# Entanglement polygon inequality in qudit systems

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

###### Abstract

Entanglement is one of important resources for quantum communication tasks.
Most of results are focused on qubit entanglement. Our goal in this work is to
characterize the multipartite high-dimensional entanglement. We firstly derive
an entanglement polygon inequality for the $q$-concurrence, which manifests
the relationship among all the “one-to-group” marginal entanglements in any
multipartite qudit system. This implies lower and upper bounds for the
marginal entanglement of any three-qudit system. We further extend to general
entanglement distribution inequalities for high-dimensional entanglement in
terms of the unified-$(r,s)$ entropy entanglement including Tsallis entropy,
Rényi entropy, and von Neumann entropy entanglement as special cases. These
results provide new insights into characterizing bipartite high-dimensional
entanglement in quantum information processing.

##  I Introduction

Entanglement as one of the most puzzling features in quantum mechanics reveals
the fundamental insights into quantum correlations. This can be quantified by
using special quantities such as the linear entropy for pure entanglement
EOF1996 ; Hill1997 ; Wootters1998 . So far, various entanglement measures have
been proposed for quantifying bipartite entangled systems Rungta2001 ;
Kim2010T ; Kim2010R ; KimBarry2011 ; Khan2019 . The study of entanglement has
inspired great applications in quantum communications Ekert1991 ;
Horodecki2009 , the features of many-body systems Eisert2010 ; Amico2008 , and
the potential limitations for quantum-enhanced technologies Dowling2003 .

The bipartite entanglement measure can be extended for multipartite
entanglement. Interestingly, varied entanglement measures may be recovered
from different bipartition of multipartite systems. Consider a tripartite
entangled state $\rho_{123}$ as an example. There are six different bi-partite
entanglements ${\cal E}_{1|23}$, ${\cal E}_{2|13}$, ${\cal E}_{3|12}$, ${\cal
E}_{1|2}$, ${\cal E}_{1|3}$ and ${\cal E}_{2|3}$. Herein, ${\cal E}$ is an
arbitrary bipartite entanglement measure, ${\cal E}_{i|jk}$ called as “one-to-
group” marginal entanglement denotes the entanglement between a single partite
$i$ and a group of partite $\\{j,k\\}$, and ${\cal E}_{i|j}$ is called “one-
to-one” entanglement between the $i$-th and $j$-th particles. According to the
spirit of the Coffman-Kundu-Wootters (CKW) inequality VCoffman , the
entanglement measure for a three-qubit system shows the following relationship
as

$\displaystyle{\cal E}_{1|23}\geq{\cal E}_{1|2}+{\cal E}_{1|3}.$ (1)

This means the summation of one-to-one entanglements is upper bounded by the
one-to-group marginal entanglement. Unfortunately, the so-called entanglement
monogamy inequality VCoffman ; Terhal2004  is invalid for general entanglement
measures. It might hold for special entangled systems. Especially, Osborne and
Verstraete proved it for an arbitrary $n$-qubit systems TJOsborne  with the
squared concurrence. Similar monogamy inequalities has been proved for
multiple qubit states with different entanglement measures such as the squared
negativity Kim2009N , the squared entanglement of formation (EOF) Oliveira2014
; Bai3 ; Bai2014 , the squared Tsallis-$q$ entropy Luo2016 , the squared
Rényi-$q$ entropy R2015 , and the squared unified-$(q,s)$ entropy Khan2019 .
This is further improved as a class of tight $\alpha$-th power monogamy
relations for multiqubit systems Fei1 ; Luo2015 ; Luo2016 .

Different from the CKW monogamy inequality, there is another polygon
inequality which gives a upper bound for quantum marginal entanglements as
Qian2018 :

$\displaystyle{\cal E}_{1|23}\leq{\cal E}_{2|13}+{\cal E}_{3|12}$ (2)

for any tripartite entangled pure state. As its stated in ref. Walter2013 ,
the collection of eigenvalues of the reduced density matrices of a pure
entanglement form a convex polytope. This results in a local criterion for
witnessing global multi-particle entanglement. The polygon inequalities are
proved to be valid for arbitrary qubit pure states with respect to generic
entanglement measures Qian2018 . Interestingly, high-dimensional entanglement
opens intriguing perspectives in quantum information science Friis2019 ,
quantum communications Cerf2002 ; Sheridan2010 ; Mafu2013 ; Mirhosseini2015 ;
Islam2017 ; Cozzolino2019  or technological advances Erhard2020 . Hence, a
natural problem is to feature high-dimensional entangled systems. Inspired by
recent results Qian2018 ; Walter2013 , we explore the polygon relationship
among all the one-to-group marginal entanglement measures of pure entangled
states. This can be regarded as resource sharing rules in distributed quantum
applications.

The outline of the rest is as follows. In Sec.II, we present an entanglement
polygon inequality in terms of the $q$-concurrence for any pure multipartite
high-dimensional entanglement. This implies a lower bound for the one-to-group
marginal entanglement of three-qudit system. In Sec.III, we establish
entanglement distribution inequalities of multi-qudit entanglement based on
the two-parametric unifed-$(r,s)$ entanglement. In Sec.IV, we construct a set
of multipartite entanglement indicators. These results are further applied for
featuring general high-dimensional entangled states and quantum networks. The
last section concludes this paper.

##  II Entanglement polygon inequalities in terms of the $q$-concurrence

In this section, our goal is to prove the entanglement polygon inequalities
for multipartite high-dimensional entangled systems in terms of the
$q$-concurrence.

###  II.1 The $q$-concurrence

In this subsection, we introduce a bipartite entanglement measure based on the
parameterized entropy. A parameterized entropy function $F_{q}(\rho)$ of a
quantum state $\rho$ on $d$-dimensional Hilbert space is recently defined
Yang2021  as

$\displaystyle F_{q}(\rho)=1-{\rm{Tr}}\rho^{q}$ (3)

for each $q\geq 2$, where $F_{q}(\rho)=0$ for any pure state $\rho$ while its
maximal value is 1 for a completely mixed state $\frac{1}{d}\mathbbm{1}$ with
the identity matrix $\mathbbm{1}$.

For any bipartite pure state $|\phi\rangle_{AB}$ on finite-dimensional Hilbert
space ${\cal H}_{A}\otimes{\cal H}_{B}$, the $q$-concurrence is defined as
Yang2021 :

$\displaystyle{\cal C}_{q}(|\phi\rangle_{AB})=F_{q}(\rho_{A}),$ (4)

where $\rho_{A}={\rm{Tr}}_{B}(|\phi\rangle_{AB}\langle\phi|)$ is the reduced
density matrix of the subsystem $A$, and $F_{q}(\rho_{A})$ is defined in
Eq.(3). For a mixed state $\rho_{AB}$ on ${\cal H}_{A}\otimes{\cal H}_{B}$,
the $q$-concurrence is defined by the convex extension of all the pure
decompositions as

$\displaystyle{\cal
C}_{q}(\rho_{AB})=\inf_{\\{p_{i},|\phi_{i}\rangle\\}}\sum_{i}p_{i}{\cal
C}_{q}(|\phi_{i}\rangle_{AB}),$ (5)

where the infimum is taken over all the possible pure state decompositions of
$\rho_{AB}$, that is,
$\rho_{AB}=\sum_{i}p_{i}|\phi_{i}\rangle_{AB}\langle\phi_{i}|$, $\\{p_{i}\\}$
is a probability distribution with $p_{i}\geq 0$ and $\sum_{i}p_{i}=1$. As its
mentioned in ref. Yang2021 , the $q$-concurrence for a large $q$ may be
applicable for featuring long-range entangled systems beyond the squared
concurrence Hill1997 .

Proposition 1. The $q$-concurrence is an entanglement measure.

It has been proved that the $q$-concurrence is an entanglement monotone Wei .
Actually, according to the result in ref. Vidal2000 , the concavity of the
entropy $F_{q}$ of an entanglement monotone ensures the monotonicity under
local operations and classical communication (LOCC). The monotonicity of mixed
states can be naturally inherited from the monotonicity of pure states via the
convex roof extension. For the later use, we give following Lemma.

###### Lemma 1

Yang2021  For a general bipartite entangled state $\rho_{AB}$ on Hilbert space
${\cal H}_{A}\otimes\mathcal{H}_{B}$, $F_{q}(\rho_{AB})$ satisfies the
inequalities:

$\displaystyle|F_{q}(\rho_{A})-F_{q}(\rho_{B})|\leq F_{q}(\rho_{AB})\leq
F_{q}(\rho_{A})+F_{q}(\rho_{B}),$ (6)

where $\rho_{A}$ and $\rho_{B}$ are reduced density matrices of the subsystem
$A$ and $B$, respectively.

