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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 {\Large {\bf Distribution of geometric quantum discord in photon-added coherent states}}\\
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{\bf  M. Daoud}$^{a,b,c}${\footnote { email: {\sf m$_-$daoud@hotmail.com}}}, {\bf
W. Kaydi}$^{d,e}$ {\footnote { email: {\sf kaydi.smp@gmail.com}}} and {\bf
	 H. El Hadfi }$^{d,e}$ {\footnote { email: {\sf hanane.elhadfi@gmail.com}}}

\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}
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We examine the influence of photon excitation on the  monogamy property of  quantum discord  in tripartite coherent states of
Greenberger-Horne-Zeilinger type. The Hilbert-Schmidt norm is used as  quantifier of pairwise quantum correlations.
The  geometric quantum discord in all bipartite subsystems are explicitly given. We show
that  the geometric discord is monogamous for any photon excitation order.
\newpage

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






Entangled coherent states have found various applications
in quantum information science (for a recent review see \cite{Sanders-rev}). In
fact, they have been recognized as a 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}.
Despite their extreme sensitivity to environmental effects, several experimental schemes were proposed
 for their production \cite{18,19,20,21,22,23}.  Many protocols for quantum information processing
use coherent states as either  continuous or discrete variables. In a discrete setting,
the information is  encoded in superpositions of coherent states \cite{Ralph2} like for instance
the encoding scheme  of logical qubits  in even and odd superpositions of Glauber
coherent states: $\vert 0 \rangle \longrightarrow |\alpha\rangle + |-\alpha\rangle $ and
$\vert 1 \rangle \longrightarrow |\alpha\rangle - |-\alpha\rangle  $  ($\alpha \in \mathbb{C}$).  In this picture,
  balanced superpositions of $n$-partite
Glauber coherent states of type \cite{Wang}
$$ |\psi_n \rangle \backsim \vert   \alpha \rangle_1 \otimes | \alpha  \rangle_2  \otimes \cdots \otimes|\alpha \rangle_n \pm \vert - \alpha \rangle_1
\otimes - | \alpha \rangle_2 \otimes \cdots \otimes |-\alpha \rangle_n$$
provide a physical tool to encode
  $n$ logical qubits system. This kind of multipartite Glauber states  includes  the bipartite states ($n=2$), commonly
termed in the literature,  quasi-Bell coherent states in analogy with
the four Bell states defined for two dimensional quantum systems \cite{Sanders}. For $n=3$, the states $\vert \psi_n \rangle$ coincide with
quasi-{\rm GHZ} coherent states \cite{Jeong} which constitute the nonorthogonal analogue of the usual
Greenberger-Horne-Zeilinger three qubit states \cite{Greenberger}. Entanglement
properties of quasi-Bell ($n=2$) and quasi-{\rm GHZ} coherent states ($n=3$) were initially
investigated using the formalism of concurrence or equivalently entanglement of
formation \cite{Wang,Sanders}. Quantum discord,  introduced in \cite{Henderson,Ollivier},  which goes beyond entanglement of formation,
was also considered to evaluate the pairwise quantum discord in multipartite
coherent states \cite{daoud1,daoud2,daoud3,daoud4}. The entropy-based quantum discord involves optimization procedure
that is, in general, challenging to achieve analytically. To avoid this difficulty and to get
computable expressions of bipartite quantum correlations a geometric
variant of quantum discord was proposed in \cite{Dakic}. It uses the Hilbert-Schmidt
distance as a measure to distinguish between quantum and classical correlations. This
geometric quantifier was employed to determine the pairwise  correlations in multipartite states
of type $\vert \psi_n \rangle$ \cite{daoud4,Yin,daoud5}.






\noindent In the other hand,  an important
issue, in investigating multipartite entangled coherent states, concerns the influence of the environment on the evolution of quantum correlations.
 In this context,  bipartite correlations in single mode excited
entangled quasi-Bell \cite{Xu,Yuan}  and quasi-{\rm GHZ} coherent states \cite{Li}  were investigated
to analyze  the effect of the photon excitations processes. Interesting results
were derived in \cite{Xu,Yuan,Li}  where the pairwise quantum correlations are quantified by means
of Wootters concurrence \cite{Wootters98}. In  this paper, we  use the Hilbert-Schmidt distance as
quantifier of bipartite quantum correlations to study
  the distribution of quantum correlation  in
photon added quasi-{\rm GHZ} coherent states.  Another important aspect is the monogamy property
of the geometric discord
which limits the free shareability and subsequently imposes severe restrictions
on the distribution of quantum correlations between the different parts of
the system. The monogamy property can be
summarized as follows. Let $D_{12}$ denote the shared geometric quantum discord
 between the modes 1 and 2. Similarly, we  denote by $D_{23}$  the
geometric measure of the quantum correlation between 2 and 3 and $D_{1\vert 23}$ the correlation
shared between the mode 1  and the composite subsystem 23
comprising 2 and 3. The geometric quantum discord is monogamous if, and
only if, the following inequality
$$ D_{1\vert 23} \geq D_{12} + D_{23}$$
is satisfied.  In this respect, to establish the conditions
under which the geometric quantum discord is monogamous requires the determination of
pairwise quantum correlations between the three modes. It worth noticing that
the concept of monogamy  of quantum correlations was first discussed by Coffman, Kundo and
Wootters in 2001 \cite{Coffman} for the entanglement of formation in three qubits. It was
generalized after to multi-qubit systems \cite{Osborne} as well as other measures like
quantum discord and its geometric variant
\cite{Giorgi,Prabhu,Sudha,Allegra,Ren,Streltsov}.











