# Strong polygamy of multi-party $q$-expected quantum correlations

Jeong San Kim<EMAIL_ADDRESS>Department of Applied Mathematics and
Institute of Natural Sciences, Kyung Hee University, Yongin-si, Gyeonggi-do
446-701, Korea

###### Abstract

We show that the polygamous nature of multi-party quantum correlations can be
characterized in a stronger form based on Tsallis $q$-entropy and
$q$-expectation value. By considering the amount of entanglement that can be
distributed in multi-party systems, we establish a class of strong polygamy
inequalities of multi-party entanglement in terms of Tsallis $q$-entropy and
$q$-expectation for $q\geq 1$. Our new class of inequalities is in fact
tighter than the usual polygamy inequalities of multi-party entanglement, and
the tightness is explicitly illustrated by an example. Moreover, our new class
of inequalities is concerned with the $q$-expected entanglement distributed
between a single party and any possible subsets of the rest parties whereas
the usual polygamy inequality only considers the entanglement between a single
party and another. We further establish the equivalence between strong
polygamy inequalities of quantum entanglement and quantum discord distributed
in multi-party quantum systems.

###### pacs:

03.67.Mn, 03.65.Ud

##  I Introduction

One distinct property making quantum mechanics fundamentally different from
classical one is in the way of sharing correlations; whereas classical
correlations can be shared among parties with no certain restriction, quantum
correlations have limited shareability and distribution in multi-party quantum
systems. For example, two parties $A$ and $B$ that are in maximally entangled
state cannot be entangled with any third party $C$. This restricted
shareability of multi-party quantum entanglement is referred to as the
monogamy of entanglement KGS .

Monogamy of entanglement was mathematically characterized in forms of an
inequality; for a three-qubit pure state ${\left|\psi\right\rangle}_{ABC}$
with its two-qubit reduced density matrices
$\rho_{AB}=\mbox{$\mathrm{tr}$}_{C}{\left|\psi\right\rangle}_{ABC}{\left\langle\psi\right|}$
and
$\rho_{AC}=\mbox{$\mathrm{tr}$}_{B}{\left|\psi\right\rangle}_{ABC}{\left\langle\psi\right|}$,
Coffman, Kundu and Wootters(CKW) established a monogamy inequality ckw

$\tau\left({\left|\psi\right\rangle}_{A(BC)}\right)\geq\tau\left(\rho_{AB}\right)+\tau\left(\rho_{AC}\right),$

where $\tau\left({\left|\psi\right\rangle}_{A(BC)}\right)=4\det\rho_{A}$ is
the tangle(or square of concurrence) of ${\left|\psi\right\rangle}_{ABC}$ with
respect to the bipartition between $A$ and $BC$, and
$\tau\left(\rho_{AB}\right)$ is the tangle of $\rho_{AB}$ defined by the
minimum average tangle

$\tau\left(\rho_{AB}\right)=\min\sum_{i}p_{i}\tau({\left|\psi_{i}\right\rangle}_{AB})$

over all possible pure-state decompositions of
$\rho_{AB}=\sum_{i}p_{i}|\psi_{i}\rangle_{AB}\langle\psi_{i}|$. Later, CKW
inequality was generalized for multi-qubit systems as well as some classes of
multi-qudit systems based on various entanglement measures ov ; kds .

CKW inequality characterizes the mutually exclusive nature of bipartite
entanglement shared in multi-party quantum systems. On the other hand,
assisted entanglement, that is a dual quantity of entanglement measure, is
known to have a dually monogamous property in multi-party quantum systems.
Using tangle of assistance

$\tau^{a}\left(\rho_{AB}\right)=\max\sum_{i}p_{i}\tau({\left|\psi_{i}\right\rangle}_{AB})$

that takes the maximum average tangle over all possible pure-state
decompositions of $\rho_{AB}$, three-qubit state
${\left|\psi\right\rangle}_{ABC}$ was shown to have the following dual
monogamy inequality GBS

$\tau\left({\left|\psi\right\rangle}_{A(BC)}\right)\leq\tau^{a}\left(\rho_{AB}\right)+\tau^{a}\left(\rho_{AC}\right).$

It is clear that tangle of assistance is a dual to tangle because one takes
the maximum while the other takes the minimum. Moreover, this dual quantity of
tangle also provides a dual monogamy(thus polygamy) inequality of three-party
entanglement. Later polygamy inequality was generalized into multi-party
quantum systems GBS ; BGK ; Kimpolys .

If two parties $A$ and $B$ are maximal entangled, it is also known that they
cannot be even classically correlated with $C$. Conversely, maximal classical
correlation between $A$ and $B$ also forbids them from sharing entanglement
with $C$ KW . Thus a perfect correlation(classical or quantum) between two
systems can even exclude the possibility of sharing different kind of
correlations with other systems. Other correlations such as Bell nonlocality
SG ; CH  and quantum discord BZYW  are also known to have some trade-off
relation in their shareability and distribution.

The limited shareability and distribution of sharing multi-party quantum
correlations is used as a fundamental ingredient in many quantum information
processing tasks. In quantum cryptography, the amount of information an
eavesdropper could possibly obtained about the secret key can be efficiently
bounded by using the mutually-exclusive nature of quantum entanglement
characterized by monogamy inequalities Paw . Entanglement monogamy also plays
an important role in condensed-matter physics such as the $N$-representability
problem for fermions anti .

Based on the concept of $q$-expectation value for real $q>0$, von Neumann
entropy can be generalized into a one-parameter family of entropy functions,
namely, Tsallis $q$-entropy tsallis ; lv . Tsallis $q$-entropy plays an
important role in various areas of physics. In nonextensive statistical
mechanics, for example, Tsallis-$q$ entropy is theoretically consistent with
the minimum cross-entropy principle(or the relative-entropy principle), which
excludes the possibility of using the ordinary expectation value from
nonextensive statistical mechanics LP ; Abe .

