# Generating multi-hops entangled network via spin Dipolar interaction

A. R. Mohammed 1111E-mail<EMAIL_ADDRESS>T. M. El-Shahat 1222E-mail
<EMAIL_ADDRESS>and N. Metwally 2,3333E-mail<EMAIL_ADDRESS>
1Math. Dept., Faculty of Science, Al-Azhar University, Assiut 71524, Egypt.
2Math. Dept., Faculty of Science, Aswan University, Aswan 81528, Egypt
. 3Department of mathematics, College of Science, Bahrain University, Bahrain

Abstract

The possibility of generating a multi-hops network between different entangled
nodes (qubits) via spin Dipolar interaction is examined. The negativity,
tangle and the non-local coherent advantage are used as quantifiers of the
generated quantum correlations. The phenomena of the sudden death/birth is
displayed for the entangled two nodes, while the sudden changes phenomena
(increasing/ decreasing) is depicted for all entangled three nodes. The amount
of correlations between the different nodes depend on the initial network
settings, where the largest amount is predicted if the network is initially
conducted via maximum entangled nodes. The generated quantum correlations
between each three nodes are more robust than those generated between two
nodes. For the generated entangled two nodes, the direction of the interaction
and its strength have a remarkable effect on the correlation behavior, while
they has a slightly effect on the correlation of the three nodes.

Keywords : Non-local coherent advantage, Negativity, Tangle, Quantum network,
Dipolar interaction.

##  1 Introduction

Quantum networks play some important roles in the quantum information
processing tasks, such as distributed quantum computations, quantum
teleportation and quantum cryptography [1, 2]. Design network protocols
depends on the efficiency of the connections between the nodes, the amount of
correlations between them and on the property of the resources to connect and
transfer quantum information between each nodes of the network.
There are some applications are implemented via these networks. For example,
Zueco et al. studied the routing of quantum information in qubit chains [3].
The possibility of using the optical photons as carriers of information
between fixed trapped atomic quantum memories to generate entangled quantum
networks is examined by Duan and Monroe [4]. Chudzicki and Strauch [5]
discussed the routing of quantum information, where they showed perfect
parallel state transfer is possible for certain networks of harmonic
oscillator modes. Metwally [6] has generated a wireless quantum network
between multi-hops by using logic quantum gates. Generating entangled network
via DzyaloshinskiiMoriya (DM)interaction is investigated in [7]. It is shown
that, that entanglement may be enhanced under a magnetic field [8, 9, 10].
Castro et al. have used of a Dipolar spin thermal system as a noisy quantum
channel to perform quantum teleportation [11].

In this contribution, we are motivated to introduce a theoretical technique to
generate entangled network by using different types of entangled states, where
it is assumed that, a source supplies the users with pairs of $X$ states. As
it is displayed in Fig.(1), it is allowed that the second and third nodes
(qubits) interact locally via Dipolar interaction [11]. Due to this
interaction, an entangled network is generated between the four nodes. We
assume that, the initial network is conducted via maximum entangle two nodes
($MM$ type), Werner-Werner entangled nodes ($WW$ type), or by using maximum-
Werner entangled nodes ($MW$ type). The amount of quantum correlations that
generated between the network’s nodes are quantified by the negativity, non-
local coherent advantage, and the tangle. The effect of the interaction
strength and its direction on the behavior of the generated quantum
correlation is discussed.

The paper is organized as follows. In Sec.(2), a model of the suggested
quantum network is described, where an explicit analytical forms of the final
state between each two and three nodes are introduced. Moreover, we examine
the possibility of generating multi-hops network between either two or three
nodes of the network. In Sec.(3), we recall, the negativity, non-local
advantage and the tangle as measures of the generated quantum correlations
between the different nodes. The behavior of these quantifiers are
investigated numerically in Sec.(4). Finally, our results are summarized in
Sec.(5).

##  2 The system and its evaluation

The spin Dipolar interaction arises from the magnetic field created by a
magnetic moment of a spin-$\frac{1}{2}$. The Hamiltonian that describes the
interaction is given by,

$\displaystyle\mathcal{H}=-\frac{1}{3}\mathord{\buildrel\lower
3.0pt\hbox{$\scriptscriptstyle\rightarrow$}\over{S}}_{1}^{T}\cdot\mathord{\begin{picture}(12.0,12.0)(-1.0,0.0)\put(-3.0,12.0){\makebox(0.0,0.0)[t]{$\scriptscriptstyle\downarrow$}}
\put(-3.0,13.0){\makebox(0.0,0.0)[l]{$\scriptscriptstyle\longrightarrow$}}
\put(5.0,0.0){\makebox(0.0,0.0)[b]{$T$}}
\end{picture}}\cdot\mathord{\buildrel\lower
3.0pt\hbox{$\scriptscriptstyle\rightarrow$}\over{S}}_{2},$ (1)

where
$\mathcal{S}_{i}=\\{\mathcal{S}_{i}^{x},\mathcal{S}_{i}^{y},\mathcal{S}_{i}^{z}\\}$,
$i=1,2$ are the spin operators, and the dyadic
$\mathord{\begin{picture}(12.0,12.0)(-1.0,0.0)\put(-3.0,12.0){\makebox(0.0,0.0)[t]{$\scriptscriptstyle\downarrow$}}
\put(-3.0,13.0){\makebox(0.0,0.0)[l]{$\scriptscriptstyle\longrightarrow$}}
\put(5.0,0.0){\makebox(0.0,0.0)[b]{$T$}} \end{picture}}$ is defined by
$3\times 3$ matrix its diagonal elements are given by
$\\{\Delta-3\epsilon,\Delta+3\epsilon,-2\Delta\\}$, with $\Delta$ and
$\epsilon$ are the Dipolar coupling constants between the spins. If $\Delta>0$
then, the interaction is switched on the $x-y$ plane, whereas if $\Delta<0$,
the spin is directed along the z-axis. In the computational basis set
$\\{00,01,10,11\\}$, the Hamiltonian (1) may be written as,

$\displaystyle\mathcal{H}=\left(\begin{array}[]{cccc}\frac{\Delta}{6}&0&0&\frac{\epsilon}{2}\\\
0&-\frac{\Delta}{6}&-\frac{\Delta}{6}&0\\\
0&-\frac{\Delta}{6}&-\frac{\Delta}{6}&0\\\
\frac{\epsilon}{2}&0&0&\frac{\Delta}{6}\\\ \end{array}\right),$ (6)

Let us assume that, Alice and Bob share an entangled hop consists of four
nodes, where each pair of these nodes may be prepared in a singlet or Werner
states. It is allowed that, the terminals of each two nodes interact via
Dipolar interaction. Due to this interaction one may generated entangled
network either between two singlet nodes ($MM$network), or between two Werner
nodes state ($WW$ network) or between singlet- Werner ($MW$ network). This
suggested network is displayed in Fig.(1a). The mathematical description of
this network is as follows: Alice and Bob share two pairs of the $X$ state as,

$\displaystyle\rho_{1234}(0)=\rho_{12}\otimes\rho_{34},$ (7)

where $\rho_{12}$ and $\rho_{34}$ are defined as,

$\displaystyle\begin{aligned}
\rho_{12}&=\frac{1}{4}(1+a_{1}\sigma_{x}^{(1)}\tau_{x}^{(2)}+b_{1}\sigma_{y}^{(1)}\tau_{y}^{(2)}+c_{1}\sigma_{z}^{(1)}\tau_{z}^{(2)}),\\\
\rho_{34}&=\frac{1}{4}(1+a_{2}\sigma_{x}^{(3)}\tau_{x}^{(4)}+b_{2}\sigma_{y}^{(3)}\tau_{y}^{(4)}+c_{2}\sigma_{z}^{(3)}\tau_{z}^{(4)}),\end{aligned}$
(8)

where $\sigma_{i},\tau_{i},i=1..3$ represent the Pauli-operators for the
second and third nodes, respectively. The time evolution of the initial
network is given by,

$\displaystyle\rho_{1234}(t)=\mathcal{U}_{23}\rho_{1234}(0)\mathcal{U}^{\dagger}_{23}$
(9)

where $\mathcal{U}_{23}=e^{-i\mathcal{H}t}$ represents the unitary operator
between the second and third nodes. In the computational basis, this unitary
may be written as,

