximal achievable classical bound of average gain in
terms of the semiquantum game $\mathbbm{G}_{sq}$. Denote
${\mathscr{A}}_{i}=\\{a_{i}\\}$ and ${\mathscr{X}}_{i}=\\{x_{i}\\}$ with
$i=1,2$. Here, we assume that the probability distributions of input random
sources $\\{\tau_{A}^{a_{1}}\\}$ and $\\{\tau_{B}^{a_{2}}\\}$ are uniform.
Otherwise, one can redefine the constants $\alpha_{a_{1}a_{2}}^{x_{1}x_{2}}$.
For all joint conditional probability distributions
$P_{c}(x_{1},x_{2}|a_{1},a_{2})$s derived from shared classical correlations
(hidden variable model Bell ; PR ) or fully separable quantum states (hidden
state model Cire , see Figure S1(b)), we have

$\displaystyle\sum_{a_{1},a_{2},x_{1},x_{2}}\alpha_{a_{1}a_{2}}^{x_{1}x_{2}}P_{c}(x_{1},x_{2}|a_{1},a_{2})\leq
c_{1},$ (A3)

where $P_{c}(x_{1},x_{2}|a_{1},a_{2})$ can be any convex combination of
independent distributions in the variables $x_{1},x_{2}$, i.e,
$P_{c}(x_{1},x_{2}|a_{1},a_{2})=\sum_{j}p_{j}P_{j}(x_{1}|a_{1})P_{j}(x_{2}|a_{2})$,
and $\\{p_{j}\\}$ is a probability distribution.

Note that the inequalities (A1) and (A3) can be regarded as generalized Bell-
type inequalities for testing bipartite entanglement $\rho_{AB}$.

Similarly, for the bipartite entanglement $\rho_{CD}$ there is another
semiquantum game $\mathbbm{G}^{\prime}_{sq}$ (see Figure S1(c)), constants
$\beta_{a_{3}a_{4}}^{x_{3}x_{4}}$, auxiliary states
$\tau_{C_{0}}^{a_{3}}\in{\cal{H}}_{C_{0}}$ and ${\mathscr{X}}_{3}$-POVM
$\\{R^{x_{3}}\\}\in{\cal{M}}_{C_{0}C;{\mathscr{X}}_{3}}$ for Alice, auxiliary
states $\tau_{D_{0}}^{a_{4}}\in{\cal{H}}_{D_{0}}$ and ${\mathscr{X}}_{4}$-POVM
$\\{S^{x_{4}}\\}\in{\cal{M}}_{DD_{0};{\mathscr{X}}_{4}}$ for Bob such that

$\displaystyle\sum_{a_{3},a_{4},x_{3},x_{4}}\beta_{a_{3}a_{4}}^{x_{3}x_{4}}P(x_{3},x_{4}|a_{3},a_{4})>c_{2},$
(A4)

where $P(x_{3},x_{4}|a_{3},a_{4})$ are joint conditional probability
distributions computed as

$\displaystyle
P(x_{3},x_{4}|a_{3},a_{4})=\textrm{Tr}[(R^{x_{3}}_{C_{0}C}\otimes
S^{x_{4}}_{DD_{0}})(\tau^{a_{3}}_{D_{0}}\otimes\rho_{CD}\otimes\tau^{a_{4}}_{D_{0}})],$
(A5)

$c_{2}$ denotes the maximal achievable classical bound of average gain in
terms of the semiquantum game $\mathbbm{G}^{\prime}_{sq}$, and
${\mathscr{A}}_{i}=\\{a_{i}\\}$. For all joint conditional probability
distributions $P_{c}(x_{3},x_{4}|a_{3},a_{4})$s derived from shared classical
resources (hidden variable model Bell ; PR ) or separable quantum states
(hidden state model Cire , see Figure S1(d)), we have

$\displaystyle\sum_{a_{3},a_{4},x_{3},x_{4}}\beta_{a_{3}a_{4}}^{x_{3}x_{4}}P_{c}(x_{3},x_{4}|a_{3},a_{4})\leq
c_{2},$ (A6)

where $P_{c}(x_{3},x_{4}|a_{3},a_{4})$ can be any convex combination of
independent distributions in the variables $x_{3}$ and $x_{4}$, i.e,
$P_{c}(x_{3},x_{4}|a_{3},a_{4})=\sum_{j}q_{j}P_{j}(x_{3}|a_{3})P_{j}(x_{4}|a_{4})$,
$\\{q_{j}\\}$ is a probability distribution.

Figure S1: (Color online) Schematically generalized Bell testing of bipartite
entangled states derived from the semiquantum nonlocal game SQ . (a)
Generalized Bell nonlocality testing of one bipartite entanglement
$\rho_{AB}$. $\tau^{a_{1}}_{A^{\prime}}$ and $\tau^{a_{2}}_{B^{\prime}}$ are
input states of Alice and Bob, respectively, and can be sent from a trusty
referee, where $A^{\prime}$ and $B^{\prime}$ are axillary systems, and
$a_{i}\in{\mathscr{A}}_{i}$, $i=1,2$. $x_{1}\in{\mathscr{X}}_{1}$ and
$x_{2}\in{\mathscr{X}}_{2}$ are outputs (depending on special POVMs) of Alice
and Bob, respectively. (b) Hidden state model for testing the locality of one
shared source $\lambda_{1}$ or separable state
$\varrho_{AB}=\sum_{i}p_{i}\varrho^{(i)}_{A}\otimes\varrho^{(i)}_{B}$, where
$\\{p_{i}\\}$ is a probability distribution and $\varrho^{(i)}_{A(B)}$ are
density operators of the system $A$ or $B$. (c) Generalized Bell nonlocality
testing of one bipartite entanglement $\rho_{CD}$. $\tau^{a_{3}}_{C^{\prime}}$
and $\tau^{a_{4}}_{D^{\prime}}$ are input states of Bob and Charlie,
respectively, and can be sent from a trusty referees, where $C^{\prime}$ and
$D^{\prime}$ are axillary systems, and $a_{i}\in{\mathscr{A}}_{i}$, $i=3,4$.
$x_{3}\in{\mathscr{X}}_{3}$ and $x_{4}\in{\mathscr{X}}_{4}$ are outputs of Bob
and Charlie, respectively. (d) Hidden state model for testing the locality of
one shared source $\lambda_{2}$ or separable state
$\varrho_{CD}=\sum_{i}q_{i}\varrho^{(i)}_{C}\otimes\varrho^{(i)}_{D}$, where
$\\{q_{i}\\}$ is a probability distribution and $\varrho^{(i)}_{C(D)}$ are
density operators of the system $C$ or $D$. The probabilities of
$\tau^{a_{1}},\cdots,\tau^{a_{4}}$ are uniform distributions in our
assumptions.

Note that the inequalities (A4) and (A6) can be regarded as generalized Bell-
type inequalities for testing bipartite entanglement $\rho_{CD}$. In what
follows, we construct a tripartite Bell inequality for testing the nonlocality
of $\Lambda$-shaped quantum network consisting of $\rho_{AB}\otimes\rho_{CD}$
by taking use of the inequalities (A1) and (A4).

Figure S2: (Color online) Schematically generalized Bell testing of
$\Lambda$-shaped network. (a) Generalized nonlocality testing of a
$\Lambda$-shaped quantum network consisting of two bipartite entangled states
$\rho_{AB}$ and $\rho_{CD}$. Assume that Alice and Bob share the bipartite
entanglement $\rho_{AB}$ while Bob and Charlie share the bipartite
entanglement $\rho_{CD}$. $\tau^{a_{1}}_{A_{0}}$,
$\tau^{a_{2}}_{B_{0}}\otimes\tau^{a_{3}}_{C_{0}}$ and $\tau^{a_{4}}_{D_{0}}$
are input states of Alice, Bob, and Charlie, respectively, and can be sent
from a trusty party, where $A_{0},B_{0},C_{0},D_{0}$ are ancillary systems,
and $a_{i}\in{\mathscr{A}}_{i}$, $i=1,\cdots,4$.
${x}_{1}\in{\mathscr{X}}_{1},{x}_{2}{x}_{3}\in{\mathscr{X}}_{2}\times{\mathscr{X}}_{3}$,
and ${x}_{4}\in{\mathscr{X}}_{4}$ are outputs of Alice, Bob, and Charlie,
respectively. (b) Hidden state model for testing the locality of a
$\Lambda$-shaped network consisting of two shared sources $\lambda_{1}$ and
$\lambda_{2}$ or separable states
$\varrho_{ABCD}=\sum_{i}t_{i}\varrho^{(i)}_{a}\otimes\varrho^{(i)}_{BC}\otimes\varrho^{(i)}_{D}$,
where $\\{t_{i}\\}$ is a probability distribution and
$\varrho^{(i)}_{A(BC,D)}$ are density operators of the system $A$ (or $BC$, or
$D$). Here, the joint system $BC$ can be entangled. (c) Generalized Bell
testing of the activated nonlocality of the subnetwork consisting of Alice and
Charlie. Bob firstly performs a proper POVM measurement
$Q^{{x}^{*}_{2}{x}^{*}_{3}}_{B_{0}BCC_{0}}$ on his system and gets special
output ${x}^{*}_{2}{x}^{*}_{3}$. And then, the standard bipartite Bell testing
is performed by Alice and Bob for the collapsed state
$\rho_{AD}=\textrm{Tr}_{B_{0}BCC_{0}}[Q^{{x}^{*}_{2}{x}^{*}_{3}}_{B_{0}BCC_{0}}(\tau^{a_{2}}_{B_{0}}\otimes\rho_{AB}\otimes\rho_{CD}\otimes\tau^{a_{3}}_{C_{0}})]$.
(d) Hidden state model for testing the locality of the subnetwork consisting
of Alice and Charlie, where Bob performs any POVM measurement before verifying
the locality. The $\Lambda$-shaped network consists of two shared sources
$\lambda_{1},\lambda_{2}$, or one source $\lambda_{1}$ and a bipartite
separable state entanglement $\varrho_{BCD}=$, or a tripartite fully separable
state $\varrho_{A\,BC\,D}$). The probabilities of input states
$\tau^{a_{1}},\cdots,\tau^{a_{4}}$ are uniform distributions.

A1: Tripartite Bell testing experiment

Now, consider a new tripartite Bell testing experiment as shown in Figure
S2(a) and Figure S2(b). Assume that the input and output sets of Bell testing
experiments for $\rho_{AB}$ and $\rho_{CD}$ are
${\mathscr{A}}_{i}=\\{1,2,\cdots,m_{i}\\}$,
${\mathscr{X}}_{i}=\\{1,2,\cdots,n_{i}\\}$, respectively, $i=1,\cdots,4$.

Assume that the input states of Alice, Bob and Charlie are
$\tau^{{a}_{1}}_{A_{0}}$,
$\tau^{{a}_{2}}_{B_{0}}\otimes\tau^{{a}_{3}}_{C_{0}}$ and
$\tau^{{a}_{4}}_{D_{0}}$, respectively. Based on the POVMs
$\\{P^{a_{1}}\\},\\{P^{a_{2}}\\}$, $\\{Q^{x_{1}}\\},\\{Q^{x_{2}}\\}$, define
new POVMs of three observers as $\\{{P}^{{x}_{1}}\\}\in{\cal
M}_{A_{0}A;{\mathscr{X}}_{1}}$, $\\{{Q}^{{x}_{2}{x}_{3}}\\}\in{\cal
M}_{B_{0}BCC_{0};{\mathscr{X}}_{2}\times{\mathscr{X}}_{3}}$, and
$\\{{S}^{{x}_{4}}\\}\in{\cal M}_{DD_{0};{\mathscr{X}}_{4}}$, respectively.

From the equations (A2) and (A5), the joint conditional probability
distribution
$P({x}_{1},{x}_{2}{x}_{3},{x}_{4}|{a}_{1},{a}_{2}{a}_{3},{a}_{4})$ is given as
follows:

$\displaystyle
P({x}_{1},{x}_{2}{x}_{3},{x}_{4}|{a}_{1},{a}_{2}{a}_{3},{a}_{4})=\textrm{Tr}[({P}^{{x}_{1}}\otimes{Q}^{{x}_{2}{x}_{3}}\otimes{S}^{{x}_{4}})(\tau^{{a}_{1}}_{A_{0}}\otimes\rho_{AB}\otimes\tau^{{a}_{2}}_{B_{0}}\otimes\tau^{{a}_{3}}_{C_{0}}\otimes\rho_{CD}\otimes\tau^{{a}_{4}}_{D_{0}}].$
(A7)

Define an average gain $\wp$ depending on all conditional probability
distributions
$P({x}_{1},{x}_{2}{x}_{3},{x}_{4}|{a}_{1},{a}_{2}{a}_{3},{a}_{4})$s as

$\displaystyle\wp=\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{1},\cdots,{x}_{4}}}\gamma^{{x}_{1}\cdots{x}_{4}}_{{a}_{1}\cdots{a}_{4}}P({x}_{1},{x}_{2}{x}_{3},{x}_{4}|{a}_{1},{a}_{2}{a}_{3},{a}_{4}),$
(A8)

where the coefficients $\gamma^{{x}_{1}\cdots{x}_{4}}_{{a}_{1}\cdots{a}_{4}}$
satisfy
$\gamma^{{x}_{1}\cdots{x}_{4}}_{{a}_{1}\cdots{a}_{4}}=\alpha^{{x}_{1}{x}_{2}}_{{a}_{1}{a}_{2}}+\beta^{{x}_{3}{x}_{4}}_{{a}_{3}{a}_{4}}$.

From the equations (A7) and (A8), consider the quantum tripartite correlations
obtained by locally measuring the joint system of $\Lambda$-shaped quantum
network (see Figure S2(a)) with local POVMs
$\\{{P}_{A_{0}A}^{{x}_{1}}\\}\in{\cal M}_{A_{0}A;{\mathscr{X}}_{1}}$ for
Alice, $\\{{Q}_{B_{0}BCC_{0}}^{{x}_{2}{x}_{3}}\\}\in{\cal
M}_{B_{0}BCC_{0};{\mathscr{X}}_{2}\times{\mathscr{X}}_{3}}$ for Bob, and
$\\{{S}_{DD_{0}}^{{x}_{4}}\\}\in{\cal M}_{DD_{0};{\mathscr{X}}_{4}}$ for
Charlie. It is easy to obtain that

$\displaystyle\wp=$
$\displaystyle\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{1},\cdots,{x}_{4}}}\gamma^{{x}_{1}\cdots{x}_{4}}_{{a}_{1}\cdots{a}_{4}}P({x}_{1},{x}_{2}{x}_{3},{x}_{4}|{a}_{1},{a}_{2}{a}_{3},{a}_{4})$
$\displaystyle=$
$\displaystyle\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{1},\cdots,{x}_{4}}}(\alpha^{x_{1}x_{2}}_{{a}_{1}a_{2}}+\beta^{x_{3}x_{4}}_{{a}_{3}a_{4}})P({x}_{1},{x}_{2}|{a}_{1},{a}_{2})P({x}_{3},{x}_{4}|{a}_{3},{a}_{4})$
$\displaystyle=$ $\displaystyle
m_{3}m_{4}\sum_{{a}_{1},{a}_{2},{x}_{1},{x}_{2}}\alpha^{x_{1}x_{2}}_{{a}_{1}a_{2}}P(x_{1},x_{2}|a_{1},a_{2})+m_{1}m_{2}\sum_{{a}_{3},{a}_{4},{x}_{3},{x}_{4}}\beta^{x_{3}x_{4}}_{{a}_{3}a_{4}}P(x_{3},x_{4}|a_{3},a_{4})$
(A9) $\displaystyle>$ $\displaystyle m_{3}m_{4}c_{1}+m_{1}m_{2}c_{2},$ (A10)

where the equation (A9) is from the equalities
$\sum_{a_{3},a_{4},x_{3},x_{4}}P(x_{3},x_{4}|a_{3},a_{4})=\sum_{a_{3},a_{4}}\sum_{x_{3},x_{4}}P(x_{3},x_{4}|a_{3},a_{4})=\sum_{a_{3},a_{4}}1=m_{3}m_{4}$
(from the normalization of conditional probabilities) and
$\sum_{a_{1},a_{2},x_{1},x_{2}}P(x_{1},x_{2}|a_{1},a_{2})=\sum_{a_{1},a_{2}}\sum_{x_{1},x_{2}}P(x_{1},x_{2}|a_{1},a_{2})=\sum_{a_{1},a_{2}}1=m_{1}m_{2}$.
The inequality (A10) is from the inequalities (A1) and (A4).