Tsallis $q$-entropy is also importantly used in quantum information theory;
the concavity of Tsallis-$q$ entropy guarantees the property of entanglement
monotone, a key requirement to construct a faithful entanglement measure vidal
. Some conditions on separability criteria of quantum states can also be found
by using Tsallis entropy ar ; tlb ; rc . There are also discussions about
characterizing classical statistical correlations inherented in quantum states
in terms of $q$-expectation and Tsallis-$q$ entropy rr ; bpcp .

In this paper, we show that the polygamy property of multi-party quantum
correlations can be characterized in a stronger form in terms of Tsallis
$q$-entropy and $q$-expectation value. By considering the amount of
entanglement that can be distributed in multi-party systems, we establish a
class of strong polygamy inequalities of multi-party entanglement in terms of
Tsallis $q$-entropy and $q$-expectation. Our new class of inequalities is in
fact tighter(or finer) than the usual polygamy inequalities of multi-party
entanglement; the tightness of new inequalities is illustrated by an example.
Moreover, our new class of inequalities is concerned with the entanglement
distributed between a single party and any possible subsets of the rest
parties whereas the usual polygamy inequality only considers the entanglement
between one and another. Thus the term strong is twofold.

We further establish the equivalence between strong polygamy inequalities of
quantum entanglement and quantum discord, a different kind of quantum
correlation, distributed in multi-party quantum systems using Tsallis
$q$-entropy and $q$-expectation. When $q$ tends to 1, Tsallis-$q$ entropy and
$q$-expectation are reduced to von Neumann entropy and ordinary expectation,
respectively. Thus, our results encapsulate previous results of strong
polygamy inequalities of quantum correlations as special cases KimEur14 .

This paper is organized as follows. In Section II, we briefly recall the
definitions and properties of $q$-expected correlations in bipartite quantum
systems. In Section III.1, we establish a class of strong polygamy
inequalities of entanglement distributed in multi-party quantum systems of
arbitrary dimensions in terms of Tsallis-$q$ entropy and $q$-expectation
values. In Section III.2, we justify the tightness of our new inequalities by
providing an explicitly example. In Section IV, we prove the equivalence
between strong polygamy inequalities of quantum entanglement and quantum
discord distributed in multi-party quantum systems with respect to Tsallis
$q$-entropy and $q$-expectation. We summarize our results in Section V.

##  II $q$-expected Correlations in Bipartite Quantum Systems

For real parameter $q\geq 0$ with $~{}q\neq 1$, Tsallis $q$-entropy of a
quantum state $\rho$ is defined as tsallis ; lv

$\displaystyle
S_{q}\left(\rho\right)=-\mbox{$\mathrm{tr}$}\rho^{q}\ln_{q}\rho,$ (1)

where

$\displaystyle\ln_{q}x$ $\displaystyle=$ $\displaystyle\frac{x^{1-q}-1}{1-q},$
(2)

is the generalized logarithmic function(or $q$-logarithm). Based on the
spectrum $\\{\lambda_{i}\\}_{i}$ of the quantum state $\rho$ spec , Tsallis
$q$-entropy in Eq. (1) can also be expressed as

$\displaystyle
S_{q}\left(\rho\right)=-\sum_{i}\lambda_{i}^{q}\ln_{q}\lambda_{i},$ (3)

which is the $q$-expectation value of the $q$-logarithm. Because $q$-logarithm
in Eq. (2) converges to the natural logarithm as $q$ tends to $1$, Tsallis
$q$-entropy is a one-parameter generalization of von Neumann entropy based on
the concept of $q$-expectation value, in the sense that it converges to the
von Neumann entropy

$\lim_{q\rightarrow
1}S_{q}\left(\rho\right)=-\mbox{$\mathrm{tr}$}\rho\ln\rho.$ (4)

Using Tsallis $q$-entropy for $q\geq 0$, $q$-expected entanglement($q$-E)
Kim19  of a bipartite pure state ${\left|\psi\right\rangle}_{AB}$ was defined
as

${E}_{q}\left({\left|\psi\right\rangle}_{AB}\right)=S_{q}(\rho_{A}),$ (5)

where
$\rho_{A}=\mbox{$\mathrm{tr}$}_{B}{\left|\psi\right\rangle}_{AB}{\left\langle\psi\right|}$
is the reduced density matrix of ${\left|\psi\right\rangle}_{AB}$ on subsystem
$A$. For a bipartite mixed state $\rho_{AB}$, its $q$-E was defined as the
minimum $q$-expectation value

$E_{q}\left(\rho_{AB}\right)=\min\sum_{i}p^{q}_{i}E_{q}({\left|\psi_{i}\right\rangle}_{AB}),$
(6)

whereas $q$-expected entanglement of assistance($q$-EOA) Kim19  was defined as
the maximum $q$-expectation value

$E^{a}_{q}\left(\rho_{AB}\right)=\max\sum_{i}p_{i}^{q}E_{q}({\left|\psi_{i}\right\rangle}_{AB}),$
(7)

over all possible pure state decompositions of $\rho_{AB}$,

$\rho_{AB}=\sum_{i}p_{i}|\psi_{i}\rangle_{AB}\langle\psi_{i}|.$ (8)

When $q$ tends to 1, Tsallis $q$-entropy converges to von Neumann entropy and
the $q$-expectation value becomes ordinary expectation value, therefore, $q$-E
and $q$-EOA converge to the entanglement of formation(EOF) bdsw  and the
entanglement of assistance(EOA) cohen , respectively. Thus, $q$-E and $q$-EOA
are one-parameter generalizations of EoF and EOA, respectively, in the full
range of non-negative $q$ based on the concept of $q$-expectation value.

Now, we recall more generalized quantum correlations based on $q$-expectation
value and Tsallis $q$-entropy. For a probability ensemble
$\mathcal{E}=\\{p_{i},\rho_{i}\\}$ of a quantum state $\rho$, the Tsallis
$q$-difference was defined as Kim16T

$\displaystyle\chi_{q}\left(\mathcal{E}\right)=S_{q}\left(\rho\right)-\sum_{i}p_{i}^{q}S_{q}\left(\rho_{i}\right).$
(9)

We nota that the concavity of Tsallis $q$-entropy guarantees the non-
negativity of Tsallis $q$-difference for $q\geq 1$, and it converges to the
Holevo quantity when $q$ tends to 1.

