# Strong entanglement distribution of quantum networks

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

###### Abstract

Large-scale quantum networks have been employed to overcome practical
constraints of transmissions and storage for single entangled systems. Our
goal in this article is to explore the strong entanglement distribution of
quantum networks. We firstly show any connected network consisting of
generalized EPR states and GHZ states satisfies strong CKW monogamy inequality
in terms of bipartite entanglement measure. This reveals interesting feature
of high-dimensional entanglement with local tensor decomposition going beyond
qubit entanglement. We then apply the new entanglement distribution relation
in entangled networks for getting quantum max-flow min-cut theorem in terms of
von Neumann entropy and Rényi-$\alpha$ entropy. We finally classify entangled
quantum networks by distinguishing network configurations under local unitary
operations. These results provide new insights into characterizing quantum
networks in quantum information processing.

##  I Introduction

Entanglement as an intriguing phenomenon has been considered to be the heart
of quantum mechanics. It provides a crucial resource for quantum information
processing, including quantum teleportation Bennett1 , quantum dense coding
Bennett2 , quantum secret sharing Hillery , and quantum cryptography Gisin .
The study of entanglement and its distribution reveals fundamental insights
into the nature of quantum correlations Ekert(1991) ; Horodecki(2009) , the
features of many-body systems Eisert(2010) ; Amico(2008) , and the potential
limitations for quantum-enhanced technologies Dowling(2003) .

One interesting feature of entanglement is the impossibility of sharing
entanglement freely in multiparty quantum systems. This without any classical
counterpart is known as the monogamy of entanglement (MOE) Terhal(2004) ;
V.Coffman , that is, entanglement satisfy special constraints on how they can
be distributed among multipartite systems. The monogamy inequality became
synonymous due to Coffman, Kundu, and Wootters (CKW) V.Coffman  as

$\displaystyle{\cal
Q}_{\textsf{A}|\textsf{B}\textsf{C}}(\rho_{\textsf{A}\textsf{B}\textsf{C}})\geq{\cal
Q}_{\textsf{A}|\textsf{B}}(\rho_{\textsf{A}\textsf{B}})+{\cal
Q}_{\textsf{A}|\textsf{C}}(\rho_{\textsf{A}\textsf{C}})$ (1)

with specific entanglement measure ${\cal Q}$. Here,
$\rho_{\textsf{A}\textsf{B}}=\textrm{Tr}_{\textsf{C}}(\rho_{\textsf{A}\textsf{B}\textsf{C}})$
denotes the reduced state of parties A and B, and analogously for
$\rho_{\textsf{A}\textsf{C}}$. $\textsf{A}|\textsf{B}\textsf{C}$ is for the
bipartite split. The first example is related the squared concurrence between
bipartitions Terhal(2004) ; V.Coffman  for tripartite states and $N$-qubit
states T.J.Osborne . Some other monogamy inequalities are built for multiqubit
systems in terms of the $\alpha$-th power of entanglement of formation (EOF)
($\alpha\geq\sqrt{2}$) Oliveira(2014) ; Bai3PRL ; Guo  or the $\alpha$-th
power of concurrence ($\alpha\geq 2$) Fei1 ; Jin . Similar results hold for
qubit systems with the $\alpha$($\alpha\geq 2$)-th powers of Tsallis entropy
Luo(2016) , Rényi entropy Kim(2010)R , and Unified entropy KimBarry(2011) .
However, most of these well-known entanglement measures T.J.Osborne ;
Oliveira(2014) ; Bai3PRL ; Bai(2014) ; Guo ; Luo(2016) ; Sara(2016) ; R(2015)
; Khan(2019)  fail to satisfy the monogamy relation (1) for qubit systems
except for the squashed entanglement Koashi(2004) . A natural problem is to
explore which entangled states allow the CKW inequality. One intriguing
example is from higher-dimensional quantum systems Ou(2007) ; Luo(2016) ;
Luo2021 .

It is difficult to characterize general high-dimensional entangled states
because of exponential parameters. One special scenario is from distributive
constructions, that is, it allows local tensor structures further regarded as
quantum networks LuoF . This kind of multipartite resources are applicable for
large-distance quantum communication Zuk ; Duan , quantum internet Kimble ;
Wehner , quantum secret sharing Luo2021 , distributed quantum computing
Cirac(1999) , blind quantum computing Broadbent (2009)  and quantum sensing
Gottesman(2012) ; P. (2014) . This intrigues an interesting problem of
universal monogamy relations for higher-dimensional systems from quantum
networks.

Our motivation in this paper is to explore entangled quantum networks by using
quantum entropy toward quantum communication. We firstly prove the standard
CWK monogamy (1) holds for generic entangled quantum networks in terms of EOF
and Rényi entropy. This shows new features of entanglement distribution for
high-dimensional quantum states going beyond qubit entanglement Terhal(2004) ;
V.Coffman ; Fei1  or single entanglement Bai3PRL ; R(2015) . We then apply the
proposed entanglement distribution to resolve the max-flow min-cut problem for
entangled networks even it fails with single multipartite entanglement
Pira(2019) . The new Max-flow Min-cut theorem holds for any quantum networks
consisting of EPR states and GHZ states. Moreover, the entanglement entropy
implies one way for classifying the configuration for entangled networks under
local unitary operations. The present method can be adapted for cyclic
networks going beyond recent results Kela(2020) ; Aberg(2020) . These results
are interesting in entanglement theory, quantum communication and quantum
information processing.

##  II Entanglement monogamy relation of quantum networks

In comparison to single entanglement, an $n$-partite quantum network consists
of various entangled states, as shown in Fig.1. Here, the local unitary
operations may entangle these independent states into a new entanglement in
global version. This generates high-dimensional quantum entangled states that
are important resources in quantum information processing. Characterizing the
general high-dimensional quantum systems is difficult because of exponential
number of parameters. Our motivation here is to feature quantum networks vias
entanglement distribution by using quantum entropy. The high-dimensional
entanglement inspired by quantum networks allows local tensor decomposition
that provides the possibility for evaluating new monogamy relations based on
von Neumann entropy Nielsen  and Rényi entropy HHH(1996) , Tsallis entropy
Tsallis(1988) , and Unified entropy Rathie(1991) .

Figure 1: (Color online) A new monogamy relation of tripartite entanglement
inspired by tripartite quantum network consisting of one GHZ state and three
EPR states. $U,V$ and $W$ are local unitary operations. ${\cal Q}$ is
entanglement measure defined in the inequality (1).

Consider an $n$-partite entangled quantum network
$\mathcal{N}_{q}(\cal{A},\xi)$, where $\cal{A}$ denotes parties
$\textsf{A}_{1},\cdots,\textsf{A}_{n}$, and $\xi$ denotes entangled states.
Assume any two parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$ share the states
$\varrho_{ij}(\theta_{1}),\cdots,\varrho_{ij}(\theta_{s})$,
$\sigma_{ij}(\varphi_{1}),\cdots,\sigma_{ij}(\varphi_{t})$, and
$\delta_{ij}(\vartheta_{1}),\cdots,\delta_{ij}(\vartheta_{k})$, where
$\varrho(\theta)$ denotes the density matrix of generalized EPR state EPR :
$|\phi(\theta)\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$, and
$\sigma(\varphi)$ is the density matrix of generalized GHZ state GHZ :
$|\phi(\varphi)\rangle=\cos\varphi|0\rangle^{\otimes
m}+\sin\varphi|1\rangle^{\otimes m}$ with any integer $m\geq 3$, and
$\delta({\vartheta})=\cos^{2}\vartheta|00\rangle\langle
00|+\sin^{2}\vartheta|11\rangle\langle 11|)$ is the reduced density matrix of
any two subsystems obtained by tracing out other subsystems in a generalized
GHZ state. Here, each party can perform any local unitary operations to
entangle the local systems. With this formation, the total system may be
regarded as an $n$-partite entanglement in high-dimensional Hilbert spaces.
Informally, we show that the entanglement distribution for quantum networks
with local unitary operations satisfies the monogamy inequality (1). One
example of tripartite quantum network is shown in Fig.1.

###### Theorem 1

The entanglement distribution of quantum network
$\mathcal{N}_{q}(\cal{A},\xi)$ is given by

$\displaystyle{\cal
Q}_{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})\geq\sum^{N}_{j=1,j\neq
i}{\cal
Q}_{\textsf{A}_{i}|\textsf{A}_{j}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}),$ (2)

where ${\cal Q}$ denotes entanglement measure of EOF or Rényi-$\alpha$
entropy, ${\cal Q}_{\textsf{X}|\textsf{Y}}$ is an entanglement measure of a
composite quantum system with respect to the bipartite cut between X and Y,
and $\overline{\textsf{A}_{i}}$ denotes all parties except for
$\textsf{A}_{i}$. The equality holds if and only if all GHZ states being
shared by two parties.

The proof of Theorem 1 is provided in Appendix A based on the additivity of
quantum entropy. It also includes entanglement distribution of quantum
networks consisting of general entanglement (see Appendix B) or in terms of
Tsallis entropy and Unified entropy (see Appendix C). Theorem 1 indicates a
mutually exclusive relation of quantum networks between any pair of parties
$\textsf{A}_{i}$ and $\textsf{A}_{j}$, which goes beyond the single
entanglement of GHZ states being ruled out the monogamy inequality (1). The
more interesting part is the present result provides the first generic
monogamous relation for high-dimensional entanglement beyond qubit states
Terhal(2004) ; V.Coffman ; T.J.Osborne ; Fei1 ; Luo(2016) ; Kim(2010)R ;
KimBarry(2011) . It is also stronger than the existence result of high-
dimensional entanglement Luo2021 . Theorem 1 may provide a better-than single-
entanglement performance in quantum tasks. One example is device-independent
quantum key distribution Horodecki(2009)  where eavesdropper’s attacks are
basically limited by the monogamy of entanglement.

Example 1. Consider a device-independent quantum key distribution model with
entanglement $\rho_{\textsf{AB}}$ Ekert(1991) . An eavesdropper Eve is
represented by E who may be correlated with $\rho_{\textsf{AB}}$. The total
state is then denoted by $\rho_{\textsf{ABE}}$ satisfying
$\textrm{Tr}_{\textsf{E}}(\rho_{\textsf{ABE}})=\rho_{\textsf{AB}}$, as shown
in Fig.2(a). Eve is assumed to control the source or measurement devices in a
relaxed model. The entanglement between Eve and Alice satisfies the monogamy
relation Fei1 :

$\displaystyle{\cal E}_{\textsf{A}|\textsf{E}}(\rho_{\textsf{AE}})\leq({\cal
E}^{\sqrt{2}}_{\textsf{A}|\textsf{BC}}(\rho_{\textsf{ABE}})-{\cal
E}^{\sqrt{2}}_{\textsf{A}|\textsf{B}}(\rho_{\textsf{AB}}))^{\frac{1}{\sqrt{2}}},$
(3)

where ${\cal E}$ represents the EOF measure. The inequality (3) provides a
monogamy relation for the information leakage in a device-independent model.

Instead, eavesdropper may hold local systems correlated with
$\rho_{\textsf{AB}}$. That is, Eve fakes entangled sources
$|\varphi_{\beta}\rangle$ and $|\varphi_{\gamma}\rangle$ between himself and
respective Alice’s device and Bob’s device, as shown in Fig.2(b). This
generates a triangle quantum network which allows from Theorem 1 the new
relation as

$\displaystyle{\cal E}_{\textsf{A}|\textsf{E}}(\rho_{\textsf{AE}})\leq{\cal
E}_{\textsf{A}|\textsf{BE}}(\rho_{\textsf{ABE}})-{\cal
E}_{\textsf{A}|\textsf{B}}(\rho_{\textsf{AB}}).$ (4)

Herein, the inequality (4) provides a stronger constraint for the information
leakage in a device-independent model, which makes it harder for attacker to
carry out the attack.

Figure 2: (Color online). Device-independent QKD with an attacker $E$. (a) The
standard attack model. (b) The new attack model based on triangle network.

##  III Communication capacities of quantum networks

Quantum networks provide new capabilities for generating and applying quantum
entanglement. One fundamental problem is to determine the maximal rate
achievable in end-to-end transmission of states. This has so far, been
addressed for single entanglement Bennett1  or quantum networks Pira(2019)
consisting of bipartite entangled states. Our goal here is to investigate the
capacities of general quantum networks ${\cal N}_{q}$ consisting of EPR states
and GHZ states.

For any bipartite entanglement $|\varphi\rangle_{\textsf{A}\textsf{B}}$, it
can be regarded as a quantum channel with the capacity Bennett1
$\mathcal{C}_{\textsf{A}\textsf{B}}=S(\rho_{\textsf{A}})=\mathcal{E}_{\textsf{A}|\textsf{B}}(|\varphi\rangle_{\textsf{A}\textsf{B}})$,
where $S(\rho_{\textsf{A}})$ is the von Neumann entropy of the party A. As a
result, the total capacity of quantum channel $\\{\xi_{ij}\\}$ connecting the
party $\textsf{A}_{i}$ and $\textsf{A}_{j}$ in quantum network
$\mathcal{N}_{q}$ is given by

$\displaystyle S(\rho^{j}_{i})$ $\displaystyle=$
$\displaystyle\mathcal{E}_{\textsf{A}_{i}|\textsf{A}_{j}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
(5) $\displaystyle=$ $\displaystyle n_{1}+n_{2},$

where $n_{1}$ and $n_{2}$ denote the respective number of EPR states and GHZ
states shared by party $\textsf{A}_{i}$ and $\textsf{A}_{j}$,
$S(\rho^{j}_{i})$ represents the von Neumann entropy of the reduced density
matrix $\rho^{j}_{i}$ by tracing out the system of $j$, i.e.,
$\rho^{j}_{i}={\rm{Tr}}_{j}(|\xi_{ij}\rangle\langle\xi_{ij}|)$. This equality
allows us to evaluate the capacity of quantum networks via von Neumann
entropy. It is noteworthy that Rényi entropy is also available for evaluating
channel capacity.

Before stating the main result, we explain the main idea inspired by classical
network theory Voloshin(2009) . Consider an $n$-partite entangled quantum
network $\mathcal{N}_{q}({\cal{A}},\Omega)$. Let s and t be the source and
sink, respectively. The capacity of one edge is a function
$\mathcal{C}:\Omega\longrightarrow\mathbb{R}^{+}$, where the edge
$(\textsf{A}_{i},\textsf{A}_{j})\in\Omega$ is connected by the channel
$\xi_{ij}$ (consisting of EPR states and GHZ states) to transmit quantum
information from the party $\textsf{A}_{i}$ to $\textsf{A}_{j}$.
$\mathcal{C}(\xi_{ij})$ gives the maximum amount of flow through the channel
$\xi_{ij}$. We also use the short-hand notation
$\mathcal{C}(i,j)=\mathcal{C}(\xi_{ij})$. Similar to classical network
Voloshin(2009) , a flow $f$ of quantum network $\mathcal{N}_{q}$ is an
assignment of weights to channels satisfying: (i) Capacity conservation:
$f_{i\rightarrow j}\leq\mathcal{C}(\xi_{ij})$ for any $\xi_{ij}\in\Omega$.
(ii) Flow conservation: flow leaving from the party $j$ is equal to flow
entering $j$ for $j\in\cal{A}\setminus\\{\textbf{s},\textbf{t}\\}$.

