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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 {\bf Quantum correlation in multipartite photon added coherent states }\\
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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

\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
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.
\newpage
\section{Introduction}
The entanglement properties of strongly correlated quantum systems
is presently attracting great attention in condensed matter physics
\cite{1},\cite{2},\cite{3},\cite{4},\cite{5}. Its understanding has
led to the development of communication protocols like quantum
teleportation \cite{6},\cite{P. Agrawal} and quantum cryptography
\cite{7},\cite{Horodecki-RMP-2009}. Entanglement is a purely quantum
mechanical property of multipartite systems and is measurable in
terms of nonclassical correlations of the subsystems. A system is
entangled if its quantum state cannot be described as a direct
product of states of its subsystems. Famous examples of entanglement
are the Einstein-Podolsky-Rosen correlations between positions and
momenta of two particles \cite{A. Einstein} and the violation of the
Bell inequalities by spin systems that are described by Bell states,
\cite{J. Bell} which means that the existence of such states cannot
be explained by any local theory. It has been known that there exist
at least two different types of multipartite entanglement, namely,
the GHZ-type \cite{D. M. Greenberger} entanglement and the W-type
entanglement \cite{10}. However, most previous investigations have
focused on the entanglement orthogonal state. Recent research shows
that nonorthogonality plays an important role in quantum theory and
quantum information such as quantum key distribution\cite{Fuchs C
A}. The bipartite entangled non-orthogonal states is studies in
\cite{Wang}. The entangled coherent state is one of the most
important nonorthogonal states,
 and can be used to encode quantum information on continuous variables\cite{[16]}.
The even and odd Glauber coherent states, termed also
$Shr\ddot{o}dinger$ cat states, can be considered as basis states of
a logical qubit \cite{[24]},\cite{[25.]} and provides a practical
way to implement experimentally optical quantum systems useful for
quantum information. On the other hand, a coherent state is the
simplest continuous-variable state which is the closest analogue to
a classical light field and exhibits a Poisson photon number
distribution. Coherent states possess well defined amplitude and
phase, whose uncertainties are the minimum permitted by the
Heisenberg uncertainty principle. Based on coherent states, two
types of continuous-variable states, called photon-added coherent
states \cite{24} and entangled coherent states \cite{26},have been
introduced and shown to have wide applications in both quantum
physics \cite{25} and quantum information processing \cite{Jeong
},\cite{11},\cite{23},\cite{27}. To study the distribution of
quantum correlations in a quantum system among its different parts
constitutes an important issue. In this scene monogamy relations are
important, as they may indicate a structure for correlation in the
multipartite setting \cite{monogam},\cite{28}. The concept of
monogamy of entanglement  was first proved by Coffman, Kundo and
Wootters in 2001 \cite{3qubit monogmy} for three qubits and latter
generalized for N qubits \cite{Nqubit monogmy}. and since then it
was extended to other measures of quantum correlations
\cite{39},\cite{40}, \cite{41},\cite{42}, \cite{43},\cite{44}.
\newline In this work we consider a superposition of two and three-system
coherent states, we focus our study to investigate the pairwise of
quantum correlation for added tripartite coherent states, with two
special measures. The entropic measure
\cite{[25]},\cite{[26]},\cite{[27]},\cite{[28]},\cite{[29]},\cite{[30]},\cite{[31]}
and the geometric measure
\cite{[32]},\cite{[33]},\cite{[34]}\cite{[35]},  as a function of
the parameters involved. Our objective is study the dynamics of
single-mode excited entangled coherent states.

This paper is organized as follows. In  Section 2 we introduce the
Glauber coherent states. We also  present the form of the
Single-mode excited entangled coherent states which are obtained by
actions of creation operator and we study the evolution of the
entanglement of Single-mode excited entangled coherent states. The
explicit expression of three modes GHZ based on Glauber coherent
states is derived in Section 3. Section 4 is devoted for study the
evolution of the quantum correlations for each bipartite subsystems
and we obtained the explicit form of quantum discord. The some
special case is considered. Similar analysis are presented in
Section 5 when the correlation are measured by means of the
geometric discord. In section 6 we study the monogamy relation of
the two measures, quantum discord and geometric discord. Finally, as
illustration, some special cases are considered for the monogamy
relation. Concluding remarks close this paper.







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Entanglement of photon added coherent states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Photon added coherent states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We start this section with a brief description of the coherent
states generated by a canonical annihilation and creation operators
$a$ and $a^{+}$ respectively. They satisfy canonical commutation
relation $[a,a^{+} ] = I$. The coherent states are generated by
translating the vacum state $|0\rangle$,
\begin{equation}|\alpha
\rangle=D(\alpha)|0\rangle .
\end{equation}
Where the operator of translation described by
\begin{equation}D(\alpha)= exp(-\alpha a^{+}-\alpha ^{*}a).
\end{equation}
 Therefore the usual Glauber coherent states is given by the
expression
\begin{equation}
|\alpha
\rangle=\exp(\frac{-|\alpha|^{2}}{2})\sum^{\infty}_{n=0}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle,
\end{equation}
with $|\alpha|^{2}$ a complex dimensionless amplitude.
\newline
We consider m-photon excitations, by repeated application of the
creation operator of m single mode $|\alpha\rangle$, which are
satisfied the following form:
\begin{equation}\label{vecteur 1}
|\alpha;m\rangle=\left( a^{+}\right) ^{m}\mid \alpha \rangle =\exp
(\frac{-\mid \alpha \mid
^{2}}{2})\sum\limits_{n=0}^{\infty }\frac{ \alpha^{n}}{n!}\sqrt{%
\left( n+m\right) !}\mid n+m \rangle,
\end{equation}
the norm of the vector (\ref{vecteur 1}) is given by
\begin{equation}
\langle\alpha|\left( a^{-}\right) ^{m}\left( a^{+}\right) ^{m}|
\alpha \rangle =\exp (-| \alpha |^{2})\sum\limits_{n=0}^{\infty }%
\frac{\mid \alpha \mid ^{2n}}{\left( n!\right) ^{2}}\left(
n+m\right) !,
\end{equation}
we can easily obtain
\begin{equation}
\langle\alpha|(a^{-})^{m}(a^{+})^{m}|\alpha\rangle=\sum_{p=0}^{m}\frac{(m!)^{2}}{p!((m-p)!)^{2}}
|\alpha|^{2(m-p)},
\end{equation}
finally $\langle\alpha|(a^{-})^{m}(a^{+})^{m}|\alpha\rangle$ and
$\langle-\alpha|(a^{-})^{m}(a^{+})^{m}|\alpha\rangle$ can be written
respectively by

\begin{equation}\label{}
\left\{
\begin{array}{ccc}
\langle\alpha|(a^{-})^{m}(a^{+})^{m}|\alpha\rangle & = & m! L_{m}(-| \alpha|^{2} ) ,\\
\langle-\alpha|(a^{-})^{m}(a^{+})^{m}|\alpha\rangle & = & \exp(-2|\alpha|^{2}) m! L_{m}(| \alpha|^{2} ) ,\\
\end{array}%
\right.