The paper is organized as follows. In section 2, we introduce
photon added coherent states of Greenberger-Horne-Zeilinger type.
In particular we discuss the different bi-partitions of this class of tripartite
states. For each bipartite subsystem a suitable
qubit mapping is defined. The pairwise quantum correlations, quantified by
Hilbert-Schmidt distance, in quasi-{\rm GHZ} coherent states
are derived in section 3. The influence of the photon addition processes  is discussed. A special
attention is devoted to the limiting case corresponding to photon added states of $W$-type $(\vert \alpha \vert \longrightarrow \infty)$.
The monogamy relation of geometric discord is examined in section 3. A special
emphasis is devoted to  the evolution of the geometric quantum discord deficit
versus photon excitations number. Concluding remarks close
this paper.

























%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Photon added coherent states of ${\rm GHZ}$ type and qubit encoding}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



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

An interesting class of nonclassical states in quantum optics consists of the
photon-added coherent states \cite{Agarwal}. They result through successive applications
of the creation operator $a^+$ on the Glauber coherent state
\begin{equation}
|\alpha
\rangle=e^{-\frac{|\alpha|^2}{2}}\sum^{\infty}_{n=0}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle.
\end{equation}
Denoting by $m$ the number of added quanta, the explicit form of a normalized $m$-photon added
coherent state in the Fock states basis $\{ \vert n \rangle, ~ n \in \mathbb{N} \}$ is
\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}
The Laguerre polynomial $L_{m}(x)$ of order $m$  is 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  coherent states (quasi-classical states)
and Fock states and 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 result
\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 $|\alpha , m \rangle$ and $|-\alpha , m \rangle$ (\ref{vecteur 2}) 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}
Now, we consider a single mode excitation of  the ${\rm GHZ}$-type entangled coherent states defined by \cite{Jeong}
\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 $k \in \mathbb{Z}$ and 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}
Mathematically, the $m$ photon excitation process of the first mode, is realized as follows
\begin{equation}\label{eq}
|| {\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}
In equation (\ref{eq}), $\mathbb{I}$ is the identity operator.  Using the expressions (\ref{valeur-moy}) and  (\ref{valeur-moy1}), the state (\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 $\kappa_m$ (\ref{kappa}) goes to unit for $m=0$ and 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 \cite{Greenberger}
\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 $\vert {\bf 1} \rangle \equiv \vert -\alpha \rangle$.


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\subsection{Qubit encoding  }
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\noindent To derive the pairwise quantum discord in the three modes system $1-2-3$
described by the state
$$\rho_{123} = | {\rm GHZ}_k(\alpha,m) \rangle \langle {\rm GHZ}_k(\alpha,m) |,$$
 one needs the
reduced density matrices corresponding to the two qubit subsystems $1-2$, $2-3$ and $1-3$. It is simple to check that the reduced density
matrices $\rho_{12}={\rm Tr}_{3}\rho_{123}$ and $\rho_{13}={\rm Tr}_{2}\rho_{123}$ are
identical. They write
\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}
where $|{\rm B}_{k}(\alpha,m)\rangle$ are  the 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}). The factor $ \mathcal{N}_{k}$  is given
\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}). The operator $Z$, in (\ref{matrixrho12}), is the third Pauli generator defined by
$$Z |{\rm B}_{k}(\alpha,m)\rangle  = \mathcal{N}_{k}(\alpha,m)
[|m, \alpha ) \otimes|\alpha\rangle - e^{ik\pi} |m , -\alpha ) \otimes|-\alpha\rangle].$$
In the absence of photon excitation $(m=0)$, the states $|{\rm B}_{k}(\alpha,m)\rangle$ (\ref{Bell-m}) reduce
to the ordinary quasi-Bell states \cite{Sanders}
\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}
Similarly, by tracing out the first 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}
According to \cite{Ralph2}, the bipartite states $\rho_{12}$, $\rho_{13}$ and $\rho_{23}$ can be converted in two
qubit states by encoding information in even and odd Glauber coherent states (Shr\"odinger cat states). To realize such encoding scheme, we
introduce  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 logical qubits defined by
\begin{eqnarray}\label{qubit2}
\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{qubit2}) in (\ref{matrixrho12}) (resp. (\ref{matrixrho23})), one can express the density matrix
$\rho_{12}=\rho_{13}$ (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.  $\{ \vert 0 \rangle_2 \otimes \vert 0 \rangle_3,
\vert 0 \rangle_2 \otimes \vert 1 \rangle_3, \vert 1 \rangle_2 \otimes \vert 0 \rangle_3 , \vert 1 \rangle_2 \otimes \vert 1 \rangle_3\}$).