$\displaystyle\begin{aligned}
\mathcal{U}_{23}(t)&=(r_{1}+r_{4})|00\rangle\langle
00|+(r_{1}-r_{4})|01\rangle\langle 01|+(r_{1}-r_{4})|10\rangle\langle
10|+(r_{2}+r_{3})(|00\rangle\langle 11|+|11\rangle\langle 00|)\\\
&+(r_{2}-r_{3})(|01\rangle\langle 10|+|10\rangle\langle
01|)+(r_{1}+r_{4})|11\rangle\langle 11|,\end{aligned}$ (10)

where

$\displaystyle\begin{aligned} r_{1}&=c_{1}c_{2}c_{3}-is_{1}s_{2}s_{3},\quad
r_{2}=c_{1}s_{2}s_{3}-ic_{2}c_{3}s_{1},\quad
r_{3}=c_{2}s_{1}s_{3}-ic_{1}c_{3}s_{2},\quad
r_{4}=c_{3}s_{1}s_{2}-ic_{1}c_{2}s_{3},\\\ c_{1}&=\cos(\kappa_{x}\tau),\quad
c_{2}=\cos(\kappa_{y}\tau),\quad c_{3}=\cos(\kappa_{z}\tau),\quad
s_{1}=\sin(\kappa_{x}\tau),\quad s_{2}=\sin(\kappa_{y}\tau),\\\
s_{3}&=\sin(\kappa_{z}\tau),\quad\kappa_{x}=1-3\tilde{\epsilon},\quad\kappa_{y}=1+3\tilde{\epsilon},\quad\kappa_{z}=-2,\mbox{where}\quad\tilde{\epsilon}=\frac{\epsilon}{\Delta}.\end{aligned}$
(11)

Figure 1:  This scheme describes the suggested network (a) Alice and Bob hold
an $N$ entangled nodes $a_{i},b_{i},i=1...N$, respectively. The even nodes of
Alice and the odd nodes of Bob interacts locally via Dipolar interaction. Due
to this interaction multi-hopes are generated between each four nodes, (b)
Shows the possibility of extending the network, where the terminal of each
hops interacts locally via Dipolar interaction to generate entangled hops of 8
nodes and so on.

###  2.1 Two entangled nodes

To discuss the entanglement robustness of the channel $\rho_{12}(t)$ between
the first and the second nodes, one traces out the other nodes from the final
total state $\rho_{1234}(t)$, namely
$\rho_{12}(t)=Tr_{34}\\{\rho_{1234}(t)\\}$. The final network state
$\rho_{12}(t)$ is defined by a $4\times 4$ matrix. Its non-zero elements, the
final state between the first and the second nodes are given by,

$\displaystyle\begin{aligned}
\rho_{11}&=2B_{1}(A_{1}r_{1}r_{1}^{\ast}+A_{2}r_{2}r_{2}^{\ast}+A_{2}r_{3}r_{3}^{\ast}+A_{1}r_{4}r_{4}^{\ast}),\rho_{14}=2B_{1}(A_{3}r_{1}r_{1}^{\ast}+A_{4}r_{2}r_{2}^{\ast}-A_{4}r_{3}r_{3}^{\ast}-A_{3}r_{4}r_{4}^{\ast}),\\\
\rho_{22}&=2B_{1}(A_{2}r_{1}r_{1}^{\ast}+A_{1}r_{2}r_{2}^{\ast}+A_{1}r_{3}r_{3}^{\ast}+A_{2}r_{4}r_{4}^{\ast}),\rho_{23}=2B_{1}(A_{4}r_{1}r_{1}^{\ast}+A_{3}r_{2}r_{2}^{\ast}-A_{3}r_{3}r_{3}^{\ast}-A_{4}r_{4}r_{4}^{\ast}),\\\
\rho_{32}&=\rho_{23},\quad\rho_{33}=\rho_{22},\quad\rho_{41}=\rho_{14},\quad\rho_{44}=\rho_{11},\quad\beta_{1}=B_{1}+B_{2},\quad\beta_{2}=A_{1}+A_{2},\end{aligned}$
(12)

where

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(c_{1}+1),\quad A_{2}=\frac{1}{4}(1-c_{1}),\quad
A_{3}=\frac{1}{4}(a_{1}-b_{1}),\quad A_{3}=\frac{1}{4}(a_{1}+b_{1}),$
$\displaystyle B_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(c_{2}+1),\quad B_{2}=\frac{1}{4}(1-c_{2}),\quad
B_{3}=\frac{1}{4}(a_{2}-b_{2}),\quad B_{3}=\frac{1}{4}(a_{2}+b_{2}).$

Due to the interaction, an entangled state is generated between the first and
the fourth nodes, $\rho_{14}=Tr_{23}(\rho_{1234}(t))$ which may be described
by the following non zero matrix of size $4\times 4$:

$\displaystyle\begin{aligned}
\rho_{11}&=A_{1}(B_{2}\gamma_{1}\gamma_{1}^{\ast}+B_{1}\gamma_{2}\gamma_{2}^{\ast}+B_{2}\gamma_{3}\gamma_{3}^{\ast}+B_{1}\gamma_{4}\gamma_{4}^{\ast})+A_{2}(B_{1}\gamma_{1}\gamma_{1}^{\ast}+B_{2}\gamma_{2}\gamma_{2}^{\ast}+B_{1}\gamma_{3}\gamma_{3}^{\ast}+B_{2}\gamma_{4}\gamma_{4}^{\ast}),\\\
\rho_{14}&=A_{3}(B_{4}\gamma_{3}\gamma_{1}^{\ast}+B_{3}\gamma_{4}\gamma_{2}^{\ast}+B_{4}\gamma_{1}\gamma_{3}^{\ast}+B_{3}\gamma_{2}\gamma_{4}^{\ast})+A_{4}(B_{3}\gamma_{3}\gamma_{1}^{\ast}+B_{4}\gamma_{4}\gamma_{2}^{\ast}+B_{3}\gamma_{1}\gamma_{3}^{\ast}+B_{4}\gamma_{2}\gamma_{4}^{\ast}),\\\
\rho_{22}&=A_{2}(B_{2}\gamma_{1}\gamma_{1}^{\ast}+B_{1}\gamma_{2}\gamma_{2}^{\ast}+B_{2}\gamma_{3}\gamma_{3}^{\ast}+B_{1}\gamma_{4}\gamma_{4}^{\ast})+A_{1}(B_{1}\gamma_{1}\gamma_{1}^{\ast}+B_{2}\gamma_{2}\gamma_{2}^{\ast}+B_{1}\gamma_{3}\gamma_{3}^{\ast}+B_{2}\gamma_{4}\gamma_{4}^{\ast}),\\\
\rho_{23}&=A_{4}(B_{4}\gamma_{3}\gamma_{1}^{\ast}+B_{3}\gamma_{4}\gamma_{2}^{\ast}+B_{4}\gamma_{1}\gamma_{3}^{\ast}+B_{3}\gamma_{2}\gamma_{4}^{\ast})+A_{3}(B_{3}\gamma_{3}\gamma_{1}^{\ast}+B_{4}\gamma_{4}\gamma_{2}^{\ast}+B_{3}\gamma_{1}\gamma_{3}^{\ast}+B_{4}\gamma_{2}\gamma_{4}^{\ast}),\\\
\rho_{32}&=\rho_{23},\quad\rho_{33}=\rho_{22},\quad\rho_{41}=\rho_{14},\quad\rho_{44}=\rho_{11},\quad\gamma_{1}=r_{2}-r_{3},\quad\gamma_{2}=r_{2}+r_{3},\quad\gamma_{3}=r_{1}-r_{4},\\\
\gamma_{4}&=r_{1}+r_{4}.\end{aligned}$ (13)

As a direct interaction between the second and third nodes, an entangled state
$\rho_{23}=Tr_{14}\\{\rho_{1234}(t)\\}$ is generated. Its non-zero elements
are given by,

$\displaystyle\begin{aligned}
\rho_{11}&=\beta_{2}\beta_{1}(\gamma_{2}\gamma_{2}^{\ast}+\gamma_{4}\gamma_{4}^{\ast}),\quad\rho_{14}=\beta_{2}\beta_{1}(\gamma_{4}\gamma_{2}^{\ast}+\gamma_{2}\gamma_{4}^{\ast}),\\\
\rho_{22}&=\beta_{2}\beta_{1}(\gamma_{1}\gamma_{1}^{\ast}+\gamma_{3}\gamma_{3}^{\ast}),\quad\rho_{23}=\beta_{2}\beta_{1}(\gamma_{3}\gamma_{1}^{\ast}+\gamma_{1}\gamma_{3}^{\ast}),\\\
\rho_{32}&=\rho_{23},\quad\rho_{33}=\rho_{22},\quad\rho_{41}=\rho_{14},\quad\rho_{44}=\rho_{11}.\end{aligned}$
(14)