\end{equation}
where $L_{m}(x)$ 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}

\subsection{Bipartite quasi-Bell states}
The quasi-Bell states are very interesting in quantum optics and
have been used in the field of quantum teleportation and many others
quantum computing operations. The Single-mode excited entangled
bipartite quasi-Bell states are obtained through actions of a
creation operator on coherent states, which are expressed by
\begin{equation}
|\psi_{\pm}(\alpha,m)\rangle=N_{\pm}(\alpha,m)[(a^{+})^{m}
|\alpha\rangle\otimes|\alpha\rangle\pm(a^{+})^{m}|-\alpha\rangle\otimes|-\alpha\rangle],
\end{equation}
one can directly obtain the factor of normalization as
$N_{\pm}(\alpha,m)^{-2}$
\begin{equation}
N_{\pm}(\alpha,m)^{-2}=2m![L_{m}(-|\alpha|)^{2}\pm\exp(-4|\alpha|^{2})L_{m}(-|\alpha|)^{2}],
\end{equation}
now, we consider the following state
\begin{equation}
|\psi_{p}(\alpha,m)\rangle=N_{p}(\alpha,m)[(a^{+})^{m}|\alpha\rangle\otimes
|\alpha\rangle+e^{ik\pi}(a^{+})^{m}|-\alpha\rangle\otimes|-\alpha\rangle],
\end{equation}
where the integer p ($p \in \mathbb{Z}$). Thus, the state
$\psi_{p}(\alpha,m)$ in terms of the photon-odded coherent states
can be rewritten as
\begin{equation}
|\psi_{p}(\alpha,m)\rangle=\mathcal{N}_{p}(\alpha,m)
[|m;\alpha\rangle\otimes|\alpha\rangle+e^{ik\pi}|m;-\alpha\rangle\otimes|-\alpha\rangle],
\end{equation}
with
\begin{equation}\label{}
\left\{
\begin{array}{ccccc}\label{eq Nrms}
 \mathcal{N}_{p}(\alpha,m)& = & N_{p}(\alpha,m)\sqrt{\langle\alpha|(a^{-})^{m}(a^{+})^{m}|\alpha\rangle}, & &\\
 |m;\pm\alpha\rangle & = &\frac{{(a^{+})^{m}|\pm}\alpha\rangle}{\sqrt{m!L_{m}(-|\alpha|^{2})}}. &\\
\end{array}%
\right.
\end{equation}
\subsection {Entanglement of bipartite quasi-Bell states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this subsection, we will calculate the entanglement of
single-mode excited entangled coherent state in the orthogonal
bases. Here we adopt the concurence to characterize the entanglement
of the state $|\psi_{p}(\alpha,m)\rangle$ described by
\begin{equation}
|\psi_{p}(\alpha,m)\rangle=\mathcal{N}_{p}(\alpha,m)
[|m;\alpha\rangle\otimes|\alpha\rangle+e^{ip\pi}|m;-\alpha\rangle\otimes|-\alpha\rangle],
\end{equation}
we will introduce two qubits $|0\rangle$ and  $|1\rangle$ for each
subsystem
\begin{equation}\label{a}
\begin{array}{ccc}
 |m,\alpha\rangle=a_{m}|0\rangle+b_{m}|1\rangle;\qquad && |m,-\alpha\rangle=a_{m}|0\rangle-b_{m}|1\rangle\\
|a|^{2}_{m}+|b|^{2}_{m}=1 ;\qquad& & |a|^{2}_{m}-|b|^{2}_{m}=\langle
m,\alpha|m,-\alpha\rangle\\
\end{array}%.
\end{equation}
The overlapping function $p_{m}$ and  $p$ are defined by
\begin{equation}
\left\{
\begin{array}{ccccc}
p_{m}& =  & \langle m,\alpha|m,-\alpha\rangle,\\
p    & =  & \langle \alpha|-\alpha\rangle,\\
\end{array}%
\right.
\end{equation}
and the two quantities $a_{m}$ and $b_{m}$ described in equation
(\ref{a}), are defined as
\begin{equation}\label{eq OVP}
 a_{m}=\sqrt{\frac{1+p_{m}}{2}}  ;\text{ \ \ }
 b_{m}=\sqrt{\frac{1-p_{m}}{2}} .
\end{equation}
The state describing the systems is defined by
\begin{equation}\label{eq C}
|\psi_{p}(\alpha,m)\rangle=\mathcal{N}_{p}(\alpha,m))[\underbrace{a_{m}
a(1+e^{ip\pi})}_{C_{00}}|0,0\rangle+\underbrace{ab_{m}(1-e^{ip\pi})}_{C_{10}}|1,0\rangle+
\underbrace{a_{m}b(1-e^{ip\pi})}_{C_{01}}|0,1\rangle+\underbrace{bb_{m}(1+e^{ip\pi})}_{C_{11}}|1,1\rangle].
\end{equation}
Following the approach of \cite{11} and considering Eqs.(\ref{eq
Nrms}), (\ref{eq OVP}) and (\ref{eq C}) for the excited bipartite
entangled coherent states of photon excitations $|\psi_{p}(a,
m)\rangle$, the concurrence can be calculated as
\begin{equation}\label{conci}
C=2\mathcal{N}_{p}(\alpha,m)^{2}|C_{00}C_{11}-C_{01}C_{10}|,
\end{equation}
it is straightforward to check that the concurrence is given by
\begin{equation}\label{concf}
C=2\mathcal{N}_{p}(\alpha,m)^{2}\sqrt{1-p^{2}}\sqrt{1-p^{2}_{m}},
\end{equation}
where
\begin{equation}\label{p}
p=\langle \alpha|-\alpha\rangle=\exp(-2|\alpha|^{2}),
\end{equation}
and
\begin{equation}\label{pm}
p_{m}= \langle
m,\alpha|m,-\alpha\rangle=\frac{\exp(-2|\alpha|^{2})L_{m}(|\alpha|^{2})}{L_{m}(-|\alpha|^{2})}.
\end{equation}
Then submitting equations (\ref{p}) and (\ref{pm}) into
(\ref{concf}), we see that
\begin{equation}\label{Cshow}
C(\alpha,m)=\frac{L_{m}(-|\alpha|^{2})\sqrt{1-\exp(-4|\alpha|^{2})}
\sqrt{1-\exp(-4|\alpha|^{2})(\frac{L_{m}(|\alpha|^{2})}{L_{m}(-|\alpha|^{2})})^{2}}}
{(L_{m}(-|\alpha|^{2})+\exp(-4|\alpha|^{2})\cos(k\pi)L_{m}(|\alpha|^{2}))}.