In what follows, we estimate the upper bound of $\wp$ defined in the equation
(A8) for classical tripartite correlations in terms of hidden state model Cire
as shown in Figure S2(b). In detail, from the inequalities (A3) and (A6), it
is reasonable to define classical tripartite correlations
$P_{c}({x}_{1},{x}_{2}{x}_{3},{x}_{4}|{a}_{1},{a}_{2}{a}_{3},{a}_{4})$ as the
joint conditional probabilities of measuring a shared tripartite fully
separable state
$\varrho=\sum_{i}t_{i}\varrho^{(i)}_{A}\otimes\varrho^{(i)}_{BC}\otimes\varrho^{(i)}_{D}$
($\\{t_{i}\\}$ is a probability distribution) with local POVMs
$\\{\hat{P}_{A_{0}A}^{{x}_{1}}\\}\in{\cal{M}}_{A_{0}A;{\mathscr{X}}_{1}},\\{\hat{Q}_{B_{0}BCC_{0}}^{{x}_{2}{x}_{3}}\\}\in{\cal{M}}_{B_{0}BCC_{0};{\mathscr{X}}_{2}\times{\mathscr{X}}_{3}},\\{\hat{S}_{DD_{0}}^{{x}_{4}}\\}\in{\cal{M}}_{DD_{0};{\mathscr{X}}_{4}}$.
We get an inequality from the equation (A10) as

$\displaystyle\wp_{c}=$
$\displaystyle\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{1},\cdots,{x}_{4}}}\gamma^{{x}_{1}\cdots{x}_{4}}_{{a}_{1}\cdots{a}_{4}}P_{c}({x}_{1},{x}_{2}{x}_{3},{x}_{4}|{a}_{1},{a}_{2}{a}_{3},{a}_{4})$
$\displaystyle=$
$\displaystyle\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{1},\cdots,\overline{x}_{4}}}\gamma^{{x}_{1}\cdots{x}_{4}}_{{a}_{1}\cdots{a}_{4}}\textrm{Tr}[(\hat{P}_{A_{0}A}^{{x}_{1}}\otimes\hat{Q}_{B_{0}BCC_{0}}^{{x}_{2}{x}_{3}}\otimes\hat{S}_{DD_{0}}^{{x}_{4}})$
$\displaystyle\times(\tau^{{a}_{1}}_{A_{0}}\otimes\tau^{{a}_{2}}_{B_{0}}\otimes\tau^{{a}_{3}}_{C_{0}}\otimes\tau^{{a}_{4}}_{D_{0}}\otimes\sum_{i}t_{i}\varrho^{(i)}_{A}\otimes\varrho^{(i)}_{BC}\otimes\varrho^{(i)}_{D})]$
$\displaystyle=$
$\displaystyle\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{1},\cdots,{x}_{4}}}(\alpha^{x_{1}x_{2}}_{{a}_{1}a_{2}}+\beta^{x_{3}x_{4}}_{{a}_{3}a_{4}})\textrm{Tr}[(\hat{P}_{A_{0}A}^{{x}_{1}}\otimes\hat{Q}_{B_{0}BCC_{0}}^{{x}_{2}{x}_{3}}\otimes\hat{S}_{DD_{0}}^{{x}_{4}})$
$\displaystyle\times(\tau^{{a}_{1}}_{A_{0}}\otimes\tau^{{a}_{2}}_{B_{0}}\otimes\tau^{{a}_{3}}_{C_{0}}\otimes\tau^{{a}_{4}}_{D_{0}}\otimes\sum_{i}t_{i}\varrho^{(i)}_{A}\otimes\varrho^{(i)}_{BC}\otimes\varrho^{(i)}_{D})]$
$\displaystyle=$
$\displaystyle\sum_{i}t_{i}\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{1},\cdots,{x}_{4}}}(\alpha^{x_{1}x_{2}}_{{a}_{1}a_{2}}+\beta^{x_{3}x_{4}}_{{a}_{3}a_{4}})P^{(i)}(x_{1}|a_{1})P^{(i)}(x_{2}x_{3}|a_{2}a_{3})P^{(i)}(x_{4}|a_{4})$
(A11) $\displaystyle=$
$\displaystyle\sum_{i}t_{i}[\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{1},\cdots,{x}_{4}}}\alpha^{x_{1}x_{2}}_{{a}_{1}a_{2}}P^{(i)}(x_{1}|a_{1})P^{(i)}(x_{2},x_{3}|a_{2},a_{3})P^{(i)}(x_{4}|a_{4})$
$\displaystyle+\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{1},\cdots,{x}_{4}}}\beta^{x_{3}x_{4}}_{{a}_{3}a_{4}}P^{(i)}(x_{1}|a_{1})P^{(i)}(x_{2},x_{3}|a_{2},a_{3})P^{(i)}(x_{4}|a_{4})]$
$\displaystyle=$
$\displaystyle\sum_{i}t_{i}[\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{1},{x}_{2}}}\alpha^{x_{1}x_{2}}_{{a}_{1}a_{2}}P^{(i)}(x_{1}|a_{1})P^{(i)}(x_{2}|a_{2},a_{3})$
$\displaystyle+\sum_{{a}_{1},\cdots,{a}_{4},\atop{{x}_{3},{x}_{4}}}\beta^{x_{3}x_{4}}_{{a}_{3}a_{4}}P^{(i)}(x_{3}|a_{2},a_{3})P^{(i)}(x_{4}|a_{4})]$
(A12) $\displaystyle=$
$\displaystyle\sum_{i}t_{i}[\sum_{a_{3},a_{4}}\sum_{{a}_{1},{a}_{2},\atop{x_{1},{x}_{2}}}\alpha^{x_{1}x_{2}}_{{a}_{1}a_{2}}P^{(i)}(x_{1}|a_{1})P^{(i)}(x_{2}|a_{2},a_{3})$
$\displaystyle+\sum_{{a}_{1},a_{2}}\sum_{a_{3},a_{4},\atop{x_{3},x_{4}}}\beta^{x_{3}x_{4}}_{{a}_{3}a_{4}}P^{(i)}(x_{3}|a_{2},a_{3})P^{(i)}(x_{4}|a_{4})]$
$\displaystyle\leq$
$\displaystyle\sum_{i}t_{i}[\sum_{a_{3},a_{4}}c_{1}+\sum_{{a}_{1},a_{2}}c_{2}]$
(A13) $\displaystyle=$ $\displaystyle m_{3}m_{4}c_{1}+m_{1}m_{2}c_{2}.$ (A14)

In the equation (A11), we have taken use of the following notations:
$P^{(i)}(x_{1}|a_{1})=\textrm{Tr}[\hat{P}_{A_{0}A}^{x_{1}}(\tau^{{a}_{1}}_{A_{0}}\otimes\varrho^{(i)}_{a})]$,
$P^{(i)}(x_{2}x_{3}|a_{2}a_{3})=\textrm{Tr}[\hat{Q}_{B_{0}BCC_{0}}^{x_{2}x_{3}}(\tau^{{a}_{2}}_{B_{0}}\otimes\tau^{{a}_{3}}_{C_{0}}\otimes\varrho^{(i)}_{BC})]$,
and
$P^{(i)}(x_{4}|a_{4})=\textrm{Tr}[\hat{S}_{DD_{0}}^{x_{4}}(\tau^{{a}_{4}}_{D_{0}}\otimes\varrho^{(i)}_{D})]$.
In order to get the equation (A12) we have used the equalities
$\sum_{x_{3}}P^{(i)}(x_{2},x_{3}|a_{2},a_{3})=P^{(i)}(x_{2}|a_{2},a_{3})$,
$\sum_{x_{2}}P^{(i)}(x_{2},x_{3}|a_{2},a_{3})=P^{(i)}(x_{3}|a_{2},a_{3})$,
$\sum_{x_{1}}P^{(i)}(x_{1}|a_{1})=\sum_{x_{4}}P^{(i)}(x_{4}|a_{4})=1$ for each
$i,j$, $a_{1},\cdots,a_{4}$. In inequality (A13),
$P^{(i)}(x_{1}|a_{1})P^{(i)}(x_{2}|a_{2},a_{3})$ are independent conditional
probabilities in terms of the variables $x_{1},x_{2}$;
$P^{(i)}(x_{3}|a_{2},a_{3})P^{(i)}(x_{4}|a_{4})$ are independent conditional
probabilities in terms of the variables $x_{3},x_{4}$. Hence, for each $i,j$,
$a_{2},a_{3},a_{4}$, we can take use of the inequalities (A3) and (A6). The
equation (A14) is from the fact that $\\{t_{i}\\}$ is a probability
distribution.

Hence, the equation (A7) has defined a generalized Bell-type inequality for
verifying the tripartite nonlocality of a $\Lambda$-shaped quantum network
consisting of bipartite entangled states $\rho_{AB}\otimes\rho_{CD}$.

A2: Activated nonlocality of $\Lambda$-shaped quantum network

To prove the activated nonlocality, it is sufficient to prove that there are
local observables for all observers such that one local measurement of one
observer can create one bipartite entanglement for other two observers. Note
that Alice and Bob, or Bob and Charlie have shared one bipartite entanglement.
It only needs to prove the activated locality of Alice and Charlie with the
help of Bob. The proof of the activated nonlocality is completed by showing
that the bipartite correlations of Alice and Charlie (after a local
measurement of Bob) are inconsistent with these from any semiseparable state
$\varrho_{ABCD}$ in terms of some generalized Bell-type inequality for two
systems.

The proof is completed by two cases:

Case 1. $\Lambda$-shaped quantum network with qubit systems

In this case, we show that there is an entanglement $\rho_{AD}$ derived from
$\rho_{AB}\otimes\rho_{CD}$ after performing a proper measurement on the joint
system $BC$ by Bob. Note that all qubit-based bipartite mixed entangled states
are distillable HHH . It means that Alice and Bob can obtain one bipartite
entangled pure state $|\Phi\rangle_{AB}$ from $\rho_{AB}^{\otimes n}$ by using
local operations and classical communication (LOCC) when $n$ is large, where
they do not need to obtain one maximally entangled pure state. Similarly, Bob
and Charlie can obtain one bipartite entangled pure state $|\Psi\rangle_{CD}$
from $\rho_{AB}^{\otimes n}$ when $n$ is large by using LOCC. These quantum
operations are reasonable because LOCC or local operations and shared
randomness cannot create entanglement between two parties initially sharing no
entanglement HHH ; SQ . For new quantum systems $ABCD$ in the state
$|\Phi\rangle_{AB}\otimes|\Psi\rangle_{CD}$, it is easy to prove that the
bipartite nonlocality of Alice and Charlie can be activated after a Bell
measurement of Bob on his systems $BC$, i.e., the entanglement swapping holds
for any bipartite entangled pure states GMT ; Luo . The activated nonlocality
can be proved by using CHSH inequality CHSH ; GMT ; Luo , where all bipartite
entangled pure states can be verified using a universal Bell inequality-CHSH
inequality.

Figure S3: (Color online) Schematic projection circuit of bipartite entangled
states. (a) Project a bipartite entangled state $\hat{\rho}_{AB}$ into the
subspace ${\cal H}_{i_{1}i_{2}}$ spanned by
$\\{|i_{1}i_{1}\rangle,|i_{1}i_{2}\rangle,|i_{2}i_{1}\rangle,|i_{2}i_{2}\rangle\\}$.
$\hat{\rho}_{AB}$ is shared by Alice and Bob. $CNOT_{i_{1}i_{2}}$ is
controlled-NOT operation performed on the joint system of $A$ and $A_{0}$ by
Alice, or $B$ and $B_{0}$ by Bob. It is defined as:
$CNOT_{i_{1}i_{2}}:|i\rangle_{A}|0\rangle_{A_{0}}\mapsto|i\rangle_{A}|1\rangle_{A_{0}}$
for $i=i_{1},i_{2}$, and
$CNOT_{i_{1}i_{2}}:|i\rangle_{A}|0\rangle_{A_{0}}\mapsto|i\rangle_{A}|0\rangle_{A_{0}}$
for all $i\not=i_{1},i_{2}$;
$CNOT_{i_{1}i_{2}}:|i\rangle_{B}|0\rangle_{B_{0}}\mapsto|i\rangle_{B}|1\rangle_{B_{0}}$
for $i=i_{1},i_{2}$, and
$CNOT_{i_{1}i_{2}}:|i\rangle_{B}|0\rangle_{B_{0}}\mapsto|i\rangle_{B}|0\rangle_{B_{0}}$
for all $i\not=i_{1},i_{2}$, where $A_{0}$ and $B_{0}$ are auxiliary systems
in the state $|0\rangle$. $P_{A_{0}}$ or $P_{B_{0}}$ denotes the projection
measurement under the basis $\\{|0\rangle,|1\rangle\\}$ on the system $A_{0}$
or $B_{0}$ respectively. $P_{i_{1}i_{2}}$ denotes the permutation operations:
$|i_{1}\rangle\mapsto|0\rangle$ and $|i_{2}\rangle\mapsto|1\rangle$.
$P_{i_{1}i_{2}}$ is performed if the measurement outcome is $1$. (b) Project a
bipartite entangled state $\hat{\rho}_{CD}$ into the subspace ${\cal
H}_{j_{1}j_{2}}$ spanned by
$\\{|j_{1}j_{1}\rangle,|j_{1}j_{2}\rangle,|j_{2}j_{1}\rangle,|j_{2}j_{2}\rangle\\}$.
$\hat{\rho}_{CD}$ is shared by Bob and Charlie. $CNOT_{j_{1}j_{2}}$ denotes
controlled-NOT operation performed on the joint system of $C$ and $C_{0}$ by
Bob, or $D$ and $D_{0}$ by Charlie. It is defined as:
$CNOT_{j_{1}j_{2}}:|j\rangle_{C}|0\rangle_{C_{0}}\mapsto|j\rangle_{C}|1\rangle_{C_{0}}$
for $j=j_{1},j_{2}$ and
$CNOT_{j_{1}j_{2}}:|j\rangle_{C}|0\rangle_{C_{0}}\mapsto|j\rangle_{C}|1\rangle_{C_{0}}$
for $i=i_{1},i_{2}$ all $j\not=j_{1},j_{2}$;
$CNOT_{j_{1}j_{2}}:|j\rangle_{D}|0\rangle_{D_{0}}\mapsto|j\rangle_{D}|1\rangle_{D_{0}}$
for $j=j_{1},j_{2}$ and
$CNOT_{j_{1}j_{2}}:|j\rangle_{D}|0\rangle_{D_{0}}\mapsto|j\rangle_{D}|1\rangle_{D_{0}}$
for $i=i_{1},i_{2}$ all $j\not=j_{1},j_{2}$, where $C_{0}$ and $D_{0}$ are
auxiliary systems in the state $|0\rangle$. $P_{C_{0}}$ or $P_{D_{0}}$ denotes
the projection measurement under the basis $\\{|0\rangle,|1\rangle\\}$ on the
system $C_{0}$ or $D_{0}$ respectively. $P_{j_{1}j_{2}}$ denotes the
permutation operations: $|j_{1}\rangle\mapsto|0\rangle$ and
$|j_{2}\rangle\mapsto|1\rangle$. $P_{j_{1}j_{2}}$ is performed if the
measurement outcome is $1$.