###  2.2 Three entangled nodes

In this subsection, we investigate the amount of correlations that may be
generated between three different nodes via the spin Dipolar interaction.
There are three possibilities of generated $3$-nodes channels, either
$\rho_{123}$, $\rho_{234}$, or $\rho_{124}$. Each channel is described by a
matrix of size $8\times 8$ elements. The three nodes channel $\rho_{123}(t)$
is given by,

$\displaystyle\begin{aligned}
\rho_{123}(t)&=\beta_{1}(A_{2}\gamma_{2}\gamma_{2}^{\ast}+A_{1}\gamma_{4}\gamma_{4}^{\ast})(|000\rangle\langle
000|+|111\rangle\langle
111|)+\beta_{1}(A_{2}\gamma_{4}\gamma_{2}^{\ast}+A_{1}\gamma_{2}\gamma_{4}^{\ast})(|011\rangle\langle
000|+|100\rangle\langle 111|)\\\
&+\beta_{1}(A_{4}\gamma_{3}\gamma_{2}^{\ast}+A_{3}\gamma_{1}\gamma_{4}^{\ast})(|101\rangle\langle
000|+|010\rangle\langle
111|)+\beta_{1}(A_{4}\gamma_{1}\gamma_{2}^{\ast}+A_{3}\gamma_{3}\gamma_{4}^{\ast})(|110\rangle\langle
000|+|001\rangle\langle 111|)\\\
&+\beta_{1}(A_{2}\gamma_{1}\gamma_{1}^{\ast}+A_{1}\gamma_{3}\gamma_{3}^{\ast})(|001\rangle\langle
001|+|110\rangle\langle
110|)+\beta_{1}(A_{2}\gamma_{3}\gamma_{1}^{\ast}+A_{1}\gamma_{1}\gamma_{3}^{\ast})(|010\rangle\langle
001|+|101\rangle\langle 110|)\\\
&+\beta_{1}(A_{4}\gamma_{4}\gamma_{1}^{\ast}+A_{3}\gamma_{2}\gamma_{3}^{\ast})(|100\rangle\langle
001|+|011\rangle\langle
110|)+\beta_{1}(A_{4}\gamma_{2}\gamma_{1}^{\ast}+A_{3}\gamma_{4}\gamma_{3}^{\ast})(|111\rangle\langle
001|+|000\rangle\langle 110|)\\\
&+\beta_{1}(A_{1}\gamma_{3}\gamma_{1}^{\ast}+A_{2}\gamma_{1}\gamma_{3}^{\ast})(|001\rangle\langle
010|+|110\rangle\langle
101|)+\beta_{1}(A_{1}\gamma_{1}\gamma_{1}^{\ast}+A_{2}\gamma_{3}\gamma_{3}^{\ast})(|010\rangle\langle
010|+|101\rangle\langle 101|)\\\
&+\beta_{1}(A_{3}\gamma_{2}\gamma_{1}^{\ast}+A_{4}\gamma_{4}\gamma_{3}^{\ast})(|100\rangle\langle
010|+|011\rangle\langle
101|)+\beta_{1}(A_{3}\gamma_{4}\gamma_{1}^{\ast}+A_{4}\gamma_{2}\gamma_{3}^{\ast})(|111\rangle\langle
010|+|000\rangle\langle 101|)\\\
&+\beta_{1}(A_{1}\gamma_{4}\gamma_{2}^{\ast}+A_{2}\gamma_{2}\gamma_{4}^{\ast})(|000\rangle\langle
011|+|111\rangle\langle
100|)+\beta_{1}(A_{1}\gamma_{2}\gamma_{2}^{\ast}+A_{2}\gamma_{4}\gamma_{4}^{\ast})(|011\rangle\langle
011|+|100\rangle\langle 100|)\\\
&+\beta_{1}(A_{3}\gamma_{1}\gamma_{2}^{\ast}+A_{4}\gamma_{3}\gamma_{4}^{\ast})(|101\rangle\langle
011|+|010\rangle\langle
100|)+\beta_{1}(A_{3}\gamma_{3}\gamma_{2}^{\ast}+A_{4}\gamma_{1}\gamma_{4}^{\ast})(|110\rangle\langle
011|+|001\rangle\langle 100|).\end{aligned}$ (15)

Similarly the three nodes channel $\rho_{234}$ is given by,

$\displaystyle\begin{aligned}
\rho_{(234)}(t)&=\beta_{2}(B_{2}\gamma_{2}\gamma_{2}^{\ast}+B_{1}\gamma_{4}\gamma_{4}^{\ast})(|000\rangle\langle
000|+|111\rangle\langle
111|)+\beta_{2}(B_{4}\gamma_{1}\gamma_{2}^{\ast}+B_{3}\gamma_{3}\gamma_{4}^{\ast})(|011\rangle\langle
000|+|100\rangle\langle 111|)\\\
&+\beta_{2}(B_{4}\gamma_{3}\gamma_{2}^{\ast}+B_{3}\gamma_{1}\gamma_{4}^{\ast})(|101\rangle\langle
000|+|010\rangle\langle
111|)+\beta_{2}(B_{2}\gamma_{4}\gamma_{2}^{\ast}+B_{1}\gamma_{2}\gamma_{4}^{\ast})(|110\rangle\langle
000|+|001\rangle\langle 111|)\\\
&+\beta_{2}(B_{1}\gamma_{2}\gamma_{2}^{\ast}+B_{2}\gamma_{4}\gamma_{4}^{\ast})(|001\rangle\langle
001|+|110\rangle\langle
110|)+\beta_{2}(B_{3}\gamma_{1}\gamma_{2}^{\ast}+B_{4}\gamma_{3}\gamma_{4}^{\ast})(|010\rangle\langle
001|+|101\rangle\langle 110|)\\\
&+\beta_{2}(B_{3}\gamma_{3}\gamma_{2}^{\ast}+B_{4}\gamma_{1}\gamma_{4}^{\ast})(|100\rangle\langle
001|+|011\rangle\langle
110|)+\beta_{2}(B_{1}\gamma_{4}\gamma_{2}^{\ast}+B_{2}\gamma_{2}\gamma_{4}^{\ast})(|111\rangle\langle
001|+|000\rangle\langle 110|)\\\
&+\beta_{2}(B_{3}\gamma_{2}\gamma_{1}^{\ast}+B_{4}\gamma_{4}\gamma_{3}^{\ast})(|001\rangle\langle
010|+|110\rangle\langle
101|)+\beta_{2}(B_{1}\gamma_{1}\gamma_{1}^{\ast}+B_{2}\gamma_{3}\gamma_{3}^{\ast})(|010\rangle\langle
010|+|101\rangle\langle 101|)\\\
&+\beta_{2}(B_{1}\gamma_{3}\gamma_{1}^{\ast}+B_{2}\gamma_{1}\gamma_{3}^{\ast})(|100\rangle\langle
010|+|011\rangle\langle
101|)+\beta_{2}(B_{3}\gamma_{4}\gamma_{1}^{\ast}+B_{4}\gamma_{2}\gamma_{3}^{\ast})(|111\rangle\langle
010|+|000\rangle\langle 101|)\\\
&+\beta_{2}(B_{4}\gamma_{2}\gamma_{1}^{\ast}+B_{3}\gamma_{4}\gamma_{3}^{\ast})(|000\rangle\langle
011|+|111\rangle\langle
100|)+\beta_{2}(B_{2}\gamma_{1}\gamma_{1}^{\ast}+B_{1}\gamma_{3}\gamma_{3}^{\ast})(|011\rangle\langle
011|+|100\rangle\langle 100|)\\\
&+\beta_{2}(B_{2}\gamma_{3}\gamma_{1}^{\ast}+B_{1}\gamma_{1}\gamma_{3}^{\ast})(|101\rangle\langle
011|+|010\rangle\langle
100|)+\beta_{2}(B_{2}\gamma_{3}\gamma_{1}^{\ast}+B_{1}\gamma_{1}\gamma_{3}^{\ast})(|110\rangle\langle
011|+|001\rangle\langle 100|).\end{aligned}$ (16)

Finally, the channel that connect the first, second and the fourth nodes is
defined as,