One can verify that the  density matrices $\rho_{12}$, $\rho_{13}$ and $\rho_{23}$ have non vanishing entries only along the diagonal and the anti-diagonal.
They resemble to the alphabet $X$ and thus belong to the family of so called $X$-states that have been of interest in a variety of contexts in the field of
quantum information \cite{Yu}.

\noindent As the main of this paper concerns the monogamy property of geometric discord in the state $| {\rm GHZ}_k(\alpha,m) \rangle$,  we
consider the bipartition
of the state (\ref{GHZm})  in which the modes 2 and 3 are grouped in a single qubit. For the first mode,
the information is encoded in the logical qubits
 $\{ \vert 0 \rangle_1 , \vert 1 \rangle_1\}$ (\ref{qubit1}) given by
defined by
\begin{equation}\label{base1}
\vert 0 \rangle_1 = \frac{ \vert m, \alpha  \rangle +  \vert m, \alpha \rangle }{\sqrt{2(1 + \kappa_m e^{-2|\alpha|^2})}}
   \qquad \vert 1 \rangle_1 = \frac{\vert m, \alpha   \rangle  -  \vert m, \alpha   \rangle }{{\sqrt{2(1-  \kappa_m e^{-|\alpha|^2})}}}.
\end{equation}
 For the  modes $(23)$, viewed as a single subsystem, we introduce the
orthogonal basis $\{ \vert  0 \rangle_{23} , \vert  1
\rangle_{23}\}$ as follows
\begin{equation}\label{base2}
\vert 0 \rangle_{23} = \frac{ \vert \alpha, \alpha \rangle +  \vert -\alpha, -\alpha
\rangle}{\sqrt{2(1 + e^{-4|\alpha|^2})}}
   \qquad \vert 1 \rangle_{23} = \frac{ \vert \alpha, \alpha \rangle -  \vert -\alpha, -\alpha
\rangle}{\sqrt{2(1 - e^{-4|\alpha|^2})}}.
\end{equation}
Inserting(\ref{base1}) and (\ref{base2}) in $\vert {\rm GHZ}_k(\alpha,m)\rangle$, one
obtains the expansion  of the pure state $\vert {\rm GHZ}_k(\alpha,m)\rangle$  in the basis
$\{ \vert 0 \rangle_{1} \otimes \vert 0 \rangle_{23} ,
 \vert 0 \rangle_{1} \otimes \vert 1 \rangle_{23} , \vert  1 \rangle_{1}
 \otimes \vert 0 \rangle_{23} , \vert  1 \rangle_{1} \otimes \vert  1
 \rangle_{23}\}$. Explicitly, one has
\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
$$c^{\pm}_1 =\sqrt{\frac{1 \pm  \kappa_m e^{-2|\alpha|^2}}{2}} \qquad  c^{\pm}_{23} =\sqrt{\frac{1 \pm e^{-4|\alpha|^2}}{2}}.$$
It is interesting to note that using the Schmidt decomposition, one can shows that the density matrix
\begin{equation}\label{rho1|23}
\rho_{123}\equiv \rho_{1|23} = | {\rm GHZ}_k(\alpha,m) \rangle \langle {\rm GHZ}_k(\alpha,m)|
\end{equation}
is also $X$ shaped. This point will be clarified in the following subsection.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Fano-Bloch representations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


By encoding information in the qubits (\ref{qubit1}) (\ref{qubit2}) and (\ref{base2}), the arising
density matrices $\rho_{12} = \rho_{23}$ (\ref{matrixrho12}) and $\rho_{1\vert23}$ (\ref{rho1|23}) take,
in  Fano-Bloch representation, the following form
\begin{equation}\label{12X}
\rho_{AB} = \frac{1}{4} \bigg[  \sigma_{0}\otimes \sigma_{0} +   T^{AB}_{03}~
\sigma_{0}\otimes \sigma_{3} +  T^{AB}_{30}~
\sigma_{3}\otimes \sigma_{0} + \sum_{k=1,2,3} T^{AB}_{kk}
~\sigma_{k}\otimes \sigma_{k} \bigg]
\end{equation}
where $\sigma_{0}$ is the identity,  $\sigma_k$ are the Pauli matrices and  the correlation matrix elements are given by
$$ T^{AB}_{\alpha \beta} = {\rm Tr} \big(\rho_{AB}~\sigma_{\alpha}\otimes\sigma_{\beta}\big).$$
Explicitly, the correlation matrix elements entering in
Fano-Bloch expression of the bipartite density matrix $\rho_{12} = {\rm Tr}_3 \rho_{123}$ (\ref{matrixrho12}) write  as
%\begin{equation}\label{Fano12X}
%\rho_{12} = \frac{1}{4} \bigg[  \sigma_{0}\otimes \sigma_{0} +   T^{12}_{03}
%\sigma_{0}\otimes \sigma_{3} +  T^{12}_{30}
%\sigma_{3}\otimes \sigma_{0} + \sum_{k=1,2,3} T^{12}_{kk}
%\sigma_{k}\otimes \sigma_{k} \bigg]
%\end{equation}
\begin{equation}\label{R12}
 T^{12}_{03}= 2{\cal C}^2_k(\alpha , m)~ (e^{-2|\alpha|^2} + \kappa_m e^{-4|\alpha|^2}\cos k\pi )$$
$$T^{12}_{30}= 2{\cal C}^2_k(\alpha , m) ~(\kappa_m  e^{-2|\alpha|^2} + e^{-4|\alpha|^2}\cos k\pi )$$
$$T^{12}_{11}= 2{\cal C}^2_k(\alpha , m) ~\sqrt{(1 -  \kappa^2_m e^{-4|\alpha|^2} )(1 - e^{-4|\alpha|^2})}$$
$$T^{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 $$