###  II.2 Entanglement polygon inequalities with one-to-group entanglement

Note that an $n$-qudit pure state on Hilbert space $\otimes_{i=1}^{n}{\cal
H}_{i}$ can be generally represented by

$\displaystyle|\psi\rangle_{1\cdots{}n}=\sum^{n}_{j=0}\sum^{d_{j}-1}_{s_{j}=0}\alpha_{s_{1}\cdots
s_{n}}|s_{1}\cdots s_{n}\rangle,$ (7)

where $\dim({\cal H}_{i})=d_{i}$ with $d_{i}\geq 2$ for $i=1,\cdots,n$;
$\alpha_{s_{1}\cdots s_{n}}$ are the coefficients satisfying the normalization
condition of $\sum_{s_{1}\cdots s_{n}}|\alpha_{s_{1}\cdots s_{n}}|^{2}=1$.

Consider a one-to-group entanglement ${\cal E}^{j|\overline{j}}$ of the pure
state (7), where the subscript $j$ refers to the system of $j$, and
$\overline{j}$ denotes all the systems except for the system $j$, and ${\cal
E}$ is an arbitrary bipartite entanglement measure. These bipartite
entanglements are also called as quantum marginal entanglements Walter2013 .
Based on the present $q$-concurrence Yang2021  we derive the relationship
among all the one-to-group marginal entanglement measures for an arbitrary
$n$-qudit system as the following theorem.

###### Theorem 1

For any $n$-qudit pure entangled state $|\psi\rangle$ on Hilbert space
$\otimes_{i=1}^{n}{\cal H}_{i}$, the following inequality holds

$\displaystyle{\cal C}^{j|\overline{j}}_{q}(|\psi\rangle)\leq\sum_{k\neq
j,\forall k}{\cal C}^{k|\overline{k}}_{q}(|\psi\rangle)$ (8)

for any $q\geq 2$.

_Proof._ From the definition (4) we obtain

$\displaystyle{\cal C}^{j|\overline{j}}_{q}(|\psi\rangle)$ $\displaystyle=$
$\displaystyle F_{q}(\rho_{j})$ (9) $\displaystyle=$ $\displaystyle
F_{q}(\rho_{\overline{j}})$ $\displaystyle\leq$ $\displaystyle\sum_{k\neq
j,\forall k}F_{q}(\rho_{k})$ $\displaystyle=$ $\displaystyle\sum_{k\neq
j,\forall k}{\cal C}^{k|\overline{k}}_{q}(|\psi\rangle).$ (10)

Here, the inequality (9) is followed by iteratively using the subadditivity
condition in Eq.(6), and $F_{q}(\rho_{j})$ is the entropy of the reduced
density matrix $\rho_{j}$ of the subsystem $j$. The inequality (10) is
obtained from the definition (4). This has proved the result. $\Box$

For an arbitrary three-qudit system on Hilbert space ${\cal H}_{1}\otimes{\cal
H}_{2}\otimes{\cal H}_{3}$, we have three one-to-other marginal entanglements,
namely, ${\cal C}_{q}^{1|23}$, ${\cal C}_{q}^{2|13}$, ${\cal C}_{q}^{3|12}$.
Theorem 1 implies special relationships among these quantities as follows.

###### Corollary 1

For an arbitrary three-qudit pure entangled state $|\psi\rangle$ on Hilbert
space ${\cal H}_{1}\otimes{\cal H}_{2}\otimes{\cal H}_{3}$, the
$q$-concurrence ${\cal C}_{q}^{i|jk}$ satisfy the following triangle
inequalities as

$\displaystyle|{\cal C}_{q}^{j|ik}(|\psi\rangle)-{\cal
C}_{q}^{k|ij}(|\psi\rangle)|$ $\displaystyle\leq$ $\displaystyle{\cal
C}_{q}^{i|jk}(|\psi\rangle)$ (11) $\displaystyle\leq$ $\displaystyle{\cal
C}_{q}^{j|ik}(|\psi\rangle)+{\cal C}_{q}^{k|ij}(|\psi\rangle).$

The right hand side of the inequality (11) is a direct consequence of Theorem
1. Moreover, from the definition (4) and the triangle inequality in Lemma 1,
we obtain

$\displaystyle{\cal C}^{i|jk}_{q}(|\psi\rangle)$ $\displaystyle=$
$\displaystyle F_{q}(\rho_{jk})$ $\displaystyle\geq$
$\displaystyle|F_{q}(\rho_{j})-F_{q}(\rho_{k})|$ $\displaystyle=$
$\displaystyle|{\cal C}^{j|ik}_{q}(|\psi\rangle)-{\cal
C}^{k|ij}_{q}(|\psi\rangle)|$ (13)

for any $i\not=j,k$. This completes the proof. $\Box$

A simple geometric interpretation for the inequality (11) is that ${\cal
C}_{q}^{j|ik}(|\psi\rangle),{\cal C}_{q}^{k|ij}(|\psi\rangle)$ and ${\cal
C}_{q}^{i|jk}(|\psi\rangle)$ consist of a triangle, that is, the one-to-group
marginal entanglement $q$-concurrences can represent the lengths of the three
edges of a triangle, as shown in Fig.1. In fact, for a generic three-qubit
system, the entanglement polygon inequality for the squared concurrence is
firstly found as Zhu2015 :

$\displaystyle{\cal C}_{2}^{i|jk}(|\psi\rangle)\leq{\cal
C}_{2}^{j|ik}(|\psi\rangle)+{\cal C}_{2}^{k|ij}(|\psi\rangle).$ (14)

Obviously, the inequality (14) includes the result Zhu2015  as a special case
for $q=2$. Moreover, the present result shows new features of long-range
entanglement going beyond qubit systems with the squared concurrence Zhu2015 .

Figure 1: (Color online) Schematic entanglement distribution relationship of
any tripartite entangled system. The length of each side represents
correspondingly to the value of the marginal entanglement ${\cal
C}^{j|ik}_{q},{\cal C}^{i|jk}_{q}$ or ${\cal C}^{k|ij}_{q}$.

###  II.3 Entanglement polygon inequalities with general bipartite
entanglement

In this subsection, we consider the relationship among the $q$-concurrence of
any bipartition of multipartite entanglement. Consider an $m+n$-partite system
in the state $|\psi\rangle$ on Hilbert space $\otimes_{i=1}^{m}{\cal
H}_{A_{i}}\otimes_{j=1}^{n}{\cal H}_{B_{j}}$. Denote ${\bf{A}}=A_{1}\cdots
A_{m}$ and ${\bf{B}}=B_{1}\cdots B_{n}$. Here, $|\psi\rangle^{\bf{A|B}}$ is
defined by $|\psi\rangle^{\bf{A|B}}:=|\psi\rangle^{A_{1}\cdots
A_{m}|B_{1}\cdots B_{n}}$, $A_{i}$’s and $B_{j}$’s are qudit subsystems. We
have the following entanglement polygon inequalities for general bipartite
entanglement.

###### Theorem 2

For any entangled pure entangled state $|\psi\rangle_{\bf{AB}}$ the
$q$-concurrence satisfies the following inequality

$\displaystyle{\cal C}_{q}^{\bf{A|B}}(|\psi\rangle)\leq\sum^{m}_{i=1}{\cal
C}_{q}^{A_{i}|\overline{A}_{i}}(|\psi\rangle),$ (15)

where ${\cal C}_{q}^{\bf{A|B}}$ denotes the $q$-concurrence under the
bipartition $\bf{A|B}$. ${\cal C}_{q}^{A_{i}|\overline{A}_{i}}$ denotes the
marginal entanglement between the subsystem $A_{i}$ and $\overline{A}_{i}$
denotes all the particles except $A_{i}$.