\end{equation}
 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}
We can also observe in figure 2. That C($|\alpha|$,m) increases when
$|\alpha|$ increase for differents 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{Single mode excited Tripartite GHZ coherent states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we are interested to study the tripartite quantum
correlations present in single mode excited entangled coherent
states (\ref{treeprticle}) and discuss their mathematical
properties.
\newline We recall that the three-mode GHZ coherent states are
expressed as
\begin{equation}
|\alpha,0\rangle=\mathcal{N}_{0}(|\alpha,\alpha,\alpha\rangle+exp(ik\pi)|-\alpha,-\alpha,-\alpha\rangle).
\end{equation}
Where the normalization constant $\mathcal{N}_{0}$ is given by
\begin{equation}
\mathcal{N}_{0}=(2+2p^{3}\cos(k\pi))^{-\frac{1}{2}},
\end{equation}
and p is the overlap between two states
\begin{equation}
p=\langle\alpha|-\alpha\rangle=\exp(-2|\alpha|^{2}).
\end{equation}
We consider m-photon excitations of the mod-1
\begin{equation}\label{treeprticle}
|\alpha,m\rangle=\mathcal{N}a^{+m}(|\alpha,\alpha,\alpha\rangle+exp(ik\pi)
|-\alpha,-\alpha,-\alpha\rangle)=\frac{\mathcal{N}}{\mathcal{N}_{0}}a^{+m}|\alpha,0\rangle,
\end{equation}
from the norm of this state $\langle\alpha,m|\alpha,m\rangle=1$, one
can easily find that the normalization factor $\mathcal{N}$ is given
\begin{equation}
\mathcal{N}=[2m!(L_{m}(-|\alpha|^{2})+p^{3}cos k\pi
(L_{m}|\alpha|^{2})]^{\frac{-1}{2}}.
\end{equation}
In general for reducing the total system  described by the matrix
$\rho_{ij}$ to its sub-system, is performed by the trace on all
other sub-systems. It is easy to see that reduced density matrices
$\rho_{12}$, $\rho_{13}$ are identical. Then, by tracing out systems
1 in the state $|\alpha,m\rangle$, we obtain the reduced density
matrix $\rho_{23}$ as follows
\begin{equation}\label{red}
\rho_{23}=tr_{1}\rho_{123},
 \end{equation}
Where
\begin{equation}\label{systemtrpt}
\rho_{123}=|\alpha,m\rangle\langle\alpha,m|,
 \end{equation}
in this  follows case, we note
\begin{equation}
a^{+m}|\alpha\rangle\equiv|\alpha;m\rangle;\qquad
a^{+m}|-\alpha\rangle\equiv|-\alpha;m\rangle.
 \end{equation}
Indeed, the reduced density matrice $\rho_{12}$ take this form

\begin{eqnarray}\label{matrixrho12}
\rho_{12}&=&\mathcal{N}^{2}(|\alpha;m,\alpha\rangle\langle\alpha;m,\alpha|+|-\alpha;m,-\alpha\rangle\langle-\alpha;m,-\alpha|\nonumber \\
&+&e^{-ik\pi}p|\alpha;m,\alpha\rangle\langle-\alpha;m,-\alpha|+e^{ik\pi}p|-\alpha;m,-\alpha\rangle\langle\alpha;m,\alpha|),
\end{eqnarray}
in the same way, we obtain that
\begin{equation}
\rho_{12}= \rho_{13}.
\end{equation}
Thus, we get
\begin{eqnarray}
\rho_{12}&=& \mathcal{N}^{2}[|\alpha,\alpha\rangle\langle\alpha,\alpha|\langle\alpha|a^{-m}a^{+m}|\alpha\rangle+|-\alpha,-\alpha\rangle\langle-\alpha,-\alpha|\langle-\alpha|a^{-m}a^{+m}|-\alpha\rangle\nonumber \\
&+&
|\alpha,\alpha\rangle\langle-\alpha,-\alpha|e^{-ik\pi}\langle-\alpha|a^{-m}a^{+m}|\alpha\rangle+|-\alpha,-\alpha\rangle\langle\alpha,\alpha|e^{ik\pi}\langle\alpha|a^{-m}a^{+m}|-\alpha\rangle],
\end{eqnarray}
note that
\begin{equation}
\langle\alpha|a^{-m}a^{+m}|\alpha\rangle=\langle-\alpha|a^{-m}a^{+m}|-\alpha\rangle=m!L_{m}|\alpha|^{2},
\end{equation}
and
\begin{equation}
\langle-\alpha|a^{-m}a^{+m}|\alpha\rangle=pm!L_{m}|\alpha|^{2},
\end{equation}
concerning the reduced density matrix $\rho_{2,3}$ can easily obtain
as follows
\begin{eqnarray}
\rho_{2,3}& = & \mathcal{N}^{2}[|
\alpha,\alpha\rangle\langle\alpha,\alpha |m!L_{m}(-|\alpha|^{2})+|
-\alpha,-\alpha\rangle\langle-\alpha,-\alpha| m!L_{m}(-|\alpha|^{2})\nonumber \\
&+ & | \alpha,\alpha\rangle\langle-\alpha,-\alpha |e^{-ik\pi} p
m!L_{m}|\alpha|^{2}+| -\alpha,-\alpha\rangle\langle\alpha,\alpha
|e^{ik\pi} p m!L_{m}|\alpha|^{2}],
\end{eqnarray}
 which can be written also as
\begin{eqnarray}\label{matrixrho23}
\rho_{2,3} & = & \mathcal{N}^{2}m!L_{m}(-|\alpha|^{2})[|\alpha,\alpha\rangle\langle\alpha,\alpha|+|-\alpha,-\alpha\rangle\langle-\alpha,-\alpha|\nonumber  \\
&+&\frac{p
L_{m}|\alpha|^{2}}{L_{m}(-|\alpha|^{2})}|\alpha,\alpha\rangle\langle-\alpha,-\alpha|e^{-ik\pi}+|-\alpha,-\alpha\rangle\langle\alpha,\alpha|e^{ik\pi}].