$$T^{12}_{33}= 2{\cal C}^2_k(\alpha , m) ~(\kappa_m e^{-4|\alpha|^2} + e^{-2|\alpha|^2}\cos k\pi)
\end{equation}
Similarly, for $\rho_{23}$ (\ref{matrixrho23}) one obtains
%\begin{equation}\label{Fano23X}
%\rho_{23} = \frac{1}{4} \bigg[  \sigma_{0}\otimes \sigma_{0} +   T^{23}_{03}
%\sigma_{0}\otimes \sigma_{3} +  T^{23}_{30}
%\sigma_{3}\otimes \sigma_{0} + \sum_{k=1,2,3} T^{23}_{kk}
%\sigma_{k}\otimes \sigma_{k} \bigg]
%\end{equation}
\begin{equation}\label{R23}
 T^{23}_{03}=2{\cal C}^2_k(\alpha , m)~(e^{-2|\alpha|^2} + \kappa_m e^{-4|\alpha|^2}\cos k\pi )$$
$$T^{23}_{30}= 2{\cal C}^2_k(\alpha , m)~ (e^{-2|\alpha|^2} + \kappa_m e^{-4|\alpha|^2}\cos k\pi ) $$
$$T^{23}_{11}=
2{\cal C}^2_k(\alpha , m) (1 -   e^{-4|\alpha|^2} )$$
$$T^{23}_{22}=
2{\cal C}^2_k(\alpha , m) (1 -   e^{-4|\alpha|^2}) \kappa_m e^{-2|\alpha|^2} \cos k\pi$$
$$T^{23}_{33}= 2{\cal C}^2_k(\alpha , m)~ (e^{-4|\alpha|^2} + \kappa_m  e^{-2|\alpha|^2}\cos k\pi)
\end{equation}
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 Schmidt
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}
In the basis $\{ \vert + \rangle_1 \otimes \vert + \rangle_{23},
\vert + \rangle_1 \otimes \vert - \rangle_{23}, \vert - \rangle_1 \otimes \vert + \rangle_{23},
\vert - \rangle_1 \otimes \vert - \rangle_{23} \}$, the density  $\rho_{1\vert 23}$ (\ref{rho1|23}) is
$X$ shaped  and takes
the form (\ref{12X}). The associated correlation matrix elements are given by
%\begin{equation}\label{Fano123X}
%\rho_{1|23} = \frac{1}{4} \bigg[  \sigma_{0}\otimes \sigma_{0} +   T^{1|23}_{03}
%\sigma_{0}\otimes \sigma_{3} +  T^{1|23}_{30}
%\sigma_{3}\otimes \sigma_{0} + \sum_{k=1,2,3} T^{1|23}_{kk}
%\sigma_{k}\otimes \sigma_{k} \bigg]
%\end{equation}
%where
\begin{equation}\label{R123}
 T^{1|23}_{33}= 1$$
$$ T^{1|23}_{03}= T^{1|23}_{30} =
2 {\cal C}^2_k(\alpha , m) e^{-2|\alpha|^2}(\kappa_m + e^{-2|\alpha|^2}\cos k\pi )$$
$$T^{1|23}_{11}= - T^{1|23}_{22} =
2 {\cal C}^2_k(\alpha , m) \sqrt{(1- \kappa^2_m e^{-4|\alpha|^2})(1- e^{-8|\alpha|^2})}.
\end{equation}


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Hilbert-Schmidt measure of quantum discord}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The qubit encoding introduced in the previous  section is of paramount importance
in deriving the geometric quantum discord  in photon added coherent states (\ref{GHZm})
and analyzing  the influence of the photon excitation process. According to the procedure developed in \cite{Dakic},
the geometric measure of quantum discord is  the distance
between the state $\rho_{AB}$ and its closest classical-quantum state
presenting zero discord
\begin{equation}\label{definition-discord}
D_{AB} = \min_{{\chi_{AB}}} \vert\vert \rho_{AB} -
\chi_{AB} \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 A, the classical states
write
\begin{equation}
\chi_{AB} = p_1 \vert \psi_1 \rangle \langle \psi_1 \vert \otimes \rho_1^{B} + p_2 \vert \psi_2 \rangle \langle \psi_2 \vert \otimes \rho_2^{B}
\end{equation}
where $\{ \vert \psi_1 \rangle , \vert \psi_2 \rangle \}$ is an
orthonormal basis related to the qubit $A$,  $p_i$ $(i=1,2)$ stands
for probability distribution and $\rho_i^{B}$ $(i=1,2)$ is the
marginal density of the qubit $B$. The classically correlated states
$\chi_{AB} $ can also be written as
\begin{equation}\label{xi12}
\chi_{AB} = \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
$$ t = p_1-p_2, \qquad e_i = \langle \psi_1 \vert \sigma_i \vert \psi_1 \rangle,  \qquad
(s_{\pm})_{j} = {\rm Tr}\big((p_1\rho_1^{B} \pm p_2\rho_2^{B})
\sigma_{j} \big).$$
It follows that the distance between the density matrix $\rho_{AB}$
and the classical state $\chi_{AB}$, as measured by Hilbert-Schmidt
norm, is then given by
\begin{equation}\label{HS12-1}
|| \rho_{AB} - \chi_{AB}||^2 = \frac{1}{4} \bigg[  (t^2 -2te_3 T^{AB}_{30} + {\big(T^{AB}_{30}\big)}^2) +  \sum_{i=1}^{3}
(T^{AB}_{0i} -  (s_+)_{i})^2 + \sum_{i,j=1}^3   ( T^{AB}_{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^{AB}_{30},\qquad (s_-)_{i} = \sum_{j=1}^3 e_j T^{AB}_{ji}, $$
$$ (s_+)_{1}= (s_+)_{2} = 0, \qquad  (s_+)_{3} = T^{AB}_{03}.
