\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 A recursive approach for geometric quantifiers of pairwise quantum discord\\
in multiqubit Shr\"odinger cat states}\\
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1.5cm} {\bf M. Daoud}$^{a,b,c}${\footnote { email: {\sf
m$_{-}$daoud@hotmail.com}}}, {\bf R. Ahl Laamara}$^{d,e}$ {\footnote
{ email: {\sf ahllaamara@gmail.com}}}  and {\bf S. Seddik}$^{d}$
{\footnote { email: {\sf sanaa1.sanaa@hotmail.fr}}} \\
\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, 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, Rabat, Morocco}\\[0.5em]
$^{e}${\it Centre of Physics and Mathematics,  Rabat, Morocco}\\[0.5em]

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






A recursive approach  for  determining the Hilbert-Schmidt measure of pairwise
quantum discord in a balanced  superposition of symmetric $n$-qubit states
(multiqubit Shr\"odinger cat states)
 is presented. A particular emphasis is devoted  to $k$ qubits reduced states
obtained  by tracing out
$n-k$ particles  $(k=2,3, \cdots ,n-1)$ from the whole system. Two bi-partition
schemes are discussed. One consists in dividing the $k$-qubits
in two subsystems containing  one and $(k-1)$ qubits respectively. The explicit
derivation of the pairwise geometric discord  uses recursive  relations
between the Fano-Bloch correlation matrices. A detailed analysis is
given for two, three and four qubit systems. In the second  scheme, the cluster
comprising the $(k-1)$ qubits is mapped into a system of two logical qubits. We show
that
these two   bi-partitioning   schemes are equivalent in  evaluating  the pairwise
geometric discord in multiqubit Shr\"odinger cat states. The explicit expressions of
classical states exhibiting zero discord are also derived.


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



Quantum correlations in multipartite systems have generated  a lot of interest
during the last two decades \cite{Horodecki-RMP-2009,
Guhne,vedral}. This is essentially motivated by their promising
 applications in the field of  quantum information that
guarantee classically unattainable efficiency  in a variety of
quantum tasks (see for instance \cite{NC-QIQC-2000,Vedral-RMP-2002}).
In this sense, the characterization of quantum correlations  is crucial  to implement
quantum protocols such as   quantum teleportation \cite{Bennett1}, superdense coding
\cite{Bennett2}
and quantum key distribution \cite{Ekert}. Several methods and different measures of
quantum correlation were exhaustively discussed in the literature
from various perspectives  and for many purposes (for a recent review see
\cite{vedral}).
They can be classified in two main categories: entropic based measures and geometric
quantifiers or
norm based measures. Entanglement of formation, linear
entropy, relative entropy and quantum discord
\cite{Rungta,Ben3,Wootters,Coffman,Vedral-et-al,Ollivier-PRL88-2001}
constitute  familiar entropic quantifiers of correlations. Probably, quantum discord,
which goes beyond entanglement in distinguishing between quantum and classical states,
is the most prominent of these correlations.   It was originally defined as the
difference
between two quantum analogues of the classical mutual information
\cite{Vedral-et-al,Ollivier-PRL88-2001}.
The explicit evaluation of based entropy measures requires optimization procedures
which are in general very complicated to achieve and constitute the main obstacle in
order to get
computable expressions of quantum correlations. To overcome such difficulties,
geometric measures, especially ones based on Hilbert-Schmidt norm, were considered
to formulate a geometric variant of quantum discord  \cite{Dakic2010}. The
Hilbert-Schmidt distance
 was used also to quantify classical correlations \cite{Bellomo1,Bellomo2}. More
recently, it has been pointed that
 1-norm distance (trace distance) offers also a geometric tool to derive the amount  of quantum and classical correlations
 in bipartite systems \cite{Dajka,Paula1,Aaronson,Paula2}.\\


In other hand, the extension of Hilbert-Schmidt measure of quantum
discord to  $d$-dimensional quantum systems (qudits) was reported
in \cite{Rana,Hassan,Rau} (see also \cite{Zhou} and references
quoted therein). Interestingly, this higher dimensional
extension can be adapted to evaluate the pairwise quantum
correlations in multi-qubit systems.  In fact in multi-qubit systems, geometric
quantum discord based on the
Hilbert-Schmidt norm is more tractable  from a
computational point of view than entropic based measures. In this
spirit, we shall employ in this work the approach proposed  by Dakic et al
\cite{Dakic2010} to
investigate the quantum correlations in mixed multi-qubit states. Specifically, we will
consider a balanced superposition of  symmetric multi-qubit states in which the
symmetry properties offer drastic simplification
in evaluating quantum correlations.\\