\end{eqnarray}
To study the tripartite quantum correlations present in
eq.(\ref{treeprticle}), a new formalism is used to describe our
system
\begin{equation}
|\psi\rangle_{123}=\mathcal{N^{'}}(|\psi_{1}\rangle\otimes|\psi_{2}\rangle\otimes|\psi_{3}\rangle
+e^{im\pi}|\phi_{1}\rangle\otimes|\phi_{2}\rangle\otimes|\phi_{3}\rangle),
\end{equation}
 the constant of normalization $\mathcal{N^{'}}$ can be written in the form
\begin{equation}
\mathcal{N^{'}}=(2+2cosp_{1}p_{2}p_{3}cos(m\pi))^{-\frac{1}{2}},
\end{equation}
recall that the general form of reduced density matrix in the
bipartite system can be written as
\begin{equation}\label{cte tree P}
\rho_{ij}=\mathcal{N}^{2}[|\psi_{i}\psi_{j}\rangle\langle\psi_{i}\psi_{j}|+|\phi_{i}\phi_{j}\rangle\langle\phi_{i}\phi_{j}|
+e^{ik\pi}q_{ij}|\phi_{i}\phi_{j}\rangle\langle\psi_{i}\psi_{j}|+e^{-ik\pi}q_{ij}|\psi_{i}\psi_{j}\rangle\langle\phi_{i}\phi_{j}|].
\end{equation}
The quantity $q_{ij}$ mentioned in the eq. (\ref{cte tree P}) is
defined by
\begin{equation}
q_{ij}=p_{1}p_{2}...\breve{p_{i}}...\breve{p_{j}}...p_{n},
\end{equation}
one can see that the density $\rho_{ij}$ take the following compact
form
\begin{equation}
\rho_{ij}=\frac{\mathcal{N}^{2}}{\mathcal{N}^{2}_{ij}}[a^{2}_{ij}|\psi_{ij}\rangle\langle\psi_{ij}|+b^{2}_{ij}Z|\psi_{ij}\rangle\langle\psi_{ij}|Z],
\end{equation}
with
\begin{equation}
|\psi_{ij}\rangle=\mathcal{N}_{ij}(|\psi_{i},\psi_{j}\rangle+
+e^{im\pi}|\phi_{i}\phi_{j}\rangle,
\end{equation}
$|\psi_{i}\rangle$ and $|\phi_{i}\rangle$ is written as
\begin{equation}
|\psi_{i}\rangle=a_{i}(|0_{i}\rangle+ b_{_{i}}|b_{i}\rangle,
\end{equation}
\begin{equation}
|\phi_{i}\rangle=a_{i}(|0_{i}\rangle+ -b_{_{i}}|b_{i}\rangle,
\end{equation}
where
\begin{equation}
a_{i}=\sqrt{\frac{1+p_{i}}{2}};\qquad
b_{i}=\sqrt{\frac{1-p_{i}}{2}}.
\end{equation}
Similarly for the subsystem j. \newline The state $\psi_{12}$ is
defined by the following expression:
\begin{equation}
|\psi_{12}\rangle=\mathcal{N}_{12}(|\alpha;m\rangle|\alpha\rangle+e^{im\pi}|-\alpha;m\rangle|-\alpha\rangle),
\end{equation}
for the subsystem 1
\begin{equation}\label{alpham}
|\alpha;m\rangle=a_{1}(|0\rangle_{1}+b_{_{1}}|1\rangle_{1};\qquad
|-\alpha;m\rangle=a_{1}(|0\rangle_{1}-b_{_{1}}|1\rangle_{1},
\end{equation}
with
\begin{equation}
a_{1}=\sqrt{\frac{1+p_{1}}{2}};\qquad
b_{1}=\sqrt{\frac{1-p_{1}}{2}},
\end{equation}
and
\begin{equation}
p_{1}=\langle\alpha;m|-\alpha;m\rangle=\langle\alpha|(a^{-})^{m}(a^{+})^{m}|-\alpha\rangle,
\end{equation}
for the subsystem 2
\begin{equation}\label{alpha}
|\alpha\rangle=a_{2}(|0\rangle_{2}+b_{2}|1\rangle_{1};\qquad
|-\alpha;m\rangle=a_{2}(|0\rangle_{2}-b_{2}|1\rangle_{2},
\end{equation}
with
\begin{equation}
a_{2}=\sqrt{\frac{1+p_{2}}{2}};\qquad
b_{2}=\sqrt{\frac{1-p_{2}}{2}},
\end{equation}
and
\begin{equation}
p_{2}=\langle\alpha|-\alpha\rangle.
\end{equation}
Substituting the eqs.(\ref{alpha}) and (\ref{alpham}) into
(\ref{matrixrho23}), we obtain the density matrix
\begin{equation} \label{rho112-matrix}
\rho_{12} = \mathcal{N}_{m}^2 \left( \begin{smallmatrix}
2a^{2}_{1}a^{2}_{2}(1+p\cos k\pi)& 0
    & 0&  2a_{1}a_{2}b_{1}b_{2}(1-p\cos k\pi )
)\\
0  & 2a^{2}_{1}b^{2}_{1}(1-p\cos k\pi ) &
2a_{1}a_{2}b_{1}b_{2}(1-p\cos k\pi )
) & 0 \\
0  &  2a_{1}a_{2}b_{1}b_{2}(1-p\cos k\pi ) &
2a^{2}_{2}b^{1}_{1}(1-p\cos k\pi )
) & 0 \\
2a_{1}a_{2}b_{1}b_{2}(1-p\cos k\pi ) & 0 & 0 &
2b^{2}_{1}b^{2}_{2}(1+p\cos k\pi)
\end{smallmatrix}
\right),
\end{equation}
with
\begin{equation}\label{ }
  \mathcal{{N}}_{m} =\sqrt{\frac{m!
L_{m(-|\alpha|^{2})}}{2m!(L_{m}(-|\alpha|^{2}+p^{3}\cos(k\pi)L_{m}(|\alpha|^{2})))}}.
