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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 {\bf Monogamy property of quantum discord in  photon added tripartite\\
  Glauber coherent states of ${\rm GHZ}$-type}\\
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\vspace{1.5cm}

{\bf xxxx}$^{a}${\footnote { email: {\sf xxxx}}}, {\bf
yyyy}$^{b,c}$ {\footnote { email: {\sf yyyy}}} and {\bf
zzzz}$^{b,c}$ {\footnote { email: {\sf zzzz}}}

\vspace{0.5cm}
$^{a}${\it Department of Physics , Faculty of Sciences, University Ibnou Zohr,\\
 Agadir ,
Morocco}\\[1em]

$^{b}${\it LPHE-Modeling and Simulation, Faculty  of Sciences,
University
Mohammed V,\\ Rabat, Morocco}\\[1em]

$^{c}${\it Centre of Physics and Mathematics,
CPM, CNESTEN,\\ Rabat, Morocco}\\[1em]



\vspace{3cm} {\bf Abstract}
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\baselineskip=18pt
\medskip

%{\bf figure 2.1} Tracer $D_{12}$ (formule (50)) en fonction de $|\alpha|^2$ pour
%$k=0$  avec $m = 0,2,4,6,10$.
%{\bf figure 2.2} Tracer $D_{12}$ (formule (50)) en fonction de $|\alpha|^2$ pour
%$k=1$  avec $m = 0,2,4,6,10$.
%{\bf figure 3.1} Tracer $D_{23}$ (formule (51)) en fonction de $|\alpha|^2$ pour
%$k=0$ avec $m = 0,2,4,6,10$.
%{\bf figure 3.2} Tracer $D_{23}$ (formule (51)) en fonction de $|\alpha|^2$ pour
%$k=1$ avec $m = 0,2,4,6,10$.
%{\bf figure 4.1} Tracer $\Delta_{123}$ (formule (57)) en fonction de $|\alpha|^2$ pour
%$k=0$ avec $m = 0,2,4,6,10$. (utiliser les formules (56) et (50), bien sur $D_{12}= D_{13}$ )
%{\bf figure 4.2} Tracer $\Delta_{123}$ (formule (57)) en fonction de $|\alpha|^2$ pour
%$k=1$ avec $m = 0,2,4,6,10$.
%NOTA BENE: Dans ces formules $\kappa_m$ est donnee par la formule (14).

%We introduce the single-mode excited entangled coherent states and
%studied their quantum entanglement characteristics. Thus, we
%investigate the influence of photon excitations on quantum
%entanglement by using the different measurement (i.e. the
%concurrence, quantum discord and its geometric version). Therefor,
%To illustrate our results, we give the explicit expressions of the
%pairwise quantum correlations presented in multipartite coherent
%states and the special case also is analyzed. We found that the
%quantum discord and its geometrized variant does not follow the
%property of the concept of the monogamy, except is some particular
%situations that we presented.

We present a complete analysis of multipartite quantum discord of photon added three-mode coherent states.
We derive the explicit forms of
pairwise quantum discord which characterize quantum correlations in bipartite subsystems, showing
the effect of the order of the photon addition process. We also investigate the violation of the monogamy property
\newpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





In the context of information processing and transmission, several theoretical and experimental
results certify the potential advantages of quantum protocols compared to  their classical counterparts (see for instance \cite{Nielsen,Mermin,Wilde}).
Quantum technology exploiting  the intriguing  phenomena of quantum world, such as entanglement, offers secure ways
for communication \cite{Bennett1,Bennett2} and potentially  powerful algorithms in quantum computation \cite{Shor}. Originally, quantum information processing
focused on discrete (finite-dimensional) entangled states like the polarizations of a photon or
discrete levels of an atom. But, the extension
from discrete to continuous variables  has been proven beneficial for various kinds of quantum tasks  in coding and manipulating efficiently quantum information.
Coherent states, which constitute the prototypical instance of  continuous-variables  states, are
expected to play a central role in this context. They are appealing for
their mathematical elegance (continuity and over-completion property)
and closeness to classical  physical states  of a quantum system (minimization of Heisenberg uncertainty relation).  Implementing
a logical qubit encoding by treating entangled
coherent states  as  qubits in a two dimensional Hilbert space has been shown a promising strategy
in performing successfully  various quantum tasks such as  quantum  teleportation
\cite{Enk,JKL01},
 quantum computation \cite{JK,Ralph,Ralph2},
entanglement purification \cite{JKpuri} and errors correction
\cite{Glancy}.  In view of these potential applications, a special
attention was paid, during the last years,  to the identification,
characterization and quantification of  quantum correlations in
bipartite coherent states systems (see for instance the papers
\cite{Sanders3,Sanders,Wang} and references therein). The bipartite
treatment was extended to superpositions of multimode coherent
states \cite{Jex,Zheng,Wang1,daoud1,daoud2} which exhibit
multipartite entanglement as for instance in {\rm GHZ}
(Greenberger-Horne-Zeilinger), {\rm W} (Werner) states
\cite{Jeong1,Li} and entangled coherent state versions of cluster
states \cite{Munhoz,Wang-WF,Becerra}. To quantify quantum
correlations beyond entanglement in coherent states systems,
measures such as bipartite quantum discord
\cite{Ollivier-PRL88-2001,Vedral-et-al} and its geometric variant
\cite{Dakic2010} were used. Explicit results were derived for
quantum discord \cite{Luo,Ali,Shi1,Girolami,Shi2,Rachid1,Rachid2}
and geometric quantum discord \cite{Adesso1,Giorda,Yin,Rachid3} for
some special sets of coherent states.\\



In other hand, decoherence is a crucial process to understand the
emergence of classicality in quantum systems. It describes the
inevitable degradation of quantum correlations due to experimental
and environmental noise. Various decoherence models were
investigated and in particular  the phenomenon of entanglement
sudden death  was considered in a number of distinct contexts (see
for instance \cite{YuEberly-2007} and reference therein). For
optical qubits based on coherent states,  the influence of the
environment,  is mainly due to energy loss or photon
 absorption. The photon loss or equivalently amplitude damping in a noisy
 environment can be modeled by assuming that some of field energy and
 information is lost after transmission through a beam splitter
 \cite{Rachid1,Wickert}. Interestingly, it has been shown that a
 beam spitting device with a coherent in the first input and a
 number state in the second input generates photon-added coherent coherent
 states. In this respect, understanding the influence of adding photons
 process might be useful to develop the adequate strategies in improving the performance of noise reduction in quantum
 processing protocols. From a purely theoretical view, some
some authors considered the entanglement measure  in  photon added
 multipartite  coherent states \cite{Xu,Li}.\\