_Proof._ According to the $q$-concurrence (4), we get

$\displaystyle{\cal C}_{q}^{\bf{A|B}}(|\psi\rangle)$ $\displaystyle=$
$\displaystyle F_{q}(\rho_{\bf{A}})$ (16) $\displaystyle\leq$
$\displaystyle\sum^{m}_{i=1}F_{q}(\rho_{A_{i}})$ $\displaystyle=$
$\displaystyle\sum^{m}_{i=1}{\cal
C}_{q}^{A_{i}|\overline{A}_{i}}(|\psi\rangle).$ (17)

Similar to the inequalities (9) and (10), the inequalities (16) and (17) hold
for any $m+n$-qudit pure state $|\psi\rangle_{\bf{AB}}$ on Hilbert space
$\otimes_{i=1}^{i}{\cal H}_{A_{i}}\otimes_{j=1}^{n}\otimes{\cal H}_{B_{j}}$.
This has completed the proof. $\square$

Inspired by ref. Qian2018 , define the distribution of a given amount of total
entanglement, that is, ${\cal C}^{T}_{q}(|\psi\rangle)=\sum^{n}_{j=1}{\cal
C}^{j|\overline{j}}_{q}(|\psi\rangle)$. It is easy to prove that

$\displaystyle{\cal C}^{j|\overline{j}}_{q}(|\psi\rangle)\leq\frac{{\cal
C}^{T}_{q}}{2}(|\psi\rangle).$ (18)

By regarding $|\psi\rangle$ as an $n$-partite entanglement shared by $n$
parties $A_{1},\cdots,A_{n}$, the present inequality (20) implies that for any
single party $A_{j}$, its information (quantified by ${\cal
C}^{j|\overline{j}}_{q}(|\psi\rangle)$) obtained from its reduced matrix
$\rho_{A_{j}}$ is no more than half of the total. Theorem 1 may provide an
appropriate proposal in various quantum tasks. The more intriguing part is the
present result shows new features of marginal entanglement distribution for
high-dimensional quantum systems beyond the qubit entanglement Qian2018 . It
also provides different inequalities beyond the linear polygon inequality
Walter2013 .

Note Theorem 2 is not a special case of Theorem 1, they are not inequivalent.
Take a 4-partite system as an example. In Theorem 2, we get new upper bounds
of the marginal entanglement of $q$-concurrence on $4$-partite system under
the partition 12 and 34, that is,

$\displaystyle{\cal C}^{12|34}_{q}\leq{\cal C}_{q}^{1|234}+{\cal
C}_{q}^{2|134}.$ (19)

In Theorem 1, we provide the restrictions among all “one-to-group”
entanglements between a single qudit $j\in\\{1,2,3,4\\}$ and the remaining
ones in an arbitrary $n$-qudit system, for example,

$\displaystyle{\cal C}^{1|234}_{q}\leq{\cal C}_{q}^{2|134}+{\cal
C}_{q}^{3|124}+{\cal C}_{q}^{4|123}.$ (20)

Obviously, Theorem 2 describes a global entangled distribution between the
partial set $ij$ and the partial set $\\{k,l\\}$ while Theorem 1 describes the
intrinsic entanglement between a single partite $i$ and the partial set
$\\{j,k,l\\}$. That is, the new partition of ${\bf{A}}=A_{1}A_{2}\cdots A_{m}$
and ${\bf{B}}=B_{1}B_{2}\cdots B_{n}$ in Theorem 2 cannot be reformed into the
partition of one qudit and all the other systems in Theorem 1.

##  III Entanglement polygon inequality in terms of unified-$(r,s)$ entropy
entanglement

In Sec.II, we have proved the polygon inequality for pure multipartite high-
dimensional entanglement by using the entanglement measure of $q$-concurrence.
In this section, we extend the result with the unified-$(r,s)$ entropy
entanglement.

For a given state $\rho$ on Hilbert space $\mathcal{H}$, the unified-$(r,s)$
entropy Rathie1991 ; Hu2006 ; Rastegin2011  is defined as

$\displaystyle S_{r,s}(\rho)=\frac{1}{(1-r)s}[({\rm{Tr}}\rho^{r})^{s}-1],$
(21)

where $r,s\geq 0$ and $r\neq 1$, $s\neq 0$. The unified-$(r,s)$ entropy
converges to the Rényi entropy Renyi1961 ; Hu2006 ; Rastegin2011 ,

$\displaystyle\lim_{s\to
0}S_{r,s}(\rho)=\frac{1}{1-r}\log_{2}{\rm{Tr}}(\rho^{r}):=R_{r}(\rho).$ (22)

The Rényi entropy is not additive Linden2013 , but weak sub-additive Dam2002 .
Especially, the Rényi entropy satisfies the following inequality

$\displaystyle R_{r}(\rho_{A})-R_{0}(\rho_{B})\leq R_{r}(\rho_{AB})\leq
R_{r}(\rho_{A})+R_{0}(\rho_{B})$ (23)

for any $r>0$ and $r\neq 1$, where $\rho_{A}$ and $\rho_{B}$ are reduced
density matrices of the subsystem $A$ and $B$, respectively. Moreover, the
unified $(r,s)$-entropy tends to the Tsallis entropy Tsallis1988 ; Hu2006 ;
Rastegin2011  as

$\displaystyle\lim_{s\to
1}S_{r,s}(\rho)=\frac{1}{1-r}({\rm{Tr}}(\rho^{r})-1):=T_{r}(\rho).$ (24)

Here, the Tsallis entropy satisfies the subadditivity and triangle inequality
as Audenaert2007 :

$\displaystyle|T_{r}(\rho_{A})-T_{r}(\rho_{B})|\leq T_{r}(\rho_{AB})\leq
T_{r}(\rho_{A})+T_{r}(\rho_{B})$ (25)

for any $r>1$.

For the case of $r\to 1$, the unified $(r,s)$-entropy converges to the von
Neumann entropy Nielsen ; Hu2006 ; Rastegin2011  as

$\displaystyle\lim_{r\to 1}S_{r,s}(\rho)={\rm{Tr}}\rho\log_{2}\rho:=S(\rho).$
(26)

Note that the von Neumann entropy satisfies the following Araki-Lieb
inequality AL  as

$\displaystyle|S(\rho_{A})-S(\rho_{B})|\leq S(\rho_{AB})\leq
S(\rho_{A})+S(\rho_{B}),$ (27)

For a given bipartite pure state $|\phi\rangle_{AB}$, the unified-$(r,s)$
entropy entanglement KimBarry2011  is given by

$\displaystyle{\cal U}_{r,s}(|\phi\rangle_{AB})=S_{r,s}(\rho_{A})$ (28)

for each $r,s\geq 0$. For a given mixed state $\rho_{AB}$, its Unified-$(r,s)$
entropy entanglement via the convex-roof extension is defined by

$\displaystyle{\cal
U}_{r,s}(\rho_{AB})=\inf_{\\{p_{i},|\phi_{i}\rangle\\}}\sum_{i}p_{i}{\cal
U}_{r,s}(|\phi_{i}\rangle_{AB}),$ (29)

where the infimum is over all the possible pure state decompositions of
$\rho_{AB}$.

For any $r\geq 1$ and $s\geq 0$, the unified $(r,s)$-entropy satisfies the
subadditivity Hu2006 ; Rastegin2011  as

$\displaystyle S_{r,s}(\rho_{AB})\leq S_{r,s}(\rho_{A})+S_{r,s}(\rho_{B})$
(30)

and the Araki-Lieb inequality

$\displaystyle|S_{r,s}(\rho_{A})-S_{r,s}(\rho_{B})|\leq S_{r,s}(\rho_{AB}).$
(31)

Both inequalities are necessary to prove entanglement polygon inequalities.

###### Theorem 3

For any pure $n$-partite entanglement $|\psi\rangle$ on Hilbert space ${\cal
H}_{A_{1}}\otimes\cdots\otimes{\cal H}_{A_{n}}$, the following inequality
holds

$\displaystyle{\cal U}_{r,s}^{j|\overline{j}}(|\psi\rangle)\leq\sum^{n}_{k\neq
j}{\cal U}_{r,s}^{k|\overline{k}}(|\psi\rangle),$ (32)

where ${\cal U}_{r,s}^{j|\overline{j}}(|\psi\rangle)$ represents the
unified-$(r,s)$ entropy entanglement under the bipartition $j|\overline{j}$
for each $r\geq 1$ and $s\geq 0$, and $\overline{j},\overline{k}$ is defined
in Theorem 1.

_Proof._ From the definition (28), for any $r\geq 1$ and $s\geq 0$ we have

$\displaystyle{\cal U}_{r,s}^{j|\overline{j}}(|\psi\rangle)$ $\displaystyle=$
$\displaystyle S_{r,s}(\rho_{j})$ (33) $\displaystyle=$ $\displaystyle
S_{r,s}(\rho_{\overline{j}})$ $\displaystyle\leq$ $\displaystyle\sum_{k\neq
j,\forall k}S_{r,s}(\rho_{k})$ $\displaystyle=$ $\displaystyle\sum_{k\neq
j,\forall k}{\cal U}_{r,s}^{k|\overline{k}}(|\psi\rangle),$ (34)

where $S_{r,s}(\rho_{j})$ is the unified-$(r,s)$ entropy for the reduced
density matrix $\rho_{j}$. Here, the inequality (33) is obtained by
iteratively using the subadditivity of the entropy $S_{r,s}(\rho)$ in the
inequality (30). The inequality (34) is followed from the definition (28).
This has completed the proof. $\Box$

###### Corollary 2

For any pure tripartite entanglement $|\psi\rangle$ on Hilbert space ${\cal
H}_{1}\otimes{\cal H}_{2}\otimes{\cal H}_{3}$, we obtain

$\displaystyle|{\cal U}_{r,s}^{j|ik}(|\psi\rangle)-{\cal
U}_{r,s}^{k|ij}(|\psi\rangle)|$ $\displaystyle\leq$ $\displaystyle{\cal
U}_{r,s}^{i|jk}(|\psi\rangle)$ (35) $\displaystyle\leq$ $\displaystyle{\cal
U}_{r,s}^{j|ik}(|\psi\rangle)+{\cal U}_{r,s}^{k|ij}(|\psi\rangle)$

for any $r\geq 1$ and $s\geq 0$, where ${\cal U}_{r,s}^{i|jk}(|\psi\rangle)$
represents the unified-$(r,s)$ entropy entanglement with respect to the
bipartition $i|jk$, $i\neq j\neq k\in\\{1,2,3\\}$.