\end{equation}
%%%Note importante: la dependance en m est contenue dans p_{1} qui est donn? par \langle\alpha|-\alpha\rangle
For the matrix $\rho_{23}$
\begin{equation}
|\psi_{23}\rangle=\frac{\mathcal{N}^{2}}{\mathcal{N}^{2}_{23}}(|\alpha\rangle|\alpha\rangle+e^{im\pi}|-\alpha\rangle|-\alpha\rangle),
\end{equation}
in this case we have
\begin{equation}
p=p_{1}=p_{2}=\langle\alpha|-\alpha\rangle,
\end{equation}
with
\begin{equation}
a=a_{1}=a_{2}=\sqrt{\frac{1+p}{2}};\qquad
b=b_{1}=b_{2}=\sqrt{\frac{1-p}{2}},
\end{equation}
for $\rho_{23}$, we obtain the density matrix
\begin{equation}\label{rho32-matrix}
\rho_{23} = \mathcal{N}_{m}^2 \left( \begin{smallmatrix}
2a^{4}(1+q_{23}\cos k\pi)& 0
    & 0&  2a^{2}b^{2}(1-q_{23}\cos k\pi )
)\\
0  & 2a^{2}b^{2}(1-q_{23}\cos k\pi ) &  2a^{2}b^{2}(1-q_{23}\cos
k\pi )
) & 0 \\
0  &  2a^{2}b^{2}(1-q_{23}\cos k\pi ) & 2a^{2}b^{2}(1-q_{23}\cos
k\pi )
) & 0 \\
2a^{2}b^{2}(1-q_{23}\cos k\pi ) & 0 & 0 & 2b^{4}(1+q_{23}\cos k\pi)
\end{smallmatrix}
\right),
\end{equation}
where
\begin{equation}
q_{23}=p\times\frac{L_{m}(|\alpha|^{2})}{L_{m}(-|\alpha|^{2})}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Quantifying the quantum discord}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It has recently been shown that quantum discord can only be
calculated for the limiting case of a quantum state has two qubits,
but for analytical expression of a more general quantum state are
not known until now. We have shown in this subsection the explicit
form of quantum discord applied on the reduced density matricx
described in (\ref{rho112-matrix}) and (\ref{rho32-matrix})
respectively, for which the discord can be evaluated in a simple
way.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Generalities }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%The concept of quantum discord was proposed to quantify
 The total correlation is usually quantified by the mutual
information, usually expressed in term of von Neumann entropy as
\begin{equation}\label{def: mutual information}
    I(\rho_{AB})=S(\rho_A)+S(\rho_B)-S(\rho_{AB}),
\end{equation}
where $\rho_{AB}$ is the state of a bipartite quantum system
composed of the subsystems $A$ and $B$, the operator
$\rho_{A(B)}={\rm Tr}_{B(A)}(\rho_{AB})$ is the reduced state of
$A$($B$) and $S(\rho)$ is the von Neumann entropy of a quantum state
$\rho$.\\
Clearly, the mutual information $I(\rho_{AB})$ contains both quantum
and classical correlations. Therefore, it is natural to decompose
total correlations as
$$ I(\rho_{AB}) =  D(\rho_{AB}) +  C(\rho_{AB}).$$
Let $\mathcal{B} = {B_{i}} $ be a POVM, then $B_{i}$ should be a
positive valued operator.
 Given the bipartite state $\rho_{AB}$ a possible
measure of the classical correlation between subsystems A and B is
\begin{equation}\label{def clas corlt}
    C_{B}(\rho_{AB})=max_{\mathcal{B}}[S(\rho_{A})-\sum_{i}p_{i}S(\rho^{i}_{A})],
\end{equation}
$\mathcal{B}$ is a $POVM$ and the sum over i runs over all its
elements $B_{i}$ . The conditional density matrix $\rho_{ A}^{ i}$
is the density matrix of A after performing the measurement $B_{i}$
on $B$:
\begin{equation}
   \rho_{A}^{i}=\frac{(\mathcal{B}^{i}\otimes II)\rho_{AB}(\mathcal{B}^{i}\otimes
   II)}{p_{i}},
\end{equation}
where $p_{i}=tr(\mathcal{B}^{i}\otimes
II)\rho_{AB}(\mathcal{B}^{i}\otimes II)$ is the probability of $A$
being in the state $\rho^{i}_{A}$. \newline It follows that, for a
bipartite quantum system, the quantum discord is defined as the
difference between total correlation and classical
correlation(\cite{L. Henderson},
 \cite{H. Ollivier},
 \cite{S. Hamieh}).
\begin{equation} \label{def: discord}
    D(\rho_{AB})= I(\rho_{AB}) -
    max_{\mathcal{B}}[S(\rho_{A})-\sum_{i}p_{i}S(\rho^{i}_{A})].
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Quantum discord}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 For the $X-$states $\rho_{AB}$ (rank 2), which are writing in the form
\begin{equation}
\rho_{AB} =  \left( \begin{matrix}\rho_{00}& 0
    & 0&\sqrt{\rho_{00}\rho_{33}} \\
0  & \rho_{11} &  \sqrt{\rho_{11}\rho_{22}}
) & 0 \\
0  & \sqrt{\rho_{11}\rho_{22}}  & \rho_{22} & 0 \\
\sqrt{\rho_{00}\rho_{33}} & 0 & 0 &\rho_{33}
\end{matrix}
\right). \label{rho12-matrix}
\end{equation}
the quantum discord $D_{AB}=D(\rho_{AB})$ is given by the main
equation
\begin{equation}
D_{AB}=H(\rho_{00}+\rho_{11})-H(\rho_{00}+\rho_{33})+H(\frac{1}{2}+\frac{1}{2}\sqrt{1-4(\sqrt{\rho_{00}\rho_{11}}-\sqrt{\rho_{22}\rho_{33}})^{2}}
),
\end{equation}
calculating the quantum discord in the states $\rho_{12}$ as well as
the states $\rho_{13}$, we resulting
\begin{eqnarray}
D_{12}&=&H(\mathcal{N}_{m}^{2}(1+p\cos k\pi)(1+pp_{1}))\\
&-&H(\mathcal{N}_{m}^{2}(1+p_{1})(1+p^{2}\cos k\pi))\\
&+&H(\frac{1}{2}+\frac{1}{2}\sqrt{1-4\mathcal{N}^{4}_{m}(1-p^{2})^{2}p^{2}_{1}}).
\end{eqnarray}
Clearly, $D_{13}=D_{12}.$ Then, the explicit expression of quantum
discord $D(\rho_{23})$in the states $\rho_{23}$ is given by the
expression
\begin{eqnarray}
D_{23}&=&H(\mathcal{N}_{m}^{2}(1+p)(1+pq_{23}\cos k\pi))\\
&-&H(\mathcal{N}_{m}^{2}(1+q_{23}\cos k\pi)(1+p^{2}))\\
&+&H(\frac{1}{2}+\frac{1}{2}\sqrt{1-4\mathcal{N}_{m}^{4}(1-p^{2})(1-q_{23}^{2})}).