To address the maximal flow of given network
$\mathcal{N}_{q}({\cal{A}},\Omega)$, the main idea is to find the cut of
associated undirected graph $\mathcal{G}=({\cal V},\Omega)$, where ${\cal V}$
consists of all vertices and $\Omega$ consists of all edges. Here, one vertex
schematically denotes one party. One edge $(i,j)$ connected two vertices
schematically denotes as an EPR state shared by the corresponding parties
$\textsf{A}_{i}$ and $\textsf{A}_{j}$. The GHZ state shared by $m$ parties is
associated with hyperedge connecting $m$ vertices. This correspondence is
reasonable because of the equality
$S(\rho_{\textsf{A}_{i}})=S(\rho_{\textsf{A}_{j}})$ for any GHZ state
$\rho_{\textsf{A}_{1}\cdots\textsf{A}_{m}}$, $1\leq i<j\leq m$. Thus the
quantum network $\mathcal{N}_{q}$ is associated with an undirected finite
hypergraph $\mathcal{G}$. With this correspondence, the weight of each edge
$(i,j)$ or hyperedge is unit, that is, the entanglement entropy of EPR state
or GHZ state. A cut $T_{cut}$ of $\mathcal{G}$ is a bipartition
$(\textbf{S}_{1},\textbf{S}_{2})$ of ${\cal V}$ such that
$\textbf{s}\in\textbf{S}_{1}$ and $\textbf{t}\in\textbf{S}_{2}$. Thus the
capacity of cut equals to the sum of weights of edges leaving from
$\textbf{S}_{1}$ to $\textbf{S}_{2}$, i.e.,
$\mathcal{C}(\textbf{S}_{1},\textbf{S}_{2})=\sum_{(i,j)|i\in\textbf{S}_{1},j\in\textbf{S}_{2}}\mathcal{C}(i,j)$,
where $\mathcal{C}(i,j)$ denotes the weight of edge $(i,j)$. Significantly,
the flow in classical network obeys the celebrated Max-flow Min-cut Theorem
Voloshin(2009) . We prove similar result for the quantum settings.

###### Theorem 2

For a given quantum network $\mathcal{N}_{q}$ associated with undirected graph
$\mathcal{G}$, the maximal flow $f_{\max}$ is equal to the minimal cut of
$\mathcal{G}$, that is,

$\displaystyle
f_{\max}=\min_{T_{cut}}\mathcal{C}(\textbf{S}_{1},\textbf{S}_{2}).$ (6)

Theorem 2 states that the maximum flow from s to t in quantum network
$\mathcal{N}_{q}$ equals to the minimum cut that separates s and t in the
corresponding graph $\mathcal{G}$. This provides a general method similar to
point-to-point quantum protocol Bennett1  for general quantum networks beyond
recent result Pira(2019) . The proof is shown in Appendix E. In applications,
for a given quantum network $\mathcal{N}_{q}$ consisting of EPR states and GHZ
states under local unitary operations, the network capacity can be evaluated
by extending Ford-Fulkerson Algorithm Ford-Fulkerson  as follows.

  * Input:

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

  * Output:

A maximal flow.

  * •

Find the associated undirected graph $\mathcal{G}=({\cal V},\Omega)$.

  * •

Initialize $f(i,j)=0$ for all edges and hyperedges.

  * •

Find a path $p_{\textbf{s},\textbf{t}}$ from s to t with
$f(i,j)\leq\mathcal{C}(i,j)$ (augmenting path) on every edge and hyperedge.

  * •

Augment flow $f$ on edges or hyperedge along the path
$p_{\textbf{s},\textbf{t}}$.

  * •

Repeat the procedure until there is no augmenting path from s to t.

Example 2. Consider the maximization of the flow from s to t as shown in
Fig.3(a) by applying to extended Ford-Fulkerson Algorithm. It is easy to check
that the maximum flow from s to t is equal to 7, which is consistent with the
minimum capacity of all source-sink cuts in a corresponding undirected graph
as shown in Fig.3(b). The iterations of Example 2 are illustrated in Appendix
E.

Figure 3: (Color online) (a) Max-flow of 6-partite quantum network ${\cal
N}_{q}$. s is source while t is sink. Three parties
$\textsf{A}_{2},\textsf{A}_{4}$ and t are connected by two GHZ states and the
remaining parties are connected by multiple EPR states. The red integer on
quantum channel denotes its capacity, i.e., the number of EPR states or GHZ
states. The blue and green lines are two maximal flows. (b) Min-cut of the
associated graph of ${\cal N}_{q}$. EPR state is denoted as one edge while GHZ
state is denoted as one hyperedge (red line). A source-sink cut achieves the
min-cut of 7.

##  IV Network topology classification

The actual quantum networks have a complex topology which significantly
affects the communication efficiency. It is natural to consider how to
identify network configuration for a given network. One method is to explore
the measurement statistics by using the covariance matrix Kela(2020) ;
Aberg(2020)  or the generalized Finner inequality Luo2021b  in a device-
independent manner. The other is from the coherence theory Kraft(2021) . These
methods can be applied for characterizing quantum networks assisted by
specific measurement. Our goal here is to consider a beyond problem of
classifying configurations of given the set of networks with lower complexity.
It is well-known that two local unitary equivalent multipartite entangled
states have the same entanglement entropy with respect to any bipartition. We
show that the converse holds for quantum networks consisting of EPR states and
GHZ states. Especially, for a given $n$-partite quantum network
$\mathcal{N}_{q}$, denote the characteristic vector of quantum network
$\mathcal{N}_{q}$ as

$\displaystyle
S_{\mathcal{N}_{q}}=(S(\rho_{\mathsf{A}_{1}}),S(\rho_{\mathsf{A}_{2}}),\cdots,S(\rho_{\mathsf{A}_{n}})),$
(7)

where $S(\rho_{\mathsf{A}_{i}})$ denotes the von Neumann entropy of the
reduced state of the party $\mathsf{A}_{i}$ of $\mathcal{N}_{q}$. We present a
configuration classification under the local unitary equivalence by virtue of
the characteristic vector of quantum networks.

###### Theorem 3

Assume $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ are two quantum networks
consisting of EPR states and GHZ state, where any two parties $\mathsf{A}_{i}$
and $\mathsf{A}_{j}$ in each network share no more than one entanglement.
Then, $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ are unitary equivalence if and
only if their characteristic vectors are equal to each other, that is,
$S_{\mathcal{N}_{1}}=S_{\mathcal{N}_{2}}$.

Figure 4: (Color online) Different networks with same characteristic vector.
(a) Triangle network consisting of three EPR states (black lines) or two GHZ
states (blue lines). (b) 4-partite network consisting of four EPR states or
two GHZ states and one EPR states.

The proof of Theorem 3 is shown in Appendix F. Theorem 3 provides simple
method to classify the given set of networks consisting of EPR states and GHZ
states. In application, it only requires to evaluate $n$ number of von Neumann
entropy $S(\rho_{\mathsf{A}_{i}})$, which has linear complexity beyond the
exponential complexity for evaluating all entanglement entropies of
$S(\rho_{\mathsf{A}_{i_{1}}\cdots\mathsf{A}_{i_{t}}})$, $1\leq
i_{1}<\cdots<i_{t}\leq n$. The drawback is from the assumption of no more than
one entanglement shared by given group of parties. Otherwise, two
counterexamples are shown in Fig.4. From Fig.4(a), the triangle network
consisting of three EPR states is inequivalent to its consisting of two GHZ
states even for the same characteristic vector $(2,2,2)$. The other is the
cyclic networks consisting four EPR states or two GHZ states and one EPR state
in Fig.4(b) with characteristic vector $(2,2,2,2)$. These can be extended for
any cyclic quantum network in Appendix G. Thus a new method beyond the
characteristic vector of von Neumann entropy should be explored to verify
these cyclic quantum networks. One possibility is from Shannon mutual
information Shannon(1998)  derived from local projection measurements in
Appendix G.

##  V Discussions and conclusion

Quantum entropy provides an easy way for characterizing entanglement
distribution of specific quantum networks. This strong monogamy relations
implies interesting applications in quantum communications and quantum network
configurations. The present method is useful for generic settings of quantum
network consisting of EPR states and GHZ states. These kind of networks are
universal for measurement based quantum computation RB . A natural problem is
for general entangled networks. Specially, it should be useful for exploring
quantum networks consisting of mixed states, which may intrigue new features
going beyond pure systems. Another is for the max-flow problem on noisy
networks including dephasing and erasure channels.

To sum up, we have proved the strong monogamy relation for quantum networks
consisting of any bipartite entangled pure states and generalized GHZ states.
These relations provide new feature of high dimensional entanglement beyond
qubit states. This implies a celebrated max-flow min-cut theorem on quantum
networks based on von Neumann entropy. Finally, we presented a new network
classification by using Shannon or von Neumann entropies of the measurement
statistics. This provides a device-independent manner for verifying quantum
network configuration under local unitary operations. The present results are
interesting in quantum entanglement, quantum networks and quantum information
processing.

## Acknowledgements

We thank the helps of Shaoming Fei, Yu Guo, Xiuyong Ding, Xiubo Chen. This
work was supported by the National Natural Science Foundation of China
(Nos.61772437,62172341), Sichuan Youth Science and Technique Foundation
(No.2017JQ0048), Fundamental Research Funds for the Central Universities
(No.2018GF07), and Shenzhen Institute for Quantum Science and Engineering.

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##  Appendix A Proof of Theorem 1

###  A.1 Entanglement measure based on quantum entropy

In this subsection, we recall necessary bipartite entanglement measures based
on quantum entropies. As a entanglement measure characterizing bipartite
entanglement through von Neumann entropy, the entanglement of formation (EOF)
${\cal{E}}$ for a pure state $|\phi\rangle_{\textsf{A}\textsf{B}}$ on Hilbert
space ${\cal H}_{\textsf{A}}\otimes{\cal H}_{\textsf{B}}$ is defined as
Bennett(1996)3824 :

$\displaystyle{\cal{E}}(|\phi\rangle_{\textsf{A}\textsf{B}})=S(\rho_{\textsf{A}})=-{\rm{Tr}}(\rho_{\textsf{A}}\log_{2}\rho_{\textsf{A}}),$
(8)

where $\rho_{\textsf{A}}={\rm
Tr}_{\textsf{B}}(|\phi\rangle_{\textsf{A}\textsf{B}}\langle\phi|)$ denotes the
density operator of the subsystem A obtained by tracing out the subsystem B,
and $S(\rho_{\textsf{A}})$ denotes the von Neumann entropy of the system A.
Note that von Neumann entropy of two density operators $\rho$ and $\sigma$
satisfies the additivityNielsen :

$\displaystyle S(\rho\otimes\sigma)=S(\rho)+S(\sigma).$ (9)

For a bipartite mixed state $\rho_{\textsf{A}\textsf{B}}$ on Hilbert space
${\cal H}_{\textsf{A}}\otimes{\cal H}_{\textsf{B}}$, EOF is given by

$\displaystyle{\cal{E}}(\rho_{\textsf{A}\textsf{B}})=\inf_{\\{p_{i},|\phi_{i}\rangle\\}}\sum_{i}p_{i}{\cal{E}}(|\phi_{i}\rangle_{\textsf{A}\textsf{B}}),$
(10)

where the infimum takes over all possible pure-state decompositions of
$\rho_{\textsf{A}\textsf{B}}=\sum_{i}p_{i}|\phi_{i}\rangle_{\textsf{A}\textsf{B}}\langle\phi_{i}|$
with $p_{i}\geq 0$, $\sum_{i}p_{i}=1$, and
$|\phi_{i}\rangle_{\textsf{A}\textsf{B}}\in{\cal H}_{\textsf{A}}\otimes{\cal
H}_{\textsf{B}}$.

As a generalization of EOF, the Rényi-$\alpha$ entanglement measure
$\mathcal{R}_{\alpha}$ can be defined by using the Rényi-$\alpha$ entropy as
Kim(2010)R :

$\displaystyle\mathcal{R}_{\alpha}(|\phi\rangle_{\textsf{A}\textsf{B}})=S_{\alpha}(\rho_{\textsf{A}})=\frac{1}{1-\alpha}\log_{2}{\rm{Tr}}(\rho^{q}_{\textsf{A}}).$
(11)

where $S_{\alpha}(\rho_{\textsf{A}})$ denotes the Rényi-$\alpha$ entropy of
the system A. Similar to the von Neumann entropy, Rényi-$\alpha$ entropy has
the additivity Dam(2002) :

$\displaystyle
S_{\alpha}(\rho\otimes\sigma)=S_{\alpha}(\rho)+S_{\alpha}(\sigma)$ (12)

for $\alpha>0$ with $\alpha\neq 1$. For a bipartite mixed state
$\rho_{\textsf{A}\textsf{B}}$, Rényi-$\alpha$ entanglement
$\mathcal{R}_{\alpha}$ is defined via the convex-roof extension as Kim(2010)R
:

$\displaystyle\mathcal{R}_{\alpha}(\rho_{\textsf{A}\textsf{B}})=\inf_{\\{p_{i},|\phi_{i}\rangle\\}}\sum_{i}p_{i}\mathcal{R}_{\alpha}(|\phi_{i}\rangle_{\textsf{A}\textsf{B}})$
(13)

with the minimum taking over all possible decompositions of
$\rho_{\textsf{A}\textsf{B}}$.

For a bipartite pure state $|\phi\rangle_{\textsf{A}\textsf{B}}$ on Hilbert
space ${\cal H}_{\textsf{A}}\otimes{\cal H}_{\textsf{B}}$, The Tsallis-$q$
entanglement measure $\mathcal{T}^{(q)}$ derived from Tsallis-$q$ entropy is
defined as Kim(2010)T :

$\displaystyle\mathcal{T}^{(q)}(|\phi\rangle_{\textsf{A}\textsf{B}})=S_{q}(\rho_{\textsf{A}})=\frac{1}{q-1}(1-{\rm{Tr}}(\rho_{\textsf{A}}^{q})),$
(14)

where $S_{q}(\rho_{\textsf{A}})$ denotes the Tsallis-$q$ entropy of the system
A, $q>0$ and $q\neq 1$. $S_{q}(\rho)$ converges to the von Neumann entropy
when $q\to 1$, that is, $\lim_{q\rightarrow 1}S_{q}(\rho)=-{\rm
Tr}\rho\log_{2}(\rho)=S(\rho)$ for any bipartite operator density $\rho$. For
a bipartite mixed state $\rho_{\textsf{A}\textsf{B}}$, Tsallis-$q$ entropy
entanglement $\mathcal{T}^{(q)}$ is defined by

$\displaystyle\mathcal{T}^{(q)}(\rho_{\textsf{A}\textsf{B}})=\inf_{\\{p_{i},|\phi_{i}\rangle\\}}\sum_{i}p_{i}\mathcal{T}_{q}(|\phi_{i}\rangle_{\textsf{A}\textsf{B}}),$
(15)

where the infimum is taken over all possible pure-state decompositions of
$\rho_{\textsf{A}\textsf{B}}$.