In this work, we  derive the analytical expression of pairwise quantum discord
between the three modes (parts) of the so-called quasi-${\rm GHZ}$ coherent states. A special
emphasis is devoted the process of adding photons  to the first mode. Mathematically,
this is represented by the action of a suitable creation operator on the the states
of the first subsystem. The influence of this excitation
on the quantum correlations is investigated.  Another important issue in photon added ${\rm GHZ}$-type coherent states
 concerns  the distribution of quantum
discord between the different parts of
 the whole system.  In fact, the free shareability of quantum
correlations  obey an restrictive inequality termed  in the literature as monogamy property \cite{Coffman} (see
also \cite{Giorgi,Prabhu,Allegra,Ren,Bruss}).\\




This paper is organized as follows. In section 2, basic definitions and
equations related to photon added coherent states are presented. In particular,
we consider the quantum correlations as measured by the concurrence in quasi-Bell states.
In particular, we introduce the  encoding mapping according with one passes from continues
variables (coherent states) to  discrete variables (logical quantum bits).  Along the same
line of reasoning, this qubit encoding is extended, in section 3, to tripartite entangled coherent states of
${\rm GHZ}$-type. The pairwise quantum discord quantifying the amount of quantum correlations
existing in the system is analytically derived. We especially study the effect of adding photon to the first mode.
In section 4,  we study the monogamy prperty of quantum discord. Numerical
illustrations of the violation of the monogamy relation  are presented for
 some special cases. Concluding remarks close this paper.