Case 2. $\Lambda$-shaped quantum network with high-dimensional systems

In this case, we show that there is a qubit-based entanglement $\rho_{AD}$
derived from high-dimensional entangled states $\rho_{AB}\otimes\rho_{CD}$
after performing proper projective measurements on the joint systems $BC$ by
Bob. Assume that
$\rho_{AB}=\sum_{i}p_{i}|\Phi_{i}\rangle_{AB}\langle\Phi_{i}|$ and
$\rho_{CD}=\sum_{j}q_{j}|\Psi_{j}\rangle_{CD}\langle\Psi_{j}|$, where
$|\Phi_{i}\rangle_{AB},|\Psi_{j}\rangle_{CD}$ are entangled pure states for
some $i,j$, $\\{p_{i}\\}$ and $\\{q_{j}\\}$ are probability distributions. To
explain the main idea clearly, we only consider the simplest case in what
follows, where similar proof can be easily followed for general cases. Assume
that $\rho_{AB}$ and $\rho_{CD}$ has the following forms:

$\displaystyle\rho_{AB}=p_{1}|\Phi_{1}\rangle_{AB}\langle\Phi_{1}|+p_{2}|\Phi_{2}\rangle_{AB}\langle\Phi_{2}|,$
(A15)
$\displaystyle\rho_{CD}=q_{1}|\Psi_{1}\rangle_{CD}\langle\Psi_{1}|+q_{2}|\Psi_{2}\rangle_{CD}\langle\Psi_{2}|,$
(A16)

where $|\Phi_{1}\rangle_{AB},|\Psi_{1}\rangle_{CD}$ are assumed to be
bipartite entangled pure states. Let the bipartite decompositions of
$|\Phi_{1}\rangle_{AB},|\Psi_{1}\rangle_{CD}$ be
$|\Phi_{1}\rangle_{AB}=\sum_{i}r_{i}|\phi_{i}\rangle_{A}|{\phi}^{\prime}_{i}\rangle_{B},|\Psi_{1}\rangle_{CD}=\sum_{j}s_{j}|\psi_{j}\rangle_{C}|{\psi}^{\prime}_{j}\rangle_{D}$,
where
$\\{|\phi_{i}\rangle_{A}\\},\\{|{\phi}^{\prime}_{i}\rangle_{B}\\},\\{|\psi_{j}\rangle_{C}\\},\\{|\psi^{\prime}_{j}\rangle_{D}\\}$
are orthogonal basis states of the system $A,\cdots,D$, respectively, and
$\\{r_{i}\\}$ and $\\{s_{j}\\}$ are probability distributions. After proper
local unitary operations performed by three parties, i.e.,
$U:|\phi_{i}\rangle_{A}\mapsto|i\rangle_{A}$ by Alice,
$V:|\phi^{\prime}_{i}\rangle_{B}\mapsto|i\rangle_{B}$ and
$W:|\psi_{j}\rangle_{C}\mapsto|j\rangle_{C}$ by Bob, and
$Q:|\psi^{\prime}_{j}\rangle_{D}\mapsto|j\rangle_{D}$ by Charlie, the systems
of $\rho_{AB},\rho_{CD}$ are changed as follows:

$\displaystyle\hat{\rho}_{AB}$ $\displaystyle=(U\otimes
V)\rho_{AB}(U^{\dagger}\otimes V^{\dagger})$ $\displaystyle=$ $\displaystyle
p_{1}(\sum_{i}r_{i}|ii\rangle_{AB})(\sum_{i}r_{i}\langle
ii|_{AB})+p_{2}|\hat{\Phi}_{2}\rangle_{AB}\langle\hat{\Phi}_{2}|,$ (A17)
$\displaystyle\hat{\rho}_{CD}$ $\displaystyle=(W\otimes
Q)\rho_{AB}(W^{\dagger}\otimes Q^{\dagger})$ $\displaystyle=$ $\displaystyle
q_{1}(\sum_{j}s_{j}|jj\rangle_{CD})(\sum_{j}r_{j}\langle
jj|_{CD})+q_{2}|\hat{\Psi}_{2}\rangle_{CD}\langle\hat{\Psi}_{2}|,$ (A18)

where $|\hat{\Phi}_{2}\rangle_{AB}=(U\otimes V)|\Phi_{2}\rangle_{AB}$ and
$|\hat{\Psi}_{2}\rangle_{CD}=(W\otimes Q)|\Psi_{2}\rangle_{CD}$.

Define ${\cal{H}}_{i_{1}i_{2}}$ as the subspaces spanned by
$\\{|i_{1}i_{1}\rangle,|i_{1}i_{2}\rangle,|i_{2}i_{1}\rangle,|i_{2}i_{2}\rangle\\}$.
By taking use of the quantum circuit (projective operations) shown in Figure
S3(a), Alice and Bob can obtain one bipartite state
$\varrho^{i_{1}i_{2}}_{AB}$ given by

$\displaystyle\varrho^{i_{1}i_{2}}_{AB}=p_{1}(r_{i_{1}}|00\rangle+r_{i_{2}}|11\rangle)_{AB}(r_{i_{1}}\langle
00|+r_{i_{2}}\langle
11|)+q_{1}|\hat{\Phi}_{i_{1}i_{2}}\rangle_{AB}\langle\hat{\Phi}_{i_{1}i_{2}}|$
(A19)

for the measurement outcome $00$ with a probability $p_{00}$, where
$|\hat{\Phi}_{i_{1}i_{2}}\rangle_{AB}\in{\cal H}_{i_{1}i_{2}}$ can be a proper
pure state. If $\varrho^{i_{1}i_{2}}_{AB}$ is entangled, Alice and Bob obtain
a qubit-based bipartite entanglement. Otherwise, one of collapsed states of
Alice and Bob for the measurement outcomes $01,10$ and $11$ is entangled
because the linear superposition of these four states, i.e.,
$\hat{\rho}_{AB}$, is entangled, where we have taken use of the fact that the
linear superposition of any separable states are separable. Hence, two
observers can obtain a collapsed entanglement by performing projective
operations and classical communications. And then they let
$\varrho^{i_{1}i_{2}}_{AB}$ as the input of the quantum circuit shown in
Figure S3(a). This procedure can be iteratively performed with LOCC and post-
selections of Alice and Bob. Hence, Alice and Bob can obtain a qubit-based
bipartite entanglement with nonzero success probability. Similarly, Bob and
Charlie can probabilistically obtain a qubit-based entanglement
$\varrho^{j_{1}j_{2}}_{CD}$ by iteratively performing LOCC and post-selection
from the quantum circuit shown in Figure S3(b).

Similar to Case 1, if $\varrho^{i_{1}i_{2}}_{AB}$ and
$\varrho^{j_{1}j_{2}}_{CD}$ are entangled pure states, the bipartite
nonlocality of Alice and Charlie can be activated after a Bell measurement of
Bob, using CHSH inequality CHSH ; Gisin1 ; Luo . Otherwise, they can obtain
entangled pure states using the entanglement distillation HHH  if multiple
copies of $\varrho^{i_{1}i_{2}}_{AB}$ and $\varrho^{j_{1}j_{2}}_{CD}$ are
available. Therefore, we have completed verifying the activated nonlocality of
Theorem 1. Another method is as follows: For a $\Lambda$-shaped network
consisting of Alice, Bob and Charlie, they can obtain two bipartite entangled
pure states. In this case, nonlinear Bell-type inequalities GMT ; Luo  can be
used to prove the $k$-partite nonlocality. $\blacksquare$

From Theorem 1, it is easy to verify the following generalized
$\Lambda$-shaped quantum networks and chain-shaped quantum networks.

Figure S4: (Color online) Schematically generalized Bell testing of a
generalized $\Lambda$-shaped network. (a) Generalized nonlocality testing of a
generalized $\Lambda$-shaped quantum network consisting of two entangled
states $\rho_{\bf A},\rho_{\bf B}$, where ${\textbf{A}}=A_{1}\cdots A_{k}$ and
${\bf B}=B_{k}\cdots B_{n}$. The observers $\texttt{P}_{1}$, $\cdots$,
$\texttt{P}_{k}$ share one $k$-partite entanglement $\rho_{\textbf{A}}$ while
the observers $\texttt{P}_{k}$, $\cdots$, $\texttt{P}_{n}$ share an
$n-k+1$-partite entanglement $\rho_{\bf B}$. $\tau^{a_{1}}_{A^{\prime}_{1}}$,
$\tau^{a_{2}}_{A^{\prime}_{2}}$, $\cdots$,
$\tau^{a_{k}}_{A^{\prime}_{k}}\otimes\tau^{b_{k}}_{B_{k}^{\prime}},\cdots,\tau^{b_{n}}_{B_{n}^{\prime}}$
are input states of the observers $\texttt{P}_{1}$, $\cdots$,
$\texttt{P}_{k}$, $\cdots$, $\texttt{P}_{n}$, respectively, and can be sent
from a trusty referee, where $a_{i}\in{\mathscr{A}}_{i}$ and
${x}_{i}\in{\mathscr{X}}_{i}$ are respective inputs and outputs of the
observer $\texttt{P}_{i}$, $i=1,\cdots,k$; $b_{j}\in{\mathscr{B}}_{j}$ and
${y}_{j}\in{\mathscr{Y}}_{j}$ are respective inputs and outputs of the
observer $\texttt{P}_{j}$, $j=k,\cdots,n$. (b) Hidden state model for testing
the locality of a generalized $\Lambda$-shaped network consisting of two
shared sources $\lambda_{1}$ and $\lambda_{2}$ (or two bipartite separable
states $\varrho_{\bf
A}=\sum_{i}p_{i}\varrho^{(i)}_{A_{1}}\otimes\cdots\otimes\varrho^{(i)}_{A_{k}}$
and $\varrho_{\bf
B}=\sum_{i}q_{i}\varrho^{(i)}_{B_{k}}\otimes\cdots\otimes\varrho^{(i)}_{B_{n}}$,
where $\\{p_{i}\\}$ and $\\{q_{j}\\}$ are probability distributions and
$\varrho^{(i)}_{A_{j}(B_{s})}$ are density operators of the system $A_{j}$ (or
$B_{s}$)). (c) Generalized Bell testing of the activated nonlocality of the
subnetwork consisting of the observers $\texttt{P}_{i}$, $\texttt{P}_{k}$ and
$\texttt{P}_{j}$ with a $\Lambda$-shaped system
$\rho_{A_{i}A_{k}}\otimes\rho_{B_{k}B_{j}}$, where $\rho_{A_{i}A_{k}}$ is the
collapsed state of $\rho_{\bf A}$ after a proper local measurement of all
observers $\texttt{P}_{1}$, $\cdots$, $\texttt{P}_{k-1}$ except for the
observer $\texttt{P}_{i}$; $\rho_{B_{k}B_{n}}$ is the collapsed state of
$\rho_{\bf B}$ after a proper local measurement of all observers
$\texttt{P}_{k+1}$, $\cdots$, $\texttt{P}_{n}$ except for the observer
$\texttt{P}_{j}$, and $\tau^{a_{s}^{*}},\tau^{b_{t}^{*}}$ and
${x}_{s}^{*},{y}_{t}^{*}$ are respective special inputs and outputs. The new
system is a standard $\Lambda$-shaped quantum network shown in Figure S2(a).
(d) Hidden state model for testing the locality of the subnetwork consisting
of three observers $\texttt{P}_{i}$, $\texttt{P}_{k}$ and $\texttt{P}_{j}$
with the reduced sources of $\lambda_{1}$ and $\lambda_{2}$ (or the collapsed
states of any semi-separable states
$\varrho_{\textbf{A}}=\sum_{s}p_{s}\varrho^{(s)}_{A_{i}}\otimes\varrho^{(s)}_{\hat{\textbf{A}}_{i}}$
and $\varrho_{\bf
B}=\sum_{t}q_{t}\varrho^{(t)}_{B_{j}}\otimes\varrho^{(t)}_{\hat{\textbf{B}}_{j}}$,
where $\\{p_{s}\\}$ and $\\{q_{t}\\}$ are probability distributions, and
$\varrho^{(s)}_{\hat{\textbf{A}}_{i}}$ and $\varrho^{(t)}_{\hat{\bf B}_{j}}$
are density operators of the joint systems $\hat{\textbf{A}}_{i}=A_{1}\cdots
A_{i-1}A_{i+1}\cdots A_{k}$ (or $\hat{\bf B}_{j}=B_{k}\cdots
B_{j-1}B_{j+1}\cdots B_{n}$)), which can be entangled. The new system is a
standard $\Lambda$-shaped network shown in Figure S2(b).

Corollary 1. For any generalized $\Lambda$-shaped quantum network ${\cal N}$
with $n$ observers $\texttt{P}_{1}$, $\texttt{P}_{2}$, $\cdots$,
$\texttt{P}_{n}$, assume that $\texttt{P}_{1},\cdots$, $\texttt{P}_{k}$ share
one $k$-partite entanglement $\rho_{A_{1}\cdots A_{k}}$ on the Hilbert space
${\cal H}_{A_{1}}\otimes\cdots\otimes{\cal H}_{A_{k}}$, and
$\texttt{P}_{k+1},\cdots,\texttt{P}_{n}$ share an $n-k+1$-partite entanglement
$\rho_{B_{k}\cdots B_{n}}$ on the Hilbert space
${\cal{H}}_{B_{k}}\otimes\cdots\otimes{\cal H}_{B_{n}}$. Then the joint system
of $\rho_{A_{1}\cdots A_{k}}\otimes\rho_{B_{k}\cdots B_{n}}$ is $n$-partite
nonlocal. Moreover, any two observers $\texttt{P}_{i}$ and $\texttt{P}_{j}$
with $1\leq i\leq k$, $k\leq j\leq n$ and $i\not=j$ has bipartite activated
nonlocality after all the other observers in ${\cal{N}}$ perform proper local
POVM.

Proof. For a generalized $\Lambda$-shaped quantum network shown in Figure S4,
from forward evaluations of Theorem 1 it is easy to prove the inconsistency of
$n$-partite quantum correlations derived from the local POVMs of a generalized
$\Lambda$-shaped quantum system $\rho_{A_{1}\cdots
A_{k}}\otimes\rho_{B_{k}\cdots B_{n}}$ shown in Figure S4(a), and $n$-partite
classical correlations from from the local measurements of any fully separable
states $\varrho_{A_{1}\cdots A_{k}}\otimes\varrho_{B_{k}\cdots
B_{n}}=(\sum_{i}p_{i}\varrho^{(i)}_{A_{1}}\otimes\cdots\otimes\varrho^{(i)}_{A_{k}})\otimes(\sum_{i}q_{i}\varrho^{(i)}_{B_{k}}\otimes\cdots\otimes\varrho^{(i)}_{B_{n}})$
shown in Figure S4(b)) in terms of the hidden state model Cire  using the
semiquantum nonlocal game SQ  and similar procedure of the proofs given in
Subsection A1 of Appendix A.