For a bipartite quantum state $\rho_{AB}$, its one-way classical
$q$-correlation ($q$-CC) Kim19  was defined as the maximum Tsallis-$q$
difference

$\displaystyle{\mathcal{J}}_{q}^{\leftarrow}(\rho_{AB})$
$\displaystyle=\max_{\mathcal{E}}\chi_{q}\left(\mathcal{E}\right)$ (10)

over all possible ensemble representations
$\mathcal{E}=\\{p_{x},\rho_{A}^{x}\\}$ of the reduced density matrix
$\rho_{A}$ induced by measurements $\\{M^{x}_{B}\\}$ on subsystem $B$ in such
a way that $p_{x}=\mbox{$\mathrm{tr}$}[(I_{A}\otimes M_{B}^{x})\rho_{AB}]$ is
the probability of the outcome $x$ and
$\rho^{x}_{A}=\mbox{$\mathrm{tr}$}_{B}[(I_{A}\otimes{M_{B}^{x}})\rho_{AB}]/p_{x}$
is the state of system $A$ when the outcome was $x$. As a dual quantity to
$q$-CC, the one-way unlocalizable $q$-entanglement($q$-UE) Kim16T  was defined
by taking the minimum Tsallis-$q$ difference

$\displaystyle{\mathbf{u}}E_{q}^{\leftarrow}(\rho_{AB})$
$\displaystyle=\min_{\mathcal{E}}\chi_{q}\left(\mathcal{E}\right),$ (11)

over all probability ensembles $\mathcal{E}$ of $\rho_{A}$ induced by rank-1
measurements on subsystem $B$. We note that $q$-CC and $q$-UE are
generalizations of the one-way classical correlation(CC) KW  and one-way
unlocalizable entanglement(UE) BGK , respectively, due to their convergence
when $q$ tends to $1$.

The following proposition shows the trade-off relations between $q$-CC and
$q$-E as well as $q$-UE and $q$-EOA distributed in three-party quantum
systems.

######  Proposition 1.

Kim19  For $q\geq 1$ and a three-party pure state
${\left|\psi\right\rangle}_{ABC}$ with its reduced density matrices
$\rho_{AB}=\mbox{$\mathrm{tr}$}_{C}{\left|\psi\right\rangle}_{ABC}{\left\langle\psi\right|}$,
$\rho_{AC}=\mbox{$\mathrm{tr}$}_{B}{\left|\psi\right\rangle}_{ABC}{\left\langle\psi\right|}$
and
$\rho_{A}=\mbox{$\mathrm{tr}$}_{BC}{\left|\psi\right\rangle}_{ABC}{\left\langle\psi\right|}$,
we have

$\displaystyle
S_{q}(\rho_{A})={\mathcal{J}}_{q}^{\leftarrow}(\rho_{AB})+E_{q}\left(\rho_{AC}\right)$
(12)

and

$\displaystyle
S_{q}(\rho_{A})={\mathbf{u}}E_{q}^{\leftarrow}(\rho_{AB})+E^{a}_{q}\left(\rho_{AC}\right).$
(13)

The concept of $q$-expectation value together with Tsallis-$q$ entropy can
also be used to generalize another kind of quantum correlation, quantum
discord discord ; based on Tsallis $q$-mutual entropy,

$\displaystyle{\mathcal{I}}_{q}\left(\rho_{AB}\right)=S_{q}\left(\rho_{A}\right)+S_{q}\left(\rho_{B}\right)-S_{q}\left(\rho_{AB}\right),$
(14)

quantum $q$-discord($q$-D)qdiscord  of $\rho_{AB}$ was defined as the
difference between its Tsallis-$q$ mutual entropy and $q$-CC,

$\displaystyle\delta_{q}^{\leftarrow}(\rho_{AB})={\mathcal{I}}_{q}\left(\rho_{AB}\right)-{\mathcal{J}}_{q}^{\leftarrow}(\rho_{AB}).$
(15)

Due to the duality between $q$-CC and $q$-UE, a dual definition to $q$-D was
also provided as

$\displaystyle{\mathbf{u}}\delta_{q}^{\leftarrow}(\rho_{AB})={\mathcal{I}}_{q}\left(\rho_{AB}\right)-{\mathbf{u}}E_{q}^{\leftarrow}(\rho_{AB}).$
(16)

Eq. (16) is referred to as the one-way unlocalizable quantum
$q$-discord($q$-UD) of $\rho_{AB}$ Kim19 , which is a generalization of one-
way unlocalizable quantum discord(UD) XFL12 .

The following proposition provides a trade-off relation between quantum
entanglement($q$-UE) and quantum discord($q$-UD) distributed in three-party
quantum systems.

######  Proposition 2.

Kim19  For $q\geq 1$ and a three-party pure state
${\left|\psi\right\rangle}_{ABC}$ with its reduced density matrices
$\rho_{AB}=\mbox{$\mathrm{tr}$}_{C}{\left|\psi\right\rangle}_{ABC}{\left\langle\psi\right|}$,
$\rho_{AC}=\mbox{$\mathrm{tr}$}_{B}{\left|\psi\right\rangle}_{ABC}{\left\langle\psi\right|}$
and
$\rho_{A}=\mbox{$\mathrm{tr}$}_{BC}{\left|\psi\right\rangle}_{ABC}{\left\langle\psi\right|}$,
we have

$\displaystyle
S_{q}(\rho_{A})={\mathbf{u}}\delta_{q}^{\leftarrow}(\rho_{BA})+{\mathbf{u}}E_{q}^{\leftarrow}(\rho_{CA}).$
(17)