For $q>1$, the quantum Tsallis entropy satisfies the subadditivity property
Raggio(1995) :

$\displaystyle S_{q}(\rho\otimes\sigma)\leq S_{q}(\rho)+S_{q}(\sigma),$ (16)

while for $0<q<1$, it satisfies

$\displaystyle S_{q}(\rho\otimes\sigma)\geq S_{q}(\rho)+S_{q}(\sigma).$ (17)

Different from all the stated entropies, there exists a generalized entropy,
that is, Unified $(q,s)$-entropy Rathie(1991) , which involves two real
parameters $q$ and $s$:

$\displaystyle S_{q,s}(\rho)=\frac{1}{(1-q)s}({\rm{Tr}}(\rho^{q})^{s}-1)$ (18)

where $q,s\geq 0$ and $q\neq 1$, $s\neq 0$. This entropy includes
Rényi-$\alpha$ entropy HHH(1996)  and Tsallis-$q$ entropy Tsallis(1988)  as
special cases of $s\rightarrow 0$ and $s\rightarrow 1$, respectively.
Moreover, for $q\to 1$, it converges to the von Neumann entropy Nielsen .
Using the Unified $(q,s)$-entropy, the Unified $(q,s)$ entanglement measure is
defined for a bipartite pure state $|\phi\rangle_{AB}$ on Hilbert space ${\cal
H}_{\textsf{A}}\otimes{\cal H}_{\textsf{B}}$ by KimBarry(2011) :

$\displaystyle\mathcal{U}^{(q,s)}(|\phi\rangle_{\textsf{A}\textsf{B}})=S_{q,s}(\rho_{\textsf{A}})$
(19)

for each $q,s\geq 0$. For a bipartite mixed state
$\rho_{\textsf{A}\textsf{B}}$, its entanglement measure is given by

$\displaystyle\mathcal{U}^{(q,s)}(\rho_{\textsf{A}\textsf{B}})=\inf_{\\{p_{i},|\phi_{i}\rangle\\}}\sum_{i}p_{i}\mathcal{U}^{(q,s)}(|\phi_{i}\rangle_{\textsf{A}\textsf{B}}),$
(20)

where the infimum is taken over all possible pure state decompositions of
$\rho_{\textsf{A}\textsf{B}}$.

For $0<q<1$ and $s<0$, or $q\geq 1$ and $s\geq 0$, the quantum Unified
$(q,s)$-entropy satisfies the subadditivity property Hu(2006) :

$\displaystyle S_{q,s}(\rho\otimes\sigma)\leq S_{q,s}(\rho)+S_{q,s}(\sigma),$
(21)

while for $q>1$ and $s<0$, or $0<q<1$ and $s>0$ it satisfies

$\displaystyle S_{q,s}(\rho\otimes\sigma)\geq S_{q,s}(\rho)+S_{q,s}(\sigma).$
(22)

From all the aforementioned bipartite entanglement measures, they are
subadditive, that is,

$\displaystyle{\cal Q}(\rho\otimes\sigma)\leq{\cal Q}(\rho)+{\cal Q}(\sigma),$
(23)

where ${\cal Q}$ can be any one of ${\cal
E},\mathcal{R}_{\alpha},\mathcal{T}^{(q)}$ and $\mathcal{U}^{(q,s)}$.

###  A.2 Proof of Theorem 1

In this subsection, we prove Theorem 1 in terms of EOF and Rényi-$\alpha$
entropy. The proof is completed by following the procedure from the chain
network in case 1, cyclic network in case 2, star network in case 3 to general
quantum network in case 4, as shown in Fig.5.

Consider a quantum network $\mathcal{N}_{q}(\cal{A},\xi)$, where $\cal{A}$
denotes parties $\textsf{A}_{1},\cdots,\textsf{A}_{n}$, and $\xi$ denotes
entangled states. Assume that any two parties $\textsf{A}_{i}$ and
$\textsf{A}_{j}$ share the generalized EPR state EPR  as

$\displaystyle|\phi(\theta_{s})\rangle=\cos\theta_{s}|00\rangle+\sin\theta_{s}|11\rangle,$
(24)

or generalized GHZ state GHZ  as

$\displaystyle|\phi(\varphi_{t})\rangle=\cos\varphi_{t}|0\rangle^{\otimes
m}+\sin\varphi_{t}|1\rangle^{\otimes m},$ (25)

or only two qubits of one generalized GHZ state as

$\displaystyle\delta({\vartheta_{k}})=\cos^{2}\vartheta_{k}|00\rangle\langle
00|+\sin^{2}\vartheta_{k}|11\rangle\langle 11|.$ (26)

Figure 5: (Color online) Schematic quantum network consisting of generalized
EPR states or generalized GHZ states. (a) A chain quantum network consisting
of $n-1$ generalized EPR states. (b) A cyclic quantum network consisting of
$n$ generalized EPR states. (c) A star quantum network consisting of
generalized EPR states and generalized GHZ states. (d) A general quantum
network consisting of generalized EPR states and generalized GHZ states.

Case 1. Chain quantum networks

From Fig.5(a), we consider a chain quantum network $\mathcal{N}^{(a)}_{q}$
consisting of $n$ parties $\mathsf{A}_{1},\cdots,\mathsf{A}_{n}$, where each
adjacent pair of $\mathsf{A}_{i}$ and $\mathsf{A}_{i+1}$ share $s_{i+1}$
generalized EPR states, i.e.,

$\displaystyle\xi_{i,i+1}=\mathop{\otimes}^{s_{i+1}}\limits_{s=1}|\phi({\theta_{s}})\rangle_{i,i+1},1\leq
i\leq n-1,$ (27)

where $|\phi(\theta_{s})\rangle$ is defined in Eq.(24). The total state
$\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}$ of $n$-partite system is given by

$\displaystyle\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}=\mathop{\otimes}^{n}_{i=1}\xi_{i,i+1}.$
(28)

It is easy to obtain

$\displaystyle{\cal
Q}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})=\sum^{n}_{j=1,j\neq
i}{\cal
Q}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}),$
(29)

where ${\cal Q}_{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}$ and ${\cal
Q}_{\textsf{A}_{i}|\textsf{A}_{j}}$ are given in Theorem 1. The proofs are as
follows. From Eq.(8), we have

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle=$ $\displaystyle S(\rho_{\textsf{A}_{i}})$ (30)
$\displaystyle=$ $\displaystyle\sum^{i+1}_{j=i-1,j\neq
i}S(\mathop{\otimes}^{s_{j}}_{s=1}\rho_{i;j}(\theta_{s}))$ $\displaystyle=$
$\displaystyle\sum^{i+1}_{j=i-1,j\neq
i}\sum^{s_{j}}_{s=1}S(\rho_{i;j}(\theta_{s})),$ (31)

where ${\cal{E}}$ denotes the EOF. Eq.(30) is derived from
$\rho_{\textsf{A}_{i}}=\mathop{\otimes}^{s_{j}}_{s=1}\rho_{i;j}(\theta_{s})$,
where $\rho_{i;j}({\theta_{s}})$ is the reduced density matrix of the systems
owned by the party $\textsf{A}_{i}$ by tracing out all the subsystems in a
generalized EPR state $|\phi({\theta_{s}})\rangle_{i,j}$ owned by
$\textsf{A}_{j}$. Eq.(31) holds because of the additivity of the von Neumann
entropy in Eq.(9). On the other hand, the entanglement between the parties
$\textsf{A}_{i}$ and $\textsf{A}_{j}$ is from entangled pairs associated with
them. This implies that

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{i-1}}}(\rho_{\textsf{A}_{i}\textsf{A}_{i-1}})$
$\displaystyle=$
$\displaystyle{\cal{E}}(\mathop{\otimes}^{s_{i-1}}_{s=1}|\phi({\theta_{s}})\rangle_{i,i-1})$
(32) $\displaystyle=$ $\displaystyle
S(\mathop{\otimes}^{s_{i-1}}_{s=1}\rho_{i;i-1}(\theta_{s}))$ $\displaystyle=$
$\displaystyle\sum^{s_{i-1}}_{s=1}S(\rho_{i;i-1}(\theta_{s}))$ (33)

and

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{i+1}}}(\rho_{\textsf{A}_{i}\textsf{A}_{i+1}})$
$\displaystyle=$
$\displaystyle{\cal{E}}(\mathop{\otimes}^{s_{i+1}}_{s=1}|\phi({\theta_{s}})\rangle_{i,i+1})$
(34) $\displaystyle=$ $\displaystyle
S(\mathop{\otimes}^{s_{i+1}}_{s=1}\rho_{i;i+1}(\theta_{s}))$ $\displaystyle=$
$\displaystyle\sum^{s_{i+1}}_{s=1}S(\rho_{i;i+1}(\theta_{s})),$ (35)

where Eqs.(32) and (34) from the definition of EOF in Eq.(A1). The additivity
of the von Neumann entropy in Eq.(9) yields to Eqs.(33) and (35).

Note that the party $\textsf{A}_{i}$ is only entangled with the adjacent
parties $\textsf{A}_{i-1}$ and $\textsf{A}_{i+1}$. We have

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})=0,j\not\in\\{i-1,i,i+1\\}.$
(36)

Therefore, from Eqs.(33) and (35) it follows that

$\displaystyle\sum^{n}_{j=1}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
$\displaystyle=$
$\displaystyle\sum^{s_{i-1}}_{s=1}S(\rho_{i;i-1}(\theta_{s}))$ (37)
$\displaystyle+\sum^{s_{i+1}}_{s=1}S(\rho_{i;i+1}(\theta_{s})).$

Combining Eqs.(31) and (37), we conclude

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})=\sum^{n}_{j=1,j\neq
i}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}).$
(38)

Similarly, from the definition of Rényi-$\alpha$ entropy entanglement in
Eq.(11) and the additivity in Eq.(12), we can prove Eq.(29) holds for
Rényi-$\alpha$ entropy entanglement.

Case 2. Cyclic quantum networks

Consider a cyclic network $\mathcal{N}^{(b)}_{q}$ as shown in Fig.5(b), where
each pair of $\mathsf{A}_{i}$ and $\mathsf{A}_{i+1}$ share $s_{i+1}$
generalized EPR states, i.e.,
$\xi_{i,i+1}=\mathop{\otimes}^{s_{i+1}}\limits_{s=1}|\phi({\theta_{s}})\rangle_{i,i+1}$($1\leq
i\leq n-1$). Moreover, the parties $\mathsf{A}_{1}$ and $\mathsf{A}_{n}$ share
the quantum state
$\xi_{1,n}=\mathop{\otimes}^{s_{n}}\limits_{s=1}|\phi({\theta_{s}})\rangle_{1,n}$.
The total state $\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}$ of $n$-partite
system is denoted as

$\displaystyle\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}=\mathop{\otimes}^{n}\limits_{i=1}(\mathop{\otimes}^{s_{n}}\limits_{s=1}|\phi({\theta_{s}})\rangle_{1,n}\mathop{\otimes}^{s_{i+1}}\limits_{s=1}|\phi({\theta_{s}})\rangle_{i,i+1}).$
(39)

Similar to the proof of Eq.(29), it follows that

$\displaystyle{\cal
Q}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})=\sum^{n}_{j=1,j\neq
i}{\cal
Q}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}).$
(40)

Case 3. Star quantum networks

Consider an $n+1$-partite star quantum network $\mathcal{N}^{(c)}_{q}$ as
shown in Fig. 5(c). Suppose the party $\textsf{A}_{0}$ and any other party
$\textsf{A}_{j}$ ($j=1,2,\cdots,n$) share the quantum state $\xi_{0j}$ given
by

$\displaystyle\xi_{0j}=\mathop{\otimes}^{s_{j}}_{s=1}\varrho_{0j}(\theta_{s})\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{0j}(\varphi_{t})\mathop{\otimes}^{k_{j}}_{k=1}\delta_{0j}(\vartheta_{k}),$
(41)

where $\varrho({\theta_{s}})$ is the reduced density matrix of generalized EPR
state EPR :
$|\phi({\theta_{s}})\rangle=\cos\theta_{s}|00\rangle+\sin\theta_{s}|11\rangle$,
$\sigma({\varphi_{t}})$ is the reduced density matrix of generalized GHZ state
GHZ : $|\phi(\varphi_{t})\rangle=\cos\varphi_{t}|0\rangle^{\otimes
m}+\sin\varphi_{t}|1\rangle^{\otimes m}$, and
$\delta({\vartheta_{k}})=\cos^{2}\vartheta_{k}|00\rangle\langle
00|+\sin^{2}\vartheta_{k}|11\rangle\langle 11|)$ is the reduced density matrix
of any two subsystems by tracing out the remaining subsystems in a
multipartite generalized GHZ state. Herein, $s_{j}$, $t_{j}$, and $k_{j}$
denote the numbers of $\varrho({\theta_{s}})$, $\sigma({\varphi_{t}})$, and
$\delta({\vartheta_{k}})$, are shared by the party $\textsf{A}_{0}$ and any
other party $\textsf{A}_{j}$, respectively.

The total state $\rho_{\textsf{A}_{0}\cdots\textsf{A}_{n}}$ of $n+1$-partite
system is given by

$\displaystyle\rho_{\textsf{A}_{0}\cdots\textsf{A}_{n}}=\mathop{\otimes}^{n}_{j=1}(\mathop{\otimes}^{s_{j}}_{s=1}\varrho_{0j}(\theta_{s})\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{0j}(\varphi_{t})\mathop{\otimes}^{k_{j}}_{k=1}\delta_{0j}(\vartheta_{k})).$
(42)

The Theorem 1 means the following inequality

$\displaystyle{\cal
Q}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{0}\cdots\textsf{A}_{n}})\geq\sum^{n}_{j=1}{\cal
Q}_{{\textsf{A}_{0}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{0}\textsf{A}_{j}}).$
(43)

where ${\cal Q}_{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}$ and ${\cal
Q}_{\textsf{A}_{i}|\textsf{A}_{j}}$ are given in Theorem 1. The proof is as
follows. From Eq.(8), we get

$\displaystyle{\cal{E}}_{\textsf{A}_{0}|\overline{\textsf{A}_{0}}}(\rho_{\textsf{A}_{0}\cdots\textsf{A}_{n}})$
(44) $\displaystyle=$ $\displaystyle S(\rho_{\textsf{A}_{0}})$
$\displaystyle=$ $\displaystyle
S(\mathop{\otimes}^{n}_{j=1}(\mathop{\otimes}^{s_{j}}_{s=1}\rho_{0;j}(\theta_{s})\mathop{\otimes}^{t_{j}}_{t=1}\rho_{0;j}(\varphi_{t})\mathop{\otimes}^{k_{j}}_{k=1}\rho_{0;j}(\vartheta_{k})))$
$\displaystyle=$
$\displaystyle\sum^{n}_{j=1}(\sum^{s_{j}}_{s=1}S(\rho_{0;j}(\theta_{s}))+\sum^{t_{j}}_{t=1}S(\rho_{0;j}(\varphi_{t}))$
$\displaystyle+\sum^{k_{j}}_{k=1}S(\rho_{0;j}(\vartheta_{k}))),$

where Eq.(44) follows from the additivity of the von Neumann entropy in
Eq.(9). $\rho_{0;j}({\theta_{s}})$ is the reduced density matrix of the
subsystems owned by $\textsf{A}_{0}$, which is obtained by tracing out all the
subsystems owned by $\textsf{A}_{j}$ in a generalized EPR state
$|\phi({\theta_{s}})\rangle_{0,j}$. $\rho_{0;j}(\varphi_{t})$ is the reduced
density operator of the subsystem owned by $\textsf{A}_{0}$ with respect to
generalized GHZ state shared by two parties $\textsf{A}_{0}$ and
$\textsf{A}_{j}$. $\rho_{0;j}(\vartheta_{k})$ is the reduced density matrix of
the subsystems owned by $\textsf{A}_{0}$, which is obtained by tracing out the
other subsystems in a generalised GHZ state shared by multipartite parties. On
the other side, the entanglement of $\textsf{A}_{0}$ and $\textsf{A}_{j}$ is
generated by generalized EPR state and GHZ states. This implies that