$\displaystyle\begin{aligned}
\rho_{124}&=(A_{2}B_{1}\gamma_{1}\gamma_{1}^{\ast}+A_{2}B_{2}\gamma_{2}\gamma_{2}^{\ast}+A_{1}B_{2}\gamma_{3}\gamma_{3}^{\ast}+A_{1}B_{1}\gamma_{4}\gamma_{4}^{\ast})|000\rangle\langle
000|\\\
&+(A_{2}B_{3}\gamma_{4}\gamma_{1}^{\ast}+A_{2}B_{4}\gamma_{3}\gamma_{2}^{\ast}+A_{1}B_{4}\gamma_{2}\gamma_{3}^{\ast}+A_{1}B_{3}\gamma_{1}\gamma_{4}^{\ast})(|011\rangle\langle
000|+|100\rangle\langle 111|+|111\rangle\langle 111|)\\\
&+(A_{4}B_{3}\gamma_{3}\gamma_{1}^{\ast}+A_{4}B_{4}\gamma_{4}\gamma_{2}^{\ast}+A_{3}B_{4}\gamma_{1}\gamma_{3}^{\ast}+A_{3}B_{3}\gamma_{2}\gamma_{4}^{\ast})(|101\rangle\langle
000|+|010\rangle\langle 111|)\\\
&+(A_{4}B_{1}\gamma_{2}\gamma_{1}^{\ast}+A_{4}B_{2}\gamma_{1}\gamma_{2}^{\ast}+A_{3}B_{2}\gamma_{4}\gamma_{3}^{\ast}+A_{3}B_{1}\gamma_{3}\gamma_{4}^{\ast})(|110\rangle\langle
000|+|001\rangle\langle 111|)\\\
&+(A_{2}B_{2}\gamma_{1}\gamma_{1}^{\ast}+A_{2}B_{1}\gamma_{2}\gamma_{2}^{\ast}+A_{1}B_{1}\gamma_{3}\gamma_{3}^{\ast}+A_{1}B_{2}\gamma_{4}\gamma_{4}^{\ast})(|001\rangle\langle
001|+|110\rangle\langle 110|)\\\
&+(A_{2}B_{4}\gamma_{4}\gamma_{1}^{\ast}+A_{2}B_{3}\gamma_{3}\gamma_{2}^{\ast}+A_{1}B_{3}\gamma_{2}\gamma_{3}^{\ast}+A_{1}B_{4}\gamma_{1}\gamma_{4}^{\ast})(|010\rangle\langle
001|+|101\rangle\langle 110|)\\\
&+(A_{4}B_{4}\gamma_{3}\gamma_{1}^{\ast}+A_{4}B_{3}\gamma_{4}\gamma_{2}^{\ast}+A_{3}B_{3}\gamma_{1}\gamma_{3}^{\ast}+A_{3}B_{4}\gamma_{2}\gamma_{4}^{\ast})(|100\rangle\langle
001|+|011\rangle\langle 110|)\\\
&+(A_{4}B_{2}\gamma_{2}\gamma_{1}^{\ast}+A_{4}B_{1}\gamma_{1}\gamma_{2}^{\ast}+A_{3}B_{1}\gamma_{4}\gamma_{3}^{\ast}+A_{3}B_{2}\gamma_{3}\gamma_{4}^{\ast})(|111\rangle\langle
001|+|000\rangle\langle 110|)\\\
&+(A_{1}B_{4}\gamma_{4}\gamma_{1}^{\ast}+A_{1}B_{3}\gamma_{3}\gamma_{2}^{\ast}+A_{2}B_{3}\gamma_{2}\gamma_{3}^{\ast}+A_{2}B_{4}\gamma_{1}\gamma_{4}^{\ast})(|001\rangle\langle
010|+|110\rangle\langle 101|)\\\
&+(A_{1}B_{2}\gamma_{1}\gamma_{1}^{\ast}+A_{1}B_{1}\gamma_{2}\gamma_{2}^{\ast}+A_{2}B_{1}\gamma_{3}\gamma_{3}^{\ast}+A_{2}B_{2}\gamma_{4}\gamma_{4}^{\ast})(|010\rangle\langle
010|+|101\rangle\langle 101|)\\\
&+(A_{3}B_{2}\gamma_{2}\gamma_{1}^{\ast}+A_{3}B_{1}\gamma_{1}\gamma_{2}^{\ast}+A_{4}B_{1}\gamma_{4}\gamma_{3}^{\ast}+A_{4}B_{2}\gamma_{3}\gamma_{4}^{\ast})(|100\rangle\langle
010|+|011\rangle\langle 101|)\\\
&+(A_{3}B_{4}\gamma_{3}\gamma_{1}^{\ast}+A_{3}B_{3}\gamma_{4}\gamma_{2}^{\ast}+A_{4}B_{3}\gamma_{1}\gamma_{3}^{\ast}+A_{4}B_{4}\gamma_{2}\gamma_{4}^{\ast})(|111\rangle\langle
010|+|000\rangle\langle 101|)\\\
&+(A_{1}B_{3}\gamma_{4}\gamma_{1}^{\ast}+A_{1}B_{4}\gamma_{3}\gamma_{2}^{\ast}+A_{2}B_{4}\gamma_{2}\gamma_{3}^{\ast}+A_{2}B_{3}\gamma_{1}\gamma_{4}^{\ast})(|000\rangle\langle
011|+|111\rangle\langle 100|)\\\
&+(A_{1}B_{1}\gamma_{1}\gamma_{1}^{\ast}+A_{1}B_{2}\gamma_{2}\gamma_{2}^{\ast}+A_{2}B_{2}\gamma_{3}\gamma_{3}^{\ast}+A_{2}B_{1}\gamma_{4}\gamma_{4}^{\ast})(|011\rangle\langle
011|+|100\rangle\langle 100|)\\\
&+(A_{3}B_{1}\gamma_{2}\gamma_{1}^{\ast}+A_{3}B_{2}\gamma_{1}\gamma_{2}^{\ast}+A_{4}B_{2}\gamma_{4}\gamma_{3}^{\ast}+A_{4}B_{1}\gamma_{3}\gamma_{4}^{\ast})(|101\rangle\langle
011|+|010\rangle\langle 100|)\\\
&+(A_{3}B_{3}\gamma_{3}\gamma_{1}^{\ast}+A_{3}B_{4}\gamma_{4}\gamma_{2}^{\ast}+A_{4}B_{4}\gamma_{1}\gamma_{3}^{\ast}+A_{4}B_{3}\gamma_{2}\gamma_{4}^{\ast})(|110\rangle\langle
011|+|001\rangle\langle 100|).\end{aligned}$ (17)

###  2.3 Extension of Network

It is assumed that, we have two networks each consists of four nodes. These
two networks are connected via spin Dipolar interaction where the terminal
node of the first network interacts with the first node of the second network
to generate an entangled hop consists of $8$ nodes. We continue to generate a
large network by allowing these hops of the eighth nodes interact to generate
a larger entangled hops and so on. In the following we consider only the
channel between the terminals of this network, namely the channel $\rho_{18}$.
The non-zero elements of the $4\times 4$ matrix are given by,