This paper is organized as follows. In section 2, we discuss the
relevance of symmetric multi-qubit ($n$-qubits) states in defining
Schr\"odinger cat states. We shall essentially focus on balanced
superpositions, symmetric or antisymmetric under  parity
transformation, which coincide with even and odd spin atomic coherent
states. A special attention, in section 3, is devoted to reduced states
describing subsystems containing $k$ qubits ($k = 2, 3,
\cdots, n-1$) obtained by tracing out $(n-k)$ qubits from a $n$-qubit
Schr\"odinger cat state. The obtained reduced density matrices are $X$ shaped and subsequently
their algebraic structures
simplify considerably the determination the quantum
correlation based on Hilbert-Schmidt (geometric quantum discord)
between one qubit and $(k-1)$ qubits contained  in a mixed $k$-qubit
state. This  is explicitly described in section 4. In particular considering
the cases of two and three qubit systems,  we provide  the general
method to determine analytically geometric discord in mixed
$k$-qubits states. We also derive the explicit forms of classical
(zero discord) states.
In section 5, we introduce another scheme according to which the second part
of the system comprising $(k-1)$ qubits is regarded as a single qubit. In this
picture the
whole system reduces to a two qubit system. Remarkably, the geometric measure
of quantum discord obtained,  in this second scheme, coincides with one derived
in the first bi-partition scheme (section 4).  As illustration, a detailed analysis
is given for $k=3$
and $k=4$. The method developed in this
paper which extends the geometric measure of two-qubit $X$ states to
embrace $k$-qubit $X$ states is useful in investigating the
global pairwise  correlation in multipartite qubit systems.
 Concluding
remarks close this paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Symmetric multi-qubit systems}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


The multi-qubit symmetric states were shown relevant  for different purposes  in
quantum information
science~\cite{solano,
mixed,usa1,usa2,markham1,markham2,gebastin,markham3}. In this paper,  we
shall mainly focus on an ensemble of $n$ spin-$1/2$ prepared in even and
 odd spin coherent states.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spin coherent as  symmetric multi-qubit systems}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We consider $n$ identical qubits. Each qubit lives in a
2-dimensional Hilbert space
$\mathcal{H}=\mathrm{span}\{\vert{0}\rangle,\vert{1}\rangle\}$. The
Hilbert space of the $n$-qubit system is given  by $n$ tensored
copies of $\mathcal{H}$
$$\mathcal{H}_n := \mathcal{H}^{\otimes n}.$$
Among the multi-partite states in $\mathcal{H}_n$, multi-qubit
states obeying exchange symmetry are of special interest from
experimental as well as mathematical point of views. An arbitrary
symmetric $n$-qubit state is commonly represented in either
Majorana~\cite{Maj32} or  Dicke ~\cite{Dic54}
representation. Any multi-qubit state, invariant under the exchange
symmetry, is specified in the Majorana description by the state (up
to a normalization factor)
\begin{equation}\label{psisym}
|\psi_s\rangle = \frac{1}{n!} \sum_{\sigma \in {\cal S}_n} \vert
{\eta_{\sigma(1)},\ldots,\eta_{\sigma(n)}}\rangle,
\end{equation}
where each single qubit state is $|\eta_i\rangle \equiv (1 + \eta_i
\bar\eta_i)^{-\frac{1}{2}}(|0\rangle + \eta_i |1\rangle)~ (i=1,\ldots, n)$
 and
the sum is over the elements of the permutation group ${\cal S}_n$
of $n$ objects. In Equation (\ref{psisym}), the vector
$\vert{\eta_{\sigma(1)},\ldots,\eta_{\sigma(n)}}\rangle$ stands for
the tensor product
$\vert{\eta_{\sigma(1)}}\rangle\otimes\ldots\otimes\vert{\eta_{\sigma(n)}}\rangle$.
The totally symmetric $n$-qubit states can be also formulated  in
Dike representation. The symmetric Dicke states with $k$ excitations
are defined by ~\cite{Dic54}
\begin{equation}\label{Dicke}
\vert n,k \rangle = \sqrt{\frac{k!(n-k)!}{n!}} \sum_{\sigma \in {\cal S}_n}
\vert{\underbrace{0,\ldots, 0}_{n-k},\underbrace{1,\ldots, 1}_{k}}\rangle,
\end{equation}
which generate an orthonormal basis of the  symmetric Hilbert subspace of dimension
$(n+1)$. Therefore, permutation invariance,  in symmetric
multi-qubit states, implies a  restriction to $n+1$ dimensional subspace from
the entire  $2^n$ dimensional Hilbert space. The Dicke states (\ref{Dicke})
constitute a special subset of
the symmetric multi-qubit states (\ref{psisym}) corresponding to the
situation where the first $k$-qubit are such that $\eta_i = 0$ for
$i = 0,1, \ldots k$ and the remaining qubits are in the states
$\vert \eta_i = 1 \rangle $ with $i = k+1, \ldots , n$.  Any symmetric state
$|\psi_s \rangle$ (\ref{psisym}) can be
expanded in terms of Dicke states (\ref{Dicke}) as follows
\begin{equation}
\label{psisymD}
|\psi_s\rangle = \frac{1}{n!} \sum_{k=0}^n c_k ~|n,k \rangle,
\end{equation}
where  the $c_k$'s ($k=0,\ldots,n$) stand for  complex expansion
coefficients. In particular, when the qubits are all identical
($\eta_i = \eta$ for all qubits), it is simple to check that the
coefficients $c_k$ are given by
\begin{equation}
\label{ck} c_k = n!  \sqrt{\frac{n!}{k!(n-k)!}} \frac{\eta^{k}}{(1 +
\eta\bar\eta)^{\frac{n}{2}}}
\end{equation}
and the symmetric multi-qubit states  (\ref{psisym}) write
\begin{equation}\label{psisym-cs}
|\psi_s\rangle :=  \vert n, \eta \rangle = (1 +
\eta\bar\eta)^{-\frac{n}{2}} \sum_{k=0}^n \sqrt{\frac{n!}{k!(n-k)!}}
\eta^{k}  ~|n,k \rangle,
\end{equation}
which are exactly the  $j=\frac{n}{2}$-spin coherent states (for
more details see for instance \cite{daoud2}). In particular, the
state $\vert n, \eta \rangle $ can be identified for $n=1$ with
spin-$\frac{1}{2}$ coherent state with $|0\rangle \equiv
|\frac{1}{2}, -\frac{1}{2}\rangle$ and $|1\rangle \equiv
|\frac{1}{2} ,+ \frac{1}{2}\rangle$)