$\displaystyle{\cal{E}}_{\textsf{A}_{0}|\textsf{A}_{j}}(\rho_{\textsf{A}_{0}\textsf{A}_{j}})$
$\displaystyle=$
$\displaystyle{\cal{E}}(\mathop{\otimes}^{s_{j}}_{s=1}\varrho_{0j}(\theta_{s})\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{0j}(\varphi_{t})\mathop{\otimes}^{k_{j}}_{k=1}\delta_{0j}(\vartheta_{k}))$
(45) $\displaystyle\leq$
$\displaystyle{\cal{E}}(\mathop{\otimes}^{s_{j}}_{s=1}\varrho_{0j}(\theta_{s}))+{\cal{E}}(\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{0j}(\varphi_{t}))$
$\displaystyle=$
$\displaystyle\sum^{s_{j}}_{s=1}S(\rho_{0;j}(\theta_{s}))+\sum^{t_{j}}_{t=1}S(\rho_{0;j}(\varphi_{t}))$
(46)

Here, by using the additivity of the entanglement measure ${\cal{E}}$ in
Eq.(23) and ${\cal{E}}(\delta_{0j}(\vartheta_{k}))=0$ iteratively, we can get
the inequality (45). Eq.(46) follows from the definition of EOF in Eq.(8).
Therefore, we have

$\displaystyle\sum^{n}_{j=1}{\cal{E}}_{\textsf{A}_{0}|\textsf{A}_{j}}(\rho_{\textsf{A}_{0}\textsf{A}_{j}})$
$\displaystyle\leq$
$\displaystyle\sum^{n}_{j=1}(\sum^{s_{j}}_{s=1}S(\rho_{0;j}(\theta_{s}))$ (47)
$\displaystyle+\sum^{t_{j}}_{t=1}S(\rho_{0;j}(\varphi_{t}))).$

Combining Eqs.(44) and (47), we get

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{0}\cdots\textsf{A}_{n}})\geq\sum^{n}_{j=1}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{0}\textsf{A}_{j}}).$
(48)

Note that for $k_{j}=0$, i.e., all $m$-qubit GHZ states are shared by two
parties, we have

$\displaystyle{\cal{E}}_{\textsf{A}_{0}|\overline{\textsf{A}_{0}}}(\rho_{\textsf{A}_{0}\cdots\textsf{A}_{n}})$
$\displaystyle=$
$\displaystyle\sum^{n}_{j=1}(\sum^{s_{j}}_{s=1}S(\rho_{0;j}(\theta_{s}))$ (49)
$\displaystyle+\sum^{t_{j}}_{t=1}S(\rho_{0;j}(\varphi_{t}))).$

Moreover, we have

$\displaystyle{\cal{E}}_{\textsf{A}_{0}|\textsf{A}_{j}}(\rho_{\textsf{A}_{0}\textsf{A}_{j}})$
$\displaystyle=$
$\displaystyle\sum^{n}_{j=1}(\sum^{s_{j}}_{s=1}S(\rho_{0;j}(\theta_{s}))$ (50)
$\displaystyle+\sum^{t_{j}}_{t=1}S(\rho_{0;j}(\varphi_{t}))).$

So, combining Eqs.(49) and (50), we get

$\displaystyle{\cal{E}}_{\textsf{A}_{0}|\overline{\textsf{A}_{0}}}(\rho_{\textsf{A}_{0}\cdots\textsf{A}_{n}})=\sum^{n}_{j=1}{\cal{E}}_{\textsf{A}_{0}|\textsf{A}_{j}}(\rho_{\textsf{A}_{0}\textsf{A}_{j}}).$
(51)

Similarly, the inequality (43) holds for Rényi-$\alpha$ entropy entanglement
defined in Eq.(11) from the additivity of Rényi-$\alpha$ entropy in Eq.(12).

Case 4. General quantum networks

Now, we prove the monogamy for general network $\mathcal{N}_{q}(\cal{A},\xi)$
as shown in Fig.5(d), where any two parties $\textsf{A}_{i}$ and
$\textsf{A}_{j}$ share the quantum state $\xi_{ij}$ given by

$\displaystyle\xi_{ij}=\mathop{\otimes}^{s_{j}}_{s=1}\varrho_{ij}(\theta_{s})\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{ij}({\varphi_{t}})\mathop{\otimes}^{k_{j}}_{k=1}\delta_{ij}(\vartheta_{k}),$
(52)

where $\varrho({\theta_{s}})$, $\sigma({\varphi_{t}})$, and
$\delta({\vartheta_{k}})$ are defined in Eq.(41). $s_{j}$, $t_{j}$, and
$k_{j}$ denote the numbers of $\varrho({\theta_{s}})$,
$\sigma({\varphi_{t}})$, and $\delta({\vartheta_{k}})$ are shared by any two
parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$, respectively.

The total state $\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}$ of $n$-partite
system is given by

$\displaystyle\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}=\mathop{\otimes}^{n}_{\begin{subarray}{c}i,j=1,\\\
i\neq j\end{subarray}}\xi_{ij},$ (53)

where $\xi_{ij}$ has the form of Eq.(52). Hence, Theorem 1 means that

$\displaystyle{\cal
Q}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})\geq\sum^{n}_{j=1,j\neq
i}{\cal
Q}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}),$
(54)

where ${\cal Q}_{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}$ and ${\cal
Q}_{\textsf{A}_{i}|\textsf{A}_{j}}$ are given in Theorem 1.

The proof of the inequality (54) is as follows. Assume that any party
$\mathsf{A}_{i}$ is entangled with $N$ parties of $\mathsf{A}_{j}$’s. From
Eq.(8), we get

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle=$ $\displaystyle S(\rho_{\textsf{A}_{i}})$ (56)
$\displaystyle=$ $\displaystyle
S(\mathop{\otimes}^{N}_{j=1}(\mathop{\otimes}^{s_{j}}_{s=1}\rho_{i;j}(\theta_{s})\mathop{\otimes}^{t_{j}}_{t=1}\rho_{i;j}(\varphi_{t})$
$\displaystyle\mathop{\otimes}^{k_{j}}_{k=1}\rho_{i;j}(\vartheta_{k})))$
$\displaystyle=$
$\displaystyle\sum^{N}_{j=1}(\sum^{s_{j}}_{s=1}S(\rho_{i;j}(\theta_{s}))+\sum^{t_{j}}_{t=1}S(\rho_{i;j}(\varphi_{t})))$
$\displaystyle+\sum^{N}_{j=1}\sum^{k_{j}}_{k=1}S(\rho_{i;j}(\vartheta_{k})),$

where Eq.(56) follows from the additivity of von Neumann entropy in Eq.(9).
Here, $\rho_{i;j}({\theta_{s}})$, $\rho_{i;j}(\varphi_{t})$, and
$\rho_{i;j}(\vartheta_{k})$ are defined in Eq.(44).

Note that the entanglement shared by the parties $\textsf{A}_{i}$ and
$\textsf{A}_{j}$ is generated by generalized EPR states and GHZ states. It is
sufficient to prove that

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
$\displaystyle=$
$\displaystyle{\cal{E}}(\mathop{\otimes}^{s_{j}}_{s=1}\varrho_{ij}(\theta_{s})\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{ij}({\varphi_{t}})\mathop{\otimes}^{k_{j}}_{k=1}\delta_{ij}(\vartheta_{k}))$
(57) $\displaystyle\leq$
$\displaystyle{\cal{E}}(\varrho_{ij}(\theta_{s}))+{\cal{E}}(\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{ij}({\varphi_{t}}))$
$\displaystyle=$
$\displaystyle\sum^{s_{j}}_{s=1}S(\rho_{i;j}(\theta_{s}))+\sum^{t_{j}}_{t=1}S(\rho_{i;j}(\varphi_{t})).$
(58)

From the additivity of the entanglement measure ${\cal{E}}$ in Eq.(23) and
${\cal{E}}(\delta_{ij}(\vartheta_{k}))=0$, it is followed the inequality (57).
Eq.(58) is derived from the definition of the entanglement measure ${\cal{E}}$
in Eq.(8). Combining Eqs. (56) with (58), it yields to

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle\geq$ $\displaystyle\sum^{N}_{j=1,j\neq
i}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
(59) $\displaystyle=$ $\displaystyle\sum^{n}_{j=1,j\neq
i}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}).$
(60)

Similar with the inequality (48), the equality (60) holds if and only if
$k_{j}=0$, i.e., all the GHZ states are shared by two parties.

Similarly, the inequality (54) holds for the Rényi-$\alpha$ entropy
entanglement in Eq.(11) from the additivity of Rényi-$\alpha$ entropy in
Eq.(12). This has completed the proof.

##  Appendix B Generalizations of Theorem 1

In this section, we extend Theorem 1 to other networks.

###  B.1 Quantum network with W states

Consider a genuine tripartite entangled W state Dur :

$\displaystyle|W\rangle=\frac{1}{\sqrt{3}}(|100\rangle+|010\rangle+|001\rangle).$
(61)

It is easy to obtain

$\displaystyle{\cal{E}}(|W\rangle_{\textsf{A}|\textsf{B}_{1}\textsf{B}_{2}})=H(\frac{1}{3})\approx
0.918,$
$\displaystyle{\cal{E}}(\rho_{\textsf{A}\textsf{B}_{i}})=H(\frac{1+\sqrt{1-C^{2}(\rho_{\textsf{A}\textsf{B}_{i}})}}{2})\approx
0.55,$ (62)

where ${\cal{E}}$ represents EOF in Eq.(8), $H(x)$ is binary Shannon entropy
given $H(x)=-x\log_{2}x-(1-x)\log_{2}(1-x)$ for $0\leq x\leq 1$ Nielsen , and
$C(\rho_{\textsf{A}\textsf{B}_{i}})=\frac{2}{3}$ ($i=1,2$) is the concurrence
Wootters(1998)  of two-qubit mixed state $\rho_{\textsf{A}\textsf{B}_{i}}$.

From Eq.(62), we have

$\displaystyle{\cal{E}}(|W\rangle_{\textsf{A}|\textsf{B}_{1}\textsf{B}_{2}})\leq{\cal{E}}(\rho_{\textsf{A}\textsf{B}_{1}})+{\cal{E}}(\rho_{\textsf{A}\textsf{B}_{2}}),$
(63)

that is, EOF ${\cal{E}}$ of W state is not monogamous. However, we prove that
the monogamy relation of ${\cal{E}}$ holds for quantum networks consisting of
GHZ states and W states. The monogamy of quantum network reaches a good
saturation in the case that quantum network consists of $|W\rangle^{\otimes
5}\otimes|GHZ\rangle$, i.e., five pairs of W state and one GHZ state are
shared by three parties, simultaneously. The main reason is as follows. For
the GHZ state, we have

$\displaystyle{\cal{E}}(|GHZ\rangle_{\textsf{A}|\textsf{B}\textsf{C}})=1,$
$\displaystyle{\cal{E}}(\rho_{\textsf{A}\textsf{B}})={\cal{E}}(\rho_{\textsf{A}\textsf{C}})=0.$
(64)

From Eqs.(62) and (64), we get the inequality

$\displaystyle{\cal{E}}(|GHZ\rangle_{\textsf{A}|\textsf{B}\textsf{C}})+n\times{\cal{E}}(|W\rangle_{\textsf{A}|\textsf{B}_{1}\textsf{B}_{2}})$
$\displaystyle\geq
n{\cal{E}}(\rho_{\textsf{A}\textsf{B}_{1}})+n{\cal{E}}(\rho_{\textsf{A}\textsf{B}_{2}})$
(65)

for $n\leq 5$. The inequality (65) is optimal in the case of $n=5$, i.e.,

$\displaystyle 1+5H(\frac{1}{3})\geq
5{\cal{E}}(\rho_{\textsf{A}\textsf{B}_{1}})+5{\cal{E}}(\rho_{\textsf{A}\textsf{B}_{2}}).$
(66)

From the inequality (66), we extend Theorem 1 to the network consisting of EPR
states, GHZ states and W states. Consider an $n$-partite entangled quantum
network $\mathcal{N}_{q_{1}}(\cal{A},\xi)$, where $\cal{A}$ denotes parties
$\textsf{A}_{1},\cdots,\textsf{A}_{n}$, and $\xi$ denotes entangled states.
Assume any two parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$ share the state
$\xi_{ij}$ given by

$\displaystyle\xi_{ij}=\mathop{\otimes}^{s_{j}}_{s=1}\varrho^{s}_{ij}\mathop{\otimes}^{t_{j}}_{t=1}\sigma^{t}_{ij}\mathop{\otimes}^{k_{j}}_{k=1}\delta^{k}_{ij}\mathop{\otimes}^{l_{j}}_{l=1}\omega^{l}_{ij},$
(67)

where $\varrho$ denotes the density matrix of EPR state EPR :
$|\Phi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$. $\sigma$ is the
density matrix of generalized GHZ state GHZ :
$|\Psi\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)$.
$\delta=\frac{1}{\sqrt{2}}(|00\rangle\langle 00|+|11\rangle\langle 11|)$ is
the reduced density matrix of any two subsystems in a GHZ state, and
$\omega=\frac{2}{3}|00\rangle\langle 00|+\frac{1}{3}|11\rangle\langle 11|$ is
the reduced density matrix of any two subsystems in W state in Eq.(61).
$s_{j}$, $t_{j}$, $k_{j}$, and $l_{j}$ denote the numbers of $\varrho$,
$\sigma$, $\delta$, and $\omega$, respectively. All these entangled states are
shared by any two parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$. Moreover,
$l_{j}$ and $k_{j}$ satisfy $l_{j}\leq 5k_{j}$. Thus, the total state
$\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}$ of $n$-partite system is given by

$\displaystyle\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}=\mathop{\otimes}^{n}_{\begin{subarray}{c}i,j=1,\\\
i\neq
j\end{subarray}}(\mathop{\otimes}^{s_{j}}_{s=1}\varrho^{s}_{ij}\mathop{\otimes}^{t_{j}}_{t=1}\sigma^{t}_{ij}\mathop{\otimes}^{k_{j}}_{k=1}\delta^{k}_{ij}\mathop{\otimes}^{l_{j}}_{l=1}\omega^{l}_{ij}).$
(68)

###### Theorem S 1

The entanglement distribution of quantum network $\mathcal{N}_{q_{1}}$
satisfies

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})\geq\sum^{n}_{j=1,j\neq
i}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}),$
(69)

where ${\cal{E}}_{\textsf{X}|\textsf{Y}}$ represents EOF of a bipartite
systems X and Y, and $\overline{\textsf{A}_{i}}$ denotes all parties except
for $\textsf{A}_{i}$.

