\documentclass[11pt,dvips]{article}
\textheight = 24truecm \textwidth = 17truecm \hoffset = -2.3truecm
\voffset = -2truecm
\renewcommand{\baselinestretch}{1.2}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{latexsym}
\usepackage{color}
\begin{document}
\thispagestyle{empty}
\begin{center}

\vspace{1.8cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 {\bf Distribution of geometric quantum discord in photon-added coherent states}\\
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\vspace{1.5cm}

{\bf  xxxx}$^{a,b,c}${\footnote { email: {\sf xxxxx}}}, {\bf
yyyyy}$^{d,e}$ {\footnote { email: {\sf yyyyy}}} and {\bf
	 zzzzzz }$^{d,e}$ {\footnote { email: {\sf zzzzzz}}}

\vspace{0.5cm}
$^{a}${\it Max Planck Institute for the Physics of Complex Systems, Dresden, Germany}\\[0.5em]
$^{b}${\it Abdus Salam  International Centre for Theoretical Physics, Miramare, Trieste, Italy}\\[0.5em]
$^{c}${\it Department of Physics , Faculty of Sciences, University Ibnou Zohr,
 Agadir ,
Morocco}\\[0.5em]
$^{d}${\it LPHE-Modeling and Simulation, Faculty  of Sciences,
University
Mohammed V, Rabat, Morocco}\\[0.5em]
$^{e}${\it Centre of Physics and Mathematics (CPM),
University
Mohammed V, Rabat, Morocco}\\[0.5em]




\vspace{3cm} {\bf Abstract}
\end{center}
\baselineskip=18pt
\medskip
We examine the monogamy relation of geometric quantum discord  in photon added coherent states of 
Greenberger-Horne-Zeilinger. The Hilbert-Schmidt norm is used as  quantifier of pairwise quantum correlations. 
The  geometric quantum discord in all bipartite subsystems are explicitly given. The behavior of geometric quantum discord 
and its monogamy property versus  the excitation photon number are discussed.
\newpage

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



Entangled coherent states have found various applications 
in quantum information science (for a recent review see \cite{Sanders}). In 
fact, they were shown to serve as valuable resource in quantum  teleportation \cite{6,7,8,9,10,11},
 quantum networks \cite{12} , quantum logical encoding \cite{13}, quantum computation
  \cite{14}, quantum information processing \cite{15}, and quantum metrology \cite{16,17}.
 The experimental production of superposed coherent states constitutes in general a challenging
 task. However, despite their extreme sensitivity to environmental effects, several schemes were proposed 
 in the literature for their generation  \cite{18,19,20,21,22}. One may quote 
 for example the generation of entangled coherent states in atomic Bose-Einstein condensates \cite{23}.  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 Glauber coherent states $|\alpha\rangle$  
 and $\vert - \alpha \rangle$ ($\alpha \in \mathbb{C}$) are
used to encode logical qubits. In this scenario, balanced superpositions of $n$-partite 
Glauber coherent states of type 
$$ \vert   \alpha , \alpha ,\cdots, \alpha \rangle \pm \vert - \alpha, - \alpha, ,\cdots, -\alpha \rangle$$
can be mapped in $n$ qubit states. They reduce for $n=2$ to bipartite states commonly
termed in the literature quasi-Bell coherent states in analogy with
the four Bell states defined for two dimensional quantum systems. Also, in the special $n=3$, one recovers the 
quasi-{\rm GHZ} coherent states which are the non orthogonal extensions of the usual
Greenberger-Horne-Zeilinger three qubit states. Entanglement 
properties of quasi-Bell and quasi-{\rm GHZ} coherent states were initially 
investigated using the formalism of concurrence or equivalently entanglement of 
formation. Quantum discord which goes beyond entanglement of formation 
was also considered to evaluate the pairwise quantum discord in multipartite 
coherent states. It is also important to emphasize that the geometric variant 
of quantum discord, based on the notion of Hilbert-Schmidt norm, was used as 
quantifier of bipartite correlations in such states. 



In the same spirit, using the formalism of photon added 
coherent states, bipartite correlations in single mode excited 
entangled quasi-Bell and quasi-{\rm GHZ} coherent states were investigated 
in ????. The pairwise quantum correlations are quantified by means 
of Wootters concurrence. In  this paper, we shall be concerned 
with  the derivation of pairwise geometric quantum discord in  photon 
added quasi-{\rm GHZ} coherent states in order 
to investigate the distribution of quantum correlation among the three modes. At 
this stage, it worth to notice that in multipartite quantum systems, one 
of the most important properties is the monogamy relation 
which limits the free shareability and subsequently imposes severe restrictions 
on the distribution of quantum correlations between the different parts of 
the system. This concept  was first discussed by Coffman, Kundo and
Wootters in 2001 \cite{3qubit monogmy} for the entanglement of formation in three qubits and latter
generalized for $N$ qubits \cite{Nqubit monogmy}. It
was also extended to other measures (quantum discord. geometric quantum discord, ...) of quantum correlations
\cite{39},\cite{40}, \cite{41},\cite{42}, \cite{43},\cite{44}.










The paper is organized as follows. In section 2, we introduce 
photon added coherent states of Greenberger-Horne-Zeilinger type which 
interpolate continuously between the . 
In particular we discuss the different bi-partitions of such tripartite 
states. Two scenario are considered: pure and mixed. In each case, a suitable 
qubit mapping is defined. The pairwise geometric discord is derived 
in section 3. The influence of the photon addition process is discussed. A special 
attention is devoted to the limiting case corresponding to photon added states of $W$-type. 
The monogamy relation of geometric discord is examined in section 3 and 
the effects of the excitation are discussed. Concluding remarks close 
this paper.