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Special case}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{itemize}
    \item Reduced density matric $\rho_{12}$
\end{itemize}
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
\begin{eqnarray}
D_{12}&=&H(\frac{1}{2}\frac{(1+p\cos k\pi)(1+p^{2})}{(1+p^{3}\cos k\pi)})\\
&-&H(\frac{1}{2}\frac{(1+p)(1+p^{2}\cos k\pi)}{1+p^{3}\cos k\pi)})\\
&+&H(\frac{1}{2}+\frac{1}{2}\sqrt{1-\frac{(1-p^{2})^{2}p^{2}}{(1+p^{3}\cos
k\pi)^{2}}}),
\end{eqnarray}
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{itemize}
\item Reduced density matric $\rho_{23}$
\end{itemize}
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{Geometric measure of quantum discord}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The main step in evaluating the quantum discord is the minimization
of conditional entropy to get an explicit expression of the quantum
discord in tripartite system. this process constitutes a hard task,
even for two-qubit systems in a general state. This motivated the
introduction of an alternative measure, which was named geometric
quantum discord. The geometric measure of quantum discord has been
recently proposed as a simple and intuitive quantifier of general
non-classical correlations in bipartite quantum states, which can be
exactly reformulated as the minimal disturbance, measured in terms
of the squared Hilbert-Schmidt distance between the given quantum
state $\rho$ and the classical-quantum state $\chi$, reading
\begin{equation}\label{}
D_{g}= min_{\chi}\|\rho-\chi\|^{2},
\end{equation}
where $\chi$ denotes the set of zero-discord states and
$\|\rho-\chi\|^{2}=Tr (\rho-\chi)^{2}$ is the Hilbert?Schmidt norm.
Here, the measurement is performed on the subsystem A, the structure
of the zero-discord state $\chi$ is defined by
\begin{equation}\label{}
\chi=\Sigma_{i=1,2}p_{i}|\psi_{i}\rangle\langle\psi_{i}|\otimes\rho_{i},
\end{equation}
$\rho_{i}$ is the density operator and $p_{i}$ is a probability
distribution. Also, $\{|\psi_{1}\rangle\langle\psi_{2}|\}$ is an
arbitrary orthonormed vector set. \newline For two-qubit systems, a
general state can be written with Fano Bloch decomposition
\begin{equation}\label{}
\rho^{AB}=\frac{1}{4}[1\otimes1+\Sigma_{i}^{3}(x_{i}\sigma_{i}^{A}\otimes1+y_{i}1\otimes1\otimes\sigma_{i}^{B})+\Sigma_{i,j}R_{ij}\sigma_{i}^{A}\otimes\sigma_{j}^{B}],
\end{equation}
with $x=x_{i}$, $y=y_{i}$ and $R_{ij}$ the real parameters and
$\sigma_{i=1,2,3}$ the Pauli matrices. We introduce a correlation
matrix define by
\begin{equation}\label{} \mathcal{R}=
 \left[%
\begin{array}{cc}
     1  & y^{T} \\
     x  &  R    \\
\end{array}%
\right],
\end{equation}
with $x=x_{i}$, $y=y_{i}$ (i=1,2,3) are the components in three
dimension of local Bloch vectors and $R$ a
 $3\times 3$ matrix with the matrix elements $\mathcal{R}_{ij}$
form tak the following form
\begin{equation}\label{}
\mathcal{R}_{ij}=Tr(\rho\sigma_{i}^{A}\otimes\sigma_{j}^{B})\text{ \
\ \ \ } for \text{ \ \ \ \ } i,j=0, 1, 2 ,3.
\end{equation}
Therefor, the concrete form of the geometric measure of quantum
discord is defined by
\begin{equation}\label{}
D_{g}(\rho)=\frac{1}{4}(\|x\|^{2}+\|R\|^{2}-k_{max}).
\end{equation}
Where $K_{max}$ is the largest eigenvalue of the matrix $k$ which
can be written as $k=xx^{T}+RR^{T}$, and the quantity given by
$\|x\|^{2}+\|R\|^{2}$ present the summation of the eigenvalues of
$k$.
\newline So, we conclude that the GMQD can be analytically computed as following
\begin{equation}\label{exp general D}
D_{g}(\rho)=\frac{1}{4}min\{\lambda_{1}+\lambda_{2},\lambda_{1}+\lambda_{3},\lambda_{2}+\lambda_{3}\}.
\end{equation}
Noting that  $\{\lambda_{1}, \lambda_{2}, \lambda_{3}\}$ present the
eigenvalues of the matrix K, and the comparison of these eigenvalues
is a key factor in the calculation of minimization procedure for
GMQD. Specially, we concentrate on the bipartite mixed density
mentioned in (\label{matrixrho23}) and (\ref{rho112-matrix}), these
kinds of states belong to the so-called X-states whose algebraic
characterization is presented in \cite{29}. Thus, it can be written
in the Bloch representation as follows
 \begin{equation}\label{ }
\rho_{ij}=\sum_{\alpha\beta}R_{\alpha\beta}\sigma_{\alpha}\otimes\sigma_{\alpha}
\end{equation}.
\begin{itemize}
    \item  Reduced density matrix $\rho_{12}$
\end{itemize}
The correlation matrix elements $R_{\alpha,\beta}$
($\alpha,\beta$=0, 1, 2, 3) are given by
\newline $R_{00}=1 $, $\text{ \ \ \ \ }$
$R_{11}=2\mathcal{N}^{2}_{m}\sqrt{(1-p_{m}^{2})((1-p^{2}))}$,$\text{
\ \ \ \ }$ $R_{22}=-2\mathcal{N}^{2}_{m}
\sqrt{(1-p_{m}^{2})((1-p^{2}))}p\cos k\pi $,$\text{ \ \ \ \ }$
$R_{33}=2\mathcal{N}^{2}_{m} p(p_{m}+\cos k\pi )$,$\text{ \ \ \ \ }$
$R_{03}=2\mathcal{N}^{2}_{m} p(1+p_{m}\cos k\pi)$,$\text { \ \ \ \
}$ $R_{30}=2\mathcal{N}^{2}_{m} (p_{m}+p^{2}\cos k\pi)$. \newline
With
\begin{equation}\label{ }
\begin{array}{ccc}
  \mathcal{{N}}_{m} & =& \sqrt{\frac{m!
L_{m(-|\alpha|^{2})}}{2m!(L_{m}(-|\alpha|^{2}+p^{3}\cos(k\pi)L_{m}(|\alpha|^{2})))}}
,\\
p_{m}& = & p\times \frac{L_{m(-|\alpha|^{2})}}{L_{m(-|\alpha|^{2})}} .\\
\end{array}%
\end{equation}
Having mapped the bipartite system ($\rho_{12}=\rho_{13}$) into a
pair of two qubit presented in the previous subsection, we obtained
the final expression of MGQD from a special comparison of the
eigenvalues given by the matrix K, which are defined by
\begin{eqnarray}
\lambda_{1}&=&4\mathcal{N}^{4}_{m}[(1+p^{2})(p^{2}_{m}+p^{2})+4p^{2}p_{m}\cos
k\pi],\\
\lambda_{2}&=&4\mathcal{N}^{4}_{m}(1-p_{m}^{2})(1-p^{2}),\\
\lambda_{3}&=&4\mathcal{N}^{4}_{m}(1-p_{m}^{2} )(1-p^{2})p^{2}.
\end{eqnarray}
In the case, we obtained  always $\lambda_{2}\geq\lambda_{3}$,n and
the maximum of the eigenvalues is $\lambda_{1}$ or $\lambda_{2}$.