%$$  (s_-)_{1} = e_1 T_{11} \quad (s_-)_{2} = e_2 T_{22} \quad (s_-)_{3} = e_3 T_{33}
\end{equation}
Inserting  the solutions (\ref{minHS12}) in (\ref{HS12-1}), one has
\begin{equation}\label{HS12}
|| \rho_{AB} - \chi_{AB}||^2 = \frac{1}{4} \bigg[  {\rm Tr}K - \vec{e}^t K \vec{e} \bigg]
\end{equation}
where the diagonal matrix $K$ is defined by
\begin{equation}\label{matrixK12}
K = {\rm diag}\big( \lambda^{AB}_1, \lambda^{AB}_2, \lambda^{AB}_3 \big)
\end{equation}
with
\begin{equation}\label{vp:lambda12}
\lambda^{AB}_1 = \big(T^{AB}_{11}\big)^2,\quad
\lambda^{AB}_2 = \big(T^{AB}_{22}\big)^2,
\qquad \lambda^{AB}_3=  \big(T^{AB}_{30}\big)^2+(T^{AB}_{33})^2.
\end{equation}
From the expressions (\ref{R12}),  we obtain
the eigenvalues
\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}, \nonumber\\
\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}\nonumber.
\end{eqnarray}
corresponding to the state $\rho_{12}$. Reporting (\ref{R23}) in (\ref{vp:lambda12}),   the eigenvalues
entering in the determination of the bipartite geometric quantum discord in the state $\rho_{23}$ are
\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,\nonumber\\
\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}\nonumber.
\end{eqnarray}
For the pure state $\rho_{1|23}$, we use the expressions (\ref{R123}) to obtain
\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},\nonumber\\
\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}\nonumber.
\end{eqnarray}
From the equation (\ref{HS12}), it is clear that the minimal value of
Hilbert-Schmidt distance is reached for the largest
eigenvalue of the matrix $K$ (\ref{matrixK12}). Hence, to get the minimal value of the Hilbert-Schmidt distance (\ref{HS12}) and subsequently the amount
of geometric quantum discord, one should compare
$\lambda^{AB}_1$, $\lambda^{AB}_2$ and $\lambda^{AB}_3$.   From the results (\ref{lambda12}), (\ref{lambda23}) and (\ref{lambda123}),
it is simply verified that
 $\lambda^{AB}_1$ is always greater than  $\lambda^{AB}_2$. This implies that
the largest eigenvalue  $\lambda^{AB}_{\rm max}$ is $\lambda^{AB}_1$ or
$\lambda^{AB}_3$. Accordingly,  the geometric discord is given by
\begin{equation}
D_{AB} = \frac{1}{4}~ {\rm min}\{ \lambda^{AB}_1 + \lambda^{AB}_2 ,
\lambda^{AB}_2 + \lambda^{AB}_3\}.\label{eq:GMQD_new}
\end{equation}
To write down the  explicit  expressions of the closest classical
state $\chi_{AB}$ to $\rho_{AB}$, 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 $\lambda_{\rm max} = \lambda^{AB}_3$) are separately
discussed. In the case where $\lambda_{\rm max} = \lambda^{AB}_1$, the corresponding eigenvector
 is given by $\vec{e}_1 = (1, 0, 0)$ and using (\ref{minHS12}), one gets the closest
classical state
\begin{equation}
\chi^1_{AB} = \frac{1}{4}\Bigg[ \sigma_{0}\otimes \sigma_{0} +
T^{AB}_{30} ~~\sigma_3 \otimes \sigma_{0} + T^{AB}_{11}~\sigma_1 \otimes \sigma_{1} \Bigg]
\end{equation}
In the second situation where  $\lambda_{\rm max} = \lambda^{AB}_3$,  the associated
eigenvector  is given by $\vec{e}_3 = (0,0,1)$. Reporting this result in the set
of  constraints (\ref{minHS12}), one shows
\begin{equation}
\chi^3_{AB} = \frac{1}{4}\Bigg[ \sigma_{0}\otimes \sigma_{0} +
T^{AB}_{30} ~~\sigma_3 \otimes \sigma_{0} + T^{AB}_{03}~\sigma_0 \otimes
\sigma_{3} + T^{AB}_{33}~\sigma_3 \otimes \sigma_{3}\Bigg]
\end{equation}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pairwise geometric discord in tripartite quasi-GHZ coherent states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To analyze the influence of the photon excitation on the pairwise
geometric discord present in the state (\ref{GHZm}), we consider first the evolution of the correlation between
the first and the second mode described by the state (\ref{matrixrho12}).  From
equation (\ref{eq:GMQD_new}), we have
\begin{equation}
D_{12} = \frac{1}{4}~ ( \lambda^{12}_2+ {\rm min}\{ \lambda^{12}_1 , \lambda^{12}_3\}).\label{D12}
\end{equation}
where $ \lambda^{12}_i (i=1,2,3)$ are given by (\ref{lambda12}). The behavior of
 $D_{12}$ in (\ref{D12})  is plotted, in figures 1 and 2, as function of $|\alpha|^2$ for
different values of photon excitation order $m$. In figure 1, corresponding
to symmetric case, it is clearly seen that the geometric quantum discord $D_{12}$ exhibits a pick.