  
   
   





















%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Adding photons to quasi-GHZ coherent states and three qubit encoding}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Excitations of quasi-GHZ coherent states }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We are interested in tripartite quantum states involving Glauber coherent states $\vert \alpha \rangle$
and  $\vert -\alpha \rangle$ with the same amplitude $\alpha$ and phases differing by $\pi$.  In
this paper, we consider the ${\rm GHZ}$-type entangled coherent states of the form
\begin{equation}\label{GHZ0}
|{\rm GHZ}_k(\alpha)\rangle=\mathcal{C}_{k} (\alpha) (|\alpha,\alpha,\alpha\rangle + e^{ik\pi} |-\alpha,-\alpha,-\alpha\rangle).
\end{equation}
where the normalization constant $\mathcal{C}_{k} $ is given by
\begin{equation}
\mathcal{C}^{-2}_{k} (\alpha) = 2 + 2 e^{-6|\alpha|^2} \cos k\pi.
\end{equation}
A single-mode of electromagnetic field is algebraically described by the Weyl-Heisenberg
algebra spanned by creation $a^+$ and annihilation $a^-$ operators.
The adding photons process is mathematically realized through successive applications
of $a^+$. Thus,
$m$ successive  actions of creation operator $a^+$  on  the  Glauber coherent states $\vert \alpha \rangle$
\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 vectors $\vert n \rangle$ denote the Fock-Hilbert states
 of harmonic oscillator. 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{laguerre}
L_{m}(x)=\sum_{n=0}^{m}\frac{(-1)^{n} 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{valeur-moy1}
\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}




The excitation of the first mode of the state (\ref{GHZ0}), by adding $m$ photon, leads to the tripartite state
\begin{equation}
|| {\rm GHZ}_k(\alpha,m) \rangle= ( (a^+)^m
\otimes \mathbb{I} \otimes \mathbb{I})~ |{\rm GHZ}_k(\alpha)\rangle,
\end{equation}
from which we introduce the normalized photon added quasi-${\rm GHZ}$ coherent states as
\begin{equation}\label{GHZm0}
| {\rm GHZ}_k(\alpha,m) \rangle=  \frac{|| {\rm GHZ}_k(\alpha,m) \rangle}{\sqrt{\langle {\rm GHZ}_k(\alpha,m) || {\rm GHZ}_k(\alpha,m) \rangle}}.
\end{equation}
Using the expressions (\ref{valeur-moy}) and  (\ref{valeur-moy1}), the vector (\ref{GHZm0}) rewrites as
\begin{equation}\label{GHZm}
| {\rm GHZ}_k(\alpha,m) \rangle= \mathcal{C}_{k} (\alpha,m) (|m , \alpha \rangle \otimes |\alpha\rangle \otimes |\alpha\rangle
 + e^{ik\pi} |m, -\alpha\rangle \otimes |-\alpha\rangle \otimes |-\alpha\rangle ) .
\end{equation}
where the normalization factor is
\begin{equation}
\mathcal{C}^{-2}_{k} (\alpha,m) = 2 + 2 \kappa_m e^{-6\vert \alpha \vert^2} \cos k\pi.
\end{equation}
with
\begin{equation}\label{kappa}
\kappa_m \equiv \kappa_m (|\alpha|^{2}) :=  \frac{L_{m}(|\alpha|^{2})}{L_{m}(-|\alpha|^{2})}.
\end{equation}

The quantity $\kappa_m$, defined by (\ref{kappa}), goes to unit for $m=0$ so that the state $| {\rm GHZ}_k(\alpha,m) \rangle$
(\ref{GHZm}) reduces to $| {\rm GHZ}_k(\alpha) \rangle$ (\ref{GHZ0}). It is also important to note
that for $|\alpha|$ large, the overlap between  Glauber coherent states $\vert \alpha \rangle$ and $\vert -\alpha \rangle$
approaches zero and then they are quasi-orthogonal. In this limiting situation the state $| {\rm GHZ}_k(\alpha) \rangle$ (\ref{GHZ0})
reduces to an usual three qubit state of {\rm GHZ}-type
\begin{equation}\label{GHZusual}
| {\rm GHZ}_k(\infty) \rangle = \frac{1}{\sqrt{3}} (|{\bf 0} \rangle \otimes |{\bf 0}\rangle \otimes |{\bf 0}\rangle
 + e^{ik\pi} |{\bf 1} \rangle \otimes |{\bf 1} \rangle \otimes |{\bf 1} \rangle ) .
\end{equation}
where $\vert {\bf 0} \rangle \equiv \vert \alpha \rangle $ and $\vert {\bf 1} \rangle \equiv \vert \alpha \rangle$.\\









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Qubit encoding  }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




\noindent In investigating the pairwise quantum discord in a tripartite system $1-2-3$
described by a quantum state of type $| {\rm GHZ}_k(\alpha,m) \rangle$, one needs the
reduced density matrices describing the two qubit subsystems $1-2$, $2-3$ and $1-3$. For the states
$| {\rm GHZ}_k(\alpha,m) \rangle$ (\ref{GHZm}), 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. The pure three mode density matrix  $\rho_{123}$ is given
$$\rho_{123} = | {\rm GHZ}_k(\alpha,m) \rangle \langle {\rm GHZ}_k(\alpha,m) |.$$
The reduced density matrices $\rho_{12}$ and $\rho_{13}$ 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 defined by
\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 quasi-Bell states
\begin{equation}\label{Bell-0}
|{\rm B}_{k}(\alpha)\rangle={\cal N}_{k}(\alpha,0)~ \big[
|\alpha\rangle\otimes|\alpha\rangle + e^{ik\pi} |-\alpha\rangle\otimes|-\alpha\rangle\big]
\end{equation}
are recovered. The operator $Z$, in (\ref{matrixrho12}), 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, by tracing out the third mode, the reduced matrix density $\rho_{23}$ takes the form
\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}
To derive the pairwise correlation between the components of the subsystems $1-2$, $2-3$ and $1-3$, we assume
that the information is encoded in even and odd Glauber coherent states (Shr\"odinger cat states). In this sense, we
introduce for the first mode the following qubit 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, we consider the qubits defined by
\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.  $\{ \vert 0 \rangle_2 \otimes \vert 0 \rangle_3,
\vert 0 \rangle_2 \otimes \vert 1 \rangle_3, \vert 1 \rangle_2 \otimes \vert 0 \rangle_3 , \vert 1 \rangle_2 \otimes \vert 1 \rangle_3\}$).
The resulting density matrices, in this encoding scheme, have non vanishing entries only along the diagonal and the anti-diagonal.






















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}

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 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.