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Photon added coherent states and qubit mapping}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The basic objects in this work are the Glauber coherent states $\vert \alpha \rangle $ and
$\vert -\alpha \rangle $ where $\alpha$ is a complex number which determines the coherent
amplitude of the electromagnetic field. Mathematically, a  single-mode quantized radiation field
is represented by the harmonic oscillator algebra spanned by the creation $a^+$ and annihilation $a^-$
operators. The process of adding $m$ photons to coherent states of type $\vert \alpha \rangle $ and
$\vert -\alpha \rangle $  is usually represented by the action  of the operator $(a^+)^m$ ($m$ is a non negative integer).
This process leads to generation of nonclassical states
from the coherent states \cite{Agarwal}. Several experimental as well theoretical studies were devoted to the generation and nonclassical
properties of photon-added coherent states \cite{Zavatta} (for a recent review see \cite{Kim}). Explicitly,
$m$ successive  actions of creation operator $a^+$  on  the  Glauber coherent states
\begin{equation}
|\alpha
\rangle=e^{-\frac{|\alpha|^2}{2}}\sum^{\infty}_{n=0}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle,
\end{equation}
leads to the un-normalized states
\begin{equation}\label{vecteur 1}
||\alpha,m\rangle=\left( a^{+}\right) ^{m}\mid \alpha \rangle = e^{-\frac{|\alpha|^2}{2}} \sum_{n=0}^{\infty }\frac{ \alpha^{n}}{n!}\sqrt{
\left( n+m\right) !}\mid n+m \rangle.
\end{equation}
The normalized $m$-photon added coherent states are defined by
\begin{equation}\label{vecteur 2}
|\alpha , m \rangle = \frac{(a^{+})^{m} \vert \alpha \rangle}
{\sqrt{\langle \alpha \vert (a^{-})^{m} (a^{+})^{m} \vert \alpha \rangle}}  ,
\end{equation}
where
\begin{equation}\label{valeur-moy}
\langle\alpha|(a^{-})^{m}(a^{+})^{m}|\alpha\rangle  =  m! L_{m}(-| \alpha|^{2} ).
\end{equation}
In the last expression $L_{m}(x)$ is the Laguerre polynomial of order $m$ defined by
\begin{equation}\label{}
L_{m}(x)=\sum_{n=0}^{m}\frac{(-1)^{m} m! x^{n}}{(n!)^{2}(m-n)!}.
\end{equation}
Photon added coherent states interpolate between  electromagnetic field coherent states (quasi-classical states)
and Fock states $\vert n \rangle$ (purely quantum states). Furthermore, photon added coherent states exhibit non-classical features such
as squeezing, negativity of Wigner distribution and sub Poissonian statistics. Their experimental generation
using parametric down conversion in a nonlinear crystal was reported in \cite{Zavatta}.  Photon-coherent states
are not orthogonal each other. Indeed using the expression
\begin{equation}\label{}
\langle-\alpha|(a^{-})^{m}(a^{+})^{m}|\alpha\rangle  =  e^{-2| \alpha |^{2}} m! L_{m}(| \alpha|^{2}),
\end{equation}
it is simply verified that the overlapping between the states $|\alpha , m )$ and $|-\alpha , m )$ is
\begin{equation}\label{}
\langle - \alpha , m  |\alpha , m \rangle = e^{-2| \alpha |^{2}} \frac{L_{m}(| \alpha|^{2}) }{L_{m}(-| \alpha|^{2})}.
\end{equation}
Usually coherent states are treated as continuous variable states. Recently, the idea of encoding quantum information on coherent states
 has led to  an interesting proposal \cite{Ralph2} in which superpositions of Glauber coherent states are
used to encode logical qubits. Accordingly, one can consider an encoding scheme of type $\vert \alpha \rangle \longrightarrow \vert 0 \rangle$
and $\vert -\alpha \rangle \longrightarrow \vert 1 \rangle$ with two non orthogonal logical qubits. Alternatively,  an orthogonal qubit
encoding scheme  involving even and odd Glauber coherent states can be be defined so that the coherent states are mapped in
in $ \mathbb{C}^2\otimes \mathbb{C}^2$ Hilbert space. Hence,  for photon added coherent of type (\ref{vecteur 2}), one introduces the  two dimensional
basis spanned by two orthogonal qubits $\vert + , m \rangle $ and $\vert - , m \rangle  $  defined as even and odd superpositions of  photon added coherent states 
(\ref{vecteur 2})
\begin{equation}\label{qubit-m}
 \vert \pm , m \rangle = \frac{1}{\sqrt{2 \pm 2 \kappa_m e^{-2|\alpha|^{2}}}} ~ (\vert \alpha , m) \pm \vert -\alpha , m))
\end{equation}
where
\begin{equation}\label{kappa}
\kappa_m :=  \frac{L_{m}(|\alpha|^{2})}{L_{m}(-|\alpha|^{2})}.
\end{equation}
Clearly, for $m = 0$, one has  $\kappa_0 = 1$ and  the logical qubits (\ref{qubit-m}) reduce to
\begin{equation}\label{qubit-0}
\vert \pm  \rangle= \frac{1}{\sqrt{2 \pm 2 e^{-2|\alpha|^{2}}}} ~ (\vert \alpha \rangle \pm \vert -\alpha \rangle),
\end{equation}
which coincide with even and odd Glauber coherent states providing the qubit encoding scheme introduced in \cite{Ralph2}.
It must be emphasized that such qubit encoding of paramount importance  
 in dealing with quantum correlation in photon added coherent states
and to investigate the influence of the photon adding excitation process.  In this, we shall first consider the
entanglement in quasi-Bell states.  The  quasi-Bell states are very interesting in quantum optics and serve as valuable resource
 for quantum teleportation and many others quantum computing operations. The quasi-Bell states
\begin{equation}\label{Bell-0}
|{\rm B}_{k}(\alpha)\rangle={\cal N}_{k}(\alpha)~ \big[
|\alpha\rangle\otimes|\alpha\rangle + e^{ik\pi} |-\alpha\rangle\otimes|-\alpha\rangle\big].
\end{equation}
with $k = 0  (~{\rm mod}~2)$ (resp.  $k = 1 (~{\rm mod}~2)$) stands for even (resp. odd) quasi-Bell states and the 
normalization factors ${\cal N}_{k}(\alpha)$ are
\begin{equation}\label{Norme-0}
{\cal N}^{-2}_{k}(\alpha) =  2 + 2  e^{-4\vert \alpha \vert^2} \cos k\pi.
\end{equation}
By  repeated actions of the creation operator on the first mode, the resulting
excited quasi-Bell states given by
\begin{equation}
||{\rm B}_{k}(\alpha,m)\rangle = {\cal N}_{k}(\alpha)~  \bigg[ \big[(a^{+})^{m}\otimes \mathbb{I}\big]
|\alpha\rangle\otimes|\alpha\rangle + e^{ik\pi} ~ \big[(a^{+})^{m} \otimes \mathbb{I}\big]|-\alpha\rangle\otimes|-\alpha\rangle\bigg].
\end{equation}
are un-normalized ($\mathbb{I}$ stands for the unity operator). In fact, one verifies
\begin{equation}
\langle {\rm B}_{k}(\alpha,m) || {\rm B}_{k}(\alpha,m) \rangle = m!~ \frac{L_{m}(-|\alpha|^2) +  e^{-4|\alpha|^{2}}L_{m}(|\alpha|^2)~\cos k\pi}{1 +  e^{-4\vert \alpha \vert^2} \cos k\pi}.
\end{equation}
It  is more appropriate for our purpose to deal with sates involving superpositions of normalized vectors. 
Thus, one introduces the normalized photon-added quasi-Bell states as
\begin{equation}\label{Bell-m}
|{\rm B}_{k}(\alpha,m)\rangle=\mathcal{N}_{k}(\alpha,m)~ 
\big[|m, \alpha \rangle \otimes|\alpha\rangle + e^{ik\pi} ~|m , -\alpha \rangle \otimes|-\alpha\rangle\big],
\end{equation}
 in terms of the normalized photon added coherent state (\ref{vecteur 2}),  with
\begin{equation}\label{Norme-m}
 \mathcal{N}^{-2}_{k}(\alpha,m) = 2 + 2 \kappa_m e^{-4\vert \alpha \vert^2} \cos k\pi.
\end{equation}
where $\kappa_m$ is defined by (\ref{kappa}). For $m=0$, the normalization factor (\ref{Norme-m}) 
reduces to (\ref{Norme-0}) and the quasi-Bell states (\ref{Bell-0}) are recovered. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection {Entanglement of bipartite quasi-Bell states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Using the qubit mapping (\ref{qubit-m}) for the first mode and (\ref{qubit-0}) for the second,
 the bipartite state (\ref{Bell-m})  writes as a two qubit state
\begin{equation}\label{eq C}
|{\rm B}_{k}(\alpha,m)\rangle=\mathcal{N}_{k}(\alpha,m)\sum_{i=\pm}\sum_{j=\pm} C_{ij} \vert i, m\rangle \otimes \vert j \rangle
\end{equation}
where the vectors $\vert i, m\rangle $ (resp. $\vert j\rangle $) are defined by (\ref{qubit-m}) (resp. (\ref{qubit-0}))
and the expansion coefficients are given by
$$C_{++} = c^+_{m}c^+(1+e^{ik\pi}),\quad
C_{-+} = c^+c^-_{m}(1-e^{ik\pi}),\quad
C_{+-} = c^+_{m}c^-(1-e^{ik\pi}),\quad
C_{--} = c^-c^-_{m}(1+e^{ik\pi}).$$
where
$$ c^{\pm}_m = \sqrt{\frac{1 \pm \kappa_m e^{-2|\alpha|^{2}}}{2}} \qquad c^{\pm} = \sqrt{\frac{1 \pm e^{-2|\alpha|^{2}}}{2}}. $$
It is well established that in a pure bipartite system, the
quantum discord coincides with entanglement of formation. Thus, to discuss the
effect of the photon excitation of quasi-Bell states, it is sufficient to  characterize
the bipartite entanglement in the states (\ref{Bell-m}) by means of the entanglement of formation.
 We recall that for $\rho_{12}$ the density
matrix for a pair of qubits~$1$ and~$2$ which may be pure or mixed, the entanglement of formation 
is defined by ~\cite{Wootters98}
\begin{equation}\label{def-EOF}
E(\rho_{12}) = H(\frac{1}{2} + \frac{1}{2} \sqrt{1 - \vert  C(\rho_{12})\vert^2})
\end{equation}
where  $H(x) = -x\log_{2} x -(1-x)\log_{2} (1-x)$ is the binary
entropy function and the concurrence $C(\rho_{12})$ is
\begin{equation}\label{def-CONC}
C(\rho_{12})=\max \left\{ \lambda _1-\lambda _2-\lambda _3-\lambda
_4,0\right\} 
\end{equation}
for~$\lambda_1\ge\lambda_2\ge\lambda_3\ge\lambda_4$ the square roots
of the eigenvalues of the "spin-flipped" density matrix
\begin{equation}
\varrho_{12}\equiv\rho_{12}(\sigma_y\otimes\sigma
_y)\rho_{12}^{\star}(\sigma_y\otimes \sigma_y). \label{eq:c2}
\end{equation}
In the last formula the star stands for complex conjugation 
and $\sigma_y $ is the usual Pauli matrix. Thus using (\ref{def-CONC}), it 
easy to check that the concurrence is
given by
\begin{equation}\label{conci}
C_{12}=2\mathcal{N}^{2}_{k}(\alpha,m)|C_{++}C_{--}-C_{+-}C_{-+}|,
\end{equation}
which rewrites explicitly as
%\begin{equation}\label{Cshow}
%C_{12}=\frac{\sqrt{1-e^{-4|\alpha|^{2}}}
%\sqrt{ \big(L_{m}(-|\alpha|^{2})\big)^2 - e^{-4|\alpha|^{2}}  \big(L_{m}(|\alpha|^{2})\big)^2}}
%{L_{m}(-|\alpha|^{2})+e^{-4|\alpha|^{2}}\cos k\pi ~L_{m}(|\alpha|^{2})}.
%\end{equation}
\begin{equation}\label{Cshow}
C_{12}=\frac{\sqrt{1-e^{-4|\alpha|^{2}}}
\sqrt{ 1 - \kappa^2_m e^{-4|\alpha|^{2}}}}
{1 + \kappa_m  e^{-4|\alpha|^{2}}\cos k\pi }
\end{equation}
in terms of the coherent states amplitude  $|\alpha|$ and the excitation order $m$. It follows that 
entanglement of formation is 
\begin{equation}\label{EOF-Bell-m}
E_{12} ~= ~ H\Bigg[ \frac{1}{2} + \frac{e^{-2|\alpha|^2}(1 + \kappa_m\cos k\pi)}{2 + 2\kappa_me^{-4|\alpha|^2}\cos k\pi} \Bigg] 
\end{equation}
For $m = 0$, one
has
\begin{equation}\label{Cshow}
C_{12} = \frac{ 1-e^{-4|\alpha|^{2}}}{1 + e^{-4|\alpha|^{2}}\cos k\pi}
\end{equation}