Now, consider the activated nonlocality of any quantum subnetwork. For any
$k$-partite quantum entanglement $\rho_{A_{1}\cdots A_{k}}$, and two observers
$\texttt{P}_{i},\texttt{P}_{j}$ with $i,j\leq k$, there are local POVMs of all
observers $\texttt{P}_{s}$ with $s\not=i,j$ and $s\leq k$ such that the
collapsed state of two observers $\texttt{P}_{i}$ and $\texttt{P}_{j}$ after a
local POVM of all the other observers $\texttt{P}_{s}$s is entangled. Similar
result holds for the $n-k+1$-partite entanglement $\rho_{B_{k}\cdots B_{n}}$.
So, for any three observers $\texttt{P}_{i}$, $\texttt{P}_{k}$ and
$\texttt{P}_{j}$ with $1\leq i<k<j\leq n$, they can consist of a standard
$\Lambda$-shaped quantum network after a proper local measurement of all the
other observers as shown in Figure S4(c) and Figure S4(d). By using Theorem 1,
it is easy to prove that the collapsed joint systems of three observers
$\texttt{P}_{i}$, $\texttt{P}_{k}$ and $\texttt{P}_{j}$ have the bipartite
activated nonlocality, i.e., the bipartite quantum correlations of
$\texttt{P}_{i}$ and $\texttt{P}_{k}$ derived from a generalized
$\Lambda$-shaped quantum network ${\cal N}$ after a local measurement of the
observer $\texttt{P}_{j}$ (see Figure S4(c)) is consistent with all classical
correlations from any separable systems (or further semi-separable states,
i.e., three observers $\texttt{P}_{i}$, $\texttt{P}_{k}$ and $\texttt{P}_{j}$
have no prior shared entangled states while other observers can share some
entangled states, see Figure S4(d)). $\blacksquare$

Figure S5: (Color online) Schematically generalized Bell testing of a chain-
shaped network. (a) Generalized nonlocality testing of a chain-shaped quantum
network ${\cal N}_{q}$ consisting of $n-1$ bipartite entangled states
$\rho_{A_{1}B_{1}},\cdots,\rho_{A_{n-1}B_{n-1}}$, where the observers
$\texttt{P}_{i}$ and $\texttt{P}_{i+1}$ share one bipartite entanglement
$\rho_{A_{i}B_{i}}$, $i=1,\cdots,n-1$. $\tau^{a_{1}}_{A^{\prime}_{1}}$ and
${x}_{1}$ are respective inputs and outputs of the observer $\texttt{P}_{1}$,
$\tau^{b_{i}}_{B^{\prime}_{i}}\otimes\tau^{a_{i+1}}_{A^{\prime}_{i+1}}$ and
${y}_{i}{x}_{i+1}$ are respective inputs and outputs of the observer
$\texttt{P}_{i}$ for $i=2,\cdots,i-2$, and $\tau^{b_{n-1}}_{B^{\prime}_{n-1}}$
and ${y}_{n-1}$ are respective inputs and outputs of the observer
$\texttt{P}_{n-1}$, where $A^{\prime}_{i}$ and $B_{i}^{\prime}$ are axillary
systems that can be chosen by a trusty referee,
$a_{i}\in{\mathscr{A}}_{i},b_{i}\in{\mathscr{B}}_{i},{x}_{i}\in{\mathscr{X}}_{i},{y}_{i}\in{\mathscr{Y}}_{i}$.
(b) Hidden state model for verifying the locality of a chain-shaped network
consisting of $n-1$ shared sources $\lambda_{1},\cdots,\lambda_{n-1}$ (or
bipartite separable states
$\varrho_{A_{i}B_{i}}=\sum_{s}p^{(i)}_{s}\varrho^{(s)}_{A_{i}}\otimes\varrho^{(s)}_{B_{i}}$,
where $\\{p^{(i)}_{s}\\}$ are probability distributions and
$\varrho^{(s)}_{A_{i}(B_{i})}$ are density operators of the system $A_{i}$ or
$B_{i}$, $i=1,\cdots,n-1$). (c) Generalized Bell testing of the activated
nonlocality of the quantum subnetwork ${\cal N}_{q}^{*}\subset{\cal{N}}_{q}$
consisting of independent observers without initially sharing entangled states
in ${\cal{N}}_{q}$. For example, ${\cal{N}}_{q}^{*}$ consists of two observers
$\texttt{P}_{1}$ and $\texttt{P}_{n-1}$. Here, all observers
$\texttt{P}_{i}\not\in{\cal N}^{*}$ perform a proper local POVM with special
inputs $\tau^{a_{i}^{*}},\tau^{b_{i}^{*}}$ and outputs
${x}_{i}^{*},{y}_{i}^{*}$. (d) Hidden state model for verifying the activated
locality of the subnetwork ${\cal{N}}^{*}$, where all parties
$\texttt{P}_{i}\not\in{\cal{N}}^{*}$ can perform any POVM with any inputs
$\tau^{a_{i}^{*}},\tau^{b_{i}^{*}}$ and outputs ${x}_{i}^{*},{y}_{i}^{*}$. The
network consists of $n-1$ shared sources $\lambda_{1},\cdots,\lambda_{n-1}$.

Corollary 2. For a chain-shaped quantum network ${\cal{N}}_{q}$ with $n$
observers $\texttt{P}_{1}$, $\texttt{P}_{2}$, $\cdots$, $\texttt{P}_{n}$,
assume that two observers $\texttt{P}_{i}$ and $\texttt{P}_{i+1}$ share one
bipartite entanglement $\rho_{A_{i}B_{i}}$ on the Hilbert space
${\cal{H}}_{A_{i}}\otimes{\cal{H}}_{B_{i}}$, $i=1,\cdots,n-1$. Then the joint
system of $\rho_{A_{1}B_{1}}\otimes\cdots\otimes\rho_{A_{n-1}B_{n-1}}$ is
$n$-partite nonlocal. Moreover, for any subnetwork
${\cal{N}}_{q}^{*}\subset{\cal{N}}_{q}$ consisting of independent observers
(without initially sharing entangled states in ${\cal{N}}_{q}$), the reduced
system of ${\cal{N}}_{q}^{*}$ is nonlocal after all observers who are not in
the subnetwork of ${\cal{N}}_{q}^{*}$ perform a proper local POVM.

Proof. For a chain-shaped quantum network shown in Figure S5, the $n$-partite
nonlocality is followed from the semiquantum nonlocal game SQ  and the proof
given in Subsection A1 of Appendix A, i.e., we can prove the inconsistency of
$n$-partite quantum correlations derived from local measurements of a chain-
shaped quantum network consisting of all bipartite entangled states
$\rho_{A_{1}B_{1}},\cdots,{\cal H}_{A_{n-1}B_{n-1}}$ shown in Figure S5(a),
and $n$-partite classical correlations from a chain-shaped network consisting
of fully separable states
$\varrho_{A_{1}B_{1}}=\sum_{i}p^{(1)}_{i}\varrho^{(i)}_{A_{1}}\otimes\varrho^{(i)}_{B_{1}}$,
$\cdots$,
$\varrho_{A_{n-1}B_{n-1}}=\sum_{i}p^{(n-1)}_{i}\varrho^{(i)}_{A_{n-1}}\otimes\varrho^{(i)}_{B_{n-1}}$,
as shown in Figure S5(b).

Now, consider the nonlocality of any quantum subnetwork consisting of
independent observers without prior sharing entangled states. By using Theorem
1 iteratively it is easy to prove that each pair of independent observers
$\texttt{P}_{i}$ and $\texttt{P}_{j}$ can share an entangled systems in the
state $\rho_{A_{i}B_{j}}$ after all observers $\texttt{P}_{k}$s with $i<k<j$
performing a proper local POVM. Generally, consider a subnetwork
${\cal{N}}_{q}^{*}$ consisting of $k$ independent observers
$\texttt{P}_{i_{1}}$, $\cdots$, $\texttt{P}_{i_{k}}$ with $1\leq
i_{1}<\cdots<i_{k}\leq n$, where an example is shown in Figure S5(c) and
Figure S5(d). For two observers $\texttt{P}_{i_{j}}$ and
$\texttt{P}_{i_{j+1}}$, they can share a bipartite entanglement
$\rho_{A_{i_{j}}B_{i_{j}}}$ after all observers $\texttt{P}_{\ell}$s with
$i_{j}<\ell<i_{j+1}$ performing a proper local POVM. The activated bipartite
nonlocality can be proved similar to the proof given in Subsection A2 of
Appendix A (the proof of Theorem 1). Furthermore, by using the proof given in
Subsection A1 of Appendix A, it is easy to prove that the joint systems of all
independent observers $\texttt{P}_{i_{1}}$, $\cdots$, $\texttt{P}_{i_{k}}$
have $k$-partite nonlocality. Another method is as follows. For the subnetwork
${\cal{N}}_{q}^{*}$, all observers can obtain bipartite entangled pure states.
In this case, the nonlinear Bell-type inequalities Luo  can be used to prove
the $k$-partite nonlocality. $\blacksquare$

Additionally, the proof of Theorem 1 provided an interesting by-product that
universal Bell inequality exists for detecting a single entanglement by using
local projection and entanglement distilling HHH . In fact, we have the
following result

Corollary 3. There exists a universal Bell inequality to detect all entangled
states with multiple copies.

Proof. The proof is derived from the procedure shown in A2. For qubit-based
entangled state $\rho_{A_{1}\cdots A_{n}}$ shared by parties
$\texttt{P}_{1},\cdots,\texttt{P}_{1}$, they can firstly perform a local
entanglement distilling HHH  to obtain a multipartite entangled pure state
$|\Phi\rangle_{A_{1}\cdots A_{n}}$. And then, $|\Phi\rangle_{A_{1}\cdots
A_{n}}$ can be verified by using a universal Bell inequality YCZ . For a high-
dimensional entangled state $\rho_{A_{1}\cdots A_{n}}$, all parties can
firstly perform a local projection shown in Figure S3 with LOCC in order to
obtain a qubit-based entangled state $\hat{\rho}_{A_{1}\cdots A_{n}}$. Here,
we have taken use of the fact that LOCC cannot create an entanglement among
parties who have no initially shared entanglement. And then, all parties can
perform a local entanglement distilling HHH  to obtain a multipartite
entangled pure state $|\Psi\rangle_{A_{1}\cdots A_{n}}$ which can be verified
by a universal Bell inequality YCZ . The only assumption of the theorem is the
multiple copies of single entanglement, which is reasonable in terms of
statistics. $\blacksquare$

## Appendix B: Proof of Theorem 2

In this section, to complete the proof of Theorem 2 we firstly verify the
nonlocality of all star-shaped quantum networks consisting of any entangled
states.

Lemma 1. For a star-shaped quantum network ${\cal{N}}_{q}$ consisting of $n+1$
observers $\texttt{P}_{1},\cdots,\texttt{P}_{n}$, and Bob, assume that two
observers $\texttt{P}_{i}$ and Bob share one bipartite entanglement
$\rho_{A_{i}B_{i}}$ on the Hilbert space
${\cal{H}}_{A_{i}}\otimes{\cal{H}}_{B_{i}}$, $i=1,\cdots,n$. Then the joint
system of $A_{1},B_{1},\cdots,A_{n},B_{n}$ is $n+1$-partite nonlocal.
Moreover, there are local observables such that a local POVM of Bob can create
an $n$-partite entanglement shared by $n$ observers
$\texttt{P}_{1},\cdots,\texttt{P}_{n}$ who are initially sharing no entangled
states.

Proof. The proof is similar to that of Theorem 1. Since $\rho_{A_{i}B_{i}}$ is
bipartite entangled, there is a semiquantum nonlocal game
$\mathbbm{G}^{i}_{sq}$ SQ , constants $\alpha^{x_{i}y_{i}}_{a_{i}b_{i}}$,
auxiliary states $\tau_{A^{\prime}_{i}}^{a_{i}}\in{\cal{H}}_{A^{\prime}_{i}}$
and ${\mathscr{X}}_{i}$-POVM
$\\{P^{x_{i}}\\}\in{\cal{M}}_{A^{\prime}_{i}A_{i};{\mathscr{X}}_{i}}$ for
$\texttt{P}_{i}$, auxiliary states
$\tau_{B^{\prime}_{i}}^{b_{i}}\in{\cal{H}}_{B^{\prime}_{i}}$ and
${\mathscr{Y}}_{i}$-POVM
$\\{Q^{y_{i}}\\}\in{\cal{M}}_{B_{i}^{\prime}B_{i};{\mathscr{Y}}_{i}}$ for Bob
such that

$\displaystyle\sum_{x_{i},y_{i},a_{i},b_{i}}\alpha^{x_{i}y_{i}}_{a_{i}b_{i}}P(x_{i},y_{i}|a_{i},b_{i})>c_{i},$
(B1)

where $P(x_{i},y_{i}|a_{i},b_{i})$ are joint conditional probability
distributions computed as

$\displaystyle
P(x_{i},y_{i}|a_{i},b_{i})=\textrm{Tr}[(P^{x_{i}}_{A^{\prime}_{i}A_{i}}\otimes
Q^{y_{i}}_{B_{i}^{\prime}B_{i}})(\tau^{a_{i}}_{A_{i}^{\prime}}\otimes\rho_{A_{i}B_{i}}\otimes\tau^{b_{i}}_{B_{i}^{\prime}})],$
(B2)

$c_{i}$ denotes the maximal achievable classical bound of average gain in
terms of the semiquantum nonlocal game $\mathbbm{G}^{i}_{sq}$,
${\mathscr{A}}_{i}=\\{a_{i}\\}$ and ${\mathscr{B}}_{i}=\\{b_{i}\\}$,
${\mathscr{X}}_{i}=\\{x_{i}\\}$ and ${\mathscr{Y}}_{i}=\\{y_{i}\\}$,
$i=1,\cdots,n$. For all joint conditional probability distributions
$P_{c}(x_{i},y_{i}|a_{i},b_{i})$s derived from shared classical correlations
or separable quantum states, we have

$\displaystyle\sum_{x_{i},y_{i},a_{i},b_{i}}\alpha^{x_{i}y_{i}}_{a_{i}b_{i}}P_{c}(x_{i},y_{i}|a_{i},b_{i})\leq
c_{i},$ (B3)

where $P_{c}(x_{i},y_{i}|a_{i},b_{i})$ can be any convex combination of
independent distributions in the variables $x_{i},y_{i}$, i.e,
$P_{c}(x_{i},y_{i}|a_{i},b_{i})=\sum_{j}p_{j}P_{j}(x_{i}|a_{i})P_{j}(y_{i}|b_{i})$,
$\\{p_{j}\\}$ is a probability distribution.

Figure S6: (Color online) Schematically generalized Bell testing of a star-
shaped network. (a) Generalized Bell nonlocality testing of a star-shaped
quantum network consisting of $n$ bipartite entangled states
$\rho_{A_{1}B_{1}},\cdots,\rho_{A_{n}B_{n}}$, where two observers
$\texttt{P}_{i}$ and Bob share one bipartite entanglement $\rho_{A_{i}B_{i}}$,
$i=1,\cdots,n$. $\tau^{a_{i}}_{A^{\prime}_{i}}$ and ${x}_{i}$ are respective
inputs and outputs of the observer $\texttt{P}_{i}$,
$\tau^{b_{1}}_{B^{\prime}_{1}}\otimes\cdots\otimes\tau^{b_{n}}_{B^{\prime}_{n}}$
and ${y}_{1}\cdots{y}_{n}$ are respective inputs and outputs of Bob, where
$A^{\prime}_{i}$ and $B_{i}^{\prime}$ are axillary systems that can be chosen
by a trusty referee,
$a_{i}\in{\mathscr{A}}_{i},b_{i}\in{\mathscr{B}}_{i},{x}_{i}\in{\mathscr{X}}_{i},{y}_{i}\in{\mathscr{Y}}_{i}$.
(b) Hidden state model for verifying the locality of a star-shaped network
consisting of $n$ shared sources $\lambda_{1},\cdots,\lambda_{n}$ (or fully
separable states
$\varrho_{A_{i}B_{i}}=\sum_{s}p^{(i)}_{s}\varrho^{(s)}_{A_{i}}\otimes\varrho^{(s)}_{B_{i}}$,
where $\\{p^{(i)}_{s}\\}$ are probability distributions and
$\varrho^{(s)}_{A_{i}(B_{i})}$ are density operators of the system $A_{i}$ or
$B_{i}$, $i=1,\cdots,n$). (c) Generalized Bell testing of the activated
nonlocality of the subnetwork consisting of all observers $\texttt{P}_{1}$,
$\cdots$, $\texttt{P}_{n}$, where Bob performs a proper POVM with special
inputs $\tau^{b_{1}^{*}}\otimes\cdots\otimes\tau^{b_{n}^{*}}$ and outputs
$\overline{y}_{1}^{*}\cdots\overline{y}_{n}^{*}$ before verifying the
nonlocality. (d) Hidden state model for verifying the locality of the
subnetwork consisting of all observers $\texttt{P}_{1}$, $\cdots$,
$\texttt{P}_{n}$, where Bob can perform any POVM with any inputs
$\tau^{b_{1}^{*}}\otimes\cdots\otimes\tau^{b_{n}^{*}}$ and outputs
$\overline{y}_{1}^{*}\cdots\overline{y}_{n}^{*}$. The network consists of $n$
shared sources $\lambda_{1},\cdots,\lambda_{n}$, where one source
($\lambda_{1}$ for example) can be an entanglement.