{\bf bipartition pure}\\




 we introduce 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  \rangle +  \vert \alpha , m \rangle }{\sqrt{2(1 + \kappa_m e^{-2|\alpha|^2})}}
   \qquad \vert 1 \rangle_1 = \frac{\vert \alpha , m   \rangle  -  \vert \alpha , m   \rangle }{{\sqrt{2(1-  \kappa_m e^{-|\alpha|^2})}}},
\end{equation}
for the first subsystem. For the  modes $(23)$, grouped into 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 ${\rm GHZ}_k(\alpha,m)$ , we
get the form of the pure state ${\rm GHZ}_k(\alpha,m)$  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 {\rm GHZ}_k(\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
$$c^{\pm}_1 =\sqrt{\frac{1 \pm  \kappa_m e^{-2|\alpha|^2}}{2}} \qquad  c^{\pm}_{23} =\sqrt{\frac{1 \pm e^{-4|\alpha|^2}}{2}} $$





























%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Geometric measure of quantum discord in photon added coherent states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Closest classical states to two qubit $X$ states }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To begin, we shall present the procedure leading to  the closest classically correlated state to the two-qubit $X$ state given by
\begin{equation}
\rho_{12} =  \left(
\begin{array}{cccc}
\rho_{11} & 0 & 0 & \rho_{14} \\
0 & \rho_{22} & \rho_{23} & 0 \\
0 & \rho_{32} & \rho_{33} & 0 \\
\rho_{41} & 0 & 0 & \rho_{44}
\end{array}
\right).
\label{eqn1}
\end{equation}
The state ({\ref{eqn1}}) reads in
 Fano-Bloch representation as
\begin{equation}\label{12X}
\rho_{12} = \frac{1}{4} \bigg[  \sigma_{0}\otimes \sigma_{0} +   T_{03}
\sigma_{0}\otimes \sigma_{3} +  T_{30}
\sigma_{3}\otimes \sigma_{0} + \sum_{kl} T_{kl}
\sigma_{k}\otimes \sigma_{l} \bigg]
\end{equation}
where the correlation matrix elements are given by  ??????.
The geometric measure of quantum discord is defined as the distance
the state $\rho_{12}$ and its closest classical-quantum state
presenting zero discord \cite{Dakic2010}
\begin{equation}\label{definition-discord}
D_{\rm g}(\rho_{12}) = \min_{\chi_{12}} \vert\vert \rho_{12} -
\chi_{12} \vert\vert^2
\end{equation}
where the Hilbert-Schmidt norm is defined by $ \vert\vert X
\vert\vert^2 = {\rm Tr} (X^{\dagger}X)$ and the minimization is
taken over the set of all classical states. When the measurement is
performed on the  qubit 1, the classical states
write
\begin{equation}
\chi_{12} = p_1 \vert \psi_1 \rangle \langle \psi_1 \vert \otimes \rho_1^{2} + p_2 \vert \psi_2 \rangle \langle \psi_2 \vert \otimes \rho_2^{2}
\end{equation}
where $\{ \vert \psi_1 \rangle , \vert \psi_2 \rangle \}$ is an
orthonormal basis related to the qubit $1$,  $p_i$ $(i=1,2)$ stands
for probability distribution and $\rho_i^{2}$ $(i=1,2)$ is the
marginal density of the qubit 2. The classically correlated states
$\chi_{12} $ can also be written as
\begin{equation}\label{xi12}
\chi_{12} = \frac{1}{4}\Bigg[ \sigma_{0}\otimes \sigma_{0} +
\sum_{i=1}^3 te_i ~~\sigma_i \otimes \sigma_{0} + \sum_{i=1}^3
(s_+)_{i} ~\sigma_0 \otimes \sigma_{i} + \sum_{i,j=1}^3  e_i
(s_-)_{j} ~\sigma_i \otimes \sigma_{j}\Bigg]
\end{equation}
where
$$ t = p_1-p_2, \qquad e_i = \langle \psi_1 \vert \sigma_i \vert \psi_1 \rangle,  \qquad
(s_{\pm})_{j} = {\rm Tr}\big((p_1\rho_1^{2} \pm p_2\rho_2^{2})
\sigma_{j} \big).$$
It follows that the distance between the density matrix $\rho_{12}$
and the classical state $\chi_{12}$, as measured by Hilbert-Schmidt
norm, is then given by
\begin{equation}\label{HS12-1}
|| \rho_{12} - \chi_{12}||^2 = \frac{1}{4} \bigg[  (t^2 -2te_3 T_{30} + T^2_{30}) +  \sum_{i=1}^{3}
(T_{0i} -  (s_+)_{i})^2 + \sum_{i,j=1}^3   ( T_{ij} -  e_i(s_-)_{j})^2  \bigg]
\end{equation}
The  minimization of the distance (\ref{HS12-1}),  with respect to the parameters $t$, $(s_+)_{i}$ and
$(s_-)_{i}$, gives
\begin{equation}\label{minHS12}
t= e_3T_{30} $$
$$ (s_+)_{1}= 0 \quad (s_+)_{2} = 0 \quad  (s_+)_{3} = T_{03} $$
$$ (s_-)_{i} = \sum_{j=1}^3 e_j T_{ji}.
%$$  (s_-)_{1} = e_1 T_{11} \quad (s_-)_{2} = e_2 T_{22} \quad (s_-)_{3} = e_3 T_{33}
\end{equation}
Inserting  these solutions in (\ref{HS12-1}), one has
\begin{equation}\label{HS12}
|| \rho_{12} - \chi_{12}||^2 = \frac{1}{4} \bigg[  {\rm Tr}K - \vec{e}^t K \vec{e} \bigg]
\end{equation}
where the matrix $K$ is defined by
\begin{equation}\label{matrixK12}
K = xx^{\dagger} + TT^{\dagger}
\end{equation}
with
$$ x^{\dagger}  = (0,0,T_{30})\qquad T = \left(%
\begin{array}{ccc}
T_{11} &  T_{12} & 0 \\
T_{21}   & T_{22} & 0 \\
0 & 0 &T_{33}  \\
\end{array}%
\right).$$
From equation (\ref{HS12}), one see that the minimal value of
Hilbert-Schmidt distance (\ref{HS12}) is reached for the largest
eigenvalue of the matrix $K$. We denote by $\lambda_1$, $\lambda_2$ and $\lambda_3$ the eigenvalues
of the matrix $K$ (\ref{matrixK12}) corresponding to the $X$ state
(\ref{eqn1})  or equivalently (\ref{12X}).  They
 are given by
\begin{equation}\label{vp:lambda12}
\lambda_1 = 4(\vert \rho_{14}\vert +\vert \rho_{23}\vert )^2,\qquad
\lambda_2 = 4(\vert \rho_{14} \vert - \vert \rho_{23}\vert )^2,
\qquad \lambda_3= 2[(\rho_{11}- \rho_{33})^2 + (\rho_{22}-
\rho_{44})^2].
\end{equation}
To get the minimal value of the Hilbert-Schmidt distance (\ref{HS12}) and subsequently the amount
of geometric quantum discord, one compares
$\lambda_1$, $\lambda_2$ and $\lambda_3$. As $\lambda_1$ is always greater than  $\lambda_2$,
the largest eigenvalue  $\lambda_{\rm max}$ is $\lambda_1$ or
$\lambda_3$. It follows that the geometric discord is given by
\begin{equation}
D_g(\rho_{12}) = \frac{1}{4}~ {\rm min}\{ \lambda_1 + \lambda_2 ,
\lambda_2 + \lambda_3\}.\label{eq:GMQD_new}
\end{equation}
To write down the  explicit  expressions of the closest classical
state $\chi_{12}$ to $\rho_{12}$, one has to determine the
eigenvector $\vec{e}_{\rm max}$ associated with the largest
eigenvalue $\lambda_{\rm max}$. In this respect, two cases ( $\lambda_{\rm max} =
\lambda_1$ and $\lambda_{\rm max} = \lambda_3$) are separately
discussed. We begin by density matrices $\rho_{12}$ (\ref{eqn1}) whose entries satisfy
the condition $\lambda_{\rm max} = \lambda_3$. The associated
eigenvector  is given by $\vec{e}_3 = (0,0,1)$. Replacing in the set
of  constraints (\ref{minHS12}), one has
\begin{equation}
\chi^3_{12} = \frac{1}{4}\Bigg[ \sigma_{0}\otimes \sigma_{0} +
T_{30} ~~\sigma_3 \otimes \sigma_{0} + T_{03}~\sigma_0 \otimes
\sigma_{3} + T_{33}~\sigma_3 \otimes \sigma_{3}\Bigg]
\end{equation}
In the second situation, the eigenvector corresponding to
$\lambda_1$ is given by $\vec{e}_1 = (\cos \frac{\phi}{2}, -
\sin\frac{\phi}{2} , 0)$ where $e^{i\phi} =
\frac{\rho_{14}\rho_{23}}{|\rho_{14}||\rho_{23}|} $. Reporting the
components of $\vec{e}_1$ in (\ref{minHS12}), one gets the closest
classical state
\begin{equation}
\chi^1_{12} = \frac{1}{4}\Bigg[ \sigma_{0}\otimes \sigma_{0} +
T_{30} ~~\sigma_3 \otimes \sigma_{0} + \sum_{i=1}^{2} \sum_{j=1}^{2}
\tilde{T}_{ij}~\sigma_i \otimes \sigma_{j} \Bigg]
\end{equation}
where
$$ \tilde{T}_{11} = \cos \frac{\phi}{2} (\cos \frac{\phi}{2} T_{11} - \sin \frac{\phi}{2} T_{21} )
\qquad  \tilde{T}_{12} =  \cos \frac{\phi}{2} (\cos \frac{\phi}{2}
T_{12} - \sin \frac{\phi}{2} T_{22} )$$
$$  \tilde{T}_{21} =  -\sin \frac{\phi}{2} (\cos \frac{\phi}{2} T_{11} - \sin \frac{\phi}{2} T_{21} )
\qquad \tilde{T}_{22} =  -\sin\frac{\phi}{2} (\cos \frac{\phi}{2}
T_{12} - \sin \frac{\phi}{2} T_{22} ).$$



