 {\bf In order to observe the influence of the photon excitation on the
quantum entanglement of a single mode excited bipartite entangled
coherent states, we need to plot the  concurrence C($|\alpha|$,m)
versus $|\alpha|$ for different values of the number of excited
photon m. Therefore, there are two case of k (even and odd) which
are shown in figure 1, 2 respectively, we can see from the figure 1,
after increasing of entanglement with $|\alpha|$, it approaches as
the maximum value unit  when $|\alpha|$ tends the infinity. In the
other hand, the entanglement degrees increases as $|\alpha|$ during
a number of photon increases.}
\begin{figure}[!htb]\label{C(K=0)}
\begin{centering}
\includegraphics[scale=0.6]{C(K=0)}
\par\end{centering}
\caption{ The concurrence C (even)  of SMEECS
$|\psi(|\alpha|,m)\rangle$ versus $|\alpha|$ for the different
number photon excitations.}
\end{figure}
{\bf We can also observe in figure 2. That C($|\alpha|$,m) increases when
$|\alpha|$ increase for different values of m (m=0, 1, 2, 3, 4 and
10) respectively. Together the concurrence C($|\alpha|$,m) tends to
unit for the larger $|\alpha|$.}
\begin{figure}[!htb]\label{C(K=1)}
\begin{centering}
\includegraphics[scale=0.6]{C(k=1)}
\par\end{centering}
\caption{ The concurrence C (odd) of SMEECS
$|\psi(|\alpha|,m)\rangle$ versus $|\alpha|$ for the different
number photon excitations.}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Photon added quasi-GHZ coherent states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The quasi-${\rm GHZ}$ coherent states are defined by
\begin{equation}\label{treeprticle0}
|\alpha,0\rangle=\mathcal{N}_{0}(|\alpha,\alpha,\alpha\rangle + e^{ik\pi} |-\alpha,-\alpha,-\alpha\rangle).
\end{equation}
Where the normalization constant $\mathcal{N}_{0}$ is given by
\begin{equation}
\mathcal{N}^{-2}_{0}= 2 + 2 e^{-6|\alpha|^2} \cos k\pi,
\end{equation}
As in the previous section, we shall consider the excitation of the first mode by adding $m$ photon. Thus,
the photon added quasi-${\rm GHZ}$ coherent states have the form
\begin{equation}\label{treeprticle1}
|\alpha,m\rangle=\mathcal{N}a^{+m}(|\alpha,\alpha,\alpha\rangle + e^{ik\pi}
|-\alpha,-\alpha,-\alpha\rangle)=\frac{\mathcal{N}}{\mathcal{N}_{0}}a^{+m}|\alpha,0\rangle,
\end{equation}
where the normalization factor is
\begin{equation}
\mathcal{N}^{-2}= 2m!\big[L_{m}(-|\alpha|^{2}) + e^{-6|\alpha|^2} \cos k\pi~
L_{m}(|\alpha|^{2})\big].
\end{equation}
Clearly, for $m=0$ the states $|\alpha,m\rangle$ (\ref{treeprticle1}) reduces to $|\alpha,m\rangle$ (\ref{treeprticle0}).


The tripartite state (\ref{treeprticle1}) can be re-equated as follows
\begin{equation}\label{treeprticle2}
|\alpha,m\rangle=\mathcal{C}_k(\alpha,m)~ (|\alpha, m ) \otimes|\alpha \rangle \otimes|\alpha\rangle + e^{ik\pi}
|-\alpha, m ) \otimes|-\alpha \rangle \otimes|-\alpha\rangle )
\end{equation}
so that the tree modes are described by normalized vectors. The factor $\mathcal{C}$ is defined by
$$ \mathcal{C}_k^{-2}(\alpha,m) = \mathcal{N}^{-2}(\alpha,m) \big[\langle \alpha | a^{-m} a^{+m}  | \alpha \rangle\big]^{-1} $$
which rewrites
$$ \mathcal{C}_k^{-2}(\alpha,m) = 2 + 2 e^{-6|\alpha|^2}~ \cos k\pi ~\kappa_m$$
with
$$ \kappa_m = \frac{L_{m}(|\alpha|^{2})}{L_{m}(-|\alpha|^{2})}.$$