##  III Strong Polygamy of $q$-expected Quantum Entanglement

###  III.1 Strong polygamy inequality of $q$-expected entanglement

Recently, the polygamous nature of quantum entanglement in multi-party systems
was shown to be characterized in terms of $q$-expectation value Kim19_2 ; for
$q\geq 1$, and any $(n+1)$-party quantum state $\rho_{AB_{1}\cdots B_{n}}$
with bipartite reduced density matrices $\rho_{AB_{i}}$ for $i=1,\cdots,n$, we
have

$\displaystyle E^{a}_{q}\left(\rho_{A(B_{1}\cdots B_{n})}\right)\leq$
$\displaystyle\sum_{i=1}^{n}E^{a}_{q}\left(\rho_{AB_{i}}\right),$ (18)

conditioned on the superadditivity of Tsallis-$q$ mutual entropy for the
classical-classical-quantum(ccq) states,

$\displaystyle{\mathcal{I}}_{q}\left(\Omega_{XY:AB_{i}}\right)\geq{\mathcal{I}}_{q}\left(\Omega_{X:AB_{i}}\right)+{\mathcal{I}}_{q}\left(\Omega_{Y:AB_{i}}\right).$
(19)

Here, the four-qudit ccq state $\Omega_{XYAB_{i}}$ is obtained from the two-
qudit reduced density matrix $\rho_{AB_{i}}$ as party_qudit

$\displaystyle\Omega_{XYAB_{i}}=\frac{1}{d^{2}}\sum_{x,y=0}^{d-1}$
$\displaystyle{\left|x\right\rangle}_{X}{\left\langle
x\right|}\otimes{\left|y\right\rangle}_{Y}{\left\langle y\right|}$
$\displaystyle\otimes(I_{A}\otimes
X^{x}_{B_{i}}Z^{y}_{B_{i}})\rho_{A{B_{i}}}(I_{A}\otimes
Z^{-y}_{B_{i}}X^{-x}_{B_{i}}),$ (20)

where

$\displaystyle
Z_{B_{i}}=\sum_{j=0}^{d-1}\omega_{d}^{j}{\left|e_{j}\right\rangle}_{B}{\left\langle
e_{j}\right|},~{}~{}X_{B_{i}}=\sum_{j=0}^{d-1}{\left|e_{j+1}\right\rangle}_{B}{\left\langle
e_{j}\right|}$ (21)

are the generalized $d$-dimensional Pauli operators in terms of the
eigenvectors $\\{{\left|e_{j}\right\rangle}_{B_{i}}\\}$ of $\rho_{B_{i}}$ and
$\omega_{d}=e^{\frac{2\pi i}{d}}$ is the $d$th-root of unity.

Now, we establish a class of strong polygamy inequalities of multi-party
quantum entanglement that deals with the $q$-EOA between between a fixed party
and any possible subset of the rest parties. Let
$\mbox{$\mathbb{B}$}=\\{B_{1},\cdots,B_{n}\\}$ be the set of subsystems
$B_{i}$’s for $i=1,~{}\cdots,n$, and
$\mbox{$\mathbb{X}$}=\\{B_{i_{1}},\cdots,B_{i_{k}}\\}$ be a non-empty proper
subset of $\mathbb{B}$ for $1\leq k\leq n-1$. Together with the complement
$\mbox{$\mathbb{X}$}^{c}=\mbox{$\mathbb{B}$}-\mbox{$\mathbb{X}$}$ of
$\mathbb{X}$ in $\mathbb{B}$, any $n+1$-party quantum state
$\rho_{AB_{1}\cdots B_{n}}$ can be considered as a three-party quantum state
$\rho_{A\mbox{$\mathbb{X}$}\mbox{$\mathbb{X}$}^{c}}$ with respect to the
tripartition $A$-$\mathbb{X}$-$\mbox{$\mathbb{X}$}^{c}$.

Conditioned on the the superadditivity of Tsallis-$q$ mutual entropy for ccq
states as in (19), the polygamy inequality of $q$-EOA leads us to

$\displaystyle E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{B}$}}\right)=$
$\displaystyle
E_{q}^{a}\left(\rho_{A(\mbox{$\mathbb{X}$}\mbox{$\mathbb{X}$}^{c})}\right)$
$\displaystyle\leq$ $\displaystyle
E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)+E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}^{c}}\right),$
(22)

where $E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)$ and
$E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}^{c}}\right)$ are $q$-EOA of reduced
density matrices $\rho_{A\mbox{$\mathbb{X}$}}$ and
$\rho_{A\mbox{$\mathbb{X}$}^{c}}$, respectively. By considering all possible
non-empty proper subsets $\mathbb{X}$ of $\mathbb{B}$, we have

$\displaystyle E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{B}$}}\right)$
$\displaystyle\leq\frac{1}{2^{n}-2}\sum_{\mbox{$\mathbb{X}$}}\left(E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)+E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}^{c}}\right)\right),$
(23)

where the summation is over all possible non-empty proper subsets
$\mathbb{X}$’s of $\mathbb{B}$. Because the set of all non-empty proper
subsets of $\mathbb{B}$ is the same with the set of their complements
subseqcomp , we have

$\displaystyle\sum_{\mbox{$\mathbb{X}$}}\left(E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)+E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}^{c}}\right)\right)=2\sum_{\mbox{$\mathbb{X}$}}E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right).$
(24)

For $1\leq k\leq n-1$ and a non-empty proper subset
$\mbox{$\mathbb{X}$}=\\{B_{i_{1}},\cdots,B_{i_{k}}\\}$ of $\mathbb{B}$ with
its complement $\mbox{$\mathbb{X}$}^{c}=\\{B_{i_{k+1}},\cdots,B_{i_{n}}\\}$,
the polygamy inequality of $q$-EOA in (18) leads us to

$\displaystyle
E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)\leq\sum_{j=1}^{k}E_{q}^{a}\left(\rho_{AB_{i_{j}}}\right)$
(25)

and

$\displaystyle
E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}^{c}}\right)\leq\sum_{j=k+1}^{n}E_{q}^{a}\left(\rho_{AB_{i_{j}}}\right).$
(26)