In what follows, we construct an $n+1$-partite Bell inequality for testing the
nonlocality of the star-shaped quantum network consisting of bipartite
entangled states $\rho_{A_{1}B_{1}}\otimes\cdots\otimes\rho_{A_{n}B_{n}}$ from
the inequalities (B1) and (B3).

B1: Constructing $n+1$-partite generalized Bell testing

Now, we construct a new $n+1$-partite generalized Bell testing shown in Figure
S6(a) and Figure S6(b). For each bipartite entanglement $\rho_{A_{i}B_{i}}$,
assume that the input set and output set of the observer $\texttt{P}_{i}$ are
${\mathscr{A}}_{i}=\\{1,2,\cdots,m_{i}\\}$,
${\mathscr{X}}_{i}=\\{1,2,\cdots,n_{i}\\}$, respectively, and the input set
and output set of Bob are ${\mathscr{B}}_{i}=\\{1,2,\cdots,\hat{m}_{i}\\}$,
${\mathscr{Y}}_{i}=\\{1,2,\cdots,\hat{n}_{i}\\}$, respectively,
$i=1,\cdots,n$. Assume that the input states of the observers
$\texttt{P}_{1}$, $\cdots$, $\texttt{P}_{n}$, and Bob are
$\tau^{a_{1}}_{A_{1}^{\prime}}$, $\cdots,\tau^{a_{n}}_{A_{n}^{\prime}}$,
$\tau^{b_{1}}_{B_{1}^{\prime}}\otimes\cdots\otimes\tau^{b_{n}}_{B_{n}^{\prime}}$,
respectively. $P(\textbf{x},\textbf{y}|\textbf{{a}},\textbf{b})$ denote the
joint conditional probability distributions defined by

$\displaystyle P(\textbf{x},\textbf{y}|\textbf{a},\textbf{b})=$
$\displaystyle\textrm{Tr}[({P}_{A_{1}^{\prime}A_{1}}^{{x}_{1}}\otimes\cdots\otimes{P}_{A_{n}^{\prime}A_{n}}^{{x}_{n}}\otimes{Q}_{B_{1}\cdots
B_{n}B_{1}^{\prime}\cdots
B_{n}^{\prime}}^{{y}_{1}\cdots{y}_{n}})(\otimes_{i=1}^{n}(\tau^{a_{i}}_{A_{i}^{\prime}}\otimes\rho_{A_{i}B_{i}}\otimes\tau^{b_{i}}_{B_{i}^{\prime}}))]$
$\displaystyle=$
$\displaystyle\textrm{Tr}[({P}_{A_{1}^{\prime}A_{1}}^{{x}_{1}}\otimes\cdots\otimes{P}_{A_{n}^{\prime}A_{n}}^{{x}_{n}}\otimes{Q}_{B_{1}^{\prime}B_{1}}^{{y}_{1}}\otimes
Q_{B_{n}^{\prime}B_{n}}^{{y}_{n}})(\otimes_{i=1}^{n}(\tau^{a_{i}}_{A_{i}^{\prime}}\otimes\rho_{A_{i}B_{i}}\otimes\tau^{b_{i}}_{B_{i}^{\prime}}))]$
$\displaystyle=$ $\displaystyle\prod_{i=1}^{n}P(x_{i},y_{i}|a_{i},b_{i}),$
(B4)

where
$P(x_{i},y_{i}|a_{i},b_{i})=\textrm{Tr}[P^{{x}_{i}}_{A_{i}^{\prime}A_{i}}\otimes
Q^{y_{i}}_{B_{i}^{\prime}B_{i}})(\tau^{a_{i}}_{A_{i}^{\prime}}\otimes\rho_{A_{i}B_{i}}\otimes\tau^{b_{i}}_{B_{i}^{\prime}})]$,
$\\{{P}^{{x}_{1}}\\}\in{\cal{M}}_{A_{1}^{\prime}A_{1};{\mathscr{X}}_{1}},\cdots,\\{{P}^{{x}_{n}}\\}\in{\cal
M}_{A_{n}^{\prime}A_{n};{\mathscr{X}}_{n}}$,
$\\{{Q}^{{y}_{1}\cdots{y}_{n}}\\}\in{\cal{M}}_{B_{1}^{\prime}B_{1}\cdots
B_{n}^{\prime}B_{n};{\mathscr{Y}}_{1}\times\cdots\times{\mathscr{Y}}_{n}}$ are
POVMs of all observers $\texttt{P}_{1}$, $\cdots$, $\texttt{P}_{n}$, and Bob,
respectively, $\textbf{x}=({x}_{1},\cdots,{x}_{n})$ ($n$ outputs of $n$
observers $\texttt{P}_{1}$, $\cdots$, $\texttt{P}_{n}$),
$\textbf{y}={y}_{1}\cdots{y}_{n}$ (one output of Bob consisting of $n$
indexes), ${\textbf{a}}=(a_{1},\cdots,a_{n})$ (the indexes of $n$ inputs of
$n$ observers $\texttt{P}_{1}$, $\cdots$, $\texttt{P}_{n}$),
$\textbf{b}=b_{1}\cdots b_{n}$ (the index of one input of Bob consisting of
$n$ indexes).

Define the average gain $\wp$ depending on all joint conditional probabilities
$P(\textbf{x},\textbf{y}|\textbf{a},\textbf{b})$s as

$\displaystyle\wp=\sum_{\textbf{{x}},\textbf{{y}},\textbf{a},\textbf{b}}\gamma^{\textbf{x}\textbf{y}}_{\textbf{a}\textbf{b}}P(\textbf{x},\textbf{y}|\textbf{a},\textbf{b}),$
(B5)

where the coefficients $\gamma^{\textbf{x}\textbf{y}}_{\textbf{a}\textbf{b}}$
are given by
$\gamma^{\textbf{x}\textbf{y}}_{\textbf{a}\textbf{b}}=\sum_{i=1}^{n}\alpha^{{x}_{i}{y}_{i}}_{a_{i}b_{i}}$.

From the equations (B4) and (B5), consider the quantum $n+1$-partite
correlations obtained from the locally measuring the shared system of a star-
shaped quantum network shown in Figure S6(a) with POVMs
$\\{{P}_{A_{1}^{\prime}A_{1}}^{{x}_{1}}\\}\in{\cal
M}_{A_{1}^{\prime}A_{1};{\mathscr{X}}_{1}}$ for the observer $\texttt{P}_{1}$,
$\cdots$,
$\\{{P}_{A_{n}^{\prime}A_{n}}^{{x}_{n}}\\}\in{\cal{M}}_{A_{n}^{\prime}A_{n};{\mathscr{X}}_{n}}$
for the observer $\texttt{P}_{n}$, and $\\{{Q}_{B_{1}^{\prime}B_{1}\cdots
B_{n}^{\prime}B_{n}}^{{y}_{1}\cdots{y}_{n}}\\}\in{\cal{M}}_{B_{1}^{\prime}B_{1}\cdots
B_{n}^{\prime}B_{n};{\mathscr{Y}}_{1}\times\cdots\times{\mathscr{Y}}_{n}}$ for
Bob. The equation (B5) is rewritten into

$\displaystyle\wp=$
$\displaystyle\sum_{\textbf{a},\textbf{b},\textbf{x},\textbf{y}}\gamma^{\textbf{x}\textbf{y}}_{\textbf{a}\textbf{b}}P(\textbf{x},\textbf{y}|\textbf{a},\textbf{b})$
$\displaystyle=$
$\displaystyle\sum_{\textbf{a},\textbf{b},\textbf{x},\textbf{y}}(\alpha^{x_{1}y_{1}}_{a_{1}b_{1}}+\cdots+\alpha^{x_{n}y_{n}}_{a_{n}b_{n}})P(\textbf{x},\textbf{y}|\textbf{a},\textbf{b})$
$\displaystyle=$
$\displaystyle\sum_{\textbf{a},\textbf{b},\textbf{x},\textbf{y}}(\alpha^{x_{1}y_{1}}_{a_{1}b_{1}}+\cdots+\alpha^{x_{n}y_{n}}_{a_{n}b_{n}})\prod_{i=1}^{n}P(x_{i},y_{i}|a_{i},b_{i})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{\hat{M}}{{m}_{i}\hat{m}_{i}}\sum_{a_{i},b_{i},x_{i},y_{i}}\alpha^{x_{i}y_{i}}_{a_{i}b_{i}}P(x_{i},y_{i}|a_{i},b_{i})$
(B6) $\displaystyle>$
$\displaystyle\sum_{i=1}^{n}\frac{\hat{M}c_{i}}{{m}_{i}\hat{m}_{i}}$ (B7)

Here, the equation (B6) is from the equalities
$\sum_{{a}_{j},{b}_{j},{x}_{j},{y}_{j}}P(x_{j},y_{j}|a_{j},b_{j})=m_{j}\hat{m}_{j}$
from the normalization of conditional probabilities, i.e.,
$\sum_{x_{s},y_{s}}P(x_{s},y_{s}|a_{s},b_{s})=1$ for each $a_{s},b_{s}$, and
$\hat{M}=\prod_{i=1}^{n}m_{i}\hat{m}_{i}$. The inequality (B7) is from the
inequalities (B1).

In what follows, we estimate the upper bound of $\wp$ defined in the equation
(B5) for classical $n+1$-partite correlations by using hidden state model Cire
, see Figure S6(b). In detail, from the inequalities (B2), define classical
$n+1$-partite correlations
$P_{c}(\textbf{x},\textbf{y}|\textbf{a},\textbf{b})$ as the conditional
probabilities of locally measuring a shared fully separable state
$\varrho=\sum_{i}t_{i}\varrho^{(i)}_{A_{1}}\otimes\cdots\otimes\varrho^{(i)}_{A_{n}}\otimes\varrho^{(i)}_{B_{1}\cdots
B_{n}}$ ($\\{t_{i}\\}$ is a probability distribution) with POVMs
$\\{\hat{P}_{A_{1}A_{1}^{\prime}}^{{x}_{1}}\\}\in{\cal{M}}_{A_{1}^{\prime}A_{1};{\mathscr{X}}_{1}},\cdots,\\{\hat{P}_{A_{n}A_{n}^{\prime}}^{{x}_{n}}\\}\in{\cal{M}}_{A_{n}^{\prime}A_{n};{\mathscr{X}}_{n}}$,
$\\{\hat{Q}_{B_{1}^{\prime}B_{1}\cdots
B_{n}^{\prime}B_{n}}^{{y}_{1}\cdots{y}_{n}}\\}\in{\cal
M}_{B_{1}^{\prime}B_{1}\cdots
B_{n}^{\prime}B_{n};{\mathscr{Y}}_{1}\times\cdots\times{\mathscr{Y}}_{n}}$. We
get an inequality from the equation (B5) as

$\displaystyle\wp_{c}=$
$\displaystyle\sum_{\textbf{a},\textbf{b},\textbf{x},\textbf{y}}\gamma^{\textbf{x}\textbf{y}}_{\textbf{a}\textbf{b}}P_{c}(\textbf{x},\textbf{y}|\textbf{a},\textbf{b})$
$\displaystyle=$
$\displaystyle\sum_{\textbf{{a}},\textbf{{b}},\textbf{{x}},\textbf{{y}}}\textrm{Tr}[(\hat{P}_{A_{1}^{\prime}A_{1}}^{{x}_{1}}\otimes\cdots\otimes\hat{P}_{A_{n}^{\prime}A_{n}}^{{x}_{n}}\otimes\hat{Q}_{B_{1}^{\prime}B_{1}\cdots
B_{n}^{\prime}B_{n}}^{{y}_{1}\cdots{y}_{n}})$
$\displaystyle\times(\tau^{a_{1}}_{A_{1}^{\prime}}\otimes\cdots\otimes\tau^{a_{n}}_{A_{n}^{\prime}}\otimes\tau^{b_{1}}_{B_{1}^{\prime}}\otimes\cdots\otimes\tau^{b_{n}}_{B_{n}^{\prime}}\otimes\sum_{i}t_{i}\varrho^{(i)}_{A_{1}}\otimes\cdots\otimes\varrho^{(i)}_{A_{n}}\otimes\varrho^{(i)}_{B_{1}\cdots
B_{n}}]$ $\displaystyle=$
$\displaystyle\sum_{i}t_{i}\sum_{\textbf{a},\textbf{b},\textbf{x},\textbf{y}}\sum_{j=1}^{n}\alpha^{x_{j}y_{j}}_{a_{j}b_{j}}\textrm{Tr}[(\hat{P}_{A_{1}^{\prime}A_{1}}^{x_{1}}\otimes\cdots\otimes\hat{P}_{A_{n}^{\prime}A_{n}}^{x_{n}}\otimes\hat{Q}_{B_{1}^{\prime}B_{1}\cdots
B_{n}^{\prime}B_{n}}^{y_{1}\cdots{y}_{n}})$
$\displaystyle\times(\tau^{a_{1}}_{A_{1}^{\prime}}\otimes\cdots\otimes\tau^{a_{n}}_{A_{n}^{\prime}}\otimes\tau^{b_{1}}_{B_{1}^{\prime}}\otimes\cdots\otimes\tau^{b_{n}}_{B_{n}^{\prime}}\otimes\varrho^{(i)}_{A_{1}}\otimes\cdots\otimes\varrho^{(i)}_{A_{n}}\otimes\varrho^{(i)}_{B_{1}\cdots
B_{n}}]$ $\displaystyle=$
$\displaystyle\sum_{i}t_{i}\sum_{\textbf{a},\textbf{b},\textbf{x},\textbf{y}}\sum_{j=1}^{n}\alpha^{x_{j}y_{j}}_{a_{j}b_{j}}\prod_{s=1}^{n}P^{(i)}({x}_{s}|a_{s})P^{(i)}({y}_{1}\cdots{y}_{n}|b_{1}\cdots
b_{n})$ (B8) $\displaystyle=$
$\displaystyle\sum_{i}t_{i}\sum_{j=1}^{n}\sum_{\textbf{a},\textbf{b},x_{j},y_{j}}\alpha^{x_{j}y_{j}}_{a_{j}b_{j}}P_{k}^{(i)}(x_{j}|a_{j})P^{(i)}(y_{j}|b_{1}\cdots
b_{n})$ (B9) $\displaystyle=$
$\displaystyle\sum_{i}t_{i}\sum_{j=1}^{n}\sum_{a_{s},b_{s},\atop{1\leq
s\not=j\leq
n}}\sum_{a_{j},b_{j},x_{j},y_{j}}\alpha^{x_{j}y_{j}}_{a_{j}b_{j}}P^{(i)}(x_{j}|a_{j})P^{(i)}(y_{j}|b_{1}\cdots
b_{n})$ $\displaystyle\leq$
$\displaystyle\sum_{i}t_{i}\sum_{j=1}^{n}\sum_{a_{s},b_{s},\atop{1\leq
s\not=j\leq n}}c_{j}$ (B10) $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{\hat{M}c_{i}}{m_{i}\hat{m}_{i}}.$ (B11)

In the equation (B8), we have taken use of the following notations:
$P^{(i)}({x}_{j}|a_{j})=\textrm{Tr}[\hat{P}_{A_{j}^{\prime}A_{j}}^{{x}_{j}}(\tau^{a_{j}}_{A_{j}^{\prime}}\otimes\varrho^{(i)}_{A_{j}})]$,
$P^{(i)}({y}_{1}\cdots{y}_{n}|b_{1}\cdots
b_{n})=\textrm{Tr}[\hat{Q}_{B_{1}^{\prime}B_{1}\cdots
B_{n}^{\prime}B_{n}}^{{y}_{1}\cdots{y}_{n}}(\tau^{b_{1}}_{B_{1}^{\prime}}\otimes\cdots\otimes\tau^{b_{n}}_{B_{n}^{\prime}}\otimes\varrho^{(i)}_{B_{1}\cdots
B_{n}})]$, $j=1,\cdots,n$. In order to get the equation (B9) we have used the
equalities $\sum_{y_{t},1\leq t\not=j\leq n}P^{(i)}(y_{1}\cdots
y_{n}|b_{1}\cdots b_{n})=P^{(i)}(y_{j}|b_{1}\cdots b_{n})$ and
$\sum_{x_{j}}P^{(i)}(x_{j}|a_{j})=1$ for each
$i,j,a_{1},\cdots,a_{n},b_{1},\cdots,b_{n}$. To get the inequality (B10), note
that $P^{(i)}(x_{j}|a_{j})$ and $P^{(i)}(y_{j}|b_{1}\cdots b_{n})$ are
independent conditional probabilities in terms of the variables $x_{j},y_{j}$
for any fixed variables $a_{j},b_{j}$. Hence, for each $i,j,k$, $x_{j},y_{j}$,
we can take use of the inequalities (B3). The equation (B11) is from the fact
that $\\{{t}_{i}\\}$ is a probability distribution.