_Proof._ The right side of the inequality (35) is a directive consequence of
Theorem 3. Similar to Corollary 1, according to the definition of the
unified-$(r,s)$ entropy entanglement (28) and the triangle inequality (31), we
obtain

$\displaystyle{\cal U}_{r,s}^{i|jk}(|\psi\rangle)$ $\displaystyle=$
$\displaystyle S_{r,s}(\rho_{jk})$ $\displaystyle\geq$
$\displaystyle|S_{r,s}(\rho_{j})-S_{r,s}(\rho_{k})|$ $\displaystyle=$
$\displaystyle|{\cal U}_{r,s}^{j|ik}(|\psi\rangle)-{\cal
U}_{r,s}^{k|ij}(|\psi\rangle)|$ (37)

for any $i\not=j,k$. This has completed the proof. $\Box$

###### Theorem 4

For any pure $m+n$-partite entanglement $|\psi\rangle_{\bf{AB}}$ on Hilbert
space $\otimes_{i=1}^{m}{\cal H}_{A_{i}}\otimes_{j=1}^{n}{\cal H}_{B_{j}}$,
the unified-$(r,s)$ entropy entanglement satisfies the following inequality

$\displaystyle{\cal U}_{r,s}^{\bf{A|B}}(|\psi\rangle)\leq\sum^{m}_{i=1}{\cal
U}_{r,s}^{A_{i}|\overline{A}_{i}}(|\psi\rangle)$ (38)

for $r\geq 1$ and $s\geq 0$, where ${\cal U}_{r,s}^{\bf{A|B}}(|\psi\rangle)$
denotes the unified-$(r,s)$ entropy entanglement under the bipartition
$\bf{A|B}$, and ${\cal U}_{r,s}^{A_{i}|\overline{A}_{i}}(|\psi\rangle)$ is the
marginal entanglement of the subsystem ${A_{i}}$.

_Proof._ From the definition of the unified-$(r,s)$ entropy entanglement (28),
we have

$\displaystyle{\cal U}_{r,s}^{\bf{A|B}}(|\psi\rangle)$ $\displaystyle=$
$\displaystyle S_{r,s}(\rho_{\bf{A}})$ (39) $\displaystyle\leq$
$\displaystyle\sum^{m}_{i=1}S_{r,s}(\rho_{A_{i}})$ $\displaystyle=$
$\displaystyle\sum^{m}_{i=1}{\cal
U}_{r,s}^{A_{i}|\overline{A}_{i}}(|\psi\rangle),$ (40)

where the inequality (39) is from the additivity of the unified-$(r,s)$
entropy (30). The equality (40) is obtained from the definition of the
unified-$(r,s)$ entropy entanglement in Eq. (28). This has completed the
proof. $\square$

From Theorem 3 a large class of polygon inequalities holds for multi-qudit
systems in terms of the unified-$(r,s)$ entropy entanglement. That is, our
result can reduce to every case of multi-qudit polygon inequalities such as
Rényi-$\alpha$ and Tsallis-$q$ entanglement and EOF for selective choices of
$r$ and $s$. This reveals a generic frame for entanglement polygon equality
concerning all well-known bipartite entanglement measures. Similar to Theorem
2. Theorem 4 also reveals the different entanglement distribution features
going beyond Theorem 3.

Since the unified-$(r,s)$ entropy converges to the Rényi entropy as the
limiting case of $s$ tends to 0, we have

$\displaystyle\lim_{s\to 0}{\cal U}_{r,s}(\rho_{AB})={\cal R}_{r}(\rho_{AB}),$
(41)

where ${\cal R}_{r}(\rho_{AB})$ is the Rényi entropy entanglement of
$\rho_{AB}$ Kim(2010)R. From Corollary 2 and inequality (23), we get the
following corollary.

###### Corollary 3

For any pure tripartite entanglement $|\psi\rangle$ on Hilbert space ${\cal
H}_{1}\otimes{\cal H}_{2}\otimes{\cal H}_{3}$ we have

$\displaystyle|{\cal R}_{r}^{j|ik}(|\psi\rangle)-{\cal
R}_{0}^{k|ij}(|\psi\rangle)|$ $\displaystyle\leq$ $\displaystyle{\cal
R}_{r}^{i|jk}(|\psi\rangle)$ (42) $\displaystyle\leq$ $\displaystyle{\cal
R}_{r}^{j|ik}(|\psi\rangle)+{\cal R}_{0}^{k|ij}(|\psi\rangle)$

for any $r\geq 0$ with $r\neq 1$, where ${\cal R}_{r}^{j|\overline{j}}$
denotes the Rényi entropy entanglement under the bipartition $j|\overline{j}$.

Moreover, the unified-$(r,s)$ entropy KimBarry2011  converges to the Tsallis
entropy as

$\displaystyle\lim_{s\to 1}{\cal U}_{r,s}(\rho_{AB})={\cal T}_{r}(\rho_{AB}),$
(43)

where ${\cal T}_{r}(\rho_{AB})$ is the Tsallis entropy entanglement of
$\rho_{AB}$ Kim2010T . From Theorem 3, we get the following corollary.

###### Corollary 4

For any pure $n$-partite entanglement $|\psi\rangle$ on Hilbert space
$\otimes_{i=1}^{n}{\cal H}_{i}$, we obtain

$\displaystyle{\cal T}_{r}^{j|\overline{j}}(|\psi\rangle)\leq\sum_{k\neq
j,\forall k}{\cal T}_{r}^{k|\overline{k}}(|\psi\rangle)$ (44)

for any $r\geq 1$, where ${\cal T}_{r}^{j|\overline{j}}$ denotes the Tsallis
entropy entanglement under the bipartition $j|\overline{j}$. Especially, for
any tripartite entanglement $|\psi\rangle$ on Hilbert space $\in{\cal
H}_{1}\otimes{\cal H}_{2}\otimes{\cal H}_{3}$ we obtain

$\displaystyle|{\cal T}_{r}^{j|ik}(|\psi\rangle)-{\cal
T}_{r}^{k|ij}(|\psi\rangle)|$ $\displaystyle\leq$ $\displaystyle{\cal
T}_{r}^{i|jk}(|\psi\rangle)$ (45) $\displaystyle\leq$ $\displaystyle{\cal
T}_{r}^{j|ik}(|\psi\rangle)+{\cal T}_{r}^{k|ij}(|\psi\rangle)$

for each $r$ with $r>1$.

As $r$ tends to 1 we have

$\displaystyle\lim_{r\to 1}{\cal U}_{r,s}(\rho_{AB})={\cal E}_{f}(\rho_{AB}),$
(46)

where ${\cal E}_{f}(\rho_{AB})$ is the EoF of $\rho_{AB}$ EOF1996 ; Hill1997 ;
Wootters1998 . From Theorem 3, we get the following corollary.

###### Corollary 5

For any pure $n$-partite entanglement $|\psi\rangle$ on Hilbert space ${\cal
H}_{1}\otimes\cdots\otimes{\cal H}_{n}$ we obtain

$\displaystyle{\cal E}_{f}^{j|\overline{j}}(|\psi\rangle)\leq\sum_{k\neq
j,\forall k}{\cal E}_{f}^{k|\overline{k}}(|\psi\rangle).$ (47)

Here, ${\cal E}_{f}^{j|\overline{j}}(|\psi\rangle)$ represent the EOF under
the bipartition $j|\overline{j}$. Especially, for any tripartite entanglement
$|\psi\rangle$ on Hilbert space ${\cal H}_{1}\otimes{\cal H}_{2}\otimes{\cal
H}_{3}$ we obtain

$\displaystyle|{\cal E}_{f}^{j|ik}(|\psi\rangle)-{\cal
E}_{f}^{k|ij}(|\psi\rangle)|$ $\displaystyle\leq$ $\displaystyle{\cal
E}_{f}^{i|jk}(|\psi\rangle)$ (48) $\displaystyle\leq$ $\displaystyle{\cal
E}_{f}^{j|ik}(|\psi\rangle)+{\cal E}_{f}^{k|ij}(|\psi\rangle).$

To sum up, we find that general bipartite entanglement measures in terms of
entanglement entropy satisfy polygon inequalities for high-dimensional
systems. For arbitrary qubit pure states, Qian et al. Qian2018  prove a set of
the polygon inequalities with respect to entanglement measures including EOF
EOF1996 ; Wootters1998 , concurrence Rungta2001 , negativity Vidal2002 .
However, we still do not know whether the entanglement polygon inequality is
valid for higher-dimensional systems apart from qubit systems in terms of the
concurrence or negativity (see Table I), although their monogamy properties
for certain special high-dimensional systems have been investigated Kim2009N ;
Ou2007 .