Thus we have

$\displaystyle E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)+$
$\displaystyle E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}^{c}}\right)$
$\displaystyle\leq\sum_{j=1}^{k}E_{q}^{a}\left(\rho_{AB_{i_{j}}}\right)+\sum_{j=k+1}^{n}E_{q}^{a}\left(\rho_{AB_{i_{j}}}\right)$
$\displaystyle=\sum_{i=1}^{n}E_{q}^{a}\left(\rho_{AB_{i}}\right).$ (27)

Using Inequalities (24), (27) and considering all possible non-empty proper
subsets $\mathbb{X}$ of $\mathbb{B}$, we have

$\displaystyle\frac{1}{2^{n-1}-1}\sum_{\mbox{$\mathbb{X}$}}E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)\leq\sum_{i=1}^{n}E_{q}^{a}\left(\rho_{AB_{i}}\right).$
(28)

From inequalities (23) and (28), we have the following theorem about strong
polygamy inequalities of $q$-expected entanglement in multi-party quantum
systems;

######  Theorem 1.

For $q\geq 1$ and any multi-party quantum state $\rho_{AB_{1}\cdots
B_{n}}=\rho_{A\mbox{$\mathbb{B}$}}$, we have

$\displaystyle
E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{B}$}}\right)\leq\frac{1}{2^{n-1}-1}\sum_{\mbox{$\mathbb{X}$}}E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)\leq\sum_{i=1}^{n}E_{q}^{a}\left(\rho_{AB_{i}}\right),$
(29)

conditioned on the superadditivity of Tsallis-$q$ mutual entropy for the ccq
state in (19). Here, the first summation is over all non-empty proper subsets
$\mathbb{X}$ of $\mathbb{B}$.

We note that Inequalities (29) of Theorem 1 is concerned with the $q$-EOA
between the single party $A$ and all possible subsets $\mathbb{X}$’s of
$\mathbb{B}$, whereas the $q$-expected polygamy inequality in (18) only deals
with $q$-EOA between $A$ and each single party ($B_{i}$’s) in $\mathbb{B}$. We
also note that Inequality (29) is in fact tighter than the usual $q$-expected
polygamy inequality in (18).

For $q=1$, the condition in (19) is reduced to the superadditivity of quantum
mutual information

$\displaystyle{\mathcal{I}}\left(\Omega_{XY:AB}\right)\geq{\mathcal{I}}\left(\Omega_{X:AB}\right)+{\mathcal{I}}\left(\Omega_{Y:AB}\right),$
(30)

which was shown to be true for any ccq state in general Kim16T . Thus
Inequalities (29) of Theorem 1 is true for $q=1$ without any condition, and
this encapsulates the result of strong polygamy inequality in terms of EOA
KimEur14 .

###  III.2 Tightness

Here, we justify the tightness of our new inequality in (29) compared with
Inequality (18) by providing an example that makes both inequalities in (29)
strict; we will consider Inequality (29) with respect to the Greenberger-
Horne-Zeilinger(GHZ) state GHZ

${\left|\rm{GHZ}\right\rangle}_{AB_{1}B_{2}B_{3}}=\frac{1}{\sqrt{2}}\left({\left|0000\right\rangle}_{AB_{1}B_{2}B_{3}}+{\left|1111\right\rangle}_{AB_{1}B_{2}B_{3}}\right),$
(31)

in four-qubit systems.

Let us first denote $\mbox{$\mathbb{B}$}=\\{B_{1},B_{2},B_{3}\\}$, so that
${\left|\rm{GHZ}\right\rangle}_{AB_{1}B_{2}B_{3}}={\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}$,
and consider its reduced density matrix on subsystem $A$

$\rho_{A}=\mbox{$\mathrm{tr}$}_{\mbox{$\mathbb{B}$}}{\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}{\left\langle\rm{GHZ}\right|}=\frac{1}{2}\left({\left|0\right\rangle}_{A}{\left\langle
0\right|}+{\left|1\right\rangle}_{A}{\left\langle 1\right|}\right).$ (32)

From the definition of $q$-E in (5), we can easily obtain the $q$-E of
${\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}$ with respect to the
bipartition between $A$ and $\mathbb{B}$ as

$\displaystyle
E_{q}\left({\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}\right)=S_{q}\left(\rho_{A}\right)=\frac{1-2^{1-q}}{q-1},$
(33)

which is nonnegative for $q\geq 1$. We denote this quantity as

$\xi_{q}=\frac{1-2^{1-q}}{q-1}.$ (34)

For the two-qubit reduced density matrices of
${\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}$, we have

$\displaystyle\rho_{AB_{i}}$ $\displaystyle=$
$\displaystyle\mbox{$\mathrm{tr}$}_{B_{j}B_{k}}{\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}{\left\langle\rm{GHZ}\right|}$
(35) $\displaystyle=$
$\displaystyle\frac{1}{2}\left({\left|00\right\rangle}_{AB_{i}}{\left\langle
00\right|}+{\left|11\right\rangle}_{AB_{i}}{\left\langle 11\right|}\right)$

with distinct $i,~{}j,~{}k\in\\{1,~{}2,~{}3\\}$. Due to the pure-state
decomposition of $\rho_{AB_{i}}$,