Thus, the equation (\ref{exp general D}) is reduced to a simpler
expression
\begin{equation}\label{formul principal}
D_{g}=\frac{1}{4}min\{\lambda_{1}+\lambda_{3},
\lambda_{2}+\lambda_{3}\}.
\end{equation}
So, we need only  to compare the two eigenvalues $\lambda_{1}$ and
$\lambda_{2}$, for this we have two explicit expression of geometric
quantum discord
\begin{equation}\label{lnda1lnda3}
D_{g}=\mathcal{N}^{2}_{m}\text{ \ \
}[2p^{2}+p_{m}^{2}(1+p^{4})+p^{2}p_{m}\cos k\pi],
\end{equation}
when the condition $\lambda_{1}<\lambda_{2}$ is satisfied.
\newline Otherwise, the form of quantum discord is given by
\begin{equation}\label{lnda2lnda3}
D_{g}=\mathcal{N}^{2}_{m}\text{ \ \ }[(1-p_{m}^{2})(1-p^{4})].
\end{equation}
We will discuss the result obtained in equations.(\ref{lnda1lnda3})
and (\ref{lnda2lnda3}) in special cases m=0 and k=0, 1 respectively.
Noting that for symmetric states (i.e k even) and m = 0, the
condition which satisfied ($\lambda_{1}<\lambda_{2}$) is $0\leq p
\leq \sqrt{2}-1$, and the form of geometric measure of quantum
discord described in equation (\ref{lnda1lnda3}) become
\begin{equation}\label{ }
D_{g}=\frac{1}{4}\text{ \ \
}\frac{p^{2}(1+p^{2})(2+(1-p^{2})^{2})}{(1+p^{3})^{2}}.
\end{equation}
In other hand, the condition obtained ($ \sqrt{2}-1\leq p \leq 1$)
in the situation ($\lambda_{2}<\lambda_{1}$) ensured that the MGQD
stated in equation (\ref{lnda2lnda3}) can be rewritten in a simple
structure.
\begin{equation}\label{ }
D_{g}=\frac{1}{4}\text{ \ \
}\frac{p^{2}(1+p^{2})(1-p^{2})^{2}}{(1+p^{3})^{2}}.
\end{equation}
Therefore, for antisymmetric states (i.e. k odd) the result produced
$(0<p <1$) when ($\lambda_{1}<\lambda_{2}$) provided to define the
MGQD as the following form
\begin{equation}\label{ }
D_{g}=\frac{1}{4}\text{ \ \ }\frac{p^{2}(2+(1+p
)^{2})(1-p)^{2}}{(1+p^{3})}.
\end{equation}
The behavior of geometric quantum discord for special case is given
in the figure 8. This figure gives a plot the geometric quantum
discord versus the overlapping p for symmetric and antisymmetric
SMEECSs. As seen from the figure (k=0), after an initial increasing,
the quantum discord decreases to vanish when p = 1. The maximum
value of quantum discord occurs when $\lambda_{1}$ and $\lambda_{2}$
coincide. It is remarkable that for antisymmetric quantum states
take the maximum value when p = 1.
\begin{figure}[!htb]\label{cas_special_dg12}
\begin{centering}
\includegraphics[scale=0.6]{cas_special_dg12}
\par\end{centering}
\caption{ The GMQD for reduced density matric ($\rho_{12}$) of
SMEECS versus $p$ for m=0.}
\end{figure}
The geometric measure of quantum discord for symmetric and
antisymmetric states versus $|\alpha|^{2}$  are plotted. From the
figure 9 symmetric SMEECSs (k even). We observe that when
$|\alpha|^{2}\rightarrow 1$ the quantum discord decreases to vanish.
Thus, in the middle region $\lambda_{1}$ coincide with
$\lambda_{2}$. Therefore, the quantum discord is maximal. We also
observe that this maximum increases as the photon excitation number
m increases.
\begin{figure}[!htb]
\begin{centering}\label{dk0geo}
\includegraphics[scale=0.6]{12discordegeometriquek0}
\par\end{centering}
\caption{ The GMQD for reduced density matric ($\rho_{12}$) of
symmetric states of SMEECS
  versus $|\alpha|$ for the different
number photon excitations.}
\end{figure}
We can also observe through the figure 10, in the case m=2 the
geometric quantum discord decreases rapidly from a maximum value to
vanish if $|\alpha|^{2}\rightarrow 1.3$. However, for the case $m
\geq 3$, the quantum discord starts increasing to reach its maximal
value ($i.e.\lambda_{1}= \lambda_{2}$) and diminished when
$|\alpha|^{2}= 1,3$. \newline The result obtained is that, for
antisymmetric quantum states containing more than particles, the
maximal value of quantum discord increases as m increases contrarily
to the symmetric states (see figure 9).
\begin{figure}[!htb]
\begin{centering}\label{dk1geo}
\includegraphics[scale=0.6]{12discordegeometriquek1}
\par\end{centering}
\caption{The GMQD for reduced density matric ($\rho_{12}$) of
antisymmetric states of SMEECS
  versus $|\alpha|$ for the different
number photon excitations.}
\end{figure}
\begin{itemize}
    \item  Reduced density matrix $\rho_{23}$
\end{itemize}
for  calculated the eigenvalue of the reduced density matric
$\rho_{23}$, we have follow the same procedure studied in reduced
density matric.($\rho_{12}$). Finaly we obtianed $\lambda_{1}$,
$\lambda_{2}$ and $\lambda_{3}$ of the  matrix $\rho_{23}$ as
\begin{eqnarray}
\lambda_{1}&=&4\mathcal{N}_{m}^{4}[(1+p^{2})+(p^{2}+q^{2}_{23})+4p^{2}p_{m}\cos k\pi],\\
\lambda_{2}&=&4\mathcal{N}_{m}^{4}(1-p^{2})(1-p^{2}),\\
\lambda_{3}&=&4\mathcal{N}_{m}^{4}(1-p^{2})(1-p^{2})q_{23}^{2}
.\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 curve of geometric
quantum discord of the reduced matric $\rho_{23}$ versus
$|\alpha|^{2}$ for symmetric and antisymmetric case is given in the
figure 11, 12 respectively. As seen from this figures, after an
initial increasing, the quantum discord decreases to vanish when
$|\alpha|^{2}\longrightarrow 1,3$. Thus, in the middle region
$\lambda_{1}$ coincide with $\lambda_{2}$. Therefore, the quantum
discord is maximal. It is clearly seen that the quantum discord
decreases with the photon excitation number is increases in the
symmetric case. Also, for the antisymmetric states, the quantum
discord is decreases when the photon excitation number increases.