From the inequalities (B7) and (B11), we have verify the $n+1$-partite
nonlocality of the star-shaped quantum network. Here, the inequalities (B7)
and (B11) have defined $n+1$-partite generalized Bell-type inequalities for
verifying the $n+1$-partite nonlocality of a star-shaped joint system
$\rho_{A_{1}B_{1}}\otimes\cdots\otimes\rho_{A_{n}B_{n}}$.

Up to now, we have proved that the inequalities (B7) and (B11) can ensure that
the $n+1$-partite quantum correlations derived from a star-shaped quantum
network with entangled states are different from the $n+1$-partite classical
correlations derived from the fully separable state.

B2: Activated nonlocality of a star-shaped quantum network

To complete the proof, we further prove that the $n$-partite nonlocality of
all independent observers, i.e., $\texttt{P}_{1}$, $\cdots$, $\texttt{P}_{n}$,
can be activated by a local measurement of Bob, see Figure S6(c) and Figure
S6(d). From the proof in subsection A2 of Appendix A, two observers
$\texttt{P}_{i}$ and Bob can probabilistically obtain a qubit-based entangled
state from their shared state $\rho_{A_{i}B_{i}}$. And then, by using the
entanglement distill HHH  with LOCC, $\texttt{P}_{i}$ and Bob can share an
entangled pure state $|\Phi\rangle_{A_{i}B_{i}}$, where LOCC cannot create an
entanglement between two parties who have not initially shared entanglement.

In what follows, we only need to prove that the $n$-partite nonlocality of $n$
observers $\texttt{P}_{1}$, $\cdots$, $\texttt{P}_{n}$, can be activated by a
local measurement of Bob for any star-shaped quantum network consisting of all
entangled pure states $\otimes_{i=1}^{n}|\Phi\rangle_{A_{i}B_{i}}$. In detail,
assume $|\Phi\rangle_{A_{i}B_{i}}=u_{i}|00\rangle+v_{i}|11\rangle$ under some
local operations, where $u_{i}^{2}+v_{i}^{2}=1$, $i=1,\cdots,n$. Define the
measurement of Bob as $n$-particle Bell basis, i.e., $\\{Q_{\pm}^{y_{1}\cdots
y_{n}}=\frac{1}{\sqrt{2}}(|y_{1}\cdots
y_{n}\rangle\pm|\overline{y}_{1}\cdots\overline{y}_{n}\rangle)\\}$, where
$\overline{y}_{i}=1\oplus y_{i}$. For each measurement $Q_{\pm}^{y_{1}\cdots
y_{n}}$, the collapsed state of all observers $\texttt{P}_{1}$, $\cdots$,
$\texttt{P}_{n}$ is given by

$\displaystyle\Omega^{\pm}_{A_{1}\cdots A_{n}}=r(u_{y_{1}\cdots
y_{n}}|y_{1}\cdots y_{n}\rangle\pm v_{y_{1}\cdots
y_{n}}|\overline{y}_{1}\cdots\overline{y}_{n}\rangle)$ (B12)

which is an $n$-partite generalized GHZ state, where $u_{y_{1}\cdots
y_{n}}=\prod_{i=1}^{n}u_{y_{i}}$, $v_{y_{1}\cdots
y_{n}}=\prod_{i=1}^{n}v_{y_{i}}$ and $r$ is the normalization constant.

For any $n\geq 2$, we can easily prove the following $n$-partite CHSH
inequality

$\displaystyle|\langle{M}^{1}_{0}\otimes\cdots\otimes{M}^{n-1}_{0}\otimes{M}^{n}_{0}\rangle+\langle{M}^{1}_{0}\otimes\cdots\otimes{M}^{n-1}_{0}\otimes{M}^{n}_{1}\rangle$
$\displaystyle+\langle{M}^{1}_{1}\otimes\cdots\otimes{M}^{n-1}_{1}\otimes{M}^{n}_{0}\rangle-\langle{M}^{1}_{1}\otimes\cdots\otimes{M}^{n-1}_{1}\otimes{M}^{n}_{1}\rangle|\leq
2,$ (B13)

where one can schematically represent
${M}_{i}:={M}^{1}_{i}\otimes\cdots\otimes{M}^{n-1}_{i}$ ($i=0,1$) and prove
the inequality by a similar procedure of CHSH inequality CHSH . This
$n$-partite Bell inequality is used to prove that $\Omega^{\pm}_{A_{1}\cdots
A_{n}}$ are entangled for all nonzero $u_{i},v_{i}$.

  * •

For an even $m$, define dichotomic observables
$M^{j}_{i}=(1-i)\sigma_{z}+i\sigma_{x}$, $i=0,1;j=1,\cdots,n-1$, and
$M^{n}_{i}=\cos\theta\sigma_{z}+(-1)^{i}\sin\theta\sigma_{x}$ with
$\cos\theta=1/\sqrt{1+4r^{2}u_{y_{1}\cdots y_{n}}^{2}v^{2}_{y_{1}\cdots
y_{n}}}$, where $\sigma_{z}$ and $\sigma_{x}$ are Pauli matrices. From
straight forward evaluations we get

$\displaystyle\langle{M}^{1}_{0}\otimes\cdots\otimes{M}^{n-1}_{0}\otimes{M}^{n}_{0}\rangle+\langle{M}^{1}_{0}\otimes\cdots\otimes{M}^{n-1}_{0}\otimes{M}^{n}_{1}\rangle$
$\displaystyle+\langle{M}^{1}_{1}\otimes\cdots\otimes{M}^{n-1}_{1}\otimes{M}^{n}_{0}\rangle-\langle{M}^{1}_{1}\otimes\cdots\otimes{M}^{n-1}_{1}\otimes{M}^{n}_{1}\rangle$
$\displaystyle=$ $\displaystyle{\rm
tr}(({M}^{1}_{0}\otimes\cdots\otimes{M}^{m-1}_{0}\otimes{M}^{n}_{0}+{M}^{1}_{0}\otimes\cdots\otimes{M}^{n-1}_{0}\otimes{M}^{n}_{1}$
$\displaystyle+{M}^{1}_{1}\otimes\cdots\otimes{M}^{n-1}_{1}\otimes{M}^{n}_{0}-{M}^{1}_{1}\otimes\cdots\otimes{M}^{n-1}_{1}\otimes{M}^{n}_{1})\Omega^{\pm}_{A_{1}\cdots
A_{n}}(\Omega^{\pm}_{A_{1}\cdots A_{n}})^{\dagger})$ $\displaystyle=$
$\displaystyle 2\sqrt{1+4r^{2}u_{y_{1}\cdots y_{n}}^{2}v^{2}_{y_{1}\cdots
y_{n}}}$ $\displaystyle>$ $\displaystyle 2,$ (B14)

which violates the multipartite CHSH inequality shown in equation (B13) for
all nonzero $u_{i},v_{i},i=1,\cdots,n$. It implies that
$\Omega^{\pm}_{A_{1}\cdots A_{n}}$ are $n$-partite nonlocal.

  * •

For an odd $m$, by defining observables
$M^{j}_{i}=(1-i)\sigma_{z}+i\sigma_{x}$, $i=0,1;j=1,\cdots,n-2$,
$M^{n-1}_{i}=(1-i){\bf I}_{2}+i\sigma_{x}$, and
$M^{n}_{i}=\cos\theta\sigma_{z}+(-1)^{i}\sin\theta\sigma_{x}$ with
$\cos\theta=1/\sqrt{1+(1-v)^{2}}$, we obtain the same inequality (B14) which
implies that $\Omega^{\pm}_{A_{1}\cdots A_{n}}$ violates the Bell inequality
given in equation (B13), where ${\bf I}_{2}$ is the identity matrix.

Consequently, we have proved the lemma 1. $\blacksquare$

The Lemma 1 can be easily extended to generalized star-type networks.

Figure S7: (Color online) Schematic quantum network. (a) Triangle (cyclic)
quantum network consisting of three observers
$\texttt{P}_{1},\texttt{P}_{2},\texttt{P}_{3}$. $\texttt{P}_{i}$ and
$\texttt{P}_{(i\mod 3)+1}$ share one bipartite entanglement, $i=1,2,3$. (b)
Multiple cyclic quantum network consisting of five observers
$\texttt{P}_{1},\texttt{P}_{2},\cdots,\texttt{P}_{5}$. The observers
$\texttt{P}_{1},\cdots,\texttt{P}_{4}$ share one 4-partite entanglement.
$\texttt{P}_{2},\texttt{P}_{4},\texttt{P}_{5}$ share one tripartite
entanglement. The observers $\texttt{P}_{3},\texttt{P}_{4},\texttt{P}_{5}$
share one tripartite entanglement. (c) Symmetric cyclic quantum network
consisting of four observers $\texttt{P}_{1},\cdots,\texttt{P}_{4}$. The
observers $\texttt{P}_{1},\texttt{P}_{2},\texttt{P}_{4}$ share one tripartite
entanglement. The observers $\texttt{P}_{2},\texttt{P}_{3},\texttt{P}_{4}$
share one tripartite entanglement. (d) Star-shaped quantum network consisting
of four observers $\texttt{P}_{1},\cdots,\texttt{P}_{4}$. Two observers
$\texttt{P}_{i}$ and $\text{P}_{3}$ share one bipartite entanglement,
$i=1,2,4$. (e) Chain-shaped quantum network consisting of three observers
$\texttt{P}_{1},\cdots,\texttt{P}_{3}$. Two observers $\texttt{P}_{i}$ and
$\texttt{P}_{2}$ share one bipartite entanglement, $i=1,3$. Networks in
subfigures (a) and (b) have no independent observers while the networks in
other subfigures have independent observers shown with blue boxes.

Corollary 4. For a quantum network ${\cal N}_{q}$ consisting of $n+k+1$
observers $\texttt{P}_{1},\cdots,\texttt{P}_{n+k}$, and Bob, assume that all
observers $\texttt{P}_{j}$ with $j\in{\cal I}_{i}$ and Bob share one
multipartite entanglement $\rho_{i}$ on the Hilbert space $(\otimes_{j\in{\cal
I}_{i}}{\cal H}_{j})\otimes{\cal H}_{B_{i}}$, where ${\cal
I}_{i}\subset\\{1,\cdots,n+k\\}$, $i=1,\cdots,s$. Then the joint system in the
state $\rho_{1}\otimes\cdots\otimes\rho_{s}$ is $n+k+2$-partite nonlocal.
Moreover, for the subnetwork consisting of $m$ observers
$\texttt{P}_{j_{1}},\cdots,\texttt{P}_{j_{m}}$, their reduced system is
$m$-partite nonlocal after all the other observers performing a proper local
POVM, where all observers of $\texttt{P}_{j_{1}},\cdots,\texttt{P}_{j_{m}}$
are independent, i.e., they have not initially shared entangled states.

Proof. The first part of Corollary 4, i.e., the $n+k+2$-partite locality is
similar to that stated in Corollary 1, which can be proved by using the
semiquantum nonlocal game for single entanglement SQ  and the procedures given
in subsections A1 of Appendix A. Moreover, consider one subnetwork ${\cal
N}_{m}$ consisting of independent observers
$\texttt{P}_{j_{1}},\cdots,\texttt{P}_{j_{m}}$, who have not initially shared
entangled states. For each observer $\texttt{P}_{j_{t}}$, there is an integer
set ${\cal I}_{l_{t}}$ satisfying $j_{t}\in{\cal I}_{l_{t}}$ and
$j_{r}\not\in{\cal I}_{l_{t}}$ for $r\not=t$, i.e., two observers
$\texttt{P}_{j_{t}}$ and Bob share an entanglement $\rho_{l_{t}}$. After a
proper local POVM of all observers $\texttt{P}_{r}$ with $r\in{\cal
I}_{l_{t}}$ and $r\not=j_{t}$, the observers $\texttt{P}_{j_{t}}$ and Bob can
share a bipartite entanglement. Based on this fact, there is a standard star-
shaped quantum subnetwork after a proper local POVM of all observers except
for $\texttt{P}_{j_{1}},\cdots,\texttt{P}_{j_{m}}$. The $m$-partite
nonlocality of the subnetwork ${\cal N}_{m}$ is followed from Lemma 1.
$\blacksquare$

In what follows, we complete the proof of Theorem 2.

Proof of Theorem 2. Assume that a connected quantum network ${\cal N}_{q}$
consists of $n$ observers $\texttt{P}_{1},\cdots,\texttt{P}_{n}$, who have
shared entangled states $\rho_{1},\cdots,\rho_{m}$, where each entanglement
$\rho_{i}$ is shared by some observers $\texttt{P}_{j}$ with $j\in{\cal
I}_{i}\subset\\{1,2,\cdots,n\\}$, $i=1,\cdots,m$. From the definition of the
connected quantum network, for each pair observers $\texttt{P}_{s}$ and
$\texttt{P}_{t}$ there is one set of observers
$\texttt{P}_{i_{1}},\cdots,\texttt{P}_{i_{k}}$ such that any adjacent two
observers of
$\texttt{P}_{s},\texttt{P}_{i_{1}},\cdots,\texttt{P}_{i_{k}},\texttt{P}_{t}$
share at least one entangled state. Based on this fact, we prove the result in
three cases as follows.