In investigating the pairwise quantum discord in a tripartite system $1-2-3$, one needs the
reduced density matrices describing the two qubit subsystems $1-2$, $2-3$ and $1-3$. For the states
$|\alpha,m\rangle$ (\ref{treeprticle2}), it is simply seen that the reduced density
matrices $\rho_{12}={\rm Tr}_{3}\rho_{123}$ and $\rho_{13}={\rm Tr}_{2}\rho_{123}$ are
identical $(\rho_{123}=|\alpha,m\rangle\langle\alpha,m|)$.  They are given by
\begin{eqnarray}\label{matrixrho12}
\rho_{12} = \rho_{13} = \frac{\mathcal{C}^2_{k}(\alpha,m)}{\mathcal{N}^2_{k}(\alpha,m)}\Bigg[ \bigg(\frac{1 + e^{-2|\alpha|^2}}{2}\bigg) |{\rm B}_{k}(\alpha,m)\rangle \langle {\rm B}_{k}(\alpha,m) |
+ \bigg(\frac{1 - e^{-2|\alpha|^2}}{2}\bigg) Z |{\rm B}_{k}(\alpha,m)\rangle \langle {\rm B}_{k}(\alpha,m) |Z \Bigg]
\end{eqnarray}
in terms of photon added quasi-Bell states (\ref{Bell-m}).  The operator $Z$ is the third Pauli generator defined by
$$Z |{\rm B}_{k}(\alpha,m)\rangle  = \mathcal{N}_{k}(\alpha,m)
[|m, \alpha ) \otimes|\alpha\rangle - e^{ik\pi} |m , -\alpha ) \otimes|-\alpha\rangle]$$
Similarly, one obtains the reduced matrix density
\begin{eqnarray}\label{matrixrho23}
 \rho_{23} = \frac{\mathcal{C}^2_{k}(\alpha,m)}{\mathcal{N}^2_{k}(\alpha,0)}\Bigg[ \bigg(\frac{1 + \kappa_m e^{-2|\alpha|^2}}{2}\bigg) |{\rm B}_{k}(\alpha,0)\rangle \langle {\rm B}_{k}(\alpha,0) |
+ \bigg(\frac{1 - \kappa_m  e^{-2|\alpha|^2}}{2}\bigg) Z |{\rm B}_{k}(\alpha,0)\rangle \langle {\rm B}_{k}(\alpha,0) |Z \Bigg]
\end{eqnarray}
Using the following mapping
\begin{eqnarray}\label{qubit1}
\vert  m , \pm \alpha )= \sqrt{\frac{1 + \kappa_m e^{-2|\alpha|^2}}{2}} ~\vert 0 \rangle_1 \pm \sqrt{\frac{1 - \kappa_m e^{-2|\alpha|^2}}{2}} ~\vert 1 \rangle_1
\end{eqnarray}
for the first mode. For the second and third modes, have
\begin{eqnarray}\label{qubits23}
\vert  \pm \alpha \rangle = \sqrt{\frac{1 + e^{-2|\alpha|^2}}{2}} ~\vert 0 \rangle_i \pm \sqrt{\frac{1 - e^{-2|\alpha|^2}}{2}}~ \vert 1 \rangle_i \qquad i = 2,3
\end{eqnarray}
Substituting (\ref{qubit1}) and (\ref{qubits23}) in (\ref{matrixrho12}) (resp. (\ref{matrixrho23})), one can express the density matrix
$\rho_{12}$ (resp. $\rho_{23}$) in the two qubit basis $\{ \vert 0 \rangle_1 \otimes \vert 0 \rangle_2,
\vert 0 \rangle_1 \otimes \vert 1 \rangle_2, \vert 1 \rangle_1 \otimes \vert 0 \rangle_2 , \vert 1 \rangle_1 \otimes \vert 1 \rangle_2\}$
(resp.  $\{ \vert 0 \rangle_2 \otimes \vert 0 \rangle_3,
\vert 0 \rangle_2 \otimes \vert 1 \rangle_3, \vert 1 \rangle_2 \otimes \vert 0 \rangle_3 , \vert 1 \rangle_2 \otimes \vert 1 \rangle_3\}$ )











%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Quantifying the quantum discord}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Bipartite measures of entanglement of formation and quantum discord}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 For the density matrix $\rho_{12}$,  the total correlation is quantified
by the mutual information 
\begin{equation}\label{def: mutual information}
    I_{12}=S_{1}+S_{2}-S_{12},
\end{equation}
where 
$\rho_{1(2)}={\rm Tr}_{2(1)}( \rho_{12})$ is the reduced state of
$1$($2$) and $S(\rho)$ is the von Neumann entropy of a quantum state
$\rho$. The mutual information $I_{12}$ contains both quantum and
classical correlations. It can be decomposed as
$$ I_{12} =  D_{12} +  C_{12}.$$
Consequently, for a bipartite quantum system, the quantum discord
$D_{12}$ is defined as the difference between total correlation
$I_{12}$ and classical correlation $C_{12}$. The classical part
$C_{12}$ can be determined by a local measurement optimization
procedure. To remove
the measurement dependence, a maximization over all possible
measurements is performed and the classical correlation writes
\begin{eqnarray}
    C_{12}  = S(\rho_2) - \widetilde{S}_{\rm min}
      \label{def: classical correlation}
\end{eqnarray}
where $\widetilde{S}_{\rm min}$  denotes the minimal value of the
conditional  entropy. 
When optimization is taken over all  measurements, the quantum
discord is
\begin{equation} \label{def: discord}
    D_{12} = I_{12} - C_{12}
    =S_{1}+\widetilde{S}_{\rm min}-S_{12}.
\end{equation}
Thus,  the derivation of quantum discord requires the minimization
of conditional entropy. This constitutes a complicated issue when
dealing with an arbitrary mixed state. The explicit analytical
expressions of quantum discord were obtained only for few
exceptional two-qubit quantum states, especially ones of rank two.
One may quote for instance the results obtained in
\cite{Ali,Adesso2} (see also \cite{Rachid1,Rachid2,Rachid3}). The 
density matrix $\rho_{12}$ is of rank two and in this case the minimization of the conditional
entropy (\ref{condit-entropy}) can be performed by purifying the
density matrix $\rho_{12}$ and  making use of Koashi-Winter relation
\cite{Koachi-Winter} (see also \cite{Shi1}). This relation
establishes the connection between the classical correlation of a
bipartite state $\rho_{12} $ and the entanglement of formation of
its complement $\rho_{23}$.   Indeed,  the minimal value of the conditional
entropy coincides with the entanglement of formation of $\rho_{23}$.
It is given by
\begin{equation}\label{stild-min}
 \widetilde{S}_{\rm min} = E(\rho_{23}). 
\end{equation}
 It  follows that the
Koaschi-Winter relation and the purification procedure provide us
with a computable expression of quantum discord
\begin{equation}
D_{12} =  S_1 - S_{12} + E_{23}
\end{equation}
when the measurement is performed on the subsystem $1$. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pairwise quantum discord}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The pairwise quantum discord present in the mixed states $\rho_{12}$,
and equivalently in $\rho_{13}$,
can be computed using the procedure presented in the previous
subsection. As result,  when  the measurement is performed on the
subsystem $A \equiv 1$,  the quantum discord is
\begin{equation}\label{D12}
 D_{12} =  S_1 - S_{12} + E_{23}
\end{equation}
where $k$ stands for the third subsystem traced out to get the
reduced matrix density $\rho_{ij}$. The von Neumann entropy of the
reduced density $\rho_i$ is
\begin{equation}\label{S1}
S_1 = H\bigg(\frac{1}{2} \frac{(1 + \kappa_m  e^{-2|\alpha|^2})(1 + e^{-4|\alpha|^2}~\cos k\pi)}{1
+\kappa_m  e^{-6|\alpha|^2}\cos k\pi}\bigg),
\end{equation}
and the entropy of the bipartite density $\rho_{12}$ is explicitly
given by
\begin{equation}\label{S12}
S_{12} = H\bigg(\frac{1}{2} \frac{(1 + \kappa_m  e^{-4|\alpha|^2} \cos k\pi)(1 +
 e^{-2|\alpha|^2})}{1 +\kappa_m  e^{-6|\alpha|^2}\cos k\pi}\bigg).
\end{equation}
It important to emphasize that the entanglement of formation
measuring the entanglement of the subsystem $2$ with the ancillary
qubit, required in the purification process to minimize the
conditional entropy, is exactly the entanglement of formation
measuring the degree of intricacy between the subsystem $2$ and the
traced out qubit $3$. The concurrence in the subsystem $2-3$
takes the following form
\begin{equation}\label{C23}
{\cal C}_{23}= \kappa_m  e^{-2|\alpha|^2}~\frac{\sqrt{(1 - e^{-4|\alpha|^2})(1- e^{-4|\alpha|^2})}}{ 1 +
\kappa_m  e^{-6|\alpha|^2} \cos k \pi}.
\end{equation}
\begin{equation}\label{E23}
E_{23}  = H\bigg(\frac{1}{2} + \frac{1}{2} \sqrt{ 1- \frac{\kappa^2_m  e^{-4|\alpha|^2}(1 -
e^{-4|\alpha|^2})(1 - e^{-4|\alpha|^2})}{(1 +\kappa_m  e^{-6|\alpha|^2}\cos k\pi)^2}}\bigg).
\end{equation}
The pairwise quantum discord present in the mixed states $\rho_{23}$
can be computed using the procedure presented in the previous
subsection. As result,  when  the measurement is performed on the
subsystem $A \equiv 2$,  the quantum discord is
\begin{equation}\label{D23}
 D_{23} =  S_2 - S_{23} + E_{13}
\end{equation}
where $k$ stands for the third subsystem traced out to get the
reduced matrix density $\rho_{23}$. The von Neumann entropy of the
reduced density $\rho_i$ is
\begin{equation}\label{S2}
S_2 = H\bigg(\frac{1}{2} \frac{(1 + e^{-2|\alpha|^2})(1 + \kappa_m  e^{-4|\alpha|^2}\cos k\pi)}{1
+\kappa_m  e^{-6|\alpha|^2}\cos k\pi}\bigg),
\end{equation}
and the entropy of the bipartite density $\rho_{23}$ is explicitly
given by
\begin{equation}\label{S23}
S_{23} = H\bigg(\frac{1}{2} \frac{(1 + e^{-4|\alpha|^2}\cos k\pi)(1 +
\kappa_m  e^{-2|\alpha|^2} )}{1 +\kappa_m  e^{-6|\alpha|^2}\cos k\pi}\bigg).
\end{equation}
It important to emphasize that the entanglement of formation
measuring the entanglement of the subsystem $3$ with the ancillary
qubit, required in the purification process to minimize the
conditional entropy, is exactly the entanglement of formation
measuring the degree of intricacy between the subsystem $3$ and the
traced out qubit $1$.
The concurrence in the subsystem $1-3$
takes the following form
\begin{equation}\label{conc-mixte}
{\cal C}_{13}= e^{-2|\alpha|^2}~\frac{\sqrt{(1 - \kappa^2_m  e^{-4|\alpha|^2})(1- e^{-4|\alpha|^2})}}{ 1 +
\kappa_m  e^{-6|\alpha|^2} \cos k \pi}.
\end{equation}
\begin{equation}\label{Ejk}
E_{13}  = H\bigg(\frac{1}{2} + \frac{1}{2} \sqrt{ 1- \frac{e^{-4|\alpha|^2}(1 -
\kappa^2_m  e^{-4|\alpha|^2})(1 - e^{-4|\alpha|^2})}{(1 +\kappa_m  e^{-6|\alpha|^2}\cos k\pi)^2}}\bigg).
\end{equation}