$\rho_{AB_{i}}=\frac{1}{2}{\left|\phi^{+}\right\rangle}_{AB_{i}}{\left\langle\phi^{+}\right|}+\frac{1}{2}{\left|\phi^{-}\right\rangle}_{AB_{i}}{\left\langle\phi^{-}\right|}$
(36)

in terms of the two-qubit Bell states

${\left|\phi^{\pm}\right\rangle}_{AB_{i}}=\frac{1}{\sqrt{2}}\left({\left|00\right\rangle}_{AB_{i}}\pm{\left|11\right\rangle}_{AB_{i}}\right),$
(37)

we can bound the $q$-EOA of $\rho_{AB_{i}}$ as

$\displaystyle 2^{1-q}\xi_{q}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}\right)^{q}E_{q}\left({\left|{\phi^{+}}\right\rangle}_{AB_{i}}\right)+\left(\frac{1}{2}\right)^{q}E_{q}\left({\left|{\phi^{-}}\right\rangle}_{AB_{i}}\right)$
(38) $\displaystyle\leq$ $\displaystyle E_{q}^{a}\left(\rho_{AB_{i}}\right)$
$\displaystyle\leq$ $\displaystyle\xi_{q},$

where the equality is due to
$\mbox{$\mathrm{tr}$}_{B_{i}}{\left|\phi^{\pm}\right\rangle}_{AB_{i}}{\left\langle\phi^{\pm}\right|}=\frac{1}{2}\left({\left|0\right\rangle}_{A}{\left\langle
0\right|}+{\left|1\right\rangle}_{A}{\left\langle 1\right|}\right)$, so that
$E_{q}\left({\left|\phi^{\pm}\right\rangle}_{AB_{i}}\right)=\xi_{q}$, the
first inequality is from the definition of
$E_{q}^{a}\left(\rho_{AB_{i}}\right)$ that takes the maximum $q$-expectation
value, and the last inequality is from the fact that $q$-EOA of any two-qubit
state is bounded above by $\xi_{q}$. Thus we have

$E_{q}^{a}\left(\rho_{AB_{1}}\right)=E_{q}^{a}\left(\rho_{AB_{2}}\right)=E_{q}^{a}\left(\rho_{AB_{3}}\right)=\xi$
(39)

for some non-zero $\xi$ such that

$2^{1-q}\xi_{q}\leq\xi\leq\xi_{q}.$ (40)

Let us now consider the three-qubit reduced density matrices

$\displaystyle\rho_{AB_{i}B_{j}}$ $\displaystyle=$
$\displaystyle\mbox{$\mathrm{tr}$}_{B_{k}}{\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}{\left\langle\rm{GHZ}\right|}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left({\left|000\right\rangle}_{AB_{i}B_{j}}{\left\langle
000\right|}+{\left|111\right\rangle}_{AB_{i}B_{j}}{\left\langle
111\right|}\right),$

with distinct $i,~{}j,~{}k\in\\{1,~{}2,~{}3\\}$. We note that
$\rho_{AB_{i}B_{j}}$ in Eq. (LABEL:GHZrhoAB1B2) can be represented as

$\rho_{AB_{i}B_{j}}=\frac{1}{2}{\left|\phi^{+}\right\rangle}_{AB_{i}B_{j}}{\left\langle\phi^{+}\right|}+\frac{1}{2}{\left|\phi^{-}\right\rangle}_{AB_{i}B_{j}}{\left\langle\phi^{-}\right|},$
(42)

where

$\displaystyle{\left|\phi^{\pm}\right\rangle}_{AB_{i}B_{j}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left({\left|000\right\rangle}_{AB_{i}B_{j}}\pm{\left|111\right\rangle}_{AB_{i}B_{j}}\right)$

are the three-qubit GHZ states.

By relabelling the orthonormal basis elements of two-qubit system $B_{i}B_{j}$
as

$\displaystyle{\left|00\right\rangle}_{(B_{i}B_{j})}={\left|\overline{0}\right\rangle}_{(B_{i}B_{j})},~{}{\left|11\right\rangle}_{(B_{i}B_{j})}={\left|\overline{1}\right\rangle}_{(B_{i}B_{j})},$
(44)

the three-qubit GHZ states in Eq. (LABEL:GHZ3) can be rewritten as

$\displaystyle{\left|\phi^{\pm}\right\rangle}_{A(B_{i}B_{j})}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left({\left|0\right\rangle}_{A}{\left|\overline{0}\right\rangle}_{(B_{i}B_{j})}\pm{\left|1\right\rangle}_{A}{\left|\overline{1}\right\rangle}_{(B_{i}B_{j})}\right),$

which are equivalent to the two-qubit bell states in Eq. (37) with respect to
the bipartition between $A$ and $B_{i}B_{j}$.

Eqs. (42) and (LABEL:BellABi2) impliy that the three-qubit reduced density
matrices in (42) are equivalent to the two-qubit states in (36) when we
consider the bipartition between $A$ and $B_{i}B_{j}$. Thus the same argument
in Inequalities (38), (40) and Eq. (39) leads us to

$\displaystyle E_{q}^{a}\left(\rho_{A(B_{1}B_{2})}\right)$ $\displaystyle=$
$\displaystyle E_{q}^{a}\left(\rho_{A(B_{1}B_{3})}\right)$ (46)
$\displaystyle=$ $\displaystyle
E_{q}^{a}\left(\rho_{A(B_{2}B_{3})}\right)=\xi.$

For the four-qubit GHZ state
${\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}$, Eq. (39) and Eq. (46)
lead the intermediate term of Inequalities (29) as

$\frac{1}{2^{n-1}-1}\sum_{\mbox{$\mathbb{X}$}}E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)=\frac{1}{2^{3-1}-1}6\xi=2\xi$
(47)

where the summation is over all possible non-empty proper subsets $\mathbb{X}$
of $\mbox{$\mathbb{B}$}=\\{B_{1},B_{2},B_{3}\\}$. We also note that the first
and last terms of Inequalities (29) for
${\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}$ are

$\displaystyle
E_{q}^{a}\left({\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}\right)=\xi_{q}$
(48)

and

$\displaystyle\sum_{i=1}^{3}E_{q}^{a}\left(\rho_{AB_{i}}\right)=3\xi,$ (49)

respectively.