$\displaystyle\begin{aligned}
\rho_{11}&=\delta_{3}^{2}((\gamma_{2}\gamma_{2}^{\ast}+\gamma_{4}\gamma_{4}^{\ast})M_{11}^{2}+(\gamma_{1}\gamma_{1}^{\ast}+\gamma_{3}\gamma_{3}^{\ast})(M_{22}+M_{33})M_{11}+(\gamma_{2}\gamma_{2}^{\ast}+\gamma_{4}\gamma_{4}^{\ast})M_{22}M_{33}),\\\
\rho_{14}&=\delta_{3}^{2}((\gamma_{4}\gamma_{2}^{\ast}+\gamma_{2}\gamma_{4}^{\ast})M_{14}^{2}+(\gamma_{3}\gamma_{1}^{\ast}+\gamma_{1}\gamma_{3}^{\ast})(M_{23}+M_{32})M_{14}+(\gamma_{4}\gamma_{2}^{\ast}+\gamma_{2}\gamma_{4}^{\ast})M_{23}M_{32}),\\\
\rho_{22}&=\delta_{3}^{2}(M_{11}(\gamma_{1}\gamma_{1}^{\ast}M_{44}+\gamma_{2}\gamma_{2}^{\ast}M_{22}+\gamma_{3}\gamma_{3}^{\ast}M_{44}+\gamma_{4}\gamma_{4}^{\ast}M_{22})+M_{22}(\gamma_{1}\gamma_{1}^{\ast}M_{22}+\gamma_{2}\gamma_{2}^{\ast}M_{44}\\\
&+\gamma_{3}\gamma_{3}^{\ast}M_{22}+\gamma_{4}\gamma_{4}^{\ast}M_{44})),\\\
\rho_{23}&=\delta_{3}^{2}(M_{14}(\gamma_{3}\gamma_{1}^{\ast}M_{41}+\gamma_{4}\gamma_{2}^{\ast}M_{23}+\gamma_{1}\gamma_{3}^{\ast}M_{41}+\gamma_{2}\gamma_{4}^{\ast}M_{23})+M_{23}(\gamma_{3}\gamma_{1}^{\ast}M_{23}+\gamma_{4}\gamma_{2}^{\ast}M_{41}\\\
&+\gamma_{1}\gamma_{3}^{\ast}M_{23}+\gamma_{2}\gamma_{4}^{\ast}M_{41})),\\\
\rho_{32}&=\delta_{3}^{2}(M_{14}(\gamma_{3}\gamma_{1}^{\ast}M_{41}+\gamma_{4}\gamma_{2}^{\ast}M_{32}+\gamma_{1}\gamma_{3}^{\ast}M_{41}+\gamma_{2}\gamma_{4}^{\ast}M_{32})+M_{32}(\gamma_{3}\gamma_{1}^{\ast}M_{32}+\gamma_{4}\gamma_{2}^{\ast}M_{41}\\\
&+\gamma_{1}\gamma_{3}^{\ast}M_{32}+\gamma_{2}\gamma_{4}^{\ast}M_{41})),\\\
\rho_{33}&=\delta_{3}^{2}(M_{11}(\gamma_{1}\gamma_{1}^{\ast}M_{44}+\gamma_{2}\gamma_{2}^{\ast}M_{33}+\gamma_{3}\gamma_{3}^{\ast}M_{44}+\gamma_{4}\gamma_{4}^{\ast}M_{33})+M_{33}(\gamma_{1}\gamma_{1}^{\ast}M_{33}+\gamma_{2}\gamma_{2}^{\ast}M_{44}\\\
&+\gamma_{3}\gamma_{3}^{\ast}M_{33}+\gamma_{4}\gamma_{4}^{\ast}M_{44})),\\\
\rho_{41}&=\delta_{3}^{2}(M_{23}(\gamma_{3}\gamma_{1}^{\ast}M_{41}+\gamma_{4}\gamma_{2}^{\ast}M_{32}+\gamma_{1}\gamma_{3}^{\ast}M_{41}+\gamma_{2}\gamma_{4}^{\ast}M_{32})+M_{41}(\gamma_{3}\gamma_{1}^{\ast}M_{32}+\gamma_{4}\gamma_{2}^{\ast}M_{41}\\\
&+\gamma_{1}\gamma_{3}^{\ast}M_{32}+\gamma_{2}\gamma_{4}^{\ast}M_{41})),\\\
\rho_{44}&=\delta_{3}^{2}(M_{22}(\gamma_{1}\gamma_{1}^{\ast}M_{44}+\gamma_{2}\gamma_{2}^{\ast}M_{33}+\gamma_{3}\gamma_{3}^{\ast}M_{44}+\gamma_{4}\gamma_{4}^{\ast}M_{33})+M_{44}(\gamma_{1}\gamma_{1}^{\ast}M_{33}+\gamma_{2}\gamma_{2}^{\ast}M_{44}\\\
&+\gamma_{3}\gamma_{3}^{\ast}M_{33}+\gamma_{4}\gamma_{4}^{\ast}M_{44})),\end{aligned}$
(18)

where,

$\displaystyle\delta_{1}$ $\displaystyle=$ $\displaystyle
N_{11}+N_{22},\quad\delta_{2}=N_{33}+N_{44},\delta_{3}=N_{11}+N_{22}+N_{33}+N_{44},$
$\displaystyle\delta_{4}$ $\displaystyle=$
$\displaystyle(\gamma_{2}\gamma_{2}^{\ast}+\gamma_{4}\gamma_{4}^{\ast})M_{11}^{2}+M_{22}((\gamma_{1}\gamma_{1}^{\ast}+\gamma_{3}\gamma_{3}^{\ast})M_{22}+(\gamma_{2}\gamma_{2}^{\ast}+\gamma_{4}\gamma_{4}^{\ast})(M_{33}+M_{44}))$
$\displaystyle+M_{11}((\gamma_{1}\gamma_{1}^{\ast}+\gamma_{3}\gamma_{3}^{\ast})(M_{33}+M_{44})+2M_{22}(r_{1}(r_{1}){}^{*}+r_{2}(r_{2}){}^{*}+r_{3}(r_{3}){}^{*}+r_{4}(r_{4}){}^{*})),$
$\displaystyle\delta_{5}$ $\displaystyle=$ $\displaystyle
M_{11}(\gamma_{1}\gamma_{1}^{\ast}M_{44}+\gamma_{2}\gamma_{2}^{\ast}M_{33}+\gamma_{3}\gamma_{3}^{\ast}M_{44}+\gamma_{4}\gamma_{4}^{\ast}M_{33})+M_{22}(\gamma_{1}\gamma_{1}^{\ast}M_{44}+\gamma_{2}\gamma_{2}^{\ast}M_{33}$
(19)
$\displaystyle+\gamma_{3}\gamma_{3}^{\ast}M_{44}+\gamma_{4}\gamma_{4}^{\ast}M_{33})+(M_{33}+M_{44})(\gamma_{1}\gamma_{1}^{\ast}M_{33}+\gamma_{2}\gamma_{2}^{\ast}M_{44}+\gamma_{3}\gamma_{3}^{\ast}M_{33}+\gamma_{4}\gamma_{4}^{\ast}M_{44}).$

and,

$\displaystyle\begin{aligned}
M_{11}&=A_{1}(B_{2}\gamma_{1}\gamma_{1}^{\ast}+B_{1}\gamma_{2}\gamma_{2}^{\ast}+B_{2}\gamma_{3}\gamma_{3}^{\ast}+B_{1}\gamma_{4}\gamma_{4}^{\ast})+A_{2}(B_{1}\gamma_{1}\gamma_{1}^{\ast}+B_{2}\gamma_{2}\gamma_{2}^{\ast}+B_{1}\gamma_{3}\gamma_{3}^{\ast}+B_{2}\gamma_{4}\gamma_{4}^{\ast}),\\\
M_{14}&=A_{3}(B_{4}\gamma_{3}\gamma_{1}^{\ast}+B_{3}\gamma_{4}\gamma_{2}^{\ast}+B_{4}\gamma_{1}\gamma_{3}^{\ast}+B_{3}\gamma_{2}\gamma_{4}^{\ast})+A_{4}(B_{3}\gamma_{3}\gamma_{1}^{\ast}+B_{4}\gamma_{4}\gamma_{2}^{\ast}+B_{3}\gamma_{1}\gamma_{3}^{\ast}+B_{4}\gamma_{2}\gamma_{4}^{\ast}),\\\
M_{22}&=A_{2}(B_{2}\gamma_{1}\gamma_{1}^{\ast}+B_{1}\gamma_{2}\gamma_{2}^{\ast}+B_{2}\gamma_{3}\gamma_{3}^{\ast}+B_{1}\gamma_{4}\gamma_{4}^{\ast})+A_{1}(B_{1}\gamma_{1}\gamma_{1}^{\ast}+B_{2}\gamma_{2}\gamma_{2}^{\ast}+B_{1}\gamma_{3}\gamma_{3}^{\ast}+B_{2}\gamma_{4}\gamma_{4}^{\ast}),\\\
M_{23}&=A_{4}(B_{4}\gamma_{3}\gamma_{1}^{\ast}+B_{3}\gamma_{4}\gamma_{2}^{\ast}+B_{4}\gamma_{1}\gamma_{3}^{\ast}+B_{3}\gamma_{2}\gamma_{4}^{\ast})+A_{3}(B_{3}\gamma_{3}\gamma_{1}^{\ast}+B_{4}\gamma_{4}\gamma_{2}^{\ast}+B_{3}\gamma_{1}\gamma_{3}^{\ast}+B_{4}\gamma_{2}\gamma_{4}^{\ast}),\\\
M_{32}&=M_{23},\quad M_{33}=M_{22},\quad M_{41}=M_{14},\quad
M_{44}=M_{11},\quad
N_{11}=\beta_{2}\beta_{1}(\gamma_{2}\gamma_{2}^{\ast}+\gamma_{4}\gamma_{4}^{\ast}),\\\
N_{14}&=\beta_{2}\beta_{1}(\gamma_{4}\gamma_{2}^{\ast}+\gamma_{2}\gamma_{4}^{\ast}),\quad
N_{22}=\beta_{2}\beta_{1}(\gamma_{1}\gamma_{1}^{\ast}+\gamma_{3}\gamma_{3}^{\ast}),\quad
N_{23}=\beta_{2}\beta_{1}(\gamma_{3}\gamma_{1}^{\ast}+\gamma_{1}\gamma_{3}^{\ast}),\\\
N_{32}&=N_{23},\quad N_{33}=N_{22},\quad N_{41}=N_{14},\quad
N_{44}=N_{11}.\end{aligned}$ (20)