\begin{figure}[!htb]
\begin{centering}\label{rho23symsymcase}
\includegraphics[scale=0.6]{rho23symsymcase}
\par\end{centering}
\caption{ The GMQD for reduced density matric  ($\rho_{23}$)of
symmetric states of SMEECS
  versus $|\alpha|^{2}$ for the different
number photon excitations.}
\end{figure}

\begin{figure}[!htb]\label{12525}
\begin{centering}
\includegraphics[scale=0.6]{imageeeeeee}
\par\end{centering}
\caption{ The GMQD for reduced density matric ($\rho_{23}$) of
antisymmetric states of SMEECS
  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
\begin{equation}\label{E(formation)}
   D(\rho_{1\mid23})
   =H(\frac{1}{2}+\frac{1}{2}\sqrt{1-C^{2}(\rho_{1\mid23}}))=E(\rho_{1\mid23}),
\end{equation}
with
\begin{equation}\label {Mgm(c)}
 C(\rho_{1\mid23}) =
 (\frac{L_{m}(-|\alpha|^{2})}{L_{m}(-|\alpha|^{2})+\cos(k\pi)e^{-6|\alpha|^{2}}L_{m}(|\alpha|^{2})})\text{ \ }(1-(e^{-8|\alpha|^{2}}))\text{ \ }(1-(e^{-2|\alpha|^{2}}
 \frac{L_{m}(|\alpha|^{2})}{L_{m}(-|\alpha|^{2})})^{2}).
\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}
D_{123}=D_{123}(m,|\alpha|^{2})=D(\rho_{1\mid23})-D(\rho_{12})-D(\rho_{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}

\begin{itemize}
    \item Monogamy of geometric quantum discord
\end{itemize}
In this part, we will examine whether the geometric quantum discord
of a tripartite state $\rho_{123}$ wich satisfy the following
 inequality
\begin{equation}\label {Mgm D}
 \mathcal{D}_{g}(\rho_{1\mid23}  \geq
 \mathcal{D}_{g}(\rho_{12})+\mathcal{D}_{g}(\rho_{13}).
\end{equation}
In the pure bi-partitioning scheme $123$, it is easy to check, using
the method presented in $\cite{28}$, that the geometric discord is
related to the concurrence of the state $\rho (1|23)$ as follows
\begin{equation}\label {Mgm vrf D}
 \mathcal{D}_{g}(\rho _{1|23}) = \frac{1}{2}\mathcal{C}^{2}_{1|23}.
\end{equation}

\begin{figure}[!htb]\label{casspecialmongamy}
\begin{centering}
\includegraphics[scale=0.6]{casspecialmongamy}
\par\end{centering}
\caption{ Monogamy $D_{g}(123)$ of geometric quantum discord as a
function of overlapping  for .}
\end{figure}

\begin{figure}[!htb]\label{monogamyk0discordegeometrique}
\begin{centering}
\includegraphics[scale=0.6]{monogamyk0discordegeometrique}
\par\end{centering}
\caption{ Monogamy of geometric quantum discord versus
$|\alpha|^{2}$ for symmetric states and for different values of m.}
\end{figure}

\begin{figure}[!htb]\label{MonogamieantisymDiscordgeometriq}
\begin{centering}
\includegraphics[scale=0.6]{MonogamieantisymDiscordgeometriq}
\par\end{centering}
\caption{ Monogamy of geometric quantum discord versus
$|\alpha|^{2}$ for antisymmetric states and for different values of
m.}
\end{figure}
Using the result obtained in the previous subsection. Thus, We treat
first the Monogamy of the geometric quantum discord for symmetric
states (k = 0, 1 respectively) and for the special case( m = 0), we
have
\begin{equation}\label{}
D_{g}(\rho_{12})=D_{g}(\rho_{13})=\frac{3p^{2}-4p^{3}+p^{6}}{4(1+p^{3})},
\end{equation}
for $0\leq p \leq\sqrt{2}-1$ and
\begin{equation}\label{}
D_{g}(\rho
_{12})=D_{g}(\rho_{13})=\frac{(1-p^{2})(1-p^{4})}{4(1+p^{3})},
\end{equation}
if the condition $\sqrt{2}-1 \leq p \leq1$ is satisfied. we have
also
\begin{equation}\label{}
D_{g}(\rho_{1|23})=\frac{1}{2}\frac{(1-p^{4})^{?2}(1-p_{m}^{2})^{2}}{(1+p^{3})^{2}},
\end{equation}
for $0\leq p \leq1$. So, we can define the quantity $D_{g}(123)$ as
the following form
\begin{equation}\label{ }
D_{g}(123)=D_{g}(\rho_{1|23})-D_{g}(\rho_{12})-D_{g}(\rho_{13}).
\end{equation}
The quantity indicate whether above is plotted in figure 16
 versus the overlap p. Clearly, the geometric quantum discord is monogamous for the
tripartite states with p such that $0\leq p \leq 0.388$.
Contrariwise, the geometric quantum discord is discord
polygamous.\newline For antisymmetric states(k = 1), the quantum
discord take the following form
\begin{equation}\label{}
D_{g}(\rho_{13})=D_{g}(\rho_{12})=\frac{1}{4}\text{ \
}[\frac{3p^{2}-4p^{3}+p^{6}}{4(1-p^{3})^{2}}],
\end{equation}
when the condition   $0 \leq p \leq 1$ is satisfied. And for
$D_{g}(\rho1/23)$, we have

\begin{equation}\label{}
D_{g}(\rho_{1|23})=\frac{1}{2}[\frac{(1-p^{2})^{2}(1-p^{4})^{2}}{4(1-p^{3})^{2}}],
\end{equation}
for  $0\leq p\leq1$. Therefor, the numerical results reported in
figure 16 show that the geometric discord is monogamous when p such
that $0\leq|\alpha|^{2}\leq 0.61649$. Otherwise, the geometric
quantum discord is polygamous.
\newline Thereafter, we plot the quantity $D_{g}(123)$ for the
different values m and investigate the influence of photon
excitations on monogamy of geometric discord. So, for symmetric case
(figure 17) the behaviour of quantum discord the monogamy inequality
is not satisfied for $0\leq|\alpha|^{2} \leq 0.524$. In the case
where $|\alpha|^{2}>0.524$ the monogamy quantity is monotonically
increasing for everything values m and the geometric quantum discord
is monogamous. In the other hand, for antisymmetric states(i.e.k=1)
with different value m (see figure 18), the monogamy is satisfied
when $|\alpha|^{2} \geq 0.53$, and this quantity becomes violated
for everything values m if $0\leq|\alpha|^{2}\leq 0.53$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.





















\newpage


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\end{document}