Proof. Assume that any party $\mathsf{A}_{i}$ is entangled with $N$ parties of
$\mathsf{A}_{j}$’s. The total numbers of EPR, GHZ and W states are $n_{1}$,
$n_{2}$, $n_{3}$, respectively. Any two parties $\mathsf{A}_{i}$ and
$\mathsf{A}_{j}$ share $s_{j}$ pairs of EPR states $\varrho$, $t_{j}$ pairs of
GHZ states $\sigma$, $k_{j}$ pairs of $\delta$, and $l_{j}$ pairs of $\omega$.
As a result, it is easy to obtain that $\sum^{N}_{j=1}s_{j}=n_{1}$,
$\sum^{N}_{j=1}t_{j}=n^{\prime}_{2}$,
$\sum^{N}_{k=1}k_{j}=2(n_{2}-n^{\prime}_{2})$, and
$\sum^{N}_{l=1}l_{j}=2n_{3}$, where $n^{\prime}_{2}$ is the number of GHZ
states shared by two parties. There are $n_{2}-n^{\prime}_{2}$ pairs of GHZ
states shared by three parties. From the inequality (66), the entanglement
distribution holds for quantum network consisting of GHZ states and W states
from $l_{j}\leq 5k_{j}$. This yields the inequality $n_{3}\leq
5(n_{2}-n^{\prime}_{2})$. From Eq.(8), we get

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle=$ $\displaystyle S(\rho_{\textsf{A}_{i}})$ (70)
$\displaystyle=$ $\displaystyle S((\frac{\mathbbm{1}}{2})^{\otimes
n_{1}}\otimes(\frac{\mathbbm{1}}{2})^{\otimes n_{2}}\otimes\omega^{\otimes
n_{3}})$ $\displaystyle=$ $\displaystyle
n_{1}S(\frac{\mathbbm{1}}{2})+n_{2}S(\frac{\mathbbm{1}}{2})+n_{3}S(\omega)$
$\displaystyle=$ $\displaystyle n_{1}+n_{2}+n_{3}H(\frac{1}{3}),$ (71)

where Eq.(70) follows from the additivity of the von Neumann entropy in
Eq.(9). In the equality (71), we have used $S(\frac{\mathbbm{1}}{2})=1$ and
$S(\omega)=H(\frac{1}{3})$.

Moreover, we have

$\displaystyle\\!\\!\\!\\!\\!\\!{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
$\displaystyle=$
$\displaystyle{\cal{E}}(\mathop{\otimes}^{s_{j}}_{s=1}\varrho^{s}_{ij}\mathop{\otimes}^{t_{j}}_{t=1}\sigma^{t}_{ij}\mathop{\otimes}^{k_{j}}_{k=1}\delta^{k}_{ij}\mathop{\otimes}^{l_{j}}_{l=1}\omega^{l}_{ij})$
(73) $\displaystyle\leq$
$\displaystyle{\cal{E}}(\mathop{\otimes}^{s_{j}}_{s=1}\varrho^{s}_{ij})+{\cal{E}}(\mathop{\otimes}^{t_{j}}_{t=1}\sigma^{t}_{ij})$
$\displaystyle+{\cal{E}}(\mathop{\otimes}^{l_{j}}_{l=1}\omega^{l}_{ij})$
$\displaystyle\leq$
$\displaystyle\sum^{s_{j}}_{s=1}S(\frac{\mathbbm{1}}{2})+\sum^{t_{j}}_{t=1}S(\frac{\mathbbm{1}}{2})$
$\displaystyle+\sum^{l_{j}}_{l=1}{\cal{E}}(\omega^{l}_{ij})$ $\displaystyle=$
$\displaystyle\sum^{s_{j}}_{s=1}1+\sum^{t_{j}}_{t=1}1+\sum^{l_{j}}_{l=1}{\cal{E}}(\omega)$
(74) $\displaystyle=$ $\displaystyle s_{j}+t_{j}+l_{j}{\cal E}(\omega),$ (75)

where the inequality (73) follows from the additivity of EOF in Eq.(23). The
inequality (73) is from the definition of EOF in Eq.(8) and the additivity of
EOF in Eq.(23). We therefore have

$\displaystyle\\!\\!\\!\\!\\!\\!\sum^{N}_{j=1}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
$\displaystyle\leq$
$\displaystyle\sum^{N}_{j=1}(s_{j}+t_{j}+l_{j}{\cal{E}}(\omega))$ (76)
$\displaystyle=$ $\displaystyle n_{1}+n^{\prime}_{2}+2n_{3}{\cal{E}}(\omega)$
$\displaystyle\leq$ $\displaystyle
n_{1}+n^{\prime}_{2}+\frac{n_{3}}{5}+n_{3}H(\frac{1}{3})$ (77)
$\displaystyle\leq$ $\displaystyle n_{1}+n_{2}+n_{3}H(\frac{1}{3}),$ (78)

where the equality (76) is derived from the equalities:
$\sum^{N}_{j=1}s_{j}=n_{1}$, $\sum^{N}_{j=1}t_{j}=n^{\prime}_{2}$, and
$\sum^{N}_{j=1}l_{j}=2n_{3}$. The inequality (77) follows from the inequality
$5\times 2{\cal{E}}(\omega)\leq 1+5H(\frac{1}{3})$ which is from the
inequality (66). The inequality (78) holds for the inequality $n_{3}\leq
5(n_{2}-n^{\prime}_{2})$. Finally, we obtain

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle\geq$ $\displaystyle\sum^{N}_{j=1,j\neq
i}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
(79) $\displaystyle=$ $\displaystyle\sum^{n}_{j=1,j\neq
i}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$

This completes the proof of Theorem S1.

###  B.2 Quantum network with arbitrary bipartite entangled pure states

In this subsection, consider an $n$-partite entangled quantum network
$\mathcal{N}_{q_{2}}(\cal{A},\xi)$, where any two parties $\textsf{A}_{i}$ and
$\textsf{A}_{j}$ share $s_{j}$ pairs of arbitrary bipartite entangled pure
states
$|\varphi^{1}\rangle_{ij},|\varphi^{2}\rangle_{ij},\cdots,|\varphi^{s}\rangle_{ij}$.
The joint state shared by two parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$ is
denoted by $\mathop{\otimes}^{s_{j}}_{s=1}|\varphi^{s}\rangle_{ij}$. The total
state $\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}$ of
$\mathcal{N}_{q_{2}}(\cal{A},\xi)$ is given by

$\displaystyle\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}=\mathop{\otimes}^{n}_{\begin{subarray}{c}i,j=1,\\\
i\neq j\end{subarray}}\mathop{\otimes}^{s_{j}}_{s=1}|\varphi^{s}\rangle_{ij}.$
(80)

We prove strong monogamy relation for quantum network $\mathcal{N}_{q_{2}}$.

###### Theorem S 2

The entanglement distribution of quantum network $\mathcal{N}_{q_{2}}$
satisfies

$\displaystyle{\cal
Q}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})=\sum^{n}_{j=1,j\neq
i}{\cal
Q}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}),$
(81)

where ${\cal Q}_{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}$ and ${\cal
Q}_{\textsf{A}_{i}|\textsf{A}_{j}}$s are defined in Theorem 1.

_Proof._ Assume that there exist $N$ parties $\textsf{A}_{j}$’s who are
entangled with the party $\textsf{A}_{i}$. Note that any two parties
$\textsf{A}_{i}$ and $\textsf{A}_{j}$ share the state of
$\mathop{\otimes}^{s_{j}}_{s=1}|\varphi^{s}\rangle_{ij}$. It is sufficient to
prove that

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle=$
$\displaystyle{\cal{E}}(\mathop{\otimes}^{N}_{j=1}\mathop{\otimes}^{s_{j}}_{s=1}|\varphi^{s}\rangle_{ij})$
(82) $\displaystyle=$ $\displaystyle S(\rho^{j}_{i})$ $\displaystyle=$
$\displaystyle
S(\mathop{\otimes}^{N}_{j=1}\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j})$
$\displaystyle=$
$\displaystyle\sum^{N}_{j=1}\sum^{s_{j}}_{s=1}S(\rho^{s}_{i;j}),$ (83)

where the equality (82) is due to the systems $\rho^{s}_{i;j}$’s are
uncorrelated, that is,
$\rho^{j}_{i}=\mathop{\otimes}^{N}_{j=1}\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j}$,
$\rho^{s}_{i;j}$ is the reduced density matrix of the subsystems owned by the
party $\textsf{A}_{i}$ by tracing out the subsystems owned by $\textsf{A}_{j}$
in arbitrary bipartite entangled pure state
$|\varphi^{s}\rangle_{ij}\langle\varphi^{s}|$. The equality (83) is obtained
by iteratively using the additivity of von Neumann entropy in Eq.(9).

Moreover, the entanglement shared by two parties $\textsf{A}_{i}$ and
$\textsf{A}_{j}$ is from entangled sources of
$\mathop{\otimes}^{s_{j}}_{s=1}|\varphi^{s}\rangle_{ij}$. According to Eq.(8),
we obtain

$\displaystyle\sum^{N}_{j=1}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
$\displaystyle=$
$\displaystyle\sum^{N}_{j=1}{\cal{E}}(\mathop{\otimes}^{s_{j}}_{s=1}|\varphi^{s}\rangle_{ij})$
(84) $\displaystyle=$ $\displaystyle\sum^{N}_{j=1}S(\rho^{j}_{i})$
$\displaystyle=$
$\displaystyle\sum^{N}_{j=1}\sum^{s_{j}}_{s=1}S(\rho^{s}_{i;j}).$

Similar to Eqs.(82) and (83), we obtain Eqs.(84). Thus, by combining Eqs.(83)
and (84), we have

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle=$
$\displaystyle\sum^{N}_{j=1}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
(85) $\displaystyle=$
$\displaystyle\sum^{n}_{j=1}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}).$

Similar to Eq.(85), we get Eq.(81) for Rényi-$\alpha$ entropy entanglement
from its definition in Eq.(11) and the additivity of Rényi-$\alpha$ entropy in
Eq.(12). This has completed the proof. $\Box$

Example 4. Consider a $4$-partite star quantum network consisting of three EPR
states. It can be regarded as a $4$-partite pure state on $8\otimes 2\otimes
2\otimes 2$ dimensional Hilbert space, i.e.,

$\displaystyle|\phi\rangle_{\textsf{A}\textsf{B}\textsf{C}\textsf{D}}$
$\displaystyle=$ $\displaystyle\frac{1}{2\sqrt{2}}(|0000\rangle+|1001\rangle$
(86) $\displaystyle+|2010\rangle+|3011\rangle+|4100\rangle$
$\displaystyle+|5101\rangle+|6110\rangle+|7111\rangle)_{\textsf{A}\textsf{B}\textsf{C}\textsf{D}}.$

The reduced state of A is given by

$\displaystyle\rho_{\textsf{A}}=\frac{1}{8}\sum_{i=0}^{7}|i\rangle\langle{}i|.$
(87)

The EOF of $|\phi\rangle_{\textsf{A}|\textsf{B}\textsf{C}\textsf{D}}$ is given
by: ${\cal{E}}(|\phi\rangle_{\textsf{A}|\textsf{B}\textsf{C}\textsf{D}})=3$.
For mixed state $\rho_{\textsf{A}\textsf{B}}$, it should take an infimum for
the EOF. Note that the reduced density matrix $\rho_{\textsf{A}\textsf{B}}$ of
$|\psi\rangle_{\textsf{A}\textsf{B}\textsf{C}\textsf{D}}$ in Eq. (86) has a
spectral decomposition

$\displaystyle\rho_{\textsf{A}\textsf{B}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(|x_{1}\rangle\langle x_{1}|+|x_{2}\rangle\langle
x_{2}|+|x_{3}\rangle\langle x_{3}|+|x_{4}\rangle\langle x_{4}|),$

where $|x_{i}\rangle$ are eigenstates given by

$\displaystyle|x_{1}\rangle_{\textsf{A}\textsf{B}}=\frac{1}{\sqrt{2}}(|00\rangle+|41\rangle),$
$\displaystyle|x_{2}\rangle_{\textsf{A}\textsf{B}}=\frac{1}{\sqrt{2}}(|10\rangle+|51\rangle),$
$\displaystyle|x_{3}\rangle_{\textsf{A}\textsf{B}}=\frac{1}{\sqrt{2}}(|20\rangle+|61\rangle),$
$\displaystyle|x_{4}\rangle_{\textsf{A}\textsf{B}}=\frac{1}{\sqrt{2}}(|30\rangle+71\rangle).$
(88)

By the Hughston-Jozsa-Wootters (HJW) Theorem Hughston(1993) , any pure state
ensemble of $\rho_{\textsf{A}\textsf{B}}$ can be realized as a superposition
of $|x_{i}\rangle$ with $i=1,\cdots,4$, that is, for arbitrary pure state
$|\varphi\rangle_{\textsf{A}\textsf{B}}=\sum_{i}c_{i}|x_{i}\rangle$ with
$\sum_{i}c^{2}_{i}=1$, its reduced density matrix $\rho_{\textsf{A}}={\rm
Tr}_{\textsf{B}}(|\varphi\rangle_{\textsf{A}\textsf{B}}\langle\varphi|)$ has
the same spectrum $\frac{1}{2}$ with double root Vidal(2002)  while other
spectrum are $0$. So, we have
${\cal{E}}(|\varphi\rangle_{\textsf{A}\textsf{B}})=1$.

Moreover, the state $\rho_{\textsf{A}\textsf{B}}$ in Eq. (B.2) can be
decomposed into

$\displaystyle\rho_{\textsf{A}\textsf{B}}=\sum_{i}p_{i}|\varphi_{i}\rangle_{\textsf{A}\textsf{B}}\langle\varphi_{i}|.$
(89)

Thus, we have

$\displaystyle{\cal{E}}(\rho_{\textsf{A}\textsf{B}})=\sum_{i}p_{i}{\cal{E}}(|\varphi_{i}\rangle_{\textsf{A}\textsf{B}})=1$
(90)

from ${\cal{E}}(|\varphi_{i}\rangle_{\textsf{A}\textsf{B}})=1$ for each state
$|\varphi_{i}\rangle_{\textsf{A}\textsf{B}}$.

From similar proofs, we show the EOFs of $\rho_{\textsf{A}\textsf{C}}$ and
$\rho_{\textsf{A}\textsf{D}}$ are

$\displaystyle{\cal{E}}(\rho_{\textsf{A}\textsf{C}})={\cal{E}}(\rho_{\textsf{A}\textsf{D}})=1.$
(91)

Consequently, we have

$\displaystyle{\cal{E}}(|\psi\rangle_{\textsf{A}|\textsf{B}\textsf{C}\textsf{D}})={\cal{E}}(\rho_{\textsf{A}\textsf{B}})+{\cal{E}}(\rho_{\textsf{A}\textsf{C}})+{\cal{E}}(\rho_{\textsf{A}\textsf{D}}).$
(92)

Similarly, for the Rényi-$\alpha$ entanglement measure in Eq.(11), the
entanglement distribution satisfies

$\displaystyle\mathcal{R}_{\alpha}(|\psi\rangle_{\textsf{A}|\textsf{B}\textsf{C}\textsf{D}})=\mathcal{R}_{\alpha}(\rho_{\textsf{A}\textsf{B}})+\mathcal{R}_{\alpha}(\rho_{\textsf{A}\textsf{C}})+\mathcal{R}_{\alpha}(\rho_{\textsf{A}\textsf{D}})$
(93)

This has completed the proof.