This pick is moving from the right to the left. In particular, the
height of the pick, reflecting the maximal amount of quantum discord, increases as
the photon excitation order $m$ increases. Also, the photon excitation  induces
a quick decreases of quantum discord as the optical strength $|\alpha|$ becomes large.
  For antisymmetric states $(k=1)$, the
quantum discord $D_{12}$, depicted in figure 2, shows that the quantum discord
corresponding to   $m=0,1,2$ does not present peak and it decreases monotonically as
$|\alpha|$ increases.  This is not the case
for $m \geq 3$. In fact, the geometric discord starts increasing to reach a maximal
value for $\alpha$, different from zero, to decrease after quickly as the Glauber coherent
state amplitude becomes large. It must be noticed that for  symmetric as well as
antisymmetric states the quantum discord $D_{12}$ vanishes when  $\alpha \to \infty$ independently of
photon excitation number $m$. This limiting case  corresponds
to usual {\rm GHZ} states (\ref{GHZusual}). Another interesting limiting case concerns the
antisymmetric states $\vert {\rm GHZ}_1(\alpha,m)
\rangle$  when $|\alpha| \to 0$. 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  three qubit ${\rm W}$ states for $m=0$ \cite{Dur}.  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 $|\alpha| \longrightarrow 0$,
the Laguerre function (\ref{laguerre}) behaves like $L_m(|\alpha|^2) \simeq 1 - m|\alpha|^2$ 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 to
\begin{eqnarray}\label{lambda12alpha0}
\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}
This shows that the quantum discord $D_{12}$ is $m$-dependent for states of ${\rm W}$-type (\ref{GHZalpha=0}) and one has
$$ D_{12} \longrightarrow  ~\frac{(m+1)^2 + 2}{2(m+3)^2} \quad {\rm for} \quad m \leq  2$$
$$ D_{12} \longrightarrow 2 ~\frac{m+1}{(m+3)^2} \quad {\rm for} \quad m > 2.$$
\begin{center}
\includegraphics[width=3.5in]{D12k0}\\
{\bf Figure 1.}  {\sf The quantum discord  $D_{12}$ versus
 $|\alpha|^2$ for $k=0$ and  different values of photon excitation number $m$.}
\end{center}
\begin{center}
\includegraphics[width=3.5in]{D12k1}\\
{\bf Figure 2.}  {\sf The quantum discord  $D_{12}$ versus
 $|\alpha|^2$ for $k=1$ and  different values of photon excitation number $m$.}
\end{center}
In the subsystem described by the density matrix (\ref{matrixrho23}), the quantum discord
(\ref{eq:GMQD_new}) gives
\begin{equation}
D_{23} = \frac{1}{4}~ ( \lambda^{23}_2+ {\rm min}\{ \lambda^{23}_1 , \lambda^{23}_3\}).\label{D23}
\end{equation}
where $ \lambda^{23}_i (i=1,2,3)$ are given by (\ref{lambda23}). In the figures 3 and 4 are depicted
the geometric measures of quantum correlation between the  modes 2 and 3  in the states
$\vert {\rm GHZ}_k(\alpha,m)
\rangle$ (\ref{GHZm}) as function of $|\alpha|^2$. From figure 3, one observes
that the height  of the quantum discord $D_{23}$ diminishes as the
photon excitation order $m$ increases, contrarily to what happens with the pairwise quantum discord
$D_{12}$ (figure 1). The peaks move from the right to the left. In increasing
the number of added photons, $D_{23}$ becomes almost identical.