##  3 Quantifying the correlations

  * •

Two nodes entanglement:Negativity
Negativity is considered as one of the most common entanglement measure of a
two-qubit system, where it satisfies all the conditions as a measure of
entanglement [12]. However, for any quantum state $\varrho_{ab}$, the
negativity is defined by means of the eigenvalues of the partial transpose of
the state $\varrho_{ab}$. Mathematically, it may be defined as,

$\mathcal{N}_{eg}=2\sum_{i}|\lambda_{i}|,$ (21)

where $\lambda_{i},i=1,2,3,4$ are the eigenvalues of the density operator
$\varrho_{ab}^{T_{b}}$.

  * •

Three nodes entanglement:Tangle
To quantify the amount of entanglement that may be generated between each
three nodes($3$ qubits) is quantified by the $\pi$-tangle which proved to be a
natural entanglement measure [13]. For a three-qubit (node) state $\rho$, a
monogamy inequality analogous to Coffman-Kundu-Wootters (CKW) inequality is
satisfies on the single qubit as the entanglement quantified by the negativity
between A and B; A and C, and, between $A$, $BC$ as following,

$\displaystyle\begin{aligned}
\mathcal{N}_{AB}^{2}+\mathcal{N}_{AC}^{2}\leq\mathcal{N}_{A(BC)}^{2}.\end{aligned}$
(22)

In a similar way, if one takes the different focus B and C, the following
monogamy inequalities are defined by,

$\displaystyle\begin{aligned}
&\mathcal{N}_{BA}^{2}+\mathcal{N}_{BC}^{2}\leq\mathcal{N}_{B(AC)}^{2},\\\
&\mathcal{N}_{CA}^{2}+\mathcal{N}_{CB}^{2}\leq\mathcal{N}_{C(AB)}^{2},\\\
\end{aligned}$ (23)

The $\pi$-tangle is defined in terms of the global negativities [14] as,

$\displaystyle\begin{aligned}
\mathcal{N}^{A}=\parallel\rho^{T_{A}}\parallel-1,\qquad\mathcal{N}^{B}=\parallel\rho^{T_{B}}\parallel-1,\qquad\mathcal{N}^{C}=\parallel\rho^{T_{C}}\parallel-1.\end{aligned}$
(24)

The superscripts $T_{A}$, $T_{B}$, and $T_{C}$ represent the partial
transposes of $\rho$. Then, the $\pi$-tangle is defined as,

$\displaystyle\begin{aligned}
\pi_{ABC}=\frac{1}{3}(\pi_{A}+\pi_{B}+\pi_{C}),\end{aligned}$ (25)

where

$\displaystyle\begin{aligned}
\pi_{A}&=\mathcal{N}_{A(BC)}^{2}-(\mathcal{N}_{(AB)}^{2}+\mathcal{N}_{(AC)}^{2}),\\\
\pi_{B}&=\mathcal{N}_{B(AC)}^{2}-(\mathcal{N}_{(AB)}^{2}+\mathcal{N}_{(BC)}^{2}),\\\
\pi_{C}&=\mathcal{N}_{C(AB)}^{2}-(\mathcal{N}_{(AC)}^{2}+\mathcal{N}_{(BC)}^{2}).\\\
\end{aligned}$ (26)

  * •

Non-local advantage
For a two-qubit state $\rho_{AB}$, based on local measurements on subsystem A
and classical communication between the bipartite, the nonlocal advantage of
quantum coherence (NAQC)[15] can be attained by the conditional state of
subsystem $B$. The one norm coherence $C_{l_{1}}^{na1}$ is defined as,

$\displaystyle\begin{aligned} C_{l_{1}}^{na1}&=\sum_{i\neq j}|\langle
i|\rho_{ij}|j\rangle|.\end{aligned}$ (27)

Thus, the criterion for achieving a NAQC on qubit $B$ can be derived via the
possible probabilistic averaging methods [16]

$\displaystyle\tilde{C}_{\alpha}^{na}=\frac{1}{2}\sum_{\begin{subarray}{c}i,j,a\\\
i\neq
j\end{subarray}}p_{a|\prod_{i}^{a}}C_{\alpha}^{\sigma_{j}}(\rho_{B|\prod_{i}^{a}})>C_{\alpha}^{m},$
(28)

where $C_{\alpha}^{\sigma_{j}}(.)$($\alpha=l_{1}$ ) represents the quantum
coherence with respect to the reference basis spanned by the eigenstates of
$\sigma_{j}$, and the two critical values are given by
$C_{l_{1}}^{m}=\sqrt{6}$. The measured state for the two-qubit state may be
described as,

$\rho_{B|\prod_{i}^{a}}=\frac{Tr_{A}[(\prod_{i}^{a}\otimes
I_{2})\rho_{AB}]}{p_{a|\prod_{i}^{a}}}.$ (29)

Based on the criterion of equation (28), Hu et. al,[17] proposed to
characterize quantitatively the degree of the non-local coherent advantage
$\mathcal{N}_{La}$ of bipartite state as,

$\mathcal{N}_{La}(\rho_{AB})=\max\\{0,\frac{\tilde{C}_{\alpha}^{na}(\rho_{AB})-C_{\alpha}^{m}}{\tilde{C}_{\alpha,\max}^{na}-C_{\alpha}^{m}}\\},$
(30)

where $\tilde{C}_{\alpha,\max}^{na}=\max_{\rho_{AB}\in D(C^{d\times
d})}\tilde{C}_{\alpha}^{na}(\rho_{AB})$ and for the two-qubit states we have
$\tilde{C}_{\alpha,\max}^{na}=3$.

##  4 Numerical results

###  4.1  Entangled two nodes

In this subsection, we investigate the behavior of negativity,
$\mathcal{N}_{eg}$ and the local advantage coherent information,
$\mathcal{N}_{La}$ of a channel between two different nodes. These nodes may
be entangled from the beginning as $\rho_{12}$ and $\rho_{34}$, or
direct/indirect interacted nodes via Dipolar interaction as $\rho_{23}$ and
$\rho_{14}$, respectively.

($a$)$\mathcal{N}_{eg}$$\tau$$\tilde{\epsilon}$($b$)$\mathcal{N}_{La}$$\tau$$\tilde{\epsilon}$

Figure 2: The behavior of the negativity, $\mathcal{N}_{eg}$ of $\rho_{12}$ of
the four qubits network as a function of $\tau$ and $\tilde{\epsilon}$, where
the network is initially of the type $MM$ network.

Fig.(2) shows the behavior of the negativity and the non-local coherent
advantage of the quantum channel $\rho_{12}(t)$, where the initial network is
prepared by using two singlet states. At $t=0$, both $\mathcal{N}_{eg}$ and
$\mathcal{N}_{La}$ are maximum, where $\rho_{12}(0)$ is a maximum entangled
state. However, as soon as the interaction is switched on, both quantifiers
decreases, suddenly to death completely. At further time, both of
$\mathcal{N}_{eg}$ and $\mathcal{N}_{La}$ rebirth again to reach their maximum
bounds periodically. From a geometrical perspective view of the these
quantifiers, they behave as peaks, where the area of the base that predicted
for the negativity is much larger than that displayed for the non-local
coherent advantage. This means that, the non-local coherent advantage
decreases faster than the negativity. Moreover, the number of peaks increases
as the interaction time increases. The negative values of $\bar{\epsilon}$,
namely $\Delta<0$ means that, the Dipolar interaction is polarized in the
$x-y$ plane, while the positive values of $\bar{\epsilon}$, i.e., $\Delta>0$
indicates the Dipolar interaction is switched in the $z-$ axis. To display the
effect of the negative and positive values of the coupling $\bar{\epsilon}$,
we discuss the behavior of the $\mathcal{N}_{eg}$ and $\mathcal{N}_{La}$ at
some different values of $\bar{\epsilon}$.