Table 1: A comparison of the entanglement polygon inequalities (EPI) in terms of various entanglement measures. Denote $\mathcal{C}_{q}$, $\mathcal{E}_{f}$, ${\cal T}_{r}$, ${\cal R}_{r}$, ${\cal U}_{r,s}$, $\mathcal{C}$, and $\mathcal{N}$ as the $q$-concurrence, entanglement of formation, Tsallis entropy entanglement and Rényi entropy entanglement, unified-$(r,s)$ entropy entanglement, the concurrence, negativity, respectively. Measure Property | EPI | System | Ref.
---|---|---|---
$\mathcal{C}_{q}$ | $\surd$ | $d^{\otimes n}(d\geq 2)$ | Ours
$\mathcal{E}_{f}$ | $\surd$ | $2^{\otimes n}$ | Ref.Qian2018
$\mathcal{E}_{f}$ | $\surd$ | $d^{\otimes n}(d\geq 2)$ | Ours
${\cal T}_{r}$ |  $\surd$($q\geq 1$) | $d^{\otimes n}(d\geq 2)$ | Ours
${\cal R}_{r}$ |  $\surd$($r\geq 0$) | $d^{\otimes 3}(d\geq 2)$ | Ours
${\cal U}_{r,s}$ |  $\surd$($r\geq 1,s\geq 0$) | $d^{\otimes n}(d\geq 2)$ | Ours
$\mathcal{C}$ | $\surd$ | $2^{\otimes n}$ | Ref.Qian2018
$\mathcal{C}$ | $?$ | $d^{\otimes n}(d\geq 2)$ | Ref.Qian2018
$\mathcal{N}$ | $\surd$ | $2^{\otimes n}$ | Ref.Qian2018
$\mathcal{N}$ | ? | $d^{\otimes n}(d\geq 2)$ | Ref.Qian2018

##  IV  A new type of entanglement indicator

For an $n$-partite entanglement $|\psi\rangle$ on Hilbert space
$\otimes_{i=1}^{n}\mathcal{H}_{i}$, define a class of the multipartite
marginal entanglement indicator as

$\displaystyle\tau_{\cal E}(|\psi\rangle)=\min_{j}\\{\sum^{n}_{k\neq
j,k=1}{\cal E}^{k|\overline{k}}(|\psi\rangle)-{\cal
E}^{j|\overline{j}}(|\psi\rangle)\\},$ (49)

where ${\cal E}$ is an arbitrary bipartite entanglement measure, and
$\overline{j}$, $\overline{k}$ are given in Theorem 1.

Similarly, for an $n+m$-partite entanglement $|\psi\rangle$ on Hilbert space
$\otimes_{i=1}^{m}\mathcal{H}_{A_{i}}\otimes_{j=1}^{n}\mathcal{H}_{B_{j}}$

$\displaystyle\hat{\tau}_{\cal
E}(|\psi\rangle_{\bf{AB}})=\min_{i}\\{\sum^{n+m}_{i=1}{\cal
E}^{A_{i}|\overline{A}_{i}}(|\psi\rangle)-{\cal
E}^{\bf{A|B}}(|\psi\rangle)\\},$ (50)

where $\bf{A}$, $\bf{B}$, $A_{i}$ and $\overline{A}_{i}$ are defined in
Theorem 2.

Note that the EOF is defined as the von Neumann entropy of subsystem, and only
one know the von Neumann entropy satisfies the additivity Nielsen :

$\displaystyle S(\rho)=S(\varrho_{A})+S(\varrho_{B})$ (51)

for any product state $\rho=\varrho_{A}\otimes\varrho_{B}$. $\varrho_{A}$ and
$\varrho_{B}$ is the density matrix of the subsystem $A$, $B$, respectively.

Inspired by the additivity (51) and the idea in Ref.Bai3 , we obtain the
following result.

###### Theorem 5

For an arbitrary tripartite pure state $|\psi\rangle$ on Hilbert space
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$, the
multipartite entanglement indicator of EOF is zero if and only if
$|\psi\rangle$ is product state, that is, $\tau_{\cal E}(|\psi\rangle)=0$ if
and only if $|\varphi\rangle_{ijk}=|\phi\rangle\otimes|\phi^{\prime}\rangle$
for some pure states $|\phi\rangle$ and $|\phi^{\prime}\rangle$, where $i\neq
j\neq k\in\\{A,B,C\\}$.

_Proof_. We first prove the necessity. Assume that the quantum state
$|\psi\rangle$ on Hilbert space
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{C}$ is bipartite
product state, that is, $|\psi\rangle$ has the following forms:

$\displaystyle|\varphi\rangle_{ijk}$ $\displaystyle=$
$\displaystyle|\phi\rangle_{i}\otimes|\phi\rangle_{jk},$ (52)
$\displaystyle|\varphi\rangle_{ijk}$ $\displaystyle=$
$\displaystyle|\phi\rangle_{i}\otimes|\phi\rangle_{j}\otimes|\phi\rangle_{k},$
(53)

where $|\phi\rangle_{i(j,k)}$ and $|\phi\rangle_{jk}$ are different pure
states, and $i\neq j\neq k\in\\{A,B,C\\}$,

For the case (52), from the definition of EoF it follows that ${\cal
E}^{i|jk}_{f}=0$, ${\cal E}^{j|ik}_{f}={\cal E}^{jk}_{f}$, and ${\cal
E}^{k|ij}_{f}={\cal E}^{jk}_{f}$. This implies that

$\displaystyle{\cal E}^{i|jk}_{f}={\cal E}^{j|ik}_{f}+{\cal E}^{k|ij}_{f},$
(54)

where ${\cal E}^{i|jk}_{f}$ denotes the bipartite entanglement under the
bipartition $i$ and $jk$, ${\cal E}^{jk}_{f}$ is the entanglement between
subsystems $i$ and $j$. This leads to $\tau_{{\cal E}_{f}}=0$. Similar result
holds for the case (53).

Now, we show the sufficiency. When the indicator $\tau_{{\cal E}_{f}}$ is
zero, we get the equality (54) for all $\\{i,j\\}\in\\{A,B,C\\}$. According to
the definition of EoF Wootters1998 , for the left side of Eq.(54) we have

$\displaystyle{\cal E}^{i|jk}_{f}(|\varphi)$ $\displaystyle=$ $\displaystyle
S(\varrho_{i})$ (55) $\displaystyle=$ $\displaystyle S(\varrho_{jk})$
$\displaystyle=$ $\displaystyle S(\varrho_{j})+S(\varrho_{k}),$ (56)

where $\varrho_{i}$ is the reduced density matrix $\rho_{i}$ of the subsystem
$i$. Here, the equality (55) is derived from the symmetry of von Neumann
entropy Nielsen . The equality (56) holds if and only if
$\varrho_{jk}=\varrho_{j}\otimes\varrho_{k}$, that is, $|\varphi\rangle_{ijk}$
is biseparable state. This has completed the proof. $\Box$

_Example 1_. Consider a generalized three-qudit GHZ state

$\displaystyle|GHZ^{g}\rangle_{ABC}$ $\displaystyle=$
$\displaystyle\sin\theta\cos\phi|000\rangle+\sin\theta\sin\phi|111\rangle$
(57) $\displaystyle+\cos\theta|222\rangle$

with $\theta\in[0,\pi]$ and $\phi\in[0,2\pi]$. According to the definition of
EOF Wootters1998 , the values of three different bi-partite entanglements
${\cal E}^{i|jk}_{f}$s for all $i\neq j\neq k\in\\{A,B,C\\}$ can be calculated
as

$\displaystyle{\cal
E}^{i|jk}_{f}=-\sum^{3}_{i=1}\lambda_{i}\log_{2}\lambda_{i},$ (58)

where $\lambda_{1}=\sin^{2}\theta\cos^{2}\phi$,
$\lambda_{2}=\sin^{2}\theta\sin^{2}\phi$, and $\lambda_{3}=\cos^{2}\theta$.
From Eq.(49), we have

$\displaystyle\tau_{{\cal E}_{f}}(|GHZ^{g}\rangle)={\cal E}^{j|ik}_{f}+{\cal
E}^{k|ij}-{\cal E}^{i|jk}_{f}.$ (59)

Combing Eqs.(58) and (59), as shown in Fig.2, we state that $\tau_{{\cal
E}_{f}}(|GHZ^{g}\rangle)$ is nonnegative for $\theta\in[0,\pi]$ and
$\phi\in[0,2\pi]$. $\tau_{{\cal E}_{f}}(|GHZ^{g}\rangle)=0$ when
$|GHZ^{g}\rangle$ is separable, that is, corresponding to two cases of (i)
$\theta=\pi$; (ii) $\theta=\frac{\pi}{2}$,
$\phi=\frac{\pi}{2},\pi,\frac{3\pi}{2}$, or $2\pi$. For example, when
$\theta=\frac{\pi}{2}$, $\phi=\frac{\pi}{2}$, or $\frac{3\pi}{2}$, the related
state $|GHZ^{g}\rangle$ is changed into a separable state $|111\rangle$.