%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pairwise geometric discord in tripartite quasi-GHZ coherent states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In this section, we are interested to study the tripartite quantum
correlations present in single mode excited entangled coherent
states (\ref{GHZm}). Using the qubit mapping (\ref{qubit-m}) and (\ref{qubit-0}),
the bipartite density matrix $\rho_{12} = {\rm Tr}_3 \rho_{123}$ writes in Fano-Bloch
form as
\begin{equation}\label{Fano12X}
\rho_{12} = \frac{1}{4} \bigg[  \sigma_{0}\otimes \sigma_{0} +   R^{12}_{03}
\sigma_{0}\otimes \sigma_{3} +  R^{12}_{30}
\sigma_{3}\otimes \sigma_{0} + \sum_{k=1,2,3} R^{12}_{kk}
\sigma_{k}\otimes \sigma_{k} \bigg]
\end{equation}


\begin{equation}\label{R12}
R^{12}_{00}= 1$$
$$ R^{12}_{03}= 2{\cal C}^2_k(\alpha , m)~ (e^{-2|\alpha|^2} + \kappa_m e^{-4|\alpha|^2}\cos k\pi )$$
$$R^{12}_{30}= 2{\cal C}^2_k(\alpha , m) ~(\kappa_m  e^{-2|\alpha|^2} + e^{-4|\alpha|^2}\cos k\pi )$$
$$R^{12}_{11}= 2{\cal C}^2_k(\alpha , m) ~\sqrt{(1 -  \kappa^2_m e^{-4|\alpha|^2} )(1 - e^{-4|\alpha|^2})}$$
$$R^{12}_{22}= -2{\cal C}^2_k(\alpha , m)~\sqrt{(1 -  \kappa^2_m e^{-4|\alpha|^2} )(1 - e^{-4|\alpha|^2})}~ e^{-2|\alpha|^2} \cos k\pi $$
$$R^{12}_{33}= 2{\cal C}^2_k(\alpha , m) ~(\kappa_m e^{-4|\alpha|^2} + e^{-2|\alpha|^2}\cos k\pi)
\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}\label{lambda12}
\lambda^{12}_{1}&=& \frac{(1-\kappa^2_m e^{-4|\alpha|^2})(1- e^{-4|\alpha|^2})}{(1+ \kappa_m e^{-6|\alpha|^2}\cos k\pi)^2},\\
\lambda^{12}_{2}&=& e^{-4|\alpha|^2}~\frac{(1-\kappa^2_m e^{-4|\alpha|^2})(1- e^{-4|\alpha|^2})}{(1+ \kappa_m e^{-6|\alpha|^2}\cos k\pi)^2},\\
\lambda^{12}_{3}&=&e^{-4|\alpha|^2}~\frac{(1+ \kappa^2_m) (1+e^{-4|\alpha|^2}) + 4 \kappa_m e^{-2|\alpha|^2}\cos k\pi}{( 1+ \kappa_m e^{-6|\alpha|^2}\cos k\pi)^2}.
\end{eqnarray}