%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Some special cases}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{itemize}
    \item Reduced density matric $\rho_{12}$
\end{itemize}

The quantum discord in the state $\rho_{12}$ is
\begin{eqnarray}
D_{12}&=&H\bigg(\frac{1}{2} \frac{(1 + \kappa_m  e^{-2|\alpha|^2})(1 + e^{-4|\alpha|^2}~\cos k\pi)}{1
+\kappa_m  e^{-6|\alpha|^2}\cos k\pi}\bigg)\\\nonumber
&-&H\bigg(\frac{1}{2} \frac{(1 + \kappa_m  e^{-4|\alpha|^2} \cos k\pi)(1 +
 e^{-2|\alpha|^2})}{1 +\kappa_m  e^{-6|\alpha|^2}\cos k\pi}\bigg)\\\nonumber
&+&H\bigg(\frac{1}{2} + \frac{1}{2} \sqrt{ 1- \frac{\kappa^2_m  e^{-4|\alpha|^2}(1 -
e^{-4|\alpha|^2})(1 - e^{-4|\alpha|^2})}{(1 +\kappa_m  e^{-6|\alpha|^2}\cos k\pi)^2}}\bigg),
\end{eqnarray}







We start with the special case $m=0$ (see figure 3), the state
$\rho_{12}$ is coincides with three-mode coherent states. At this
point, we have $p_{1}=p$ and $L_{0}(\mid\alpha^{2}\mid)=1$, then the
explicit expression of quantum discord for the density $\rho_{12}$
is in term of the overlap $p$. This expression of discord is equivalent
with the discord given by \cite{[35]} for n = 3, and the discord for
symmetric (antisymmetric) states  $m$ even ($m$ odd) as shown in the
figure $1$. Gives a plot of quantum discord versus the overlap p for
the mixed state $\rho_{12}$. It must be noticed that for $k=0$ , we
find the maximum of discord obtained for the value p equal to $1/2$.
However, for $k=1$, the quantum discord takes the maximum value for
$p\rightarrow 1$ and the discord increase with p increase.\newline
The figure 4 gives a plot of quantum discord versus the number of
photon for symmetric added entangled coherent states (i.e.k even)
and for different value of $m$. As seen from the figure, after an
initial increasing, the quantum discord decreases to vanish when the
amplitude of photon $\mid\alpha^{2}\mid\rightarrow 2$. The minimum
value of discord is obtained when has no photon excitation  $(i.e.m
= 0)$ and the maximum of quantum discord is obtained for m=10,
 we also found that the discord is depends on the number of the
excitation photon. However, this maximum increases as $m$ increases.
In figure 5, we give a plot of the quantum discord for the
antisymmetric case and for different excitation of photon number. It
is remarkable that for antisymmetric quantum states the maximal
value of quantum discord increases as $m$ decreased contrarily to
the symmetric states.
\begin{figure}\label{speciallycas}
\begin{centering}
\includegraphics[scale=0.6]{D12casspecial}
\par\end{centering}
\caption{ The quantum discord of SMEECS $|\psi(|\alpha|,m)\rangle$
versus $|\alpha|^{2}$ for (even) and (odd) states.}
\end{figure}

\begin{figure}[!htb]\label{012}
\begin{centering}
\includegraphics[scale=0.6]{d12k00}
\par\end{centering}
\caption{ The quantum discord (even)  of reduced density matric
$\rho_{12}$ versus $|\alpha|^{2}$ for the different number photon
excitations.}
\end{figure}

\begin{figure}[!htb]
\begin{centering}\label{112}
\includegraphics[scale=0.6]{d12k1}
\par\end{centering}
\caption{ The quantum discord (odd)  of reduced density matric
$\rho_{12}$ versus $|\alpha|^{2}$ for the different number photon
excitations.}
\end{figure}