Figure 2: (Color online) The indicator $\tau_{{\cal E}_{f}}$ for a generalized
three-qutrit GHZ state in Example 1.

According to Theorem 5, the indicator is nonzero if a three-qudit pure state
contains genuine tripartite entanglement. Howbeit, whether or not there exist
similar results for the aforementioned bipartite entanglement measures is
still unknown.

Next, we present some examples to show Theorems 1-4.

_Example 2_. Consider a three-qutrit W-class state Wstate ,

$\displaystyle|W_{c}\rangle_{123}=\frac{1}{\sqrt{6}}\sum^{2}_{i=1}(|i00\rangle+|0i0\rangle+|00i\rangle).$
(60)

According to the $q$-concurrence (4), we get

$\displaystyle{\cal
C}_{q}^{i|jk}(|W_{c}\rangle)=1-\frac{2^{q}}{3^{q}}-\frac{2}{6^{q}}$ (61)

for each $i\neq j\neq k\in\\{1,2,3\\}$. Thus, from Eq.(49) and the symmetry of
$|W_{c}\rangle$, we have

$\displaystyle\tau_{{\cal C}_{q}}(|W_{c}\rangle)={\cal
C}_{q}^{i|jk}(|W_{c}\rangle)=-1+\frac{2^{q}}{3^{q}}+\frac{2}{6^{q}}.$ (62)

Similarly, from Eqs.(28), (49), and the symmetry of $|W_{c}\rangle$ we obtain

$\displaystyle\tau_{{\cal U}_{r,s}}(|W_{c}\rangle)$ $\displaystyle=$
$\displaystyle{\cal U}_{r,s}^{i|jk}(|W_{c}\rangle)$ (63) $\displaystyle=$
$\displaystyle\frac{1}{(1-r)s6^{r}}(4^{r}+2)^{s}-\frac{1}{(1-r)s}.$

Note $\tau_{{\cal C}_{q}}(|W_{c}\rangle)$ and $\tau_{{\cal
U}_{r,s}}(|W_{c}\rangle)$ are always positive. These implies Theorems 1 and 3
hold for a three-qutrit W-class state.

_Example 3_. Consider an $m$-particle $d$-dimensional GHZ state GHZ :

$\displaystyle|GHZ\rangle_{12\cdots m}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}|j\rangle^{\otimes m}.$ (64)

According to Eqs.(4) and (28), from the symmetry of $|GHZ\rangle_{12\cdots m}$
we get

$\displaystyle{\cal C}^{j|\overline{j}}_{q}(|GHZ\rangle)=1-\frac{1}{d^{q-1}}$
(65) $\displaystyle{\cal
U}^{j|\overline{j}}_{r,s}(|GHZ\rangle)=\frac{1-d^{rs-s}}{(1-r)sd^{rs-s}}$ (66)

for any partite $j$ with $1\leq j\leq m$. These lead to the following
equalities:

$\displaystyle\tau_{{\cal C}_{q}}(|GHZ\rangle)=(m-2){\cal
C}^{j|\overline{j}}_{q}(|GHZ\rangle),$ (67) $\displaystyle\tau_{{\cal
U}_{r,s}}(|GHZ\rangle)=(m-2){\cal U}^{j|\overline{j}}_{r,s}(|GHZ\rangle),$
(68)

where ${\cal C}^{j|\overline{j}}_{q}$ is defined in Eq.(65) and ${\cal
U}^{j|\overline{j}}_{r,s}$ is defined in Eq.(66). It follows that $\tau_{{\cal
C}_{q}}(|GHZ\rangle)$ and $\tau_{{\cal U}_{r,s}}(|GHZ\rangle)$ are nonnegative
from the nonnegativity of ${\cal C}^{j|\overline{j}}_{q}(|GHZ\rangle)$ and
${\cal U}^{j|\overline{j}}_{r,s}(|GHZ\rangle)$.

_Example 4_. Consider an $n$-partite complete graph-like network state
$|G\rangle_{12\cdots n}$ Luo2021 , that is, any pair of two parties shares an
EPR state, each party shares $n-1$ EPR states with others. There are
$n(n-1)/2$ number of EPR states. Thus, from Eqs.(4) and (28) we have

$\displaystyle{\cal
C}^{j|\overline{j}}_{q}(|G\rangle)=\frac{(n-1)(2^{q-1}-1)}{2^{q-1}}$ (69)
$\displaystyle{\cal
U}^{j|\overline{j}}_{r,s}(|G\rangle)=\frac{(n-1)(1-2^{(r-1)s})}{2^{(r-1)s}(1-r)s}$
(70)

for each partite $j$ with $1\leq j\leq n$. Similar to Eqs.(67) and (68), from
the symmetry of the complete graph-like network state, it is easy to show that
Theorems 1 and 3 hold for the $n$-partite complete graph-like network state
with respect to the $q$-concurrence and unified-$(r,s)$ entropy entanglement.

_Example 5_. Consider an $n$-partite entangled quantum network
$\mathcal{N}_{q}(\cal{A},\xi)$ Luo2021  consisting of EPR states and
$m$-particle $d$-dimensional GHZ states, where $\cal{A}$ denotes the 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
$\varrho^{1}_{ij},\cdots,\varrho^{s}_{ij}$,
$\sigma^{1}_{ij},\cdots,\sigma^{t}_{ij}$, and
$\delta^{1}_{ij},\cdots,\delta^{k}_{ij}$, where $\varrho$ denotes the density
matrix of the EPR state EPR :
$|\phi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$, and $\sigma$ is the
density matrix of $m$-particle $d$-dimensional GHZ state GHZ :
$\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}|j\rangle^{\otimes m}$ with any integer
$m\geq 3$, and $\delta=\frac{1}{d}\sum_{j=0}^{d-1}(|jj\rangle\langle jj|)$ is
the reduced density matrix of any two subsystems obtained by tracing out other
subsystems in an $m$-particle $d$-dimensional GHZ state. Here, any two parties
$\textsf{A}_{i}$ and $\textsf{A}_{j}$ share the quantum state $\xi_{ij}$ given
by

$\displaystyle\xi_{ij}=\mathop{\otimes}^{s_{j}}_{s=1}\varrho^{s}_{ij}\mathop{\otimes}^{t_{j}}_{t=1}\sigma^{t}_{ij}\mathop{\otimes}^{k_{j}}_{k=1}\delta^{k}_{ij},$
(71)

where $s_{j}$, $t_{j}$, and $k_{j}$ denote the numbers of $\varrho$, $\sigma$,
and $\delta$, respectively. The total state
$\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}$ of the $n$-partite system is given
by

$\displaystyle\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}=\mathop{\otimes}^{n}_{\begin{subarray}{c}i,j=1,\\\
i\neq j\end{subarray}}\xi_{ij}.$ (72)

Here, the total system may be regarded as an $n$-partite entanglement in high-
dimensional Hilbert space. From Lemma 1, the local $F_{q}$ entropy of each
party satisfies the subadditivity Yang2021 . Hence, from Eq.(4) we get

$\displaystyle{\cal
C}^{\textsf{A}_{j}|\overline{\textsf{A}}_{j}}_{q}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})$
$\displaystyle=$ $\displaystyle F_{q}(\rho_{\overline{\textsf{A}}_{j}})$ (73)
$\displaystyle\leq$ $\displaystyle\sum_{k\neq j,\forall k}F_{q}(\rho_{k})$
$\displaystyle=$ $\displaystyle\sum_{k\neq j,\forall k}{\cal
C}^{k|\overline{k}}_{q}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}).$

Analogously, from the unified $(r,s)$-entropy entanglement (28) and the
additivity (30) we obtain

$\displaystyle{\cal
U}^{j|\overline{j}}_{r,s}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}})\leq\sum_{k\neq
j}{\cal U}^{k|\overline{k}}_{r,s}(\rho_{\textsf{A}_{1}\cdots\textsf{A}_{n}}).$
(74)

These implies Theorems 1 and 3 hold for entangled quantum networks.
Specifically, we consider the following chain network for showing the detail.