\begin{equation}\label{Fano23X}
\rho_{23} = \frac{1}{4} \bigg[  \sigma_{0}\otimes \sigma_{0} +   R^{23}_{03}
\sigma_{0}\otimes \sigma_{3} +  R^{23}_{30}
\sigma_{3}\otimes \sigma_{0} + \sum_{k=1,2,3} R^{23}_{kk}
\sigma_{k}\otimes \sigma_{k} \bigg]
\end{equation}


\begin{equation}\label{R23}
R^{23}_{00}= 1$$
$$ R^{23}_{03}=2{\cal C}^2_k(\alpha , m)~(e^{-2|\alpha|^2} + \kappa_m e^{-4|\alpha|^2}\cos k\pi )$$
$$R^{23}_{30}= 2{\cal C}^2_k(\alpha , m)~ (e^{-2|\alpha|^2} + \kappa_m e^{-4|\alpha|^2}\cos k\pi ) $$
$$R^{23}_{11}= 2{\cal C}^2_k(\alpha , m)~ \sqrt{(1 -   e^{-4|\alpha|^2} )(1 - e^{-4|\alpha|^2})}$$
$$R^{23}_{22}= 2{\cal C}^2_k(\alpha , m)~ \sqrt{(1 -  e^{-4|\alpha|^2} )(1 - e^{-4|\alpha|^2})}~ \kappa_m e^{-2|\alpha|^2} \cos k\pi$$
$$R^{23}_{33}= 2{\cal C}^2_k(\alpha , m)~ (e^{-4|\alpha|^2} + \kappa_m  e^{-2|\alpha|^2}\cos k\pi)
\end{equation}
















\begin{eqnarray}\label{lambda23}
\lambda^{23}_{1}&=& \bigg(\frac{1- e^{-4|\alpha|^2}}{1+ \kappa_m e^{-6|\alpha|^2}\cos k\pi}\bigg)^2,\\
\lambda^{23}_{2}&=& \kappa^2_m e^{-4|\alpha|^2}~\bigg(\frac{1-e^{-4|\alpha|^2}}{1+ \kappa_m e^{-6|\alpha|^2}\cos k\pi}\bigg)^2,\\
\lambda^{23}_{3}&=& e^{-4|\alpha|^2}~\frac{(1+ \kappa^2_m) (1+e^{-4|\alpha|^2}) + 4 \kappa_m e^{-2|\alpha|^2}\cos k\pi}{( 1+ \kappa_m e^{-6|\alpha|^2}\cos k\pi)^2}.
\end{eqnarray}











Now we consider the case of the pure bi-partitioning scheme of type $1\vert 23$.
Using the Schmidt decomposition,  the state (\ref{mapping1}) rewrites as

\begin{equation}
\vert {\rm GHZ}_k(\alpha,m)
\rangle_{1\vert23} = \sqrt{\lambda_+}
\vert + \rangle_1 \otimes \vert +
\rangle_{23} + \sqrt{\lambda_-} \vert - \rangle_1 \otimes \vert
-\rangle_{23}
\end{equation}
where $\vert \pm \rangle_1$ (resp. $\vert \pm \rangle_{23}$) denotes the
eigenvectors of the reduced density matrix $\rho_1$ (resp. $\rho_{23}$ viewed as a single qubit state).  The
eigenvalues $\lambda_+$ and $\lambda_-$ are given by

\begin{equation}
\lambda_{\pm} = \frac{1}{2}\bigg[ 1 \pm e^{-2|\alpha|^2} \frac{\kappa_m + e^{-2|\alpha|^2}\cos k\pi }{1 + \kappa_m  e^{-6|\alpha|^2}\cos k\pi}\bigg]
\end{equation}


The Fano-Bloch representation of the state $\rho_{1\vert 23}$ takes the form

\begin{equation}\label{Fano123X}
\rho_{1|23} = \frac{1}{4} \bigg[  \sigma_{0}\otimes \sigma_{0} +   R^{1|23}_{03}
\sigma_{0}\otimes \sigma_{3} +  R^{1|23}_{30}
\sigma_{3}\otimes \sigma_{0} + \sum_{k=1,2,3} R^{1|23}_{kk}
\sigma_{k}\otimes \sigma_{k} \bigg]
\end{equation}

where

\begin{equation}\label{R123}
R^{1|23}_{00}= R^{1|23}_{33}= 1$$
$$ R^{1|23}_{03}= R^{1|23}_{30} = e^{-2|\alpha|^2} \frac{\kappa_m + e^{-2|\alpha|^2}\cos k\pi }{1 + \kappa_m  e^{-6|\alpha|^2}\cos k\pi}$$
$$R^{1|23}_{11}= - R^{1|23}_{22} = \frac{\sqrt{1- \kappa^2_m e^{-4|\alpha|^2}}~\sqrt{1- e^{-8|\alpha|^2}}}{1 + \kappa_m  e^{-6|\alpha|^2}\cos k\pi}
\end{equation}

The eigenvalues

\begin{eqnarray}\label{lambda123}
\lambda^{1|23}_{1}&=& \frac{(1 - \kappa^2_m  e^{-4|\alpha|^2})(1 - e^{-8|\alpha|^2})}{(1 + \kappa_m  e^{-6|\alpha|^2}\cos k\pi)^2},\\
\lambda^{1|23}_{2}&=& \frac{(1 - \kappa^2_m  e^{-4|\alpha|^2})(1 - e^{-8|\alpha|^2})}{(1 + \kappa_m  e^{-6|\alpha|^2}\cos k\pi)^2},\\
\lambda^{1|23}_{3}&=& 2 - \frac{(1 - \kappa^2_m  e^{-4|\alpha|^2})(1 - e^{-8|\alpha|^2})}{(1 + \kappa_m  e^{-6|\alpha|^2}\cos k\pi)^2}.
\end{eqnarray}