The behavior of $D_{23}$ for symmetric states (figure 3)
is different from one in the antisymmetric states (figure 4). In particular, the
quantum discord decreases monotonically  for $m \leq 1$. The first peak appears for $m=2$
and it moves to the right for $m \geq 3$. The quantum discord vanishes quickly, as $|\alpha|$
increases, under the effect of photon excitation.  The quantum discord $D_{23}$, in antisymmetric
states $( k=1)$, depends on the photon excitation order $m$ when $|\alpha|$ approaches zero. Indeed,
from equations (\ref{lambda23}), one shows that
\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}
and subsequently one obtains the results
$$ D_{23} \longrightarrow  ~\frac{m^2 + 3}{2(m+3)^2} \quad {\rm for} \quad m \leq  2$$
$$ D_{23} \longrightarrow  ~\frac{2}{(m+3)^2} \quad {\rm for} \quad m > 2$$
which give the amount of pairwise quantum discord between the modes 2 and 3 in the
photon added $W$ states (\ref{GHZalpha=0}) and confirms analytically the results reported in the figure 4.
\begin{center}
\includegraphics[width=3.5in]{D23k0}\\
{\bf Figure 3.}  {\sf The quantum discord  $D_{23}$ versus
 $|\alpha|^2$ for $k=0$ and  different values of photon excitation number $m$.}
\end{center}
\begin{center}
\includegraphics[width=3.5in]{D23k1}\\
{\bf Figure 4.}  {\sf The quantum discord  $D_{23}$ versus
 $|\alpha|^2$ for $k=1$ and  different values of photon excitation number $m$.}
\end{center}
Finally, we discuss the evolution of $D_{1|23}$ in the  bi-partitioning scheme (\ref{rho1|23}).  From (\ref{lambda123})
it is simple to verify that the maximal eigenvalue is $\lambda^{1|23}_{3}$.
 Thus, the quantum geometric measure of quantum discord between the first qubit and the qubit grouping the modes 2 and 3 is given by
\begin{eqnarray}\label{D123}
D_{1|23} =  \frac{1}{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}
The behavior of  $D_{1|23}$ versus $|\alpha|^2$ for symmetric ${\rm GHZ}$ states for different values of
added photons is reported in figure 5.  In the weak regime, the geometric quantum discord increases
quickly as $m$ increases. For high values of $|\alpha|$, one obtain  $D_{1|23} = 0.5$
independently of $m$. This result is easily verified from (\ref{D123}). Similar
behavior, when $|\alpha|$ is large, is obtained with antisymmetric ${\rm GHZ}$ states (see figure 6).
From figure 6, it is clearly seen that in the weak regime, the
pairwise quantum $D_{1|23}$ is strongly influenced by the photon excitation. In particular,
when $|\alpha| \to 0$, the eigenvalues (\ref{lambda123}) become
\begin{eqnarray}\label{lambda123alpha0}
\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 explicit expression of quantum discord $D_{1|23}$ as function of $m$  is
\begin{eqnarray}
D_{1|23} &\longrightarrow & 4 ~\frac{m+1}{(m+3)^2}.
\end{eqnarray}
\begin{center}
\includegraphics[width=3.5in]{D123k0}\\
{\bf Figure 5.}  {\sf The quantum discord  $D_{1|23}$ versus
 $|\alpha|^2$ for $k=0$ and  different values of photon excitation number $m$.}
\end{center}
\begin{center}
\includegraphics[width=3.5in]{D123k1}\\
{\bf Figure 6.}  {\sf The quantum discord  $D_{1|23}$ versus
 $|\alpha|^2$ for $k=1$ and  different values of photon excitation number $m$.}
\end{center}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Monogamy of geometric quantum discord in quasi-GHZ states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The
quantum discord,
in the states $\vert {\rm GHZ}_k(\alpha,m)
\rangle$ (\ref{GHZm}), is monogamous if and only if the
quantum monogamy deficit defined by
\begin{equation}\label{Delta123}
\Delta_{123}=\Delta_{123}(m,|\alpha|^{2})=D_{1\mid23}- D_{12} - D_{13},
\end{equation}
is positive. In other words,   the monogamy property is satisfied when the quantum discord  $D_{1\mid23}$ (\ref{D123}) between the first mode and the modes 2-3 (viewed as
a single subsystem) exceeds the sum of pairwise quantum discord  $D_{12}$ and $D_{13}$ (\ref{D12}). Using (\ref{lambda12}) and (\ref{D12}),
one shows that
\begin{equation}\label{D12-}
D_{12}=D_{13} = \frac{1}{4} ~\frac{(1-\kappa_m^2 e^{-4|\alpha|^2})(1 -  e^{-8|\alpha|^2})}{(1 + \kappa_m e^{-6|\alpha|^2}\cos k\pi)^2},
\end{equation}
when $\lambda^{12}_1 \leq \lambda^{12}_3$. Conversely, for photon excitation order $m$ and optical strength $\vert \alpha \vert$ satisfying
the condition   $\lambda^{12}_1 \geq \lambda^{12}_3$, one gets
\begin{equation}\label{D12+}
D_{12}=D_{13} = \frac{1}{4} ~e^{-4|\alpha|^2} ~\frac{2 +\kappa_m^2 e^{-8|\alpha|^2}+ 4\kappa_m  e^{-2|\alpha|^2}\cos k\pi}{(1 + \kappa_m e^{-6|\alpha|^2}\cos k\pi)^2}.