###  B.3 Quantum network with arbitrary multipartite entangled pure states

Generally, it is interesting to show the equality (81) for the multipartite
entangled pure state shared by two parties. Consider quantum network
$\mathcal{N}_{q_{3}}$, where any two parties $\textsf{A}_{i}$ and
$\textsf{A}_{j}$ share the states
$\tilde{\varrho}^{1}_{ij},\cdots,\tilde{\varrho}^{s}_{ij}$,
$\sigma_{ij}(\varphi_{1}),\cdots,\sigma_{ij}(\varphi_{t})$, and
$\delta_{ij}(\vartheta_{1}),\cdots,\delta_{ij}(\vartheta_{k})$. The joint
state shared by any two parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$ is given
by

$\displaystyle\xi_{ij}=\mathop{\otimes}^{s_{j}}_{s=1}\widetilde{\varrho}^{s}_{ij}\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{ij}({\varphi_{t}})\mathop{\otimes}^{k_{j}}_{k=1}\delta_{ij}(\vartheta_{k}),$
(94)

where $\tilde{\varrho}^{s}_{ij}$ denotes the density matrix of the
multipartite entangled pure state $|\psi\rangle_{ij}$ with bipartition,
$\sigma(\varphi)$ is the density matrix of generalized GHZ state GHZ :
$|\phi(\varphi)\rangle=\cos\varphi|0\rangle^{\otimes
m}+\sin\varphi|1\rangle^{\otimes m}$ with any integer $m\geq 3$, and
$\delta({\vartheta})=\cos^{2}\vartheta|00\rangle\langle
00|+\sin^{2}\vartheta|11\rangle\langle 11|)$ is the reduced density matrix of
any two subsystems obtained by tracing out other subsystems in a generalized
GHZ state. Here, $s_{j}$, $t_{j}$, and $k_{j}$ denote the numbers of
$\widetilde{\varrho}$, $\sigma(\varphi)$, and $\delta(\vartheta)$ shared by
two parties $\textsf{A}_{i}$ and $\textsf{A}_{j}$.

Thus, the total state of $\mathcal{N}_{q_{3}}$ is written as

$\displaystyle\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}=\mathop{\otimes}^{n}_{\begin{subarray}{c}i,j=1\\\
i\neq
j\end{subarray}}\mathop{\otimes}^{s_{j}}_{s=1}\widetilde{\varrho}^{s}_{ij}\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{ij}({\varphi_{t}})\mathop{\otimes}^{k_{j}}_{k=1}\delta_{ij}(\vartheta_{k}).$
(95)

We prove that the monogamy of entanglement measure ${\cal Q}$ is valid for
$\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}$ from $\mathcal{N}_{q_{3}}$.

###### Theorem S 3

The entanglement distribution of quantum network $\mathcal{N}_{q_{3}}$
satisfies

$\displaystyle{\cal
Q}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})\geq\sum^{n}_{j=1,j\neq
i}{\cal
Q}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}),$
(96)

where ${\cal Q}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}$ and ${\cal
Q}_{{\textsf{A}_{i}|\textsf{A}_{j}}}$ are defined in Theorem S2.

_Proof._ Suppose that there are $N$ parties $\textsf{A}_{j}$’s who are
entangled with the party $\textsf{A}_{i}$. According to Eq.(8), by iteratively
using the additivity of von Neumann entropy in Eq.(12), we have

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle=$ $\displaystyle S(\rho_{\textsf{A}_{i}})$ (97)
$\displaystyle=$ $\displaystyle
S(\mathop{\otimes}^{N}_{j=1}(\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j}\mathop{\otimes}^{t_{j}}_{t=1}\rho_{i;j}(\varphi_{t})\mathop{\otimes}^{k_{j}}_{k=1}\rho_{i;j}(\vartheta_{k})))$
$\displaystyle=$
$\displaystyle\sum^{N}_{j=1}S(\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j}\mathop{\otimes}^{t_{j}}_{t=1}\rho_{i;j}(\varphi_{t})\mathop{\otimes}^{k_{j}}_{k=1}\rho_{i;j}(\vartheta_{k}))$
$\displaystyle=$
$\displaystyle\sum^{N}_{j=1}(\sum^{s_{j}}_{s=1}S(\rho^{s}_{i;j})+\sum^{t_{j}}_{t=1}S(\rho_{i;j}(\varphi_{t}))$
$\displaystyle+\sum^{k_{j}}_{k=1}S(\rho_{i;j}(\vartheta_{k}))).$

Moreover, it is easy to calculate that

$\displaystyle{\cal
E}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
$\displaystyle=$ $\displaystyle{\cal
E}(\mathop{\otimes}^{s_{j}}_{s=1}\tilde{\varrho}^{s}_{ij}\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{ij}({\varphi_{t}})\mathop{\otimes}^{k_{j}}_{k=1}\delta_{ij}(\vartheta_{k}))$
(98) $\displaystyle\leq$ $\displaystyle{\cal
E}(\mathop{\otimes}^{s_{j}}_{s=1}\tilde{\varrho}^{s}_{ij})+{\cal
E}(\mathop{\otimes}^{t_{j}}_{t=1}\sigma_{ij}({\varphi_{t}}))$ $\displaystyle=$
$\displaystyle
S(\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j})+S(\mathop{\otimes}^{t_{j}}_{t=1}\rho_{i;j}(\varphi_{t}))$
(99) $\displaystyle=$
$\displaystyle\sum^{s_{j}}_{s=1}S(\rho^{s}_{i;j})+\sum^{t_{j}}_{t=1}S(\rho_{i;j}(\varphi_{t})).$
(100)

From the additivity of EOF and ${\cal
E}(\mathop{\otimes}^{k_{j}}_{k=1}\delta_{ij}(\vartheta_{k}))=0$, we get the
inequality (98). The equality (99) follows from the definition of EOF in
Eq.(8). The equality (100) is obtained by using the additivity of von Neumann
entropy in Eq.(9) iteratively. Combining Eqs. (97) and (100), it yields

$\displaystyle{\cal{E}}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle\geq$ $\displaystyle\sum^{N}_{j=1,j\neq
i}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
(101) $\displaystyle=$ $\displaystyle\sum^{n}_{j=1,j\neq
i}{\cal{E}}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}).$

Similar to the inequality (101), from Eq.(11) and the additivity of
Rényi-$\alpha$ entropy in Eq.(12), it is easy to get the result in Theorem S3.

##  Appendix C Extension of Theorem 1 with other entropies

In this section, we investigate the entanglement distribution of quantum
network $\mathcal{N}_{q_{2}}(\cal{A},\xi)$ given in Appendix $\textsf{B}.2$ in
terms of Tsallis-$q$ entanglement and Unified $(q,s)$-entropy entanglement.

###  C.1 Entanglement distribution of quantum network in terms of Tsallis-$q$
entropy

###### Theorem S 4

Assume that a quantum network $\mathcal{N}_{q_{2}}(\cal{A},\xi)$ is connected
by arbitrary multipartite entangled pure states.

  * (i)

For $0<q<1$, the entanglement distribution of $\mathcal{N}_{q_{2}}$ satisfies

$\displaystyle\mathcal{T}^{(q)}_{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})\geq\sum^{n}_{j=1,j\neq
i}\mathcal{T}^{(q)}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}).$
(102)

  * (ii)

For $q>1$, the entanglement distribution of $\mathcal{N}_{q_{2}}$ satisfies

$\displaystyle\mathcal{T}^{(q)}_{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})\geq\sum^{n}_{j=1,j\neq
i}\mathcal{T}^{(q)}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}),$
(103)

where $\mathcal{T}^{(q)}_{\textsf{X}|\textsf{Y}}$ is Tsallis-$q$ entanglement
in Eq.(14) with respect to the bipartition X and Y, and
$\overline{\textsf{A}_{i}}$ denotes all parties except for $\textsf{A}_{i}$.

_Proof._ Suppose that there exist $N$ parties $\textsf{A}_{j}$’s who are
entangled with the party $\textsf{A}_{i}$. From Eq.(14), for the case $0<q<1$,
we have

$\displaystyle\mathcal{T}^{(q)}_{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle=$ $\displaystyle S_{q}(\rho_{\textsf{A}_{i}})$ (104)
$\displaystyle=$ $\displaystyle
S_{q}(\mathop{\otimes}^{N}_{j=1}(\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j}))$
$\displaystyle\leq$
$\displaystyle\sum^{N}_{j=1}S_{q}(\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j}),$

where the inequality (104) is derived from using the additivity of Tsallis-$q$
entropy in Eq.(16) iteratively, $\rho^{s}_{i;j}$ is the reduced density matrix
of the subsystems owned by the party $\textsf{A}_{i}$ by tracing out the
subsystems owned by the party $\textsf{A}_{j}$ in a multipartite entangled
pure state $|\psi^{s}\rangle$, i.e,
$\rho^{s}_{i;j}={\rm{Tr}}_{j}(|\psi^{s}\rangle_{ij}\langle\psi^{s}|)$.

In addition, we have

$\displaystyle\sum^{N}_{j=1}\mathcal{T}^{(q)}_{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})$
$\displaystyle=$
$\displaystyle\sum^{N}_{j=1}\mathcal{T}^{(q)}(\mathop{\otimes}^{s_{j}}_{s=1}|\psi^{s}\rangle_{ij})$
(105) $\displaystyle=$ $\displaystyle\sum^{N}_{j=1}S_{q}(\rho^{j}_{i})$
$\displaystyle=$
$\displaystyle\sum^{N}_{j=1}S_{q}(\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j}).$
(106)

Here, the equality (105) follows from the definition of Tsallis-$q$
entanglement in Eq.(14), and the equality (106) is due to
$\rho^{j}_{i}=\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j}$, i.e., all the
states of $\rho^{s}_{i;j}$’s are not entangled. The inequality (104) and
equality (106) yield to the inequality (102).

Similarly, we obtain the inequality (103) for $q>1$. This completes the proof.

###  C.2 Entanglement distribution of quantum networkd in terms of Unified
$(q,s)$-entropy

Similar to Theorem S4, we obtain the following Theorem.

###### Theorem S 5

Assume that a quantum network $\mathcal{N}_{q_{2}}(\cal{A},\xi)$ consists of
arbitrary multipartite entangled pure states.

  * (i)

For $0<q<1$ and $s<0$, or $q\geq 1$ and $s\geq 0$, the entanglement
distribution of $\mathcal{N}_{q_{2}}$ satisfies

$\displaystyle\mathcal{U}^{(q,s)}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})\geq\sum^{n}_{j=1,j\neq
i}\mathcal{U}^{(q,s)}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}).$
(107)

  * (ii)

For $q>1$ and $s<0$, or $0<q<1$ and $s>0$, the entanglement distribution of
$\mathcal{N}_{q_{2}}$ satisfies

$\displaystyle\mathcal{U}^{(q,s)}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})\geq\sum^{n}_{j=1,j\neq
i}\mathcal{U}^{(q,s)}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}}),$
(108)

where $\mathcal{U}^{(q,s)}_{\textsf{X}|\textsf{Y}}$ is Unified $(q,s)$-entropy
entanglement in Eq.(19) with respect to the bipartition X and Y, and
$\overline{\textsf{A}_{i}}$ denotes all parties except for $\textsf{A}_{i}$.

_Proof._ The proof is similar to its for Theorem S4. Similar to the inequality
(104), from Eq.(19) and the subadditivity of Unified $(q,s)$-entropy for
$0<q<1$ and $s<0$, or $q\geq 1$ and $s\geq 0$, we get

$\displaystyle\mathcal{U}^{(q,s)}_{{\textsf{A}_{i}|\overline{\textsf{A}_{i}}}}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle=$ $\displaystyle S_{q,s}(\rho_{\textsf{A}_{i}})$
$\displaystyle=$ $\displaystyle
S_{q,s}(\mathop{\otimes}^{N}_{j=1}(\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j}))$
$\displaystyle\leq$
$\displaystyle\sum^{N}_{j=1}S_{q,s}(\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j}).$
(110)

Similar to Eq.(106), from Eq.(19) we have

$\displaystyle\mathcal{U}^{(q,s)}_{{\textsf{A}_{i}|\textsf{A}_{j}}}(\rho_{\textsf{A}_{i}\textsf{A}_{j}})=\sum^{N}_{j=1}S_{q,s}(\mathop{\otimes}^{s_{j}}_{s=1}\rho^{s}_{i;j}).$
(111)

Thus, Eqs.(110) and (111) imply the inequality (107) for $0<q<1$ and $s<0$, or
$q\geq 1$ and $s\geq 0$.

Similarly, we can obtain the inequality (108) for $q>1$ and $s<0$, or $0<q<1$
and $s>0$. This completes the proof.

##  Appendix D Proof of Theorem 2

###  D.1 Communication model on quantum networks

In this subsection, we present necessary notions for featuring the
communication on quantum networks.

Consider an $n$-partite entangled quantum network
$\mathcal{N}_{q}({\cal{A}},\Omega)$. Let s and t be the source and sink,
respectively. Define the _flow_ $f_{i\rightarrow j}$ in a quantum network
$\mathcal{N}_{q}$ as the number of qubits which are reliably transmitted from
$\textsf{A}_{i}$ to $\textsf{A}_{j}$ along the channel $\xi_{ij}$, that is,

$\displaystyle f_{i\rightarrow j}\geq 0.$ (112)

For the quantum channel consisting of EPR states EPR  and GHZ states GHZ , the
capacity between $\textsf{A}_{i}$ and $\textsf{A}_{j}$ is defined in Eq.(5),
which means the maximal number of qubits transmitted. Inspired by the method
in ref.Pira(2019) , the flow is bounded by the two-way quantum capacity of the
associated channel $\xi_{ij}$, i.e.,

$\displaystyle f_{i\rightarrow j}\leq\mathcal{C}(\xi_{ij})=S(\rho^{j}_{i}),$
(113)

where $S(\rho^{j}_{i})$ represents the von Neumann entropy of the reduced
density matrix $\rho^{j}_{i}$ by tracing out the system of $j$, i.e.,
$\rho^{j}_{i}={\rm{Tr}}_{j}(|\xi_{ij}\rangle\langle\xi_{ij}|)$ . In addition,
at any intermediate party, the number of qubits simultaneously received must
be equal to the number of qubits simultaneously transmitted through all the
party-to-party communications with neighbor parties. That means for any
$i\in\cal{A}\setminus\\{\textbf{s},\textbf{t}\\}$, we have

$\displaystyle\sum_{j\in\cal{A}}f_{i\rightarrow j}=0.$ (114)

This does not hold for the source s and the sink t, for which we impose

$\displaystyle\sum_{j\in\cal{A}}f_{\textbf{s}\rightarrow
j}=-\sum_{j\in\cal{A}}f_{\textbf{t}\rightarrow j}=f,$ (115)

where $f$ is known as the value of the flow. This is an achievable end-to-end
rate since it represents the total number of qubits which are transmitted by
the source and correspondingly received by the sink via all the end-to-end
communication. Thus, analogous to classical network, we define a _flow_ in
quantum network as follows.

###### Definition 1

A flow of a quantum network $\mathcal{N}_{q}(\mathcal{A},\Omega)$ from s to t
is given by a function $f:\Omega\longrightarrow\mathbb{R}^{+}$, such that $f$
satisfies:

  * (i)

Capacity constraint: the flow is bounded by the capacity of channel, that is,
for any $\xi_{ij}\in\Omega$, $f_{i\rightarrow j}$ satisfies

$\displaystyle f_{i\rightarrow j}\leq\mathcal{C}(\xi_{ij}).$ (116)

  * (ii)

Conservation of flow: the flow leaving the party $j$ is equal to the flow
entering the party $j$, that is,

$\displaystyle\sum_{\\{i:\xi_{ij}\in\Omega\\}}f_{i\rightarrow
j}=\sum_{\\{k:\xi_{jk}\in\Omega\\}}f_{j\rightarrow k},$ (117)

for any party $j\in\mathcal{A}$, other than the sender(source) s and
receiver(sink) t.