{\bf Limiting cases}\\

The behavior of the pairwise geometric quantum discord is very interesting in
antisymmetric states $\vert {\rm GHZ}_1(\alpha,m)
\rangle$ in the limiting case $|\alpha| \longrightarrow 0$. In this case, it
simply verified that the state (\ref{GHZm}) reduces to
\begin{eqnarray}\label{GHZalpha=0}
| {\rm GHZ}_1(0,m) \rangle = \frac{1}{\sqrt{m+3}} (\sqrt{m+1} \vert m+1, 0, 0 \rangle + \vert m, 1, 0 \rangle + \vert m, 0, 1 \rangle )
\end{eqnarray}
which coincides with the usual three qubit ${\rm W}$ states for $m=0$ \cite{Dur}. The state (\ref{GHZalpha=0}) is expressed in the Fock-Hilbert basis. In this
limit the first qubit is encoded in the Fock states $\vert m \rangle$ and $\vert m+1 \rangle$ and the two other modes are
encoded in the states $\vert 0 \rangle$ and $\vert 1 \rangle$. In the limit $|\alpha| \longrightarrow 0$, one has $L_m(|\alpha|^2) \simeq 1 - m|\alpha|^2$ and
the  quantity $\kappa_m$ (\ref{kappa}) writes
\begin{eqnarray}\label{kappaalpha0}
\kappa_m \simeq 1 - 2m|\alpha|^2.
\end{eqnarray}
It follows that the eigenvalues (\ref{lambda12}) reduce, in this limit, to

\begin{eqnarray}
\lambda^{12}_{1}&\longrightarrow & 4 ~\frac{m+1}{(m+3)^2},\\
\lambda^{12}_{2}&\longrightarrow & 4 ~\frac{m+1}{(m+3)^2},\\
\lambda^{12}_{3}&\longrightarrow & 2 ~\frac{m^2+1}{(m+3)^2}.
\end{eqnarray}

Similarly, for the reduced density matrix $\rho_{23}$ (\ref{matrixrho23}) with $k=1$, it is simple
to check that the eigenvalues (\ref{lambda23}) are given by

\begin{eqnarray}
\lambda^{23}_{1}&\longrightarrow & \frac{4}{(m+3)^2},\\
\lambda^{23}_{2}&\longrightarrow & \frac{4}{(m+3)^2},\\
\lambda^{23}_{3}&\longrightarrow & 2 ~\frac{m^2+1}{(m+3)^2}.
\end{eqnarray}


Finally, for the state $\rho_{1|23}$ (\ref{Fano123X}),  viewed as a two qubit system, the
corresponding take the following form
\begin{eqnarray}
\lambda^{1|23}_{1}&\longrightarrow & 8 ~ \frac{m+1}{(m+3)^2},\\
\lambda^{1|23}_{2}&\longrightarrow & 8~ \frac{m+1}{(m+3)^2},\\
\lambda^{1|23}_{3}&\longrightarrow & 2- 8 ~\frac{m+1}{(m+3)^2},
\end{eqnarray}
and the pairwise quantum discord is
\begin{eqnarray}
D_{1|23} &\longrightarrow & 4 ~\frac{m+1}{(m+3)^2},
\end{eqnarray}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Monogamy of geometric quantum discord in quasi-GHZ states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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).





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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


\begin{thebibliography}{99}


 \bibitem{Sanders} Barry. C. Sanders, J. Phys. A: Math. Theor. {\bf 45} (2012) 244002.

 \bibitem{6}  S.J. van Enk and O. Hirota, Phys. Rev. A {\bf  64} (2001) 022313.

\bibitem{7}  C.H. Bennett, G. Brassard, C. Cr\'epeau, R. Jozsa, A. Peres and W.K. Wootters ,  Phys. Rev. Lett. {\bf 70} (1993) 1895.

\bibitem{8} T.J. Johnson, S.D. Bartlett and B.C. Sanders, Phys. Rev. A {\bf 66} (2002)  042326.

\bibitem{9} X. Wang, Phys. Rev. A {\bf 64} (2001) 022302.

\bibitem{10} J. Janszky, A. Gabris, M. Koniorczyk, A. Vukics and J.K. Asb\'oth,  J. Opt. B, Quantum Semiclass. Opt.{\bf  4} (2002) S213.

\bibitem{11} H. Jeong, M.S. Kim and J. Lee, Phys. Rev. A {\bf 64} (2001) 052308.

\bibitem{12} P. van Loock, N. L\"utkenhaus, W.J. Munro and K. Nemoto,  Phys. Rev. A {\bf 78} (2008) 062319.

\bibitem{13} P.T. Cochrane, G.J. Milburn and W.J. Munro, Phys. Rev. A {\bf 59} (1999) 2631.

\bibitem{14} M.C. de Oliveira and W.J. Munro, Phys. Rev. A  {\bf 61} (2000) 042309.

\bibitem{15} H. Jeong and M.S. Kim, Phys. Rev. A {\bf 65} (2002) 042305.

\bibitem{16} N.A. Ansari, L.D. Fiore, M.A. Man'ko and V.I. Man'ko, S. Solimeno and F. Zaccaria, Phys. Rev. A {\bf 49} (1994) 2151.

\bibitem{17} J. Joo, W.J. Munro and T.P. Spiller, Phys. Rev. Lett. {\bf 107} (2011) 083601.


\bibitem{18} H. Jeong and N.B. An, Phys. Rev. A {\bf 74} (2006) 022104.

\bibitem{19} H.M. Li, H.C. Yuan and H.Y. Fan, Int. J. Theor. Phys.{\bf 48} (2009) 2849.

\bibitem{20} N.B. An, K. Kim and J. Kim, Quantum Inf. Comp. {\bf 11} (2011) 124

\bibitem{21} W.J. Munro, G.J. Milburn and B.C. Sanders, Phys. Rev. A {\bf 62} (2000) 052108.

\bibitem{22} L.M. Kuang and L. Zhou, Phys. Rev. A {\bf 68} (2003) 043606.

\bibitem{23} L.M. Kuang, Z.B. Chen and J.W. Pan, Phys. Rev. A  {\bf 76} (2007) 052324.
