_Example 6_. 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
$\otimes_{i=1}^{4}\mathcal{H}_{i}$, as illustrated in Fig.3, where
$\mathcal{H}_{1}$ is an $8$-dimensional space with basis
$\\{|0\rangle,\cdots,|7\rangle\\}$ while $\mathcal{H}_{i}$ ($i\geq 2$) is a
$2$-dimensional space with basis $\\{|0\rangle,|1\rangle\\}$.

Figure 3: (Color online). A $4$-partite star network state consisting of three
EPR states, where the subsystem 1 is regarded as an $8$-dimensional Hilbert
space with local tensor decomposition.

Thus, the total state is given by

$\displaystyle|\varphi\rangle_{1234}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{2}}(|0000\rangle+|1001\rangle+|2010\rangle+|3011\rangle$
(75)
$\displaystyle+|4100\rangle+|5101\rangle+|6110\rangle+|7111\rangle)_{1234}.$

The reduced state of the subsystems $1$ and $2$ is given by

$\displaystyle\rho_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{8}(|00\rangle\langle 00|+|41\rangle\langle
41|+|00\rangle\langle 41|+|41\rangle\langle 00|$ (76)
$\displaystyle+|10\rangle\langle 10|+|51\rangle\langle 51|+|10\rangle\langle
51|+|51\rangle\langle 10|$ $\displaystyle+|20\rangle\langle
20|+|61\rangle\langle 61|+|20\rangle\langle 61|+|61\rangle\langle 20|$
$\displaystyle+|30\rangle\langle 30|+|71\rangle\langle 71|+|30\rangle\langle
71|+|71\rangle\langle 30|$

which has four nonnative spectrum $\frac{1}{4}$. Therefore, according to the
$q$-concurrence in Eq.(4) we get

$\displaystyle{\cal
C}_{q}^{12|34}(|\varphi\rangle)=\frac{4^{q-1}-1}{4^{q-1}},$ (77)
$\displaystyle{\cal
C}_{q}^{1|234}(|\varphi\rangle)=\frac{8^{q-1}-1}{8^{q-1}},$ (78)
$\displaystyle{\cal
C}_{q}^{2|134}(|\varphi\rangle)=\frac{2^{q-1}-1}{2^{q-1}}.$ (79)

Besides, we have

$\displaystyle{\cal
C}_{q}^{34|12}(|\varphi\rangle)=\frac{4^{q-1}-1}{4^{q-1}},$ (80)
$\displaystyle{\cal C}_{q}^{3|124}(|\varphi\rangle)={\cal
C}_{q}^{4|123}(|\varphi\rangle)=\frac{2^{q-1}-1}{2^{q-1}}.$ (81)

Thus, taking the minimum of all possible bipartitions, the indicator
$\hat{\tau}_{{\cal C}_{q}}$ from Eq.(50) is given by

$\displaystyle\hat{\tau}_{{\cal C}_{q}}(|\varphi\rangle)={\cal
C}_{q}^{3|124}(|\varphi\rangle)+{\cal C}_{q}^{4|123}(|\varphi\rangle)-{\cal
C}_{q}^{34|12}(|\varphi\rangle),$ (82)

Moreover, from Eq.(28) we get that

$\displaystyle{\cal
U}_{r,s}^{12|34}(|\varphi\rangle)=\frac{1-4^{rs-s}}{(1-r)s4^{rs-s}},$ (83)
$\displaystyle{\cal
U}_{r,s}^{1|234}(|\varphi\rangle)=\frac{1-8^{rs-s}}{(1-r)s8^{rs-s}},$ (84)
$\displaystyle{\cal
U}_{r,s}^{2|134}(|\varphi\rangle)=\frac{1-2^{rs-s}}{(1-r)s2^{rs-s}}.$ (85)

On the other hand, we also obtain

$\displaystyle{\cal
U}_{r,s}^{34|12}(|\varphi\rangle)=\frac{1-4^{rs-s}}{(1-r)s4^{rs-s}},$ (86)
$\displaystyle{\cal
U}_{r,s}^{3|124}(|\varphi\rangle)=\frac{1-2^{rs-s}}{(1-r)s2^{rs-s}},$ (87)
$\displaystyle{\cal
U}_{r,s}^{4|123}(|\varphi\rangle)=\frac{1-2^{rs-s}}{(1-r)s2^{rs-s}}.$ (88)

Hence, taking the minimum of all possible bipartitions, we get

$\displaystyle\hat{\tau}_{{\cal U}_{r,s}}(|\varphi\rangle)={\cal
U}_{r,s}^{{3|124}}(|\varphi\rangle)+{\cal
U}_{r,s}^{4|123}(|\varphi\rangle)-{\cal U}_{r,s}^{12|34}(|\varphi\rangle).$
(89)

Analogously, we illustrate the indicator $\hat{\tau}_{{\cal
C}_{q}}(|\varphi\rangle)$ and $\hat{\tau}_{{\cal U}_{r,s}}(|\varphi\rangle)$
in Fig.4. Theorems 2 and 4 hold for star network state in Eq.(75).

(a)

(b)

Figure 4: (Color online) The indicator $\hat{\tau}_{{\cal
C}_{q}}(|\varphi\rangle)$ and $\hat{\tau}_{{\cal U}_{r,s}}(|\varphi\rangle)$
in Example 6. (a) The indicator $\hat{\tau}_{{\cal C}_{q}}(|\varphi\rangle)$
for $q\geq 2$. (b) The indicator $\hat{\tau}_{{\cal
U}_{r,s}}(|\varphi\rangle)$ for $1\leq r\leq 9$ and $0\leq s\leq 10$.

##  V Conclusion

The distribution of multipartite entanglement is an important problem in
entanglement theory and quantum information processing. Entanglement polygon
inequality provides a distribution relationship of all the one-to-group
marginal entanglements for arbitrary pure multi-partite entanglement. Based on
the entanglement polygon inequality in qubit systems, a new genuine tripartite
entanglement measure called the concurrence is recently proposed in
ref.Xie2021 . It turned out that for any three-qubit pure state
$|\psi\rangle_{123}$, the quantities of ${\cal C}^{2}_{1|23}(|\psi\rangle)$,
${\cal C}^{2}_{2|13}(|\psi\rangle)$ and ${\cal C}^{2}_{3|12}(|\psi\rangle)$
make up a triangle in geometry. In this case, the area of this squared
concurrence triangle is useful for quantifying the genuine tripartite
entanglement for tripartite qubit systems. A natural problem is whether the
method is feasible for any higher dimensional systems by virtue of the
inequality (11). Besides, quantum coherence is also an essential ingredient in
quantum information processing. Whether or not the coherence can be quantified
by $q$-concurrence is valuable for consideration. Based on constraint
relations among three bipartite coherences, a genuine tripartite coherence
measure has been defined Qian2020 . Similar to the quantification of genuine
multipartite entanglement, a natural question arises: How to define a genuine
multipartite coherence measure with respect to all possible bipartite
coherences.

To sum up, we have proved the entanglement polygon inequalities in terms of
all the one-to-group $q$-concurrences between a single qudit and the others in
any $n$-particle qudit system. Further, we showed that for a three-qudit
system each entanglement polygon inequality provides an upper bound for a
corresponding one-to-group marginal entanglement, while the triangle relation
provides its lower bound. Additionally, similar properties are proved for a
two-parameter class of bipartite entanglement measures in terms of the
unified-$(r,s)$ entropy entanglement. These results may intrigue new features
of high-dimensional entanglement going beyond qubit systems.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China
(No.62172341) and Fundamental Research Funds for the Central Universities
(No.2018GF07).

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# Two-Sided Fairness in Non-Personalised Recommendations

Aadi Swadipto Mondal, Rakesh Bal$*$, Sayan Sinha$*$, Gourab K Patro
The first three authors have contributed equally.