Herein, $f_{i\rightarrow j}$ denotes the number of qubits transmitted to from
the party $i$ to $j$ by the quantum channel $\xi_{ij}$. If $f_{i\rightarrow
j}=\mathcal{C}({\xi_{ij}})$, then the quantum channel $\xi_{ij}$ is saturated.
Otherwise, $\xi_{ij}$ is unsaturated.

###### Definition 2

A maximum flow $MF(\mathcal{N})$ of quantum network
$\mathcal{N}_{q}(\mathcal{A},\Omega)$ is the maximum amount of flow $f$ from
the source s to the sink t, that is,

$\displaystyle f_{\max}=\max\\{f\\}.$ (118)

Now, we transform the quantum network $\mathcal{N}_{q}(\mathcal{A},\Omega)$
into a associated undirected graph $\mathcal{G}=({\cal V},\Omega)$, where
${\cal V}$ consists of all vertices corresponding to all the parties in
$\mathcal{A}$, and $\Omega$ consists of all edges corresponding to all the
entangled states in $\Omega$, in which one EPR state is schematically
represented by one edge while the $m$-partite GHZ state is schematically
represented by hyper edge connected by $m$ nodes Voloshin(2009) . We then
adopt the standard definition of cut-set for graph Voloshin(2009) . A cut
$T_{cut}$ of $\mathcal{G}$ is a bipartition $(\textbf{S}_{1},\textbf{S}_{2})$
of ${\cal V}$ such that $\textbf{s}\in\textbf{S}_{1}$ and
$\textbf{t}\in\textbf{S}_{2}$. The cut-set $\widetilde{T}_{cut}$ of $T_{cut}$
is the set of edges so that the removal of these edges disconnects the source
s and the sink t.

###### Definition 3

An edge cut set $\widetilde{T}_{cut}$ consists of edges such that there exists
a bipartition $(\textbf{S}_{1},\textbf{S}_{2})$ of all the vertexes in
$\cal{V}$ satisfying $\textbf{s}\in\textbf{S}_{1}$ and
$\textbf{t}\in\textbf{S}_{2}$, that is,
$\widetilde{T}_{cut}=\\{(i,j):i\in\textbf{S}_{1},j\in\textbf{S}_{2}\\}$. Then
the capacity of an edge cut set $T_{cut}$ is given by

$\displaystyle\mathcal{C}(\textbf{S}_{1},\textbf{S}_{2})=\sum_{(i,j)\in\widetilde{T}_{cut}}\mathcal{C}(i,j),$
(119)

where $\mathcal{C}(i,j)$ denotes the weight of edge $(i,j)$.

###### Definition 4

If $\mathcal{C}(T_{cut})\geq\mathcal{C}(T^{\prime}_{cut})$ holds for any cut
$T_{cut}$ in graph $\mathcal{G}$, then $T^{\prime}_{cut}$ is a minimal cut of
$\mathcal{G}$, that is, a minimum cut is a cut with the minimum capacity with
the capacity given by

$\displaystyle\mathcal{C}(T^{\prime}_{cut})=\min_{T_{cut}}\mathcal{C}(\textbf{S}_{1},\textbf{S}_{2}).$
(120)

###  D.2 Proof of Theorem 2

Firstly, we resort to following Lemma 1 in classical network Elias(1956) ;
Ford(1956) .

###### Lemma 1

Let $f$ be a flow on a network $\mathcal{N}$, and $T_{cut}$ be a cut
$T_{cut}=(\textbf{S}_{1},\textbf{S}_{2})$ with the capacity
$\mathcal{C}(T_{cut})$. Then the flow satisies

$\displaystyle f\leq\mathcal{C}(T_{cut}),$ (121)

where the equality holds if and only if each edge $(i,j)$ in the cut
$T_{cut}=(\textbf{S}_{1},\textbf{S}_{2})$ is saturated, that is,
$f_{i\rightarrow j}=\mathcal{C}(i,j)$, and each edge $(j,i)$ in
$T_{cut}=(\textbf{S}_{2},\textbf{S}_{1})$ has zero flow with $f_{j\rightarrow
i}=0$.

_Proof of Lemma 1_. Denote
$f=f(\textbf{S}_{1},\textbf{S}_{2})-f(\textbf{S}_{2},\textbf{S}_{1})$. On the
one hand, the flow conservation law implies that
$f(\textbf{S}_{1},\textbf{S}_{2})\leq\mathcal{C}(T_{cut})$ and
$f(\textbf{S}_{2},\textbf{S}_{1})\geq 0$. Note that
$f=f(\textbf{S}_{1},\textbf{S}_{2})-f(\textbf{S}_{2},\textbf{S}_{1})\leq\mathcal{C}(T_{cut})$,
that is,

$\displaystyle f\leq\mathcal{C}(T_{cut}).$ (122)

For $f(\textbf{S}_{1},\textbf{S}_{2})\leq\mathcal{C}(T_{cut})$, the equality
holds iff each edge $(i,j)$ in cut $T_{cut}=(\textbf{S}_{1},\textbf{S}_{2})$
is saturated. Besides, $f(\textbf{S}_{2},\textbf{S}_{1})\geq 0$, where the
equality holds if and only if each edge $(j,i)$ in
$(\textbf{S}_{2},\textbf{S}_{1})$ has zero flow. So, $f=\mathcal{C}(T_{cut})$
holds if and only if each edge $(i,j)$ in the cut
$T_{cut}=(\textbf{S}_{1},\textbf{S}_{2})$ is saturated and each edge $(j,i)$
in $(\textbf{S}_{2},\textbf{S}_{1})$ has zero flow. $\Box$

Now, consider a quantum network $\mathcal{N}_{q}({\cal{A}},\Omega)$ consists
of quantum channels defined as EPR states and GHZ states. There exists a cut
$T_{cut}$ in $\widetilde{T}_{cut}$ of the associated graph $\cal{G}$
separating $\textbf{S}_{1}$ and $\textbf{S}_{2}$. Similar to Lemma 1, we get

$\displaystyle f(\textbf{S}_{1},\textbf{S}_{2})\leq\mathcal{C}(T_{cut}),$
(123)

where the equality holds if and only if each channel $\xi_{ij}$ in the cut
$T_{cut}=(\textbf{S}_{1},\textbf{S}_{2})$ is saturated, and $f_{i\rightarrow
j}=\mathcal{C}(\xi_{ij})=\mathcal{C}(i,j)$. As its shown in refs.Bennett1 ;
Karlsson(1998) , one can perfectly transmit one qubit by using quantum channel
of EPR state EPR  or GHZ state GHZ . Thereby, it is feasible to achieve
$f_{i\rightarrow j}=\mathcal{C}(i,j)$ from the point of view of quantum
communication. Note that the transmission of information in quantum
communication is one-way, i.e., $f(\textbf{S}_{2},\textbf{S}_{1})=0$. Further,
we get the following result.

###### Lemma 2

If a flow $f$ and cut $T_{cut}=(\textbf{S}_{1},\textbf{S}_{2})$ on a quantum
network $\mathcal{N}_{q}({\cal{A}},\Omega)$ satisfy $f=\mathcal{C}(T_{cut})$,
then $f$ is a maximum flow and the cut $T_{cut}$ is a minimum cut.

Note that any flow $f^{\prime}$ of the network $\mathcal{N}_{q}$ satisfies
$f^{\prime}\leq\mathcal{C}(T_{cut})=f$. On the other hand, for any cut
$T_{cut}^{\prime}$ of the associated graph $\cal{G}$, it follows
$\mathcal{C}^{\prime}(T_{cut})\geq f=\mathcal{C}(T_{cut})$. Thus $f$ is the
maximum flow and the cut $T_{cut}$ is the minimum cut. As a result, we obtain
the claim in Theorem 2 for the quantum network
$\mathcal{N}_{q}({\cal{A}},\Omega)$. It means that the maximal flow of
$\mathcal{N}_{q}({\cal{A}},\Omega)$ is equal to the minimum-capacity cut of
the associated graph $\mathcal{G}$. This completes the proof of Theorem 2.

##  Appendix E The iteration of Example 2

In this section, embarking on presenting the example, we recall following
Lemma.

###### Lemma 3

Ford(1956)  Let $f$ be a _flow_ on a classical network $\mathcal{N}$. Then $f$
is the maximum flow only if $f$ has no augmenting paths.

This Lemma is also available for quantum network $\mathcal{N}_{q}$ consisting
of EPR states and GHZ states. Hence, the main idea for the maximal flow of
quantum network $\mathcal{N}_{q}$ is to find augmenting paths. We apply the
Ford-Fulkerson algorithm Ford(1956)  that sends the maximal possible amount of
flow at each iteration, that is, the flow equals to the capacity of the path
under consideration, where $f_{i}$ denotes the amount of flow sent in the
iteration $i$.

Iteration 1. We find a path $s\rightarrow 1\rightarrow 2\rightarrow t$ that
can carry a positive flow, then the maximum flow we can send along this path
is $f_{1}=\min\\{4,3,2\\}=2$, as shown in Fig.6(a). In Fig.6(b), we obtain a
residual network with updated link capacities resulting from pushing the flow
along the path.

Figure 6: (Color online) The iterations for finding the maximum flow in
Example 2. (a) The augmenting path $s\rightarrow 1\rightarrow 2\rightarrow t$.
(b) The residual network in the first iteration by adding the maximum flow
$f_{1}$. (c) The augmenting path $s\rightarrow
1\rightarrow\\{2,4\\}\rightarrow t$. (d) The residual network in the second
iteration by adding a maximum flow $f_{2}$. (e) The augmenting path
$s\rightarrow 1\rightarrow 3\rightarrow t$. (f) The residual network in the
third iteration by adding a maximum flow $f_{3}$. (g) The augmenting path
$s\rightarrow 2\rightarrow 1\rightarrow 3\rightarrow t$. (h) The residual
network in the fourth iteration by adding a maximum flow $f_{4}$. (i) The
augmenting path $s\rightarrow\\{2,4\\}\rightarrow t$. (j) The residual network
in the fifth iteration by adding a maximum flow $f_{5}$.

Iteration 2. Similarly, we find an augmenting path $s\rightarrow
1\rightarrow\\{2,4\\}\rightarrow t$ to send a positive flow. The maximum flow
along this path is then given by $f_{2}=\min\\{4-2,3-2,2\\}=1$, as shown in
Fig.6(c). We get a residual network with updated capacities shown in Fig.6(d).

Iteration 3. We find a path $s\rightarrow 1\rightarrow 3\rightarrow t$ which
sends a maximum flow along this path given by $f_{3}=\min\\{4-3,3,5\\}=1$, as
shown in Fig.6(e). A residual network is updated as shown in Fig.6(f).

Iteration 4. For the path $s\rightarrow 2\rightarrow 1\rightarrow 3\rightarrow
t$, a maximum flow along this path is given by $f_{4}=\min\\{6,3,3-1,5-1\\}=2$
as shown in Fig.6(g). A residual network with updated capacities resulting
from pushing the flow along the path is given by Fig.6(h).

Iteration 5. We find a path $s\rightarrow\\{2,4\\}\rightarrow t$ for sending a
positive flow. The maximum flow we can send along this path is given by
$f_{5}=\min\\{6-2,2-1\\}=1$, as shown in Fig.6(i). We obtain a residual
network with updated link capacities resulting from pushing the flow along the
path, as shown in Fig.6(j).

##  Appendix F Proof of Theorem 3

In this section, we prove Theorem 3 by induction $n$, that is, the number of
parties in quantum network ${\cal N}_{q}$. Consider the following two cases.

Case 1. $n=3$.

In this case, there are three types of quantum networks $\mathcal{N}_{1}$,
$\mathcal{N}_{2}$, and $\mathcal{N}_{3}$, as shown in Fig. 7. The vectors of
von Neumann entropy of quantum networks are given by

$\displaystyle S_{\mathcal{N}_{1}}=(1,2,1),$ $\displaystyle
S_{\mathcal{N}_{2}}=(2,2,2),$ $\displaystyle S_{\mathcal{N}_{3}}=(1,1,1).$
(124)

It follows that the characteristic vectors of von Neumann entropy for
tripartite quantum networks consisting of EPR states or GHZ states are
different from each other, that is, they are unique for each kind of
tripartite quantum network. Note that the von Neumann entropy is invariant
under the local unitary operations. This has proved the result.

Figure 7: (Color online) Three types of tripartite quantum network
$\mathcal{N}_{1}$, $\mathcal{N}_{2}$, and $\mathcal{N}_{3}$ consisting of EPR
states or GHZ states, where any two parties share no more than an
entanglement. (a) A chain quantum network consisting of two EPR states. (b) A
triangle quantum network consisting of three EPR states. (c) A triangle
quantum network consisting of one GHZ state.

Case 2. $n>3$.

Assume that the result holds for $n\leq k-1$, that is, for any pair of quantum
networks ${\cal N}_{1}$ and ${\cal N}_{2}$ there exist local unitary
operations $U_{j}$ such that

$\displaystyle\otimes_{j=1}^{n}U_{j}\rho_{\mathcal{N}_{1}}U^{\dagger}_{j}=\rho_{\mathcal{N}_{2}}$
(125)

if $S_{\mathcal{N}_{1}}=S_{\mathcal{N}_{2}}$, where $S_{\mathcal{N}_{j}}$
denotes the characteristic vector defined by von Neumann entropies for quantum
network $\mathcal{N}_{j}$ consisting of EPR states or GHZ states. Here, each
party in networks may perform local unitary operations. It means that the
quantum network consisting of EPR states and GHZ states is unique under local
unitary operations.

In what follows, we will prove the result for quantum networks ${\cal N}_{3}$
with $n=k$. In fact, this will be proved by two subcases. The main idea is
that the new network ${\cal N}_{3}$ can be regarded as an extended network of
${\cal N}_{1}$.

Figure 8: (Color online) Schematic quantum network ${\cal N}_{3}$ decomposed
into two subnetworks ${\cal N}_{3;1}$ and ${\cal N}_{3;2}$. (a) ${\cal N}_{3}$
consisting of
$\textsf{A}_{1},\textsf{A}_{2},\cdots,\textsf{A}_{k-1},\textsf{A}_{k}$. (b)
Subnetwork ${\cal N}_{3;1}$ consisting of $k-1$ parties
($\textsf{A}_{2},\cdots,\textsf{A}_{k}$ for examples) of ${\cal N}_{1}$ and
the party ${\textsf{A}}_{k}$. (c) ${\cal N}_{3;2}$ consisting of at least one
party $\textsf{A}_{1}$ and some parties
$\textsf{A}_{s_{1}},\cdots,\textsf{A}_{s_{m}}$ entangled with $\textsf{A}_{1}$
of ${\cal N}_{3}$.

Assume that ${\cal N}_{3}$ consists of $k$ parties
$\textsf{A}_{1},\textsf{A}_{2},\cdots,\textsf{A}_{k}$ who share EPR states and
GHZ states, as shown in Fig.8(a). The quantum network ${\cal N}_{3}$ will be
decomposed into two subnetworks ${\cal N}_{3;1}$ as shown in Fig.8(b) and
${\cal N}_{3;2}$ as shown in Fig.8(c), where ${\cal N}_{3;1}$ consists of
$k-1$ parties ($\textsf{A}_{2},\cdots,\textsf{A}_{k}$ for examples) of ${\cal
N}_{3}$, and ${\cal N}_{3;2}$ which has at most $k-1$ parties consists of at
least one party $\textsf{A}_{1}$ and some parties
$\textsf{A}_{s_{1}},\cdots,\textsf{A}_{s_{m}}$ entangled with $\textsf{A}_{1}$
of ${\cal N}_{3}$.