.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Monogamy
%%\bibitem{Nqubit monogmy}. Tobias J. Osborne and Frank Verstraete, Phys.Rev. Lett. 96, 220503 (2006).
%%\bibitem{Bartosz Regula} Strong monogamy of four-qubit entanglement Bartosz Regula,1 Sara Di Martino,2, 1 Soojoon Lee,3, 1 and Gerardo Adesso1
%%\bibitem{R. Prabhu}Conditions for Monogamy of Quantum Correlation:
%%Greenberger-Horne-Zeilinger versus W states R. Prabhu, Arun Kumar
%%Pati, Aditi Sen(De), and Ujjwal Sen Harish-Chandra Research
%%Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
%%\bibitem{Gian Luca Giorgi}Monogamy properties of quantum and
%%classical correlations Gian Luca Giorgi IFISC (UIB-CSIC), Instituto
%%de F??sica Interdisciplinar y Sistemas Complejos, Campus Universitat
%%de les Illes Balears, E-07122 Palma de Mallorca, Spain (Dated:
%%November 14, 2011)
%%\bibitem{Alexander Streltsov}Are general quantum correlations monogamous?
%%Alexander Streltsov,1 Gerardo Adesso,2 Marco Piani,3 and Dagmar
%%Bru?1
%%\bibitem{ A.K. Pati} R. P, A.K. Pati, A. Sen(De), and U. Sen,
%%arXiv:1108.5168
%%\bibitem{Huan He} Disentangling Theorem and Monogamy for Entanglement Negativity
%%Huan He1 and Guifre Vidal1



\end{thebibliography}


\end{document}







\bibitem{1} G. Vidal, J. I. Latorre, E. Rico and A. Kitaev, Phys. Rev. Lett.,
90 227902 (2003);J. I. Latorre, E. Rico and G. Vidal, Quant. Inf.
and Comp. 4 048 (2004).
\bibitem{2} P. Calabrese and J. Cardy, JSTAT P06002 (2004)
\bibitem{3} G. Refael and J.E. Moore, Phys. Rev. Lett., 93 260602 (2004)
\bibitem{4} L. Amico, R. Fazio, A. Osterloh and V. Vedral, preprint, quant-ph/0703044.
\bibitem{5} M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, U.K., 2000.))))
\bibitem{6} C. H. Bennett, G. Brassard, C. Cr?epeau, R. Jozsa, A. Peres, and
W. K. Wooters, Phys. Rev. Lett. 70, 1895 (1993)
\bibitem{P. Agrawal} P. Agrawal and
A. Pati, Phys. Rev. A 74, 062320 (2006).

\bibitem{7} A. K.
Ekert, Phys. Rev. Lett. 67, 661 (1991).
\bibitem{Horodecki-RMP-2009} R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Rev. Mod. Phys. {\bf 81}(2009) 865
\bibitem{A. Einstein} A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47 (1935) 777.)
\bibitem{J. Bell} J. Bell, Physics (Long Island City, N. Y.)
1 (1964) 195.
\bibitem{D. M. Greenberger} D. M. Greenberger , M. A. Horne, and A. Zeilinger,
Bell?s theorem, Quantum theory, and Conceptions of the the Universe,
ed. M. Kafatos, Kluwer, Dordrecht, 69 (1989);
\bibitem{10} W. D?ur, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000).
\bibitem{Fuchs C A} Fuchs C A 1997 Phys. Rev. Lett. 79 1162
\bibitem{Wang}  X G 2002 J. Phys. A: Math. Gen. 35 165; Rungta P et al 2001
Phys. Rev. A 64 042315; Xiaoguang Wang J. Phys. A: Math. Gen. 35
(2002) 165?173
\bibitem{[16]} Gerardo Adesso, Sammy Ragy, Antony R. Lee Open Syst. Inf. Dyn. 21, 1440001 (2014)

\bibitem{[24]} P.T. Cochrane, G.J. Milburn and W.J. Munro, Phys. Rev. A 59
(1999) 2631.
 \bibitem{[25.]} M.C. de Oliveira and W.J. Munro, Phys.
Rev. A 61 (2000) 042309

\bibitem{24} Agarwal G S and Tara K 1991 Phys. Rev. A 43 492
\bibitem{26} Zavatta A, Viciani S and Bellini M 2004 Science 306 660 Zavatta A, Viciani S and Bellini M
2005 Phys. Rev. A 72 023820.
\bibitem{25} Sanders B C 1992 Phys. Rev. A 45 6811
\bibitem{Jeong } Jeong H and Kim M S 2002 Phys. Rev. A 65 042305

\bibitem{11} van Enk S L and Hirota O
2001 Phys. Rev. A 64 022313
\bibitem{23} Munro W J, Milburn G J and Sanders B C 2000
Phys. Rev. A 62 052108

\bibitem{27} Wang X G 2002 J. Phys. A: Math. Gen. 35 165 Wang X G 2001 Phys. Rev. A 64 022302
\bibitem{monogam} F. F. Fanchini, M. C. de Oliveira, L. K. Castelano, M. F. Cornelio, Phys. Rev. A 87, 032317 (2013)
\bibitem{28}  M. Daoud, R. Ahl Laamara, R. Essaber, International J. Quantum information Vol.11, No. 6 (2013) 1350057.
\bibitem{3qubit monogmy}  V. Coffman, J. Kundu and W.K. Wootters, Phys. Rev. A 61
(2000)052306.
\bibitem{Nqubit monogmy} Tobias J. Osborne and Frank Verstraete, Phys. Rev.
Lett. 96, 220503 (2006).
\bibitem{39} G.L. Giorgi, Phys. Rev. A 84 (2011) 054301.
\bibitem{40} R. Prabhu, A.
K. Pati, A.S. De and U. Sen, Phys. Rev. A 86 (2012) 052337.
\bibitem{41} Sudha, A.R. Usha Devi and A.K. Rajagopal, Phys. Rev. A
85 (2012) 012103.
\bibitem{42} M. Allegra, P. Giorda and A.
Montorsi, Phys. Rev. B 84 (2011) 245133.
\bibitem{43} X.-J. Ren and
H. Fan, Quant. Inf. Comp. Vol. 13 (2013) 0469.
\bibitem{44} A.
Streltsov, G. Adesso, M. Piani and D. Bruss, Phys. Rev. Lett. 109
(2012) 050503.
\bibitem{[25]} S. Luo, Phys. Rev. A 77 (2008) 042303; Phys. Rev. A 77 (2008)
022301.