Case 1. There is no independent observers in ${\cal N}_{q}$, i.e., any two
observers have shared at least one entanglement, where some examples are shown
in Figures S7(a) and S7(b). In this case, there are semiquantum nonlocal games
$\mathbbm{G}^{(1)}_{sq},\cdots,\mathbbm{G}^{(m)}_{sq}$ for entangled states
$\rho_{1},\cdots,\rho_{m}$, respectively SQ . The $n$-partite nonlocality of
${\cal N}_{q}$ can be proved by iteratively using the procedure given in
subsections A1 of Appendix A. The main steps are shown as follows: we can
firstly redefine a new semiquantum nonlocal game $\mathbbm{G}^{(1,2)}_{sq}$ to
verify the joint system $\rho_{1}\otimes\rho_{2}$ from the semiquantum
nonlocal games $\mathbbm{G}^{(1)}_{sq}$ and $\mathbbm{G}^{(2)}_{sq}$ using
Corollary 1. And then, we redefine a new semiquantum nonlocal game
$\mathbbm{G}^{(1,2,3)}_{sq}$ to verify the joint system
$(\rho_{1}\otimes\rho_{2})\otimes\rho_{3}$ from the semiquantum nonlocal games
$\mathbbm{G}^{(1,2)}_{sq}$ and $\mathbbm{G}^{(3)}_{sq}$ using Corollary 1,
where $\rho_{1}\otimes\rho_{2}$ is regarded as an entangled system. The
$n$-partite nonlocality of $\rho_{1}\otimes\cdots\otimes\rho_{m}$ can be
proved by repeating this procedure.

Case 2. There are some independent observers $\texttt{P}_{i}$ with $i\in{\cal
I}\subset\\{1,2,\cdots,n\\}$ in the network ${\cal N}_{q}$, i.e., any two
observers $\texttt{P}_{i}$ with $i\in{\cal I}$ have not initially shared
entanglement, where some examples are shown in Figures S7(c)-S7(e). Similar to
Case 1, we can prove the $n$-partite nonlocality of ${\cal N}_{q}$ by
repeating the procedure given in subsections A1 of Appendix A. The reason is
from the definition of the connected quantum network.

Figure S8: (Color online) Schematic example of reducing a connected quantum
network. (a) A connected quantum network consisting of ten observers
$\texttt{P}_{1},\cdots,\texttt{P}_{10}$. $\texttt{P}_{i}$ and $\texttt{P}_{9}$
share one bipartite entanglement, $i=2,4,5$. The observers $\texttt{P}_{2}$
and $\texttt{P}_{7}$ share one bipartite entanglement. The observers
$\texttt{P}_{3},\texttt{P}_{7}$ and $\texttt{P}_{8}$ share one tripartite
entanglement. The observers $\texttt{P}_{1},\texttt{P}_{6},\texttt{P}_{7}$ and
$\texttt{P}_{10}$ share one 4-partite entanglement. (b) An equivalent quantum
network. Here, the observers $\texttt{P}_{1},\cdots,\texttt{P}_{5}$ are
independent observers, who have not initially shared entangled states. In the
equivalent network, all the other observers
$\texttt{P}_{6},\cdots,\texttt{P}_{10}$ are placed above the dot line. (c)
Reduced quantum network of the network shown in Figure S8(b). To obtain this
quantum network, we consider the quantum subnetwork consisting of four
observers $\texttt{P}_{1},\texttt{P}_{2},\texttt{P}_{6}$, $\texttt{P}_{7}$
($j_{1}=6$, $j_{2}=7$ for the proof in Case 3 shown in Figure S8(b). Since
$j_{1}=6\not=j_{2}=7$, there is a chain-shaped quantum subnetwork ${\cal
N}_{1,2}$ consisting of two observers $\texttt{P}_{6},\texttt{P}_{7}$ as shown
in Figure S8(b). Note that there is another observer $\texttt{P}_{10}$ who
shares the same system with the observers $\texttt{P}_{6}$ and
$\texttt{P}_{7}$. So, we obtain a new quantum network after the observer
$\texttt{P}_{10}$ performing a local POVM to disentangle his shared 4-partite
entanglement into a tripartite entanglement. (d) Reduced quantum network of
the network shown in Figure S8(c). The new quantum network is obtained after
the observer $\texttt{P}_{6}$ performing a local POVM to disentangle his
shared tripartite entanglement into a bipartite entanglement. We cannot
require the observer $\texttt{P}_{7}$ to disentangle the tripartite
entanglement shared with three observers $\texttt{P}_{1},\texttt{P}_{6}$ and
$\texttt{P}_{7}$. The reason is that the reduced quantum network should be
connected. (e) Reduced quantum network of the network shown in Figure S8(d).
Now, we consider a new quantum subnetwork ${\cal N}_{2,3}$ of the network
shown in Figure S8(d), which consists of four observers
$\texttt{P}_{2},\texttt{P}_{3},\texttt{P}_{7},\texttt{P}_{8}$. The reduced
quantum network is obtained after the observer $\texttt{P}_{8}$ performing a
local POVM to disentangle the tripartite entanglement shared with three
observers $\texttt{P}_{7},\texttt{P}_{8}$ and $\texttt{P}_{3}$ into a
bipartite entanglement, where $\texttt{P}_{8}\not\in{\cal
N}_{2,3}\cup\\{\texttt{P}_{4}\\}$. (f) Equivalent quantum network of the
network shown in Figure S8(e). It is a hybrid quantum network of a chain-
shaped quantum subnetwork consisting of three observers
$\texttt{P}_{7},\texttt{P}_{2}$, and $\texttt{P}_{9}$, and two star-shaped
quantum subnetworks consisting of four observers
$\texttt{P}_{1},\texttt{P}_{2}$, $\texttt{P}_{3}$, and $\texttt{P}_{7}$, or
$\texttt{P}_{2},\texttt{P}_{4}$, $\texttt{P}_{5}$, and $\texttt{P}_{9}$. All
independent observers are shown with blue boxes.

Case 3. There are some independent observers $\texttt{P}_{i}$ with $i\in{\cal
I}\subset\\{1,2,\cdots,n\\}$ in the quantum network ${\cal N}_{q}$. Different
from Case 2, we can further prove the activated nonlocality of any subnetwork
consisting of all observers $\texttt{P}_{i}$ with $i\in{\cal
I}^{\prime}\subseteq{\cal I}$, where an example is shown in Figure S8(a). In
detail, there is a subnetwork ${\cal N}_{q}^{\prime}$ consisting of all
observes $\texttt{P}_{i}$ with $i\in{\cal I}^{\prime}$, which is obtained by
measuring the joint system $\rho_{1}\otimes\cdots\otimes\rho_{m}$ with a
proper local POVM of all the other observers. Moreover, the reduced subnetwork
consists of several chain-shaped and star-shaped quantum subnetworks. The
proof of this fact is iteratively followed by using an equivalent schematic
network, where an example is shown in Figure S8(b). For convenience, assume
that all observers $\texttt{P}_{1},\cdots,\texttt{P}_{k}$ are independent,
where ${\cal I}^{\prime}=\\{1,\cdots,k\\}$ with $k<n$. Moreover, assume that
each pair observers $\texttt{P}_{i}$ and $\texttt{P}_{j_{i}}$ share an
entanglement $\rho_{i}$, where some integers $j_{s},j_{t}$ may satisfy
$j_{s}=j_{t}$ for $s\not=t$, see the star-shaped quantum network shown in
Figure S6. Note that the assumption of shared entangled states is reasonable
from the independence of all observers $\texttt{P}_{1},\cdots,\texttt{P}_{k}$.
Now, we can reduce this subnetwork ${\cal N}^{\prime}$ as follows.

Take the subnetwork consisting of the observers
$\texttt{P}_{1},\texttt{P}_{2},\texttt{P}_{j_{1}},\texttt{P}_{j_{2}}$ as an
example.

  * S1.

When $j_{1}=j_{2}$, the joint system $\rho_{1}\otimes\rho_{2}$ consists of a
generalized $\Lambda$-shaped quantum network. In this subcase, after all
observers who share the systems $\rho_{1}\otimes\rho_{2}$ except for
$\texttt{P}_{1},\texttt{P}_{2},\texttt{P}_{j_{1}}$ performing a proper local
POVM on the joint system $\rho_{1}\otimes\rho_{2}$, the reduced system shared
by the observer $\texttt{P}_{1},\texttt{P}_{2},\texttt{P}_{j_{1}}$ is a
standard $\Lambda$-shaped quantum network consisting of entangled states. Note
that all these local POVMs do not change the joint system
$\rho_{3}\otimes\cdots\otimes\rho_{m}$ (see the subnetwork consisting of three
observers $\texttt{P}_{4}$, $\texttt{P}_{5}$, and $\texttt{P}_{j_{4}}$ shown
in Figures S8(c) and S8(d) as an example).

  * S2.

When $j_{1}\not=j_{2}$, there is a generalized chain-shaped quantum subnetwork
${\cal N}_{1,2}$ consisting of the observers
$\texttt{P}_{j_{1}},\texttt{P}_{t_{1}},\cdots,\texttt{P}_{t_{s-1}},\texttt{P}_{j_{2}}$
from the assumption of the connected quantum network, where any adjacent two
observers in ${\cal N}_{1,2}$ share at least one entangled state. Assume that
${\cal N}_{1,2}$ consists of a joint system
$\rho_{t_{1}}\otimes\cdots\otimes\rho_{t_{s}}$, where the observers
$\texttt{P}_{j_{1}}$ and $\texttt{P}_{t_{1}}$ share the entanglement
$\rho_{t_{1}}$, $\texttt{P}_{t_{l}}$ and $\texttt{P}_{t_{l+1}}$ share the
entanglement $\rho_{t_{l+1}}$, and $\texttt{P}_{t_{s}}$ and
$\texttt{P}_{j_{2}}$ share the entanglement $\rho_{t_{s}}$, and
$t_{l}\not=1,2$ for all $l$s. For each entanglement $\rho_{t_{l}}$ that is
also shared by some observer $\texttt{P}_{i}$ with $i>k$, the observer
$\texttt{P}_{t_{l}}$ or $\texttt{P}_{i}$ can perform a disentangling operation
such that the reduced joint system is entangled, where the reduced quantum
network should be connected after the disentangling operations. Moreover, for
each entanglement $\rho_{t_{l}}$ that is shared by some observer
$\texttt{P}_{i}$ with $i\leq k$, the observer $\texttt{P}_{t_{l}}$ can perform
a disentangling operation such that the reduced joint system is entangled,
where the reduced quantum network should be connected. After all observers who
do not belong to ${\cal
N}_{1,2}\cup\\{\texttt{P}_{3},\cdots,\texttt{P}_{k}\\}$ performing a local
POVM on the joint system $\rho_{t_{1}}\otimes\cdots\otimes\rho_{t_{s}}$, the
reduced subnetwork ${\cal N}_{1,2}$ is a standard chain-shaped quantum network
shown in Figure S5. If $t_{l}\not\in\\{3,\cdots,m\\}$ for all $l$s, all these
local POVMs do not change the joint system
$\rho_{3}\otimes\cdots\otimes\rho_{m}$. Otherwise, each entanglement
$\rho_{j}$ with $t_{l}=j$ for some $l$ is changed into a new entanglement
$\hat{\rho}_{j}$, which is shared by two observers $\texttt{P}_{j}$ and
$\texttt{P}_{t_{l-1}}$, where $t_{0}=j_{1}$ and $t_{s}=j_{2}$. One example is
shown in Figures S8.

Now, similar reducing procedure can be performed for the subnetwork consisting
of the subnetwork ${\cal N}_{1,2}$ and the parties
$\texttt{P}_{3},\texttt{P}_{j_{3}}$. By iteratively repeating these reducing
procedures of subcases 1 and 2 for all observers
$\texttt{P}_{1},\texttt{P}_{1},\texttt{P}_{j_{1}},\texttt{P}_{j_{2}}$, we can
obtain a generalized subnetwork ${\cal N}^{\prime}$ consisting of star-shaped
quantum subnetworks and chain-shaped quantum subnetworks, seeing one example
shown in Figures S8(e) and S8(f), where all entangled states thate are not
involved in these reducing procedures can be measured with any local POVMs.

In what follows, we show the $k$-partite nonlocality of each generalized
hybrid quantum network (one example is shown in Figure S8(f)). It can be
completed by combining Corollary 2 and Lemma 1. Specially, we can firstly
prove the multipartite nonlocality for all star-shaped quantum subnetworks
from Lemma 1. And then we can complete the proof for the hybrid quantum
network shown in Figure S8(f) from Corollary 2, where the procedure given in
subsections A1 of Appendix A can be iteratively used. Consequently, we have
proved Theorem 2. $\blacksquare$

## Appendix C: Proof of Theorem 3

In this section, we prove the nonlocality of a hybrid network, see Figure 3.
In detail, we consider an arbitrary quantum network ${\cal N}_{q}$ consisting
of $n$ observers $\texttt{P}_{1},\cdots,\texttt{P}_{n}$ with $n\geq 3$. Assume
that the quantum resources of ${\cal N}$ consist of
$\rho_{1}\otimes\cdots\otimes\rho_{m}$. When all $\rho_{i}$s are bipartite or
multipartite entangled states that are shared by two or multiple observers.
${\cal N}_{q}$ is reduced to the network shown in Theorem 2 if it is
connected. Now, we consider the following case. Note that ${\cal N}_{q}$ can
be regarded as a classical network that is local when all $\rho_{i}$s are
fully separable. So, in what follows, assume that there are some $\rho_{i}$s
that are entangled. We can prove that ${\cal N}_{q}$ is $n$-partite nonlocal
even if ${\cal N}_{q}$ is disconnected. Formally, we prove the result shown in
Theorem 3. The proof is followed from Theorem 2. In fact, assume that ${\cal
N}_{q}$ consists of $k$ connected quantum subnetworks $\tilde{\cal
N}_{1},\cdots,\tilde{\cal N}_{k}$, and classical network $\hat{\cal N}$, i.e.,

$\displaystyle{\cal N}=(\cup_{i=1}^{k}\tilde{\cal N}_{i})\cup\hat{\cal N},$
(C1)

where $\tilde{\cal N}_{i}$ and $\tilde{\cal N}_{j}$ have no shared observer
for any $i\not=j$, and $\tilde{\cal N}_{i}$ and $\hat{\cal N}$ have some
shared observers. One example is shown in Figure 3(a). Here, $\tilde{\cal
N}_{i}$ consists of all entangled states for all $i$s while $\hat{\cal N}$
consists of all fully separable states.

For each connected quantum subnetwork $\tilde{\cal N}_{i}$ consisting of
$n_{i}$ observers with a quantum state $\varrho_{i}$, from Theorem 2 there is
a semiquantum nonlocal game $\mathbbm{G}^{(i)}_{sq}$ (or generalized Bell-type
inequalities) such that the average expect $\wp_{i}$ satisfies

$\displaystyle\wp^{(i)}_{q}(\varrho_{i})>c_{i},$ (C2)
$\displaystyle\sup_{\hat{Q}_{1},\cdots\hat{Q}_{n_{i}}}\wp^{(i)}_{c}(\varrho^{c}_{i})\leq
c_{i},$ (C3)

where $\varrho^{c}_{i}$ denotes the classical resource of the subnetwork
$\tilde{\cal N}_{i}$ (or the fully separable states), $c_{i}\geq 0$ denotes
the available classical upper bound of the average gain
$\wp^{(i)}_{c}(\varrho^{c}_{i})$ which depends on the multipartite classical
correlations from $\tilde{\cal N}_{i}$ in terms of all local POVMs
$\hat{Q}_{1},\cdots,\hat{Q}_{n_{i}}$ for all observers in $\tilde{\cal
N}_{i}$. The inequality (C2) means that there are some local POVMs
$Q_{1},\cdots Q_{n_{i}}$ for all observers in $\tilde{\cal N}_{i}$ such that
$\wp^{(i)}_{q}(\varrho_{i})>c_{i}$ for the quantum state $\varrho_{i}$.