Consider the subnetwork ${\cal N}_{3;1}$ with $k-1$ parties. The
characteristic vector of ${\cal N}_{3;1}$ is given by $S_{{\cal N}_{3;1}}$.
For the subnetwork ${\cal N}_{3;2}$, the characteristic vector is given by
$S_{{\cal N}_{3;2}}$. Since these networks consist of EPR states and GHZ
states. Moreover, the local von Neumann entropy of each party satisfies the
additivity, namely,

$\displaystyle S_{j}=m^{(j)}_{EPR}+m^{(j)}_{GHZ},$ (126)

where $m_{EPR}$ and $m_{GHZ}$ denote the respective number of EPR states and
GHZ states shared by the party $\textsf{A}_{j}$. Note that the von Neumann
entropy is invariant under the local unitary operations. It follows that the
characteristic vector of quantum networks consisting of EPR and GHZ states
satisfies the additivity, that is,

$\displaystyle
S_{\mathcal{N}_{3}}=S_{\mathcal{N}_{3;1}}+S_{\mathcal{N}_{3;2}}.$ (127)

Moreover, $\mathcal{N}_{3;1}$ and $\mathcal{N}_{3;2}$ have at most $k-1$
parties. Hence, we get that the new network generated by combinations of the
subnetworks $\mathcal{N}_{3;1}$ and $\mathcal{N}_{3;2}$ is equivalent to the
network $\mathcal{N}_{3}$. From the assumption, it follows that the
characteristic vectors of $\mathcal{N}_{3;1}$ and $\mathcal{N}_{3;2}$ are
unique under the local unitary operations. It means that for each other
networks $\mathcal{N}^{\prime}_{3;1}$ and $\mathcal{N}^{\prime}_{3;2}$
consisting of EPR and GHZ states we have

$\displaystyle\otimes_{j}U_{j}\rho_{\mathcal{N}_{3;s}}U^{\dagger}_{j}=\rho_{\mathcal{N}^{\prime}_{3;s}},s=1,2$
(128)

if $S_{\mathcal{N}_{3;s}}=S_{\mathcal{N}^{\prime}_{3;s}}$. It means the
characteristic vectors of $\mathcal{N}_{3;s}$ and $\mathcal{N}^{\prime}_{3;s}$
equal to each other, where $U_{j}$ is unitary operation. This implies that

$\displaystyle S_{\mathcal{N}_{3}}=S_{\mathcal{N}^{\prime}_{3}},$ (129)

where

$\displaystyle
S_{\mathcal{N}^{\prime}_{3}}=S_{\mathcal{N}^{\prime}_{3;1}}+S_{\mathcal{N}^{\prime}_{3;2}}.$
(130)

Note Eq.(128) means that there exist local unitary operations satisfying

$\displaystyle\otimes_{j}U_{j}\rho_{\mathcal{N}_{3}}U^{\dagger}_{j}=\rho_{\mathcal{N}^{\prime}_{3}}.$
(131)

So, $\mathcal{N}_{3}$ is LU equivalent to the network
$\mathcal{N}^{\prime}_{3}$, that is, the characteristic vector of quantum
network $\mathcal{N}_{3}$ is unique. This completes the proofs of Theorem 3.
$\Box$

According to Theorem 3, we present a new quantum network classification method
by virtue of the characteristic vector of von Neumann entropy. We present some
examples to explain the main idea.

Example 3. Consider a simple network with $n$ parties
${\mathsf{A}_{1}},{\mathsf{A}_{2}},\cdots,{\mathsf{A}_{n}}$. Here, adjacent
parties share no more than one EPR-pair as shown in Fig.5(a) for chain quantum
network, (b) for cyclic network and (c) for star quantum network. The von
Neumann entropy vector of these networks are given by $(1,2,\cdots,2,1)$,
$(2,\cdots,2)$, and $(n-1,1,\cdots,1)$, respectively, which are different.
This shows that these networks are inequivalent to each other under local
unitary operations.

Example 4. Consider an $n$-partite complete graph-like network
$\mathcal{N}_{\mathbf{A}}$, as shown in Fig.9. Any pair of two parties shares
an EPR states. Each party shares $n-1$ EPR states with others. There are
$n(n-1)/2$ number of EPR states. The characteristic vector of
$\mathcal{N}_{\mathbf{A}}$ is given by $(n-1,n-1,\cdots,n-1)$, which is
different its for the networks in Fig.5.

Figure 9: (Color online) A multi-partite complete graph-like network
consisting of EPR states.

##  Appendix G Classifying networks based on the mutual information

In this section, we classify quantum networks based on the mutual information.
We firstly recall the definition of mutual information.

Shannon mutual information Shannon(1998)  describes how much information can
be determined about a random variable X, by knowing the value of a correlated
random variable Y. The Shannon mutual information for tripartite system can be
written as

$\displaystyle I(\textsf{X}:\textsf{Y}:\textsf{Z})$ $\displaystyle=$
$\displaystyle
H(\textsf{X}\textsf{Y}\textsf{Z})+H(\textsf{X})+H(\textsf{Y})+H(\textsf{Z})$
(132)
$\displaystyle-H(\textsf{X}\textsf{Y})-H(\textsf{Y}\textsf{Z})-H(\textsf{X}\textsf{Z}),$

where
$H(\textsf{X})=-\sum_{\textsf{x}\in\textsf{X}}p(\textsf{x})\log_{2}p(\textsf{x})$
is the entropy of X, and
$H(\textsf{X}\textsf{Y})=-\sum_{\textsf{x}\in\textsf{X},\textsf{y}\in\textsf{Y}}p(\textsf{x},\textsf{y})\log_{2}p(\textsf{x},\textsf{y})$
is the joint entropy of X and Y, and $p(\textsf{x},\textsf{y})$ is the joint
probability distribution of X and Y.

###### Lemma 4

Two types of cyclic networks in Fig.4(a) and (b) can be verified by Shannon
mutual information derived from local projection measurements.

_Proof_. For quantum networks $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ shown in
Fig.4(a), they are inequivalent under local unitary operations. The total
state of system for the network $\mathcal{N}_{1}$ is given by

$\displaystyle|\mu^{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)_{\textsf{A}_{1}\textsf{B}_{1}}\otimes\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)_{\textsf{A}_{1}\textsf{C}_{1}}$
(133)
$\displaystyle\otimes\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)_{\textsf{B}_{1}\textsf{C}_{1}},$

which may be written into

$\displaystyle|\mu^{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2}}(|000\rangle+|201\rangle+|012\rangle+|213\rangle$
(134)
$\displaystyle+|120\rangle+|321\rangle+|132\rangle+|333\rangle)_{\textsf{A}_{1}\textsf{B}_{1}\textsf{C}_{1}}$

where we regard the local two-qubit systems as a $4$-dimensional system with
the computation basis $\\{|0\rangle,\cdots,|4\rangle\\}$. After projection
measurements being performed on quantum state $|\mu^{1}\rangle$ in Eq.(134),
we get a joint probability distribution

$\displaystyle p(x_{1}x_{2}x_{3})=\frac{1}{8},$ (135)

where $x_{1}x_{2}x_{3}\in\\{000,201,012,213,120,321,132,333\\}$.

For the network $\mathcal{N}_{2}$ as shown in Fig.4(b) the total state is
given by

$\displaystyle|\mu^{2}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle)_{\textsf{A}_{1}\textsf{B}_{1}\textsf{C}_{1}}^{\otimes
2},$ (136)

that is,

$\displaystyle|\mu^{2}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle+|222\rangle+|333\rangle)_{\textsf{A}_{1}\textsf{B}_{1}\textsf{C}_{1}}.$
(137)

For quantum state $|\mu^{2}\rangle$ in Eq.(137), after projection measurements
being performed, a joint probability distribution is obtained as

$\displaystyle p(y_{1}y_{2}y_{3})=\frac{1}{4},$ (138)

where $y_{1}y_{2}y_{3}\in\\{000,111,222,333\\}$.

From Eq.(132), it is easy to check that the Shannon mutual information of
subsystems in $\mathcal{N}_{1}$ and $\mathcal{N}_{2}$ are given by

$\displaystyle I(\textsf{A}_{1}:\textsf{B}_{1}:\textsf{C}_{1})=3,$ (139)
$\displaystyle I(\textsf{A}_{2}:\textsf{B}_{2}:\textsf{C}_{2})=2.$ (140)

Similarly, the Shannon mutual information of subsystems in $\mathcal{N}_{3}$
and $\mathcal{N}_{4}$ in Fig.4 (b) are given by

$\displaystyle
I(\textsf{A}_{3}:\textsf{B}_{3}:\textsf{C}_{3}:\textsf{D}_{3})=4,$ (141)
$\displaystyle
I(\textsf{A}_{4}:\textsf{B}_{4}:\textsf{C}_{4}:\textsf{D}_{4})=3,$ (142)

which are different from its in Eqs.(139) and (140). This means that the
Shannon mutual information as a character can be used to distinguish two
networks. $\Box$

Note that the von Neumann entropy is unitary invariant. Another interesting
method for distinguishing network topologies is to use the von Neumann entropy
of extended system by copying the original quantum state into two copies.
Specially, by introducing an auxiliary qubit in the state $|0\rangle$ locally,
each party performs a controlled Not operation on local systems, where the
original qubit is controlling qubit while the auxiliary qubit is target. The
total system $|\mu^{1}\rangle$ in Eq.(133) is then changed into

$\displaystyle|\nu^{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|0\rangle^{\otimes 4}+|1\rangle^{\otimes
4})_{\textsf{A}_{1}\textsf{a}_{1}\textsf{B}_{1}\textsf{b}_{1}}$ (143)
$\displaystyle\otimes\frac{1}{\sqrt{2}}(|0\rangle^{\otimes
4}+|1\rangle^{\otimes
4})_{\textsf{A}_{1}\textsf{a}_{1}\textsf{C}_{1}\textsf{c}_{1}}$
$\displaystyle\otimes\frac{1}{\sqrt{2}}(|0\rangle^{\otimes
4}+|1\rangle^{\otimes
4})_{\textsf{B}_{1}\textsf{b}_{1}\textsf{C}_{1}\textsf{c}_{1}}.$

Similarly, for the network $\mathcal{N}_{2}$ in Fig.4(b), the state
$|\mu^{2}\rangle$ in Eq.(136) is changed into

$\displaystyle|\nu^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(|0\rangle^{\otimes 6}+|1\rangle^{\otimes
6})_{\textsf{A}_{2}\textsf{a}_{2}\textsf{B}_{2}\textsf{b}_{2}\textsf{C}_{2}\textsf{c}_{2}}$
(144) $\displaystyle\otimes\frac{1}{\sqrt{2}}(|0\rangle^{\otimes
6}+|1\rangle^{\otimes
6})_{\textsf{A}_{2}\textsf{a}_{2}\textsf{B}_{2}\textsf{b}_{2}\textsf{C}_{2}\textsf{c}_{2}}.$

For quantum states in Eqs.(143) and (144), we obtain

$\displaystyle S(\rho_{\textsf{A}_{1}\textsf{B}_{1}\textsf{C}_{1}})=3,$ (145)
$\displaystyle S(\rho_{\textsf{A}_{2}\textsf{B}_{2}\textsf{C}_{2}})=2,$ (146)

which are different from each other, where $S(\textsf{X})$ is the von Neumann
entropy of the subsystem X. It means that we can identify two different
quantum networks by using the von Neumann entropy for the system
$\textsf{A}_{i}\textsf{B}_{i}\textsf{C}_{i}$ with $i=1,2$.

Similarly, the von Neumann entropy of the subsystems
$\textsf{A}_{i},\textsf{B}_{i},\textsf{C}_{i}$ and $\textsf{D}_{i}$ ($i=3,4$)
for $\mathcal{N}_{3}$ and $\mathcal{N}_{4}$ are respectively given by

$\displaystyle
S(\rho_{\textsf{A}_{3}\textsf{B}_{3}\textsf{C}_{3}\textsf{D}_{3}})=4,$ (147)
$\displaystyle
S(\rho_{\textsf{A}_{4}\textsf{B}_{4}\textsf{C}_{4}\textsf{D}_{4}})=3,$ (148)

which elucidates two inequivalent networks $\mathcal{N}_{3}$ and
$\mathcal{N}_{4}$ under local unitary operations. This provides another method
for classifying quantum networks.

Figure 10: (Color online) Inequivalent networks with the same characteristic
vector defined by von Neumann entropies. (a) Quantum network
$\mathcal{N}_{\textsf{A}}$ consisting of EPR states.(b) Quantum network
$\mathcal{N}_{\textsf{B}}$ consisting of EPR states and GHZ states.

From Lemma 4, we obtain the following Theorem.

###### Theorem S 6

For any quantum network $\mathcal{N}_{\textsf{A}}$ consisting of EPR states
contains some cyclic subnetworks, there is another quantum network
$\mathcal{N}_{\textsf{B}}$ with the same characteristic vector of von Neumann
entropy as $\mathcal{N}_{\textsf{A}}$ such that they have different network
topologies under local operations.

_Proof_. As its illustrated in Fig.10(a), quantum network
$\mathcal{N}_{\textsf{A}}$ contains four subnetworks $\mathcal{N}_{1}$,
$\mathcal{N}_{2}$, $\mathcal{N}_{3}$, and $\mathcal{N}_{4}$. Here, three-
partite subnetwork $\mathcal{N}_{1}$ has parties $\textsf{A}_{1}$,
$\textsf{A}_{3}$ and $\textsf{A}_{5}$. $\mathcal{N}_{2}$ contains the parties
$\textsf{A}_{2}$, and $\textsf{A}_{4}$ and $\textsf{A}_{6})$.
$\mathcal{N}_{3}$ contains the parties $\textsf{A}_{n-2},\textsf{A}_{n-1}$ and
$\textsf{A}_{n}$. All of these subnetworks consist of three EPR states. A
four-partite cyclic subnetwork $\mathcal{N}_{4}$ consists of the parties
$\textsf{A}_{1}$, $\textsf{A}_{2}$, $\textsf{A}_{3}$, and $\textsf{A}_{4}$ who
share an EPR state with adjacent parties. By replacing EPR states with GHZ
states, we can construct another quantum network $\mathcal{N}_{\textsf{B}}$ as
shown in Fig.10(b), where the subnetwork $\mathcal{N}_{i}$ is transformed into
the corresponding subnetwork $\mathcal{N^{\prime}}_{i}$, $i=1,\cdots,4$.

From Lemma 4, it is easy to show that the subnetworks $\mathcal{N}_{i}$ are
inequivalent to the subnetworks $\mathcal{N^{\prime}}_{i}$. This follows that
two networks $\mathcal{N}_{\textsf{A}}$ and $\mathcal{N}_{\textsf{B}}$ are
inequivalent even if they have the same characteristic vector of von Neumann
entropies as
$S(\rho_{\textsf{A}_{i}})=S(\rho_{\textsf{B}_{i}})=(4,4,5,5,2,2,\cdots,3,3,2)$.

Thus, for any quantum network containing a triangle network or four-partite
cyclic subnetwork consisting of EPR states, there is another quantum network
with the same characteristic vector of von Neumann entropies such that they
are inequivalent under local unitary operations. This completes the proof.
$\Box$

Theorem S6 shows the characteristic vector of von Neumann entropies can not be
used for classifying all quantum networks. Fortunately, this problem can be
resolved by using Lemma 4.