###### Abstract

Recommender systems are one of the most widely used services on several online
platforms to suggest potential items to the end-users. These services often
use different machine learning techniques for which fairness is a concerning
factor, especially when the downstream services have the ability to cause
social ramifications. Thus, focusing on the non-personalised (global)
recommendations in news media platforms (e.g., top-$k$ trending topics on
Twitter, top-$k$ news on a news platform, etc.), we discuss on two specific
fairness concerns together (traditionally studied separately)—user fairness
(Chakraborty et al. 2019) and organisational fairness (Burke et al. 2020).
While user fairness captures the idea of representing the choices of all the
individual users in the case of global recommendations, organisational
fairness tries to ensure politically/ideologically balanced recommendation
sets. This makes user fairness a user-side requirement and organisational
fairness a platform-side requirement. For user fairness, we test with methods
from social choice theory, i.e., various voting rules known to better
represent user choices in their results. Even in our application of voting
rules to the recommendation setup, we observe high user satisfaction scores
(table 2). Now for organisational fairness, we propose a bias metric (eq-1)
which measures the aggregate ideological bias of a recommended set of items
(articles). Analysing the results obtained from voting rule-based
recommendation, we find that while the well-known voting rules are better from
the user side, they show high bias values and clearly not suitable for
organisational requirements of the platforms. Thus, there is a need to build
an encompassing mechanism by cohesively bridging ideas of user fairness and
organisational fairness. In this abstract paper, we intend to frame the
elementary ideas along with the clear motivation behind the requirement of
such a mechanism.

Table 1: Some of the seed words having high PMI values Left-biased | motstandernes, djevlene, motstander, problem, motstand, scorer, møte, sjef, kontret, balanse, maktet, rød, igjen
---|---
Right-biased | bakgrunnen, grunn, etablerte, opprettelsen, døpt, alternativ, offensivt, defensiv, renter, tilværelsen, høyre, erna, siv

## Two-Sided Fairness

### Dataset

The dataset used for this research is a publicly available news dataset that
includes news articles (in Norwegian) in connection with anonymised users
(Gulla et al. 2017). From this dataset, we use the 865 articles with publicly
available news contents and the news consumption history (how much time a user
spends reading different articles) of 63260 anonymised users.

### Non-personalised Recommendation

Let $\mathcal{U}$ and $\mathcal{A}$ be the set of users and articles (items)
respectively. Most of the recommender systems rely on some form of scoring for
user-item pairs (e.g., ratings). Here we derive such scores from the “Active
Time” of a user on an article (the total amount of time in seconds the user
spends on the article). We derive the score for user $u\in\mathcal{U}$ and
article $a\in\mathcal{A}$ pair as:
$Score(u,a)=\dfrac{ActTime(u,a)}{\sum_{u^{\prime}\in\mathcal{U}}ActTime(u^{\prime},a)}\cdot
N_{a}$. Here $ActTime(u,a)$ is the total active time of the user $u$ on
article $a$ and $N_{a}$ is the number of users who have viewed the article at
least once. Next, we linearly scale all the scores of each user separately to
floating-point numbers in the range (1 to 10), in order to ensure similar
scales for fast and slow readers. This choice of scores is purely dataset-
specific and can be replaced with other suitable score metric as per the
requirement. Here, a higher score signifies more affinity of a user towards
the article. However, there are many user-article pairs for which a score does
not exist, owing to the fact that the user might not have viewed that
particular article. We find these missing values using Non-Negative Matrix
Factorisation (NMF) (Koren, Bell, and Volinsky 2009).

Non-negative Matrix Factorisation (NMF): We obtain a sparse matrix $V$ from
the normalised scaled scores obtained earlier. The rows of $V$ represent the
users, and the columns represent the articles. We then perform NMF on $V$ and
obtain $\hat{V}$ which provides us with the predicted scores for the missing
values in $V$. $V^{*}$ denotes the final score matrix, which comprises the
original scores from $V$, as well as the predicted scores from $\hat{V}$ (for
the missing entries in $V$). NMF was set on training with hyperparameters
$\alpha=0.0002$, and $\beta=0.02$. Convergence was assumed when $\Delta
cost<0.001$ (around $2,500$ epochs).

The Recommendation: As we talk about non-personalised recommendation, the goal
is to find $\kappa$-sized recommendation which would be liked by most of the
users. For this purpose, it is wiser to use $V^{*}$ as it—being a complete
score matrix—can provide insights on all the user-article pairs. Next, we
discuss two kinds of fairness desirable in such recommendations and also the
obstacles in achieving them.

### User Fairness

As the recommendation is global, it should fairly represent the interests of
the users on the platform. Thus, we can re-purpose and use multi-winner voting
rules in our setup as voting rules are known to fairly represent voters’
choices while choosing winners. To map our setup to an election we do the
following; election $\mathcal{E}=(\mathcal{C},\mathcal{V})$, where candidate
set $\mathcal{C}=\mathcal{A}=(a_{1},...,a_{m})$ is the set of articles, voter
set $\mathcal{V}=\mathcal{U}=(u_{1},...,u_{n})$ is the set of users, and we
consider scores $V^{*}$ as the corresponding votes. The winners of the
election are denoted by the set $\mathcal{W}=(w_{1},...,w_{\kappa})$ which is
a subset of $\mathcal{C}$ and are elected using various voting rules. In this
work, we experimented with six famous voting rules, namely SNTV, STV,
$k$-Borda, Bloc voting, Chamberlin Courant and Monroe (Elkind et al. 2017).
Next, we find the average satisfaction of all users for the results obtained
from each election method like: $Satisfaction$ $=$
$\dfrac{1}{\mathcal{U}}\sum_{u\in\mathcal{U}}\frac{\mathcal{W}\cap
Top\text{-}\kappa_{u}}{\kappa}$. Here $Top\text{-}\kappa_{u}$ is the top
$\kappa$ articles for user $u$ as per the scores in $V^{*}$. The satisfaction
score for each election method has been presented in Table 2 column 2. A high
satisfaction score would ensure that the recommendation results (election
winners) hence obtained follow the general principle of user fairness. In all
our experiments, the value of $\kappa$ has been taken as 10.

### Organisational Fairness

For organisational fairness (Burke et al. 2020), the set of recommended
articles needs to be balanced in terms of political/ideological bias. Before
finding political bias of the articles in our dataset, first, we analyse
articles from politically-biased media houses – namely, Klassekampen (left)
and Aftenposten (right) (Nor 2020), as well as articles talking about eminent
people involved in politics pertaining to either of these ideologies. Using
co-occurrence analysis of words in the articles of biased media houses, we
shortlist some of the words with the highest Pointwise Mutual Information
(PMI) scores. Some of the words having high PMI values—which we later use as
seed words—have been mentioned in Table 1. We consider that their presence in
an article indicates a higher likelihood of it being politically biased. Thus,
we designate the articles in our dataset as either left-biased or right-biased
based on whether they contain the majority of seed words from the left
category or right category. Next, we find the reference bias ($\rho$) of the
user population on the platform as the ratio of total time spent on the left-
biased articles to that on the right-biased articles111In our experiments,
$\rho$ was found to be $1.423$.. Finally, we design a bias metric 222More
sophisticated methods for bias analysis can also be used with the availability
of relevant annotated resources in Norwegian. However this is not under the
purview of this abstract. which measures how far are the global
recommendations from reference $\rho$.

$Bias=\dfrac{-leftCount+\rho*rightCount}{leftCount+\rho*rightCount}$ (1)

Here $leftCount$ and $rightCount$ are the number of occurrences of left and
right biased seed words in a set of articles. Hence, $Bias<0$ indicates left-
bias and $Bias>0$ indicates right-bias. Closer the value of $Bias$ to zero,
better is the method in terms of satisfying organisation fairness. The range
of $Bias$ is [-1, 1]. We calculated the bias values of the global
recommendations obtained earlier using voting rules. The results have been
shown in Table 2 column 3.

Table 2: User Satisfaction and Organisational Bias Election Method | Satisfaction | Bias
---|---|---
SNTV | 0.878 | -0.115
$k$-Borda | 0.800 | 0.033
Bloc | 0.856 | -0.240
STV | 0.894 | -0.117
CC | 0.883 | -0.169
Monroe | 0.825 | -0.103

### Discussion

For better user fairness, we repurposed well-known voting rules from fair
voting theory, and implemented the same in our recommendation setup. Our
evaluation shows that they achieve good user satisfaction scores (column 2 of
table 2). However, they show high bias values (column 3 of table 2). This is
happening as the voting rules do not consider the article content while
choosing the global recommendations. On the other hand, organisational
fairness—relating to balance in the content distribution—requires the bias to
be close to zero. Therefore, there is a need for designing a mechanism which
simultaneously cares for user fairness and organisational fairness while
designing non-personalised recommendations. All of our resources can be found
at https://github.com/americast/electoral.

### Acknowledgement

We would like to thank Prof Niloy Ganguly (CSE, IIT Kharagpur) and Rishabh
Joshi (LTI, CMU) for their continued support throughout the development of
this work. Gourab K Patro is supported by a fellowship from Tata Consultancy
Services.

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