\bibitem{[26]}  M. Ali, A.R.P. Rau and G. Alber, Phys. Rev. A 81 (2010)
042105.
\bibitem{[27]} M. Shi, W. Yang, F. Jiang and J. Du, J. Phys. A:
Mathematical and Theoretical 44 (2011) 415304.
\bibitem{[28]} D. Girolami
and G. Adesso, Phys. Rev. A 83 (2011) 052108.
\bibitem{[29]} M. Shi, F. Jiang,
C. Sun and J. Du, New Journal of Physics 13 (2011) 073016. 19
\bibitem{[30]}
M. Daoud and R. Ahl Laamara, J. Phys. A: Math. Theor. 45 (2012)
325302.
\bibitem{[31]} M. Daoud and R. Ahl Laamara, International Journal of
Quantum Information 10 (2012) 1250060.
\bibitem{[32]} G. Adesso and A. Datta,
Phys. Rev. Lett. 105 (2010) 030501; G. Adesso and D. Girolami, Int.
J. Quant. Info. 9 (2011) 1773.
\bibitem{[33]} P. Giorda and M.G.A. Paris,
Phys. Rev. Lett. 105 (2010) 020503.
\bibitem{[34]} X. Yin, Z. Xi, X-M Lu, Z.
Sun and X. Wang, J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 245502.
\bibitem{[35]} M.Daoud and R. Ahl Laamara, Phys. Lett. A 376 (2012) 2361.


%%% povm

 \bibitem{L. Henderson}  L. Henderson and V. Vedral, J. Phys. A. Math. Gen. 34, 6899 (2001).
 \bibitem{H. Ollivier}  H. Ollivier and W. Zurek, Phys. Rev. Lett. 88, 017901 (2002).
 \bibitem{S. Hamieh}  S. Hamieh, R. Kobes, and H. Zaraket, Phys. Rev. A 70 (2004) 052325
\bibitem{29}  Xiaolei Yin, Zhengjun Xi, Xiao-Ming Lu, Zhe Sun and Xiaoguang Wang, Geometric measure of quantum discord for
superpositions of Dicke states, J. Phys. B: At. Mol. Opt. Phys. 44
(2011) 245502




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nonlinear coherent states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection{{\bf Nonlinear coherent states}}\label{sec-nl}

  The nonlinear coherent states has been extensively investigated
  in the literature from different perspectives and for
  various purposes. They are especially of fundamental importance
  in quantum optics.  All nonlinear coherent states are defined
  by extending the basic relations of the usual Weyl-Heisenberg algebra
  spanned by the creation $a^+$ , annihilation $a^-$ and number $N$ operators
   \begin{equation}\label{WH}
[a^+ , a^-] = \mathbb{I}\qquad [ a^{\pm} , N ]= \pm a^{\pm}.
  \end{equation}
  A scheme unifying the different
  variants of nonlinear coherent states was proposed by Manko by
  introducing the concept of ${\rm f}$-oscillators. In this scheme, two
  new operators $b^+$ and $b^-$ are defined by
   \begin{equation}\label{new-oper}
     b^- = f(N)~ a^- \qquad b^+ =  a^+~f(N)
  \end{equation}
   where the Hermitian operator valued function $f$ defined in terms
   of the number operator $N=a^+a^-$ and describes the nonlinearity.
      In this work, we shall focus on $f$-oscillator algebras charchterized by the one parameter function
   \begin{equation}\label{f(n)}
    f(N)\equiv f_m(N) = 1 - \frac{|m|}{N+1}
\end{equation}
where $m \in \mathbb{Z}$. The ordinary Weyl-Heisenberg algebra is recovered or $m=0$.
The $f$-oscilator algebra hence introduced satisfy the following commutation relations
\begin{equation}\label{comut}
    [b^-, b^+]=(N+1)f^2(N)-Nf^2(N-1)
\end{equation}

The associated nonlinear coherent states $\vert \alpha \rangle$ are defined as the eigen-states
of the annihilation operator $b^-$
\begin{equation}\label{eq-val-pro}
b^- \vert \alpha \rangle =  \alpha ~\vert \alpha \rangle
\end{equation}
where $ \alpha \in  \mathbb{C}$.  Expanding the state  $\vert \alpha \rangle$   in the
Fock space ${\cal F} = \{ \vert n \rangle, n\in \mathbb{N}\}$ as
\begin{equation}\label{etatalpha}
\vert \alpha \rangle = \sum_{n=0}^{\infty} a_n \alpha^n \vert n \rangle,
\end{equation}
it is simple to check that the coefficients $a_n$ satisfy the recurrence relation
\begin{equation}\label{rela-rec}
\sqrt{n+1}f(n) ~a_{n+1} = a_n.
\end{equation}



For $n = m-1$,  $f(m-1)= 0$ and from the relation (\ref{rela-rec}) we have $a_{m-1}=0$ leading to  $a_{m-2}=a_{m-3}= \cdots=a_{0}=0$.
Hence for $m$ positive, the recurrence relation (\ref{rela-rec})  gives
\begin{equation}\label{sol-rela-rec}
a_n = a_m \frac{\sqrt{n!}}{\sqrt{m!}(n-m)!} \qquad n = m, m+1, \cdots, +\infty.
\end{equation}
Reporting (\ref{sol-rela-rec}) in (\ref{etatalpha}) one obtains
\begin{equation}\label{etatalpha}
\vert \alpha \rangle = a_m \sum_{n=0}^{\infty} \frac{\sqrt{(n+m)!}}{\sqrt{m!} n!} \alpha^{n+m} \vert n \rangle,
\end{equation}
where $a_m$ is determined by the normalization condition. As result, we get
 \begin{equation}\label{PACS}
\vert \alpha \rangle = ???? \sum_{n=0}^{\infty} \frac{\sqrt{(n+m)!}}{ n!} \alpha^{n} \vert n \rangle,
\end{equation}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Generalities}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Geometric discord in 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 concurrence 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}



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{matrixrho12xx}
\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}


\newpage
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}

\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.($\rho_{12}$). Finally we obtained $\lambda_{1}$,
$\lambda_{2}$ 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