%\begin{center}
%  \includegraphics[width=3in]{fig00a.eps}\\
%{\bf Figure 3.}  {\sf The three tangle  $\tau_{A,B,E}$ versus the
%overlapping $p$ for $t^2= \frac{1}{2}$ and   $m=0$.}
%\end{center}
%\begin{center}
%  \includegraphics[width=3in]{fig00b.eps}\\
%{\bf Figure 4.}  {\sf The three tangle  $\tau_{A,B,E}$ versus the
%overlapping $p$ for $t^2= \frac{1}{2}$ and  $m=1$.}
%\end{center}





\begin{itemize}
\item Reduced density matric $\rho_{23}$
\end{itemize}
For the state $\rho_{23}$, the quantum discord is

\begin{eqnarray}
D_{23}&=&H\bigg(\frac{1}{2} \frac{(1 + e^{-2|\alpha|^2})(1 + \kappa_m  e^{-4|\alpha|^2}\cos k\pi)}{1
+\kappa_m  e^{-6|\alpha|^2}\cos k\pi}\bigg)\\\nonumber
&-&H\bigg(\frac{1}{2} \frac{(1 + e^{-4|\alpha|^2}\cos k\pi)(1 +
\kappa_m  e^{-2|\alpha|^2} )}{1 +\kappa_m  e^{-6|\alpha|^2}\cos k\pi}\bigg)\\\nonumber
&+&H\bigg(\frac{1}{2} + \frac{1}{2} \sqrt{ 1- \frac{e^{-4|\alpha|^2}(1 -
\kappa^2_m  e^{-4|\alpha|^2})(1 - e^{-4|\alpha|^2})}{(1 +\kappa_m  e^{-6|\alpha|^2}\cos k\pi)^2}}\bigg),
\end{eqnarray}




The explicit expression of quantum discord of reduced matric
$\rho_{23}$ in the special case $m=0$ take the same result obtained
for $\rho_{12}$.\newline The behavior of quantum discord of the
reduced matric $\rho_{23}$ versus $|\alpha|^{2}$ for symmetric
(antisymmetric case )respectively is given in the figure 6 and 7
respectively. As seen from this figures, after an initial
increasing, the quantum discord decreases to vanish when
$|\alpha|^{2}\longrightarrow 2$. It is clearly seen that the quantum
discord increases with the photon excitation number is increases in
the symmetric case. Otherwise, for the antisymmetric states, the
quantum discord is decreases when the photon excitation number
increases.\newline We observed that, the minimum value $(m=2)$ of
quantum discord for the reduced matric $\rho_{12}$ is higher than
the maximum value $(m=10)$ obtained for the reduced matric
$\rho_{23}$.

\begin{figure}[!htb]
\begin{centering}\label{023}
\includegraphics[scale=0.6]{d23k0}
\par\end{centering}
\caption{ The quantum discord (odd) of reduced density matric
$\rho_{23}$ versus $|\alpha|^{2}$ for the different number photon
excitations.}
\end{figure}

\begin{figure}[!htb]
\begin{centering}\label{123}
\includegraphics[scale=0.6]{discorde23pourk1}
\par\end{centering}
\caption{ The quantum discord (odd)  reduced density matric
$\rho_{23}$ versus $|\alpha|^{2}$ for the different number photon
excitations.}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Monogamy of quantum discord for a three-qubit entangled
state}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Monogamy of quantum correlations is a property satisfied by certain
entanglement measures in a multipartite scenario. Given a tripartite
state $\rho_{123}$, the monogamy condition for a bipartite quantum
correlation measure $ \mathcal{Q}$ assures that the bipartite
quantum correlations in the density operator $\rho_{1\mid23}$ are
distributed in such a way that the following inequality is satisfied
\begin{equation}\label{Mgm}
 \mathcal{Q} (\rho_{1\mid23}) \geq  \mathcal{Q}(\rho_{12})+ \mathcal{Q}(\rho_{13}).
\end{equation}
Violation of the above inequality will imply that the quantity
$\mathcal{Q}$ is polygamous for the corresponding state. Otherwise,
this inequality is sufficient for quantum discord to be
monogamous.\newline To illustrate the above analysis, we will
investigate the properties of quantum discord monogamy in two
different ways (quantum discord and geometric quantum discord using
norm 2).
\begin{itemize}
    \item  Monogamy of quantum discord
\end{itemize}
To investigate the monogamy relation of quantum discord measured in
quantum systems involving three qubits, Coffman et al \cite{3qubit
monogmy} introduced the so called tripartite state equation
(\ref{systemtrpt}). It is defined as
\begin{equation}\label{MgmD}
 D(\rho_{1\mid23}) \geq  D(\rho_{ 12})+ D(\rho_{ 13}),
\end{equation}
where 1, 2 and 3 mean the respective parts of a tripartite system.
Note that here $D_{1\mid23}$ is given by the entanglement between
qubit (1) and the joint qubits (23). The quantum discord coincides
with the entanglement of formation.