($a$)$\mathcal{N}_{eg}$$\tau$($b$)$\mathcal{N}_{eg}$$\tau$
($c$)$\mathcal{N}_{La}$$\tau$($d$)$\mathcal{N}_{La}$$\tau$

Figure 3: The behavior of the negativity $\mathcal{N}_{eg}$ is represented by
(a,b) and the non-local coherent advantage, $\mathcal{N}_{La}$ is shown in
(c,d)of $\rho_{12}$. The solid, dash, the dot and the dash-dot lines are
evaluated at $\tilde{\epsilon}=-0.2,0,0.1,0.3$, respectively. It is assumed
that, the network is initially of type $MM$ network while it is of $WW$
network in $(a,c)$.

In Fig.(3), we display the sensitive of both quantifiers to the
positive/negative values of the coupling $\bar{\epsilon}$, where it is assumed
that, the initial network is constructed either from maximum or partial
entangled nodes. The general behavior of both quantifiers displays the
phenomena of the sudden death/birth of the negativity and the non-local
coherent advantage. The death time that is shown for the negativity is smaller
than that displayed for the non-local coherent advantage. Moreover, the
maximum values of $\mathcal{N}_{eg}$ and $\mathcal{N}_{La}$ that predict in
Figs.(a,c) are much larger than those displayed in Figs.(b,d), where the
initial network is of $MM$ and $WW$ types, respectively.

###  4.2 Indirect entangled nodes

As it is mentioned above, due to the interaction there are some entangled two
nodes channels are generated. In this subsection, we quantify the amount of
entanglement, as well as, the non-local coherent advantage that contained in
the quantum channel between the first and the forth nodes, namely $\rho_{14}$.

($a$)$\mathcal{N}_{eg}$$\tau$$\tilde{\epsilon}$($a$)$\mathcal{N}_{La}$$\tau$$\tilde{\epsilon}$

Figure 4:  The same as Fig.(2), but for the two nodes $\rho_{14}.$

Fig.(4), describes the behavior of both quantifiers for the generated
entangled two nodes channels $\rho_{14}$, where we assume that, the initial
network is of $MM$ type. As it is displayed from Fig.(4a), the behavior of the
negativity $\mathcal{N}_{eg}$ is similar to that displayed in Fig.(2a).
However, the number of picks are larger than those displayed for the channel
nodes $\rho_{12}$. Moreover, the maximum bounds of entanglement that displayed
for $\rho_{14}$ are smaller than those displayed for the channel $\rho_{12}$.
Fig.(4b), describes the behavior of the non-local coherent advantage for the
channel $\rho_{14}$. The behavior of $\mathcal{N}_{La}$ is similar to that
shown in Fig.(2a). However, the number of peaks is larger than those displayed
for the channel $\rho_{12}$. Moreover, the behavior of both quantifiers shows
that, the peaks of $\rho_{14}$ appear faster than those displayed for the
channel $\rho_{12}$.

From Fig.(2) and (4), one may conclude that, the entanglement that generated
indirectly is more stable than that between the initial entangled nodes, where
the vanishing time of both quantifiers that depicted for the indirect
connected nodes is much smaller than that displayed for the initially
entangled.

($a$)$\mathcal{N}_{eg}$$\tau$($b$)$\mathcal{N}_{eg}$$\tau$($c$)$\mathcal{N}_{eg}$$\tau$

Figure 5: The negativity behavior of $\rho_{14}$, where different values of
the coupling constant are considered. The solid, dash, the dot and the dash-
dot curves represents the $\mathcal{N}_{eg}$ at
$\tilde{\epsilon}=-0.2,0,0.1,0.3$, respectively (a) $MM$ network, (b) $WW$
network, and (c)$MW$ network.

($a$)$\mathcal{N}_{La}$$\tau$($b$)$\mathcal{N}_{La}$$\tau$($c$)$\mathcal{N}_{La}$$\tau$

Figure 6: The same as Fig.(4) but for the non-local coherent advantage,
$\mathcal{N}_{La}$ of $\rho_{14}$.

In Fig.(5), we examine the effect considered some values of the coupling
strength $\tilde{\epsilon}$, and different settings of the initial networks on
the negativity of the entangled nodes $\rho_{14}$. The behavior is similar to
that shown in Fig.(3). However, at $t>0$, the entanglement reaches its maximum
values faster and it appears for a short time. The maximum bounds of
$\mathcal{N}_{eg}$ that depicted at the different values of $\tilde{\epsilon}$
are similar. It is clear that, at $\tilde{\epsilon}=0$, and $\Delta$ could be
arbitrary, the sudden time death of the entanglement is very short and the
entanglement oscillates only between $0$ and the maximum values
$(\mathcal{N}_{eg}\simeq 0.6)$. However, the smallest values of the
entanglement are displayed when the Dipolar interaction is switched on the
$z-$ axis, namely $\Delta>0$. The initial entangled network has a clear effect
on the behavior of the generated entangled nodes. It is clear that, if the
network is initially is of $MM$ type, the amount of entanglement that
generated between the second and the third nodes is the largest one. However,
the amount of entanglement on the channel $\rho_{23}$ is the smallest one,
when the initial network is prepared by using partially entangled nodes
$WW-$type. These results may be seen clearly, by comparing Figs.(5a)(5b) and
(5c), where the network is initially prepared by using maximum-maximum
entangled nodes (MM), Werner-Werner-entangled nodes (WW) or maximum-Werner
entangled nodes (MW), respectively.

The behavior of the non-local coherent advantage of the channel between the
first and the fourth node, $\rho_{14}$ is shown in Fig.(6). The behavior shows
that, it fluctuates between its maximum and minimum values. The sudden changes
of $\mathcal{N}_{La}$ (death/rebirth) appear clearly, where the death time of
this phenomena depends on the strength of the Dipolar interaction. Moreover,
the amount of correlation that quantified by $\mathcal{N}_{La}$ depends on the
initial type of the network. It is clear that, there is no any correlations
are predicted during the same period of interaction time, when the network is
initially prepared by using $WW$ entangled nodes. However, the maximum values
of the non-local coherent advantage are displayed when the initial network is
conducted by using maximum entangled nodes ($MM$-type).

###  4.3 Direct entangled nodes

($a$)$\mathcal{N}_{eg}$$\tau$$\tilde{\epsilon}$($a$)$\mathcal{N}_{La}$$\tau$$\tilde{\epsilon}$

Figure 7: (a) The behavior of the negativity,and (b) the non-local coherent
advantage of the state $\rho_{23}$ of the four qubits network as a function of
$\tau$ and $\tilde{\epsilon}$, where the initial network is $MM$\- type.

The behavior of the entanglement and the non-local coherent advantage that are
displayed for the entangled nodes state $\rho_{23}$ by using direct
interaction is shown in Fig.(7), where it is assumed that, the initial network
is superimposed from maximum entangled nodes ($MM$-type). The behavior is
similar to that displayed in Figs.(4), where the sudden death phenomena of the
non-local coherent advantage appears periodically during a large period of
interaction time. Moreover, the sudden death phenomenon of the non local
coherent advantage decreases as the interaction time increases.

($a$)$\mathcal{N}_{eg}$$\tau$($b$)$\mathcal{N}_{La}$$\tau$

Figure 8: (a) The behavior of the negativity $\mathcal{N}_{eg}$ and (b) The
non-local coherent advantage $\mathcal{N}_{La}$ of the channel $\rho_{23}$.
The solid, dash, the dot and the dash-dot lines represent the
$\mathcal{N}_{eg}$ and $\mathcal{N}_{La}$ at
$\tilde{\epsilon}=-0.2,0,0.1,0.3$, respectively. It is assumed that the
initial network via two pairs of maximum entangled nodes ($MM$-type.

In Fig.(8), we show the behavior of both quantifiers at different values of
the coupling constant $\tilde{\epsilon}$. It is clear that, the sudden death
phenomenon of both quantifiers appears periodically. However, the sudden death
time of the non-local coherent advantage is much larger than that displayed
for the negativity. The upper bounds of $\mathcal{N}_{eg}$ and
$\mathcal{N}_{La}$ depend on the direction of the interaction, where when the
interaction is switched in $x-y$ plane, the negativity is much larger than the
non-local coherent advantage. On the other hand, the smallest values of the
non-local coherent advantage are displayed at zero coupling, namely,
$\epsilon=0$, while $\Delta$ is arbitrary.