Now we have

$\displaystyle\frac{1}{2^{n-1}-1}\sum_{\mbox{$\mathbb{X}$}}E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)$
$\displaystyle\geq$ $\displaystyle 2^{2-q}\xi_{q}$ (50) $\displaystyle>$
$\displaystyle\xi_{q}$ $\displaystyle=$ $\displaystyle
E_{q}^{a}\left({\left|\rm{GHZ}\right\rangle}_{A\mbox{$\mathbb{B}$}}\right)$

where the first inequality is due to Eq. (47) as well as Inequality (40), and
the second inequality is guaranteed to be strict for $1\leq q<2$. Moreover,
Eqs. (47) and (49) lead us to the strict inequality

$\displaystyle\sum_{i=1}^{3}E_{q}^{a}\left(\rho_{AB_{i}}\right)>\frac{1}{2^{n-1}-1}\sum_{\mbox{$\mathbb{X}$}}E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)$
(51)

for any $q\geq 1$. Thus Inequality (50) and Inequality (51) imply that every
inequality in (29) is strict for the four-qubit GHZ state in Eq. (31) in the
range of $1\leq q<2$.

##  IV Equivalence in Strong Polygamy of multi-party $q$-expected quantum
correlations

Recently, the equivalence of polygamy inequalities based on different kind of
$q$-expected quantum correlations was established in three-party quantum
systems;

######  Proposition 3.

Kim19  For $q\geq 1$, and any three-party pure state
${\left|\psi\right\rangle}_{ABC}$, the polygamy inequality of $q$-EoA is
equivalent to the polygamy inequality of $q$-UD, that is,

$\displaystyle E_{q}\left({\left|\psi\right\rangle}_{A(BC)}\right)\leq$
$\displaystyle E_{q}^{a}(\rho_{AB})+E_{q}^{a}(\rho_{AC})$ (52)

if and only if

$\displaystyle{\mathbf{u}}\delta_{q}^{\leftarrow}\left({\left|\psi\right\rangle}_{A(BC)}\right)\leq$
$\displaystyle{\mathbf{u}}\delta_{q}^{\leftarrow}(\rho_{AB})+{\mathbf{u}}\delta_{q}^{\leftarrow}(\rho_{AC}),$
(53)

where
${\mathbf{u}}\delta_{q}^{\leftarrow}\left({\left|\psi\right\rangle}_{A(BC)}\right)$
is the $q$-UD of the pure state ${\left|\psi\right\rangle}_{ABC}$ with respect
to the bipartition between $A$ and $BC$.

Here, we generalize Proposition 3 in a stronger form that is valid in any
multi-party quantum systems.

From the definition of $q$-UD in Eq. (16) and the Tsallis $q$-mutual entropy
in Eq. (14) as well as the trade-off relation between $q$-EU and $q$-EOA in
Eq. (13) of Proposition 1, we have

$\displaystyle
E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)={\mathbf{u}}{\delta}_{q}^{\leftarrow}\left(\rho_{A\mbox{$\mathbb{X}$}^{c}}\right)+S_{q}\left(\rho_{A|\mbox{$\mathbb{X}$}^{c}}\right).$
(54)

for any multi-party pure state
${\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}={\left|\psi\right\rangle}_{AB_{1}\cdots
B_{n}}$ and a non-empty proper subset $\mathbb{X}$ of $\mathbb{B}$, where
$S_{q}\left(\rho_{A|{\mbox{$\mathbb{X}$}}^{c}}\right)=S\left(\rho_{A{\mbox{$\mathbb{X}$}}^{c}}\right)-S\left(\rho_{{\mbox{$\mathbb{X}$}}^{c}}\right)$
is the Tsallis $q$-conditional entropy of $\rho_{A{\mbox{$\mathbb{X}$}}^{c}}$.
Thus we have

$\displaystyle\sum_{\mbox{$\mathbb{X}$}}E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)=$
$\displaystyle\sum_{\mbox{$\mathbb{X}$}}{\mathbf{u}}\delta_{q}^{\leftarrow}\left(\rho_{A\mbox{$\mathbb{X}$}^{c}}\right)+\sum_{\mbox{$\mathbb{X}$}}S_{q}\left(\rho_{A|\mbox{$\mathbb{X}$}^{c}}\right)$
$\displaystyle=$
$\displaystyle\sum_{\mbox{$\mathbb{X}$}}{\mathbf{u}}\delta_{q}^{\leftarrow}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)+\sum_{\mbox{$\mathbb{X}$}}S_{q}\left(\rho_{A|\mbox{$\mathbb{X}$}}\right),$
(55)

where the summations are over all possible non-empty proper subsets
$\mathbb{X}$ of $\mathbb{B}$, and the last equality is due to subseqcomp .

Because of the complementary property of Tsallis $q$-conditional entropy
compqcond , we have

$\displaystyle
S_{q}\left(\rho_{A|\mbox{$\mathbb{X}$}}\right)+S_{q}\left(\rho_{A|\mbox{$\mathbb{X}$}^{c}}\right)=0$
(56)

for any multi-party pure state
${\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}$ and any non-empty proper
subset $\mathbb{X}$ of $\mathbb{B}$. Thus we have

$\displaystyle\sum_{\mbox{$\mathbb{X}$}}S_{q}\left(\rho_{A|\mbox{$\mathbb{X}$}}\right)=0,$
(57)

with the summation over all non-empty proper subsets $\mathbb{X}$ of
$\mathbb{B}$, and this leads Eqs. (55) to

$\displaystyle\sum_{\mbox{$\mathbb{X}$}}E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right)=\sum_{\mbox{$\mathbb{X}$}}{\mathbf{u}}\delta_{q}^{\leftarrow}\left(\rho_{A\mbox{$\mathbb{X}$}}\right),$
(58)

for any multi-party pure state
${\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}$.