For the classical subnetwork $\hat{\cal N}$ consisting of $n^{\prime}$
observers with a quantum state $\hat{\varrho}$, for each semiquantum nonlocal
game $\hat{\mathbbm{G}}_{sq}$ there are some local POVMs
$\hat{Q}^{\prime}_{1},\cdots,\hat{Q}^{\prime}_{n^{\prime}}$ such that

$\displaystyle\hat{\wp}_{c}(\hat{\varrho})=\hat{c}$ (C4)

where
$\hat{c}=\sup_{\hat{Q}_{1},\cdots\hat{Q}_{n^{\prime}}}\hat{\wp}_{c}(\hat{\varrho})\geq
0$. From forward computations using the similar reconstruction procedure shown
in subsection A1 of Appendix A, it is easy to prove that there is a global
semiquantum nonlocal game $\mathbbm{G}_{sq}$ such that

$\displaystyle\wp_{q}(\rho_{1}\otimes\cdots\otimes\rho_{m})>c,$ (C5)
$\displaystyle\sup_{\hat{Q}_{1},\cdots\hat{Q}_{n}}\wp_{c}(\varrho_{1}\otimes\cdots\otimes\varrho_{m})\leq
c$ (C6)

where $\wp$ is the average gain SQ  depending on the joint conditional
probabilities of the measurement outcomes of all observers, $\rho_{i}$s are
multipartite states satisfying that there is at least one entangled state
$\rho_{i}$, all $\varrho_{j}$s are fully separable states. The inequality (C5)
means that there are some local POVMs ${Q}_{1},\cdots Q_{n}$ of all observers
such that the average gain $\wp$ from quantum multipartite correlations of a
hybrid quantum network is no less than a constant $c$. The inequality (C6)
means that for all local POVMs $\hat{Q}_{1},\cdots\hat{Q}_{n}$ of all
observers the maximal average gain $\wp$ from multipartite correlations of a
classic network consisting of all fully separable states is no more than the
constant $c$. Here, the inequalities (C5) and (C6) can be regarded as
generalized Bell-type inequalities. It follows that any hybrid quantum network
consisting of at least one entangled state has multipartite nonlocality. We
have proved Theorem 3. $\blacksquare$

## Appendix D: Verifying entanglement swapping without LOCC

In this section, we present some discussions related to verifying entanglement
swapping with one copy of joint system. In detail, consider a $\Lambda$-shaped
quantum network consisting of two bipartite entangled states
$\rho_{AB}\otimes\rho_{CD}$ shared by three parties, where Alice and Bob share
the entanglement $\rho_{AB}$ while Bob and Charlie share the entanglement
$\rho_{CD}$. Theorem 1 shows that there is a generalized Bell-type inequality
for verifying the bipartite activated nonlocality of Alice and Charlie after a
local POVM of Bob from $\rho_{AB}^{\otimes n}\otimes\rho_{CD}^{\otimes n}$
with large $n$. In this section, we present some result for
$\rho_{AB}\otimes\rho_{CD}$.

To explain the main idea, assume that $\rho_{AB}$ and $\rho_{CD}$ has the
following forms:

$\displaystyle\rho_{AB}=p_{1}|\Phi_{1}\rangle_{AB}\langle\Phi_{1}|+p_{2}|\Phi_{2}\rangle_{AB}\langle\Phi_{2}|,$
(D1)
$\displaystyle\rho_{CD}=q_{1}|\Psi_{1}\rangle_{CD}\langle\Psi_{1}|+q_{2}|\Psi_{2}\rangle_{CD}\langle\Psi_{2}|,$
(D2)

where $|\Phi_{i}\rangle_{AB},|\Psi_{i}\rangle_{CD}$ are assumed to be
bipartite pure states.

Case 1. $|\Phi_{i}\rangle_{AB},|\Psi_{i}\rangle_{CD}$ are all bipartite
entangled states with $i=1,2$. In this case, assume that
$|\Phi_{1}\rangle_{AB}=\sum_{i}a_{i}|ii\rangle_{AB}$ and
$|\Psi_{1}\rangle_{AB}=\sum_{j}b_{j}|jj\rangle_{CD}$ which are formal
bipartite entangled states, and $|\Phi_{2}\rangle_{AB}=(U_{A}\otimes
U_{B})\sum_{i}a^{\prime}_{i}|ii\rangle_{AB}$ and
$|\Psi_{2}\rangle_{CD}=(U_{C}\otimes
U_{D})\sum_{j}b^{\prime}_{j}|jj\rangle_{CD}$ which are formal bipartite
entangled states under the local unitary transformations $U_{A}\otimes
U_{B},U_{C}\otimes U_{D}$, respectively. Here, $U_{A},U_{B},U_{C},U_{D}$ are
local unitary operations performed on the systems $A,B,C,D$, respectively.
Note that for each joint system:
$\rho_{ij}=|\Phi_{i}\rangle_{AB}|\Psi_{j}\rangle_{CD}\langle\Psi_{j}|\langle\Phi_{i}|$,
CHSH inequality is useful for verifying the bipartite activated nonlocality of
Alice and Charlie after a Bell measurement of Bob CHSH ; Gisin1 , where we can
take use of the equality
$\textrm{Tr}[M_{BC}\rho_{ij}]=\textrm{Tr}[M_{BC}(W_{A}\otimes W_{BC}\otimes
W_{D})\rho_{ij}(W_{A}\otimes W_{BC}\otimes
W_{D})^{\dagger}]=\textrm{Tr}[M_{BC}W_{BC}\rho_{ij}W_{BC}^{\dagger}]$,
$W_{A}$, $W_{BC}$, and $W_{D}$ are unitary operations on the systems $A,BC,D$,
respectively, and $M_{BC}$ is a local POVM on the systems $BC$. Combining the
equality $\rho_{AB}\otimes\rho_{CD}=\sum_{i,j}p_{i}q_{j}\rho_{ij}$, it follows
that the bipartite activated nonlocality of Alice and Charlie for the joint
system $\rho_{AB}\otimes\rho_{CD}$ may be verified by using CHSH inequality
CHSH ; Gisin1  after a Bell measurement of Bob. Generally, we cannot obtain
the deterministic detecting of the bipartite activated nonlocality because the
local unitary operations $U_{A},U_{B},U_{C},U_{D}$ can affect the upper bounds
$c_{ij}$s, see Example 1. Similar result holds for general bipartite entangled
states $\rho_{AB}\otimes\rho_{CD}$, where
$\rho_{AB}=\sum_{i}p_{i}|\Phi_{i}\rangle_{AB}\langle\Phi_{i}|$,
$\rho_{CD}=\sum_{j}q_{j}|\Psi_{j}\rangle_{CD}\langle\Psi_{j}|$, and
$|\Phi_{i}\rangle_{AB},|\Psi_{i}\rangle_{CD}$ are bipartite entangled states
for all $i,j$s.

Case 2. $|\Phi_{1}\rangle_{AB},|\Psi_{i}\rangle_{CD}$ are all bipartite
entangled states with $i=1,2$. Here, $|\Phi_{2}\rangle_{AB}$ is separable
state. In this case, assume that
$|\Phi_{1}\rangle_{AB}=\sum_{i}a_{i}|ii\rangle_{AB}$ and
$|\Psi_{1}\rangle_{AB}=\sum_{j}b_{j}|jj\rangle_{CD}$ which are formal
bipartite entangled states, and $|\Psi_{2}\rangle_{CD}=(U_{C}\otimes
U_{D})\sum_{j}b^{\prime}_{j}|jj\rangle_{CD}$ which is formal bipartite
entanglement under the local unitary transformations $U_{C}\otimes U_{D}$,
respectively. Here, $U_{C},U_{D}$ are local unitary operations performed on
the system $C,D$, respectively. Define

$\displaystyle\rho_{ij}=|\Phi_{i}\rangle_{AB}|\Psi_{j}\rangle_{CD}\langle\Psi_{j}|\langle\Phi_{i}|,i,j=1,2.$
(D4)

Assume that a Bell-type inequality which is linearly depending on bipartite
correlations $P(\textbf{x}|\textbf{a})$ is given as follows

$\displaystyle\wp(P_{q}(\textbf{x}|\textbf{a}))>c,$ (D5)
$\displaystyle\wp(P_{c}(\textbf{x}|\textbf{a}))\leq c,$ (D6)

where $P_{q}(\textbf{x}|\textbf{a})$ and $P_{c}(\textbf{x}|\textbf{a})$ are
the joint conditional probability distributions from the local measurements of
an entangled system $\rho_{AD}$, and fully separable systems or classical
states $\varrho_{AD}$, respectively. $c$ denotes the maximal classical bound
over any possible conditional probabilities $P_{c}(\textbf{x}|\textbf{a})$s.
Assume that there are local POVMs of three observers, i.e.,
${\mathscr{X}}_{1}$-POVM $\\{P^{x_{1}}\\}\in{\cal M}_{A;{\mathscr{X}}_{1}}$
for Alice, ${\mathscr{X}}_{2}$-POVM $\\{Q^{x_{2}}\\}\in{\cal
M}_{BC;{\mathscr{X}}_{2}}$ for Bob, and ${\mathscr{X}}_{3}$-POVM
$\\{R^{x_{3}}\\}\in{\cal M}_{D;{\mathscr{X}}_{3}}$ for Charlie. For each joint
system $\rho_{ij}$, define

$\displaystyle\wp(P_{ij}(\textbf{x}|\textbf{a}))=c_{ij}$ (D7)

where $P_{ij}(\textbf{x}|\textbf{a})$ denotes the joint conditional
probability distributions from the local POVMs of Alice and Charlie after a
local POVM by Bob on his systems $BC$ of $\rho_{ij}$. It follows that

$\displaystyle\wp(P_{q}(\textbf{x}|\textbf{a}))=$
$\displaystyle\wp(\sum_{ij}p_{i}q_{j}P_{ij}(\textbf{x}|\textbf{a}))$
$\displaystyle=$ $\displaystyle\sum_{ij}p_{i}q_{j}c_{ij},$ (D8)

where the equation (D8) is from the linearity of $\wp$. So, in order to verify
the bipartite activated nonlocality, we have

$\displaystyle\sum_{ij}p_{i}q_{j}c_{ij}>c$ (D9)

for some POVMs of all observers, where $c$ is given in the equation (D6).
Here, from assumptions $\rho_{11}$ and $\rho_{12}$ are useful joint systems.
$\rho_{21}$ and $\rho_{22}$ are useless even if they are not classical system,
where $|\Psi_{i}\rangle_{CD}$ are entangled. So, we can obtain the following
equivalent inequality of (D9) as

$\displaystyle
q_{1}c_{11}+q_{2}c_{12}>\frac{1}{p_{1}}(p_{2}(q_{1}c_{11}+q_{2}c_{12})-c)$
(D10)

which should be satisfied by a joint system $\rho_{AB}\otimes\rho_{CD}$ if it
is bipartite activated nonlocal in terms of a Bell inequality shown in the
equation (D6). Similar result holds for general bipartite entangled states
$\rho_{AB}\otimes\rho_{CD}$ consisting of multiple bipartite entangled pure
states.

In what follows, we present some examples.

Example 1. Assume that a $\Lambda$-shaped quantum network consists of two
bipartite entangled states $\rho_{AB}$ and $\rho_{CD}$, where

$\displaystyle\rho_{AB}=p_{1}|\Phi_{1}\rangle\langle\Phi_{1}|+p_{2}|\Phi_{2}\rangle\langle\Phi_{2}|,$
(D11)
$\displaystyle\rho_{CD}=q_{1}|\Psi_{1}\rangle\langle\Psi_{1}|+q_{2}|\Psi_{2}\rangle\langle\Psi_{2}|,$
(D12)

$|\Phi_{i}\rangle$ and $|\Psi_{i}\rangle$ are bipartite entangled pure states
which are defined by

$\displaystyle|\Phi_{1}\rangle=$ $\displaystyle
a_{1}|00\rangle+b_{1}|11\rangle,$ (D13) $\displaystyle|\Psi_{1}\rangle=$
$\displaystyle a_{2}|00\rangle+b_{2}|11\rangle,$ (D14)
$\displaystyle|\Phi_{2}\rangle=$ $\displaystyle
c_{1}|01\rangle+c_{1}|10\rangle,$ (D15) $\displaystyle|\Psi_{2}\rangle=$
$\displaystyle d_{2}|01\rangle+d_{2}|10\rangle,$ (D16)

$p_{1}+p_{2}=q_{1}+q_{2}=a_{i}^{2}+b_{i}^{2}=c_{i}^{2}+d_{i}^{2}=1$, $0\leq
p_{i},q_{i}\leq 1$, $0<a_{i},b_{i},c_{i},d_{i}<1$, $i=1,2$. Now, consider
bipartite CHSH inequality CHSH  as

$\displaystyle\wp(P(\textbf{x}|\textbf{a})):=$
$\displaystyle\langle{M}^{1}_{0}\otimes{M}^{2}_{0}\rangle+\langle{M}^{1}_{0}\otimes{M}^{2}_{1}\rangle+\langle{M}^{1}_{1}\otimes{M}^{2}_{0}\rangle-\langle{M}^{1}_{1}\otimes{M}^{2}_{1}\rangle$
$\displaystyle\leq$ $\displaystyle 2,$ (B17)

where ${M}^{1}_{i},{M}^{2}_{j}$ denote dichotomic quantum observables with
$\pm 1$ outputs. Now, define dichotomic quantum observables
$M^{1}_{i}=(1-i)\sigma_{z}+i\sigma_{x}$, $i=0,1$, and
$M^{2}_{i}=\cos\theta\sigma_{z}+(-1)^{i}\sin\theta\sigma_{x}$, where
$\sigma_{z}$ and $\sigma_{x}$ are Pauli matrices. Assume that Bob performs
Bell measurement on the system $BC$ under the basis
$\\{\frac{1}{\sqrt{2}}(|00\rangle\pm|11\rangle),\frac{1}{\sqrt{2}}(|01\rangle\pm|10\rangle)\\}$.
In the following, we only compute one local measurement of Bob with POVM
$M_{BC}=\frac{1}{2}\sum_{i,j=0,1}|ii\rangle\langle jj|$. It is easy to follow
that

$\displaystyle c_{11}=$
$\displaystyle\textrm{Tr}[({M}^{1}_{0}\otimes{M}^{2}_{0}+{M}^{1}_{0}\otimes{M}^{2}_{1}+{M}^{1}_{1}\otimes{M}^{2}_{0}-{M}^{1}_{1}\otimes{M}^{2}_{1})\hat{\rho}_{11}]$
$\displaystyle=$ $\displaystyle
2\cos(\theta)+4\frac{a_{1}b_{1}a_{2}b_{2}}{a_{1}^{2}a_{2}^{2}+b_{1}^{2}b_{2}^{2}}\sin(\theta),$
(D18) $\displaystyle c_{12}=$
$\displaystyle\textrm{Tr}[({M}^{1}_{0}\otimes{M}^{2}_{0}+{M}^{1}_{0}\otimes{M}^{2}_{1}+{M}^{1}_{1}\otimes{M}^{2}_{0}-{M}^{1}_{1}\otimes{M}^{2}_{1})\hat{\rho}_{12}]$
$\displaystyle=$
$\displaystyle-2\cos(\theta)+4\frac{a_{1}b_{1}c_{2}d_{2}}{a_{1}^{2}c_{2}^{2}+b_{1}^{2}d_{2}^{2}}\sin(\theta),$
(D19) $\displaystyle c_{21}=$
$\displaystyle\textrm{Tr}[({M}^{1}_{0}\otimes{M}^{2}_{0}+{M}^{1}_{0}\otimes{M}^{2}_{1}+{M}^{1}_{1}\otimes{M}^{2}_{0}-{M}^{1}_{1}\otimes{M}^{2}_{1})\hat{\rho}_{21}]$
$\displaystyle=$