In this context, it is important to mention that, the behavior of the
negativity and the non-local coherent advantage between the second and the
third nodes is similar for any initial entangled network. Moreover, our
calculations show that, there is no any quantum correlations are generated
between the other nodes as, the first and the third nodes namely $\rho_{13}$,
and between the second and fourth nodes $\rho_{24}$. These encourage to
discuss the possibility of finding quantum correlations between three
different nodes. The next subsection is devoted for this aim, where we
quantify the amount of entanglement that may be generated between each three
nodes by using the tangle.

###  4.4 Three entangled nodes

In this subsection, we investigate the possibility of generating quantum
correlation between any three different nodes via the spin Dipolar
interaction. The amount of the quantum correlations are quantified by means of
the tangle between all the three possible partitions, namely
$\rho_{123},\rho_{124}$ and $\rho_{234}$. Our results show that, the behavior
of the tangle is similar for all the three partitions. Therefore, we consider
the behavior of $\mathcal{T}_{123}$ only against the interaction strength
$\tilde{\epsilon}$.

($a$)$\tau_{123}$$\tau$$\tilde{\epsilon}$($b$)$\tau_{123}$$\tau$

Figure 9: (a)The behavior of the tangle, $\tau_{123}(\tau,\tilde{\epsilon})$
of $\rho_{123}$, where the initial network of $MM$ type, and (b) represents
$\mathcal{T}_{123}(\tau)$, where the solid, dash, the dot and the dash-dot
lines are evaluated at $\tilde{\epsilon}=-0.2,0,0.1,0.3$, respectively.

Fig.(9), describes the behavior of the amount of entanglement that generated
between the first, second and the third nodes by using the tangle. As it is
displayed from Fig.(9a), the tangle $\mathcal{T}_{123}(\tau,\tilde{\epsilon})$
behaves as peaks, where it oscillates between its maximum and minimum bounds.
Moreover, the phenomenon of the sudden death doesn’t depicted. However, the
sudden changes of the tangle are observed periodically, but never vanishes and
its minimum values are larger than $0.65$. The effect of some values of the
interaction strength is displayed in Fig.(9b), where the amplitudes of
oscillations are small and consequently, the minimum values are much better
than those displayed for the entangled two nodes. As it is shown from
Fig.(9b), the effect of interaction strengths on the tangle is almost similar.

Comparing the behavior of the tangle, which quantifies the amount of quantum
correlation between any three different nodes, and the negativity, which
represents the entanglement between two nodes, one may conclude that, although
the negativity reaches its maximum values at some cases, but it suffers from
sudden death during the interaction time. However, the amount of quantum
correlation between any three nodes are more robust and never vanishes.
Moreover, the behavior of the tangle shows that, the upper bounds of quantum
correlation between a thee qubits are much larger than that displayed between
any two nodes. Also, the initial tangle depends on the initial entangled
nodes, namely it is larger than zero at $\tau=0$.

###  4.5 Extension network

In this subsection, we extend our work to includes 8 nodes, where each 4 nodes
conduct a hop. It is assumed that, the both terminals of the two hops interact
locally via the spin Dipolar interaction and consequently an entangled network
consists of $8$ nodes is generated. By tracking all the intermediate states,
one obtains a final channel between the first and eighth nodes, $\rho_{18}$.

($a$)$\mathcal{N}_{eg}$$\tau$($b$)$\mathcal{N}_{La}$$\tau$

Figure 10:  The behavior of (a) the negativity $\mathcal{N}_{eg}$ and (b)the
non- local coherent advantage, $\mathcal{N}_{La}$ of the channel $\rho_{18}$.
The solid, dash, the dot and the dash-dot lines represent the
$\mathcal{N}_{eg}$ and $\mathcal{N}_{La}$ at
$\tilde{\epsilon}=-0.2,0,0.1,0.3$, respectively, where the initial two hops
are prepared via maximum entangled nodes ($MM$-type).

The behavior of entanglement $\mathcal{N}_{eg}$ and the non local coherent
advantage $\mathcal{N}_{La}$ is displayed in Fig.(10). The non-zero values of
both quantifiers are depicted when both hops are prepared initially by using
maximum entangled nodes. The amount of entanglement is very small compared
with those between $\rho_{14}$ and $\rho_{23}$ and $\rho_{12}$. Moreover, the
upper bounds of the non-local coherent advantage are much larger than those
displayed for the entanglement. Switching on the interaction in $z$\-
direction predicted a large amount of $\mathcal{N}_{La}$, where it vanishes if
it is switched in $x-y$ plane

##  5 Conclusion

The possibility of using the spin Dipolar interaction to generate quantum
entangled network by using multi-nodes is discussed. It is assumed that, each
two nodes are initially prepared either in a maximally or partially
correlated, namely prepared in a singlet or Werner states, respectively. The
suggested network is constructed by four nodes, where the connection either
direcet/inderict via spin Dipolar interaction. Moreover, it is allowed that,
the terminals of the two hops interact directly, to generate an entangled hop
consists of eight nodes. By using the same technique, this network may be
extended to multi-hops entangled network.

Due to the interaction, there is a quantum(classical) correlation is generated
between each two or three nodes. The predictable quantum correlations are
investigated via nonlocal coherent advantage and quantified by the negativity
for each entangled two nodes and by the tangle for the entangled three nodes.
The effect of the interaction strength and the initial entangled network on
the generated network is discussed. Our results show that, due to the
symmetric, each initial entangled nodes behave similarly under the effect of
the interaction strength, namely the quantum correlations between the first
and the second nodes, is the same as that between the third and the fourth
nodes. The predicted amount of quantum correlations between each three nodes
is affected in a similar way by the interaction strength. Moreover, there is
only classical correlations are generated between the first-third and the
second-fourth nodes.

It is shown that, the phenomena of the sudden death/birth is displayed for all
the entangled two nodes, either generated via direct or indirect interaction.
However, for the entangled three nodes, the phenomena of the sudden changes
(increasing or decreasing) is displayed. The number of peaks of the negativity
and the non-local coherent advantage that has predicted between all the nodes
increases as the interaction increase. Starting from the $MM$ initial network,
the upper bounds that predicted is much larger than those displayed if the
initial nodes are prepared in a partial entangled nodes.

However, the behavior of the negativity and the non-local coherent advantage
depends on the connection method, initially entangled, directly or non-
directly interaction. It is shown that, for initially maximum entangled nodes
network, the upper bounds of the negativity and the non-local coherent
advantage reach their maximum values periodically, depending on the
interaction strength. However, when the interaction is switched in $z-$
direction with large interaction strength, the number of oscillations of both
quantifiers is larger than those displayed in $x-y$ plane. Moreover, at zero
coupling, the oscillations that displayed for the negativity and the non-local
coherent advantage have maximum values and large death time. The survival
behavior of both measures, decreases when the network is initially constructed
from partially entangled nodes, ($WW$ network). For the indirect interaction
nodes, the behavior of both phenomena is similar to those displayed for
initially entangled nodes, but the oscillations are faster and their maximum
bounds are smaller. The smallest upper bounds are predicted when the initial
entangled nodes are prepared in partially entangled nodes, ($WW$ network). On
the other hand, for the direct connected nodes, the entanglement never reaches
its maximum values even the initial network is prepared via maximum entangled
nodes.

The behavior of the generated quantum correlations between each three nodes is
examined, where we assume that the initial network is of type $MM$ network. It
is shown that, the amount of correlations between each three qubits is almost
equalize. The sudden changes (decreasing/ increasing) of the tangle are
displayed during the interaction time. The amplitudes of the tangle
oscillations are smaller and consequently the minimum values are improved. The
direction of the interaction strength almost similar, where the oscillations
of the tangle has the same amplitudes.

The possibility of extending the network into multi-hops network is examined,
where we quantify the amount of entanglement and the non-local coherent
advantage that are generated between the initial terminal of the first hop and
the last node on the second hop. It is shown that, the amount of correlation
decreases and the time of the sudden death increases.

In conclusion, the spin Dipolar interaction could be used to generated quantum
correlations either between two or three nodes via direct or indirect
interaction. The amount of the generated correlations depend on the initial
network settings. The interaction strength and its direction has a remarkable
effect on the behavior of the generated correlations between each two nodes,
while there is a slightly effect of that generated between the three qubits.

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