The pure tripartite state $ \vert \alpha , m
\rangle$ can be partitioned as   bipartite system and can be expressed by means of two logical qubits.
This scenario  can be
achieved as follows. We introduce, for the first subsystem, the
orthogonal basis $\{ \vert 0 \rangle_1 , \vert 1 \rangle_1\}$
defined by
\begin{equation}\label{base1}
\vert 0 \rangle_1 = \frac{ \vert \alpha , m  ) +  \vert \alpha , m ) }{\sqrt{2(1 + \kappa_m e^{-2|\alpha|^2})}}
   \qquad \vert 1 \rangle_1 = \frac{\vert \alpha , m  )  -  \vert \alpha , m  ) }{{\sqrt{2(1-  \kappa_m e^{-|\alpha|^2})}}}.
\end{equation}
For the  modes $(23)$, viewed as a single subsystem, we introduce the
orthogonal basis $\{ \vert  0 \rangle_{23} , \vert  1
\rangle_{23}\}$ given by
\begin{equation}\label{base2}
\vert 0 \rangle_{23} = \frac{ \vert \alpha, \alpha \rangle +  \vert -\alpha, -\alpha
\rangle}{\sqrt{2(1 + e^{-4|\alpha|^2})}}
   \qquad \vert 1 \rangle_{23} = \frac{ \vert \alpha, \alpha \rangle +  \vert -\alpha, -\alpha
\rangle}{\sqrt{2(1 + e^{-4|\alpha|^2})}}.
\end{equation}
Inserting(\ref{base1}) and (\ref{base2}) in $ \vert \alpha , m
\rangle$ , we
get the form of the pure state $ \vert \alpha , m
\rangle$  in the basis
$\{ \vert 0 \rangle_{1} \otimes \vert 0 \rangle_{23} ,
 \vert 0 \rangle_{1} \otimes \vert 1 \rangle_{23} , \vert  1 \rangle_{1}
 \otimes \vert 0 \rangle_{23} , \vert  1 \rangle_{1} \otimes \vert  1
 \rangle_{23}\}$. Explicitly, it is given by
\begin{equation}
\vert \alpha , m
\rangle = \sum_{\alpha= 0,1} \sum_{\beta= 0,1} C_{\alpha,\beta}
\vert \alpha \rangle_1 \otimes \vert \beta
\rangle_{23}\label{mapping1}
\end{equation}
where the coefficients $C_{\alpha,\beta}$ are
$$ C_{0,0} = {\cal C}_k(\alpha , m)(1 + e^{ik\pi}) c^+_{1}c^+_{23}  , \qquad  C_{0,1} =  {\cal C}_k(\alpha , m) (1 -e^{ik\pi}) c^+_{1}c^-_{23} $$
$$ C_{1,0} = {\cal C}_k(\alpha , m) (1 - e^{ik\pi}) c^+_{23}c^-_{1}  , \qquad  C_{1,1} =  {\cal C}_k(\alpha , m) (1 + e^{ik\pi}) c^-_{1}c^-_{23}. $$
in terms of the quantities
$$c^{\pm}_1 =\sqrt{\frac{1 \pm  \kappa_m e^{-2|\alpha|^2}}{2}} \qquad  c^{\pm}_{23} =\sqrt{\frac{1 \pm e^{-4|\alpha|^2}}{2}} $$

The concurrence between the two qubits $1$ and $23$ is given by



\begin{equation}\label {C123}
 {\cal C}_{1\mid23} = \frac{\sqrt{(1- \kappa^2_m e^{-4|\alpha|^{2}})(1- e^{-8|\alpha|^{2}})}}{1+ \kappa_m e^{-6|\alpha|^{2}}\cos k\pi},
\end{equation}
from which we obtain
\begin{equation}\label{D123}
   D_{1\mid23} =E_{1\mid23}
   =H\bigg(\frac{1}{2}+\frac{1}{2}\frac{\kappa_m e^{-2|\alpha|^{2}} + e^{-4|\alpha|^{2}}\cos k\pi}{1+ \kappa_m e^{-6|\alpha|^{2}}\cos k\pi}\bigg),
\end{equation}

We will now present the conditions that signal whether a tripartite
quantum state is monogamous in nature with respect to quantum
discord. In figure 13, the monogamy quantity $D_{123}$ is plotted as
functions of overlapping p, for symmetric state the inequality
mentioned as the above is satisfied. Otherwise, for the
antisymmetric state (i.e.k=1) the monogamy relation is violated
during the interval $0\leq p\leq 1$.
\newline In the figures 14 and 15, corresponding respectively to symmetric and antisymmetric added coherent states,
 we plot the monogamy quantity as a function of
$|\alpha|^{2}$. We examine the positivity of the following
inequality
\begin{equation}\label{qty}
\Delta_{123}=D_{123}(m,|\alpha|^{2})=D_{1\mid23}- D_{12} - D_{13},
\end{equation}
defined in terms of the bipartite quantum discord. We shall restrict
our discussion in what follows to the interval
$0\leq|\alpha|^{2}\leq1,3$. Clearly, for symmetric states (see
figure 14) the function described by (\ref{qty})is non positive when
$0\leq|\alpha|^{2}\leq 0.644$, consequently the quantum discord is
non monogamous. Otherwise, if $|\alpha|^{2}\geq0.644$ the quantum
discord is monogamous whatever the photon excitations number m.
Thus, for the antisymmetric states (figure 15) the plotted curve
indicates that the inequality of monogamy $(\ref{qty} )$ is not
satisfied if $0\leq|\alpha|^{2}\leq 0.4$, Otherwise, for
$|\alpha|^{2}\geq 0.4$ the quantity $D_{123}$ is positive and the
quantum discord is monogamous for everything values m.
\begin{figure}[!htb]
\begin{centering}
\includegraphics[scale=0.6] {casspecialentropicmonogamy}
\par\end{centering}
\caption{Monogamy $D_{123}$ versus the overlapping p when m=0}
\end{figure}


\begin{figure}[!htb]
\begin{centering}\label{0}
\includegraphics[scale=0.6]{monogamyk0discordeentropique}
\par\end{centering}
\caption{Monogamy $D_{123}$ of symmetric states versus
$|\alpha|^{2}$ for different number photon excitations.}
\end{figure}

\begin{figure}[!htb]
\begin{centering}\label{1}
\includegraphics[scale=0.6]{monogamydiscordeentropiquek1}
\par\end{centering}
\caption{Monogamy $D_{123}$ of antisymmetric states versus
$|\alpha|^{2}$ for different number photon excitations. }
\end{figure}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Concluding remarks}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Summarizing, we have presented in the early of this paper a class of
the single mode excited entangled coherent states
(SMEECSs)$|\psi_{p}(\alpha,m)\rangle$ , which are obtained through
actions of creation operator on the entangled coherent states. Then,
we have exhibited the important properties of quantum entanglement
by using different ways  (specially, the concurrence, quantum
discord and its version geometric). The first way, we have studied
the concurrence for bipartite systems and  investigated the
influence of phonon excitations numbers on quantum entanglement. We
also employed the other process for studied the quantum correlation
of add coherent states for tripartite quantum states (see the
equations (\ref{rho32-matrix}) and (\ref{rho112-matrix}) by the
quantum discord. Thus, we found two explicit analytic expressions of
this measure and the results obtained are discussed. Another way
which treated the quantum correlations by introducing the geometric
version of quantum discord, at this stage we derived a necessary and
sufficient condition. Specially, for the case of three-qubit states
we have proposed in the our discussion, two version(i.e.symmetric,
antisymmetric states, respectively) and the result obtained is
explained in terms of different number photon excitations(i.e.the
influence of m on geometric quantum discord). To close our work, we
have employed the concept of quantum monogamy corresponding to
quantum discord and its geometric version. In particular, we have
investigated the relation between discord monogamy and a genuine
tripartite entanglement measure for three-qubit pure states.
Therefor, We have demonstrated that the quantum correlations
examined by the entropic measure, geometric measure respectively
does not satisfy the monogamy relation(\ref{Mgm}). A very important
result is derived in this work, from a value determined of
$|\alpha|^{2}$, we see that no effect of the addition of the photon
can be found on the measurement of monogamy.
 The analysis presented in this letter can be extended to the effect of subtracting the
photon of tripartite GHZ coherent states.














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