\end{equation}
Using the expressions  (\ref{D123}),  (\ref{D12-}) and (\ref{D12+}), it is easy to check that the quantum monogamy deficit (\ref{Delta123}) vanishes
\begin{equation}\label{Delta123-}
\Delta_{123}= 0,
\end{equation}
for $\lambda^{12}_1 \leq \lambda^{12}_3$ and it is given  by
\begin{equation}\label{Delta123+}
\Delta_{123}= \frac{1}{2} ~\frac{1-  \big[ (2-3\kappa_m^2) + (e^{-2|\alpha|^2} + 2\kappa_m \cos k\pi)^2\big]  e^{-4|\alpha|^2} }{(1 + \kappa_m e^{-6|\alpha|^2}\cos k\pi)^2},
\end{equation}
when $\lambda^{12}_1 \geq \lambda^{12}_3$.\\
\noindent The quantum discord
deficit is depicted in figures 7 and 8. It is always positive reflecting that the geometric quantum discord is monogamous. In
figure 7 corresponding to the symmetric states $(k=0)$, $\Delta_{123}$  is initially zero (see equation (\ref{Delta123-}))
reflecting that the condition  $\lambda^{12}_1 \leq \lambda^{12}_3$ is satisfied. By increasing the optical strength $|\alpha|$,
$\lambda^{12}_1$  becomes greater than  $\lambda^{12}_3$ and the quantum discord
deficit increases to reach its maximal value 0.5 for $|\alpha|$ large. In this situation, it is intersecting to note
that, according to the equation (\ref{D12+}), $D_{12}=D_{13}$ vanishes and  $\Delta_{123}$ coincides with the pairwise quantum discord $D_{1|23}$
independently of photon excitation order $m$.
\begin{center}
\includegraphics[width=3.5in]{Delta123k000}\\
{\bf Figure 7.}  {\sf The geometric quantum discord deficit $\Delta_{123}$ versus  $|\alpha|^2$ for $k=0$ and  different values of photon excitation number $m$.}
\end{center}
%\begin{center}
%\includegraphics[width=3.5in]{delta123k00}\\
%{\bf Figure 7.}  {\sf The geometric quantum discord deficit $\Delta_{123}$ versus  $|\alpha|^2$ for $k=0$ and  different values of photon excitation number $m$.}
%\end{center}
%\begin{center}
%\includegraphics[width=3.5in]{delta123k1}\\
%{\bf Figure 8.}  {\sf The geometric quantum discord deficit $\Delta_{123}$ versus  $|\alpha|^2$ for $k=1$ and  different values of photon excitation number $m$.}
%\end{center}
It is clearly seen from figure 8 that the quantum discord
deficit  is non zero for $m = 0,1,2$,  in particular when $|\alpha| \longrightarrow 0$ in contrast with the states with $m \geq 3$. This
 result can be confirmed analytically. Indeed,
when $|\alpha|$ approaches zero, the antisymmetric states (\ref{GHZm}) reduces to photon added tripartite states of ${\rm W}$-type (\ref{GHZalpha=0})
and using the results (\ref{lambda12alpha0}) and (\ref{lambda123alpha0}), the geometric quantum discord deficit $\Delta_{123}$
tends to zero for $m > 2$ and
$$\Delta_{123} \longrightarrow \frac{2-(m-1)^2}{(m+3)^2} $$
for $m=0,1,2$. The photon addition process does not affect the monogamy property ($\Delta_{123}$ is positive for any value of $m$), but
modifies the distribution of geometric quantum discord among the three modes in the states (\ref{GHZm}). In fact, in the
case when $m \geq 3$ and for small values of $|\alpha|$, one have $\Delta_{123} = 0$ reflecting that
the geometric discord $D_{1|23}$ is exactly the sum of the pairwise quantum correlations  $D_{12}$  and  $D_{13}$
in contrast with the cases where $m \leq 2$ for which we  have $\Delta_{123} > 0$ for any value of $|\alpha|$.

\begin{center}
\includegraphics[width=3.5in]{Delta123k111}\\
{\bf Figure 8.}  {\sf The geometric quantum discord deficit $\Delta_{123}$ versus  $|\alpha|^2$ for $k=1$ and  different values of photon excitation number $m$.}
\end{center}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Concluding remarks}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In conclusion, we have addressed the question of the influence of photon
excitation on pairwise quantum correlations  in balanced superpositions
of tripartite Glauber coherent states of type ${\rm GHZ}$. This family of
nonorthogonal  states include   Greenberger-Horne-Zeilinger and
$W$ states. By encoding information in even and odd coherent states,
we derived the pairwise quantum discord $D_{12}$ (\ref{D12}),  $D_{23}$ (\ref{D23})
and $D_{1|23}$ (\ref{D123}). We have used the Hilbert-Schmidt distance as quantifier
of quantum correlations. We have analyzed their behaviors as functions of the
coherent states amplitude $\alpha$.  A special attention were
devoted to photon added states of $W$ type corresponding to the
limiting case $\alpha \longrightarrow 0$ for which the explicit
expressions of the pairwise geometric discord are derived. We have shown
that the photon excitation does not affect the monogamy property of geometric quantum discord
in the quasi-${\rm GHZ}$ states (\ref{GHZm}). But, it modifies the distribution
of quantum discord among the three optical modes, especially when the Glauber coherent
states have small amplitudes. Finally, as prolongation of this work, we think that it would be interesting
to analyze the distribution of  quantum correlations in photon added coherent states
comprising four or more modes. We hope to report on this issue in a forthcoming work.


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