Now let us consider ${\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}$
together with its environmental system ${\left|\phi\right\rangle}_{C}$. Due
the the universality of Eq. (17) in Proposition 2 considered on
${\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}\otimes{\left|\phi\right\rangle}_{C}$,
we have

$\displaystyle S_{q}(\rho_{\mbox{$\mathbb{B}$}})=$
$\displaystyle{\mathbf{u}}\delta_{q}^{\leftarrow}({\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}})+{\mathbf{u}}E_{q}^{\leftarrow}\left({\left|\phi\right\rangle}_{C}{\left\langle\phi\right|}\otimes\rho_{\mbox{$\mathbb{B}$}}\right),$
(59)

where
$\rho_{\mbox{$\mathbb{B}$}}=\mbox{$\mathrm{tr}$}_{A}{\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}{\left\langle\psi\right|}$.
As the subsystems ${\mbox{$\mathbb{B}$}}C$ of
${\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}\otimes{\left|\phi\right\rangle}_{C}$
is in a product state, any measurement on subsystem $\mathbb{B}$ leaves the
subsystem $C$ intact, that is,
${\left|\phi\right\rangle}_{C}{\left\langle\phi\right|}$. Thus, the definition
of $q$-UE in (11) leads us to

$\displaystyle{\mathbf{u}}E_{q}^{\leftarrow}\left({\left|\phi\right\rangle}_{C}{\left\langle\phi\right|}\otimes\rho_{\mbox{$\mathbb{B}$}}\right)=$
$\displaystyle\min\left[S_{q}({\left|\phi\right\rangle}_{C})-\sum_{x}p_{x}^{q}S_{q}({\left|\phi\right\rangle}_{C})\right]$
$\displaystyle=$ $\displaystyle 0.$ (60)

From Eqs. (59) and (60), we have

$\displaystyle{\mathbf{u}}\delta_{q}^{\leftarrow}({\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}})=S_{q}(\rho_{\mbox{$\mathbb{B}$}}),$
(61)

therefore

$\displaystyle{\mathbf{u}}\delta_{q}^{\leftarrow}({\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}})=S_{q}(\rho_{\mbox{$\mathbb{B}$}})=S_{q}(\rho_{A})=E_{q}\left({\left|\psi\right\rangle}_{A{\mbox{$\mathbb{B}$}}}\right),$
(62)

for any multi-party pure state
${\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}$ with respect to the
bipartition between $A$ and $\mathbb{B}$..

From Eqs. (58) and (62) together with Inequality (29), we have the following
theorem about the equivalence of strong polygamy inequalities in terms of
$q$-EOA and $q$-UD in multi-party quantum systems.

######  Theorem 2.

For $q\geq 1$ and any multi-party quantum state
${\left|\psi\right\rangle}_{AB_{1}\cdots
B_{n}}={\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}$, the strong polygamy
inequality of $q$-EOA is equivalent to that of $q$-UD, that is,

$\displaystyle
E_{q}\left({\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}\right)\leq\frac{1}{2^{n-1}-1}\sum_{\mbox{$\mathbb{X}$}}E_{q}^{a}\left(\rho_{A\mbox{$\mathbb{X}$}}\right),$
(63)

if and only if

$\displaystyle{\mathbf{u}}\delta_{q}^{\leftarrow}\left({\left|\psi\right\rangle}_{A\mbox{$\mathbb{B}$}}\right)\leq\frac{1}{2^{n-1}-1}\sum_{\mbox{$\mathbb{X}$}}{\mathbf{u}}\delta_{q}^{\leftarrow}\left(\rho_{A\mbox{$\mathbb{X}$}}\right),$
(64)

where the summations are over all non-empty proper subsets $\mathbb{X}$ of
$\mathbb{B}$.

Theorem 2 is a generalization of Proposition 3 into multi-party quantum
systems in a stronger form that deals with quantum correlations between the
single party $A$ and all possible non-empty proper subsets $\mathbb{X}$’s of
$\mathbb{B}$. We further note that, due to Theorem 1, Inequality (63) itself
is guaranteed on the condition of the superadditivity of Tsallis-$q$ mutual
entropy for the ccq states in (19), therefore Inequality (64) itself is also
guaranteed on the same condition. However, Theorem 2 says that the equivalence
of the strong polygamy inequalities in terms of $q$-EOA and $q$-UD is always
guaranteed with no condition.

##  V Summary

Using Tsallis $q$-entropy and $q$-expectation value, we have shown that the
polygamous nature of multi-party quantum correlations can be characterized in
a stronger form; we have established a class of strong polygamy inequalities
of multi-party entanglement in terms of $q$-EOA. Our new class of inequalities
was shown to be tighter than the usual polygamy inequalities of multi-party
entanglement, which was illustrated by an explicit example. We also note that
our new class of inequalities is concerned with the entanglement distributed
between a single party and any possible subsets of the rest parties whereas
the usual polygamy inequality only considers the entanglement between one and
another. Thus the term ‘strong’ can be doubly implied here.

We have also established the equivalence between strong polygamy inequalities
of different kind of quantum correlations, quantum entanglement and quantum
discord, in terms of $q$-EOA and $q$-UD. When $q$ tends to 1, Tsallis-$q$
entropy and $q$-expectation are reduced to von Neumann entropy and ordinary
expectation, respectively. Thus, our results encapsulate previous results of
strong polygamy inequalities of quantum correlations as special cases.

As the limited shareability of quantum correlations in multi-party quantum
systems is the key ingredient in many quantum information tasks, it is
important and even necessary to have proper mathematical characterizations of
these trade-off relations such as monogamy and polygamy inequalities of
quantum correlations. We also note that study of quantum mechanical properties
in higher-dimensional systems is sometimes preferred in many quantum
cryptographical protocols, because it provides us with higher coding density
and stronger security.

Our results here strengthen the characterization of the polygamous nature in
distributing quantum correlations in multi-party high-dimensional quantum
systems, which can be useful methods for the foundation of many secure quantum
information and communication protocols. Noting the importance of the study on
multi-party quantum correlations, our results here can also provide a rich
reference for future work to understand the nature of multi-party quantum
correlations.

## Acknowledgments

This work was supported by Basic Science Research
Program(NRF-2020R1F1A1A010501270) and Quantum Computing Technology Development
Program(NRF-2020M3E4A1080088) through the National Research Foundation of
Korea(NRF) grant funded by the Korea government(Ministry of Science and ICT).

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