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Multi-qubit "Shro\"odinger cat" states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The prototypical multi-qubit "Shr\"odinger cat" states, the basic objects
in this work, are defined  as a balanced superpositions of the
$n$-qubit states $\vert n, \eta \rangle$ and $\vert n, -\eta
\rangle$ given by (\ref{psisym-cs}). They write
\begin{equation}\label{ncs}
 \vert \eta, n , m \rangle  =  {\cal N}( \vert n, \eta \rangle + e^{im\pi} \vert n,
- \eta
 \rangle)
\end{equation}
where
$$ \vert n, \pm \eta \rangle = \vert \pm \eta \rangle \otimes \vert \pm \eta \rangle
\cdots \otimes \vert \pm \eta \rangle, $$
and the integer $m \in \mathbb{Z}$ takes the values $m = 0 ~({\rm
mod}~2)$ and $m = 1~ ({\rm mod}~2)$. The normalization factor ${\cal
N}$ is
$$ {\cal N}= \big[ 2 + 2 p^{n} \cos m \pi\big]^{-1/2}$$
where  $p$ denotes the overlap between the states $\vert \eta
\rangle$ and $\vert  -\eta \rangle$. It is given by
\begin{equation}\label{overlap}
 p = \langle \eta \vert  - \eta \rangle = \frac{1 - \bar\eta \eta}{1 + \bar\eta \eta}.
\end{equation}
 Experimental creation of cat states
comprising multiple particles was reported in the literature
\cite{Wineland,Haroche}. Due to their experimental implementation,
"Shr\"odinger cat" states are expected to serve as an useful resource for
quantum computing as well as quantum communications. Also, in view
of their mathematical elegance, multi-qubit states obeying exchange
symmetry offer simplification  in investigating various
aspects of quantum correlations in particular the geometric measure
of quantum discord as we shall discuss in the present work.
Furthermore,  the multi-qubit symmetric states (\ref{ncs}) include
Greenberger-Horne-Zeilinger(${\rm GHZ}$)~\cite{ghz}, ${\rm W}$
\cite{Dur00} and Dicke states~\cite{Dic54}. The multi-qubits states
$ \vert n, \eta , 0 \rangle$ ($m=0 ~ {\rm mod}~ 2$) and $ \vert n,
\eta , 1 \rangle$ ($m=1 ~ {\rm mod}~ 2$) behave like a multipartite
state of Greenberger-Horne-Zeilinger (${\rm GHZ}$) type \cite{ghz}
in the limiting case $p \rightarrow 0 $. Indeed,
 the  states $|\eta \rangle $ and $|
-\eta \rangle $ approach orthogonality and an orthogonal basis can
be defined such that $\vert {\bf 0}\rangle\equiv \vert \eta \rangle$
and $\vert{\bf 1}\rangle \equiv \vert  -\eta \rangle$. Thus, the
state $\vert n , \eta, m \rangle$ becomes of ${\rm GHZ}$-type:
\begin{equation}
\vert \eta , n, m \rangle \sim \vert {\rm GHZ}\rangle_{n} = \frac
1{\sqrt{2}}(\vert {\bf 0}\rangle \otimes |{\bf 0}\rangle \otimes
        \cdots \otimes\vert {\bf 0}\rangle
    +e^{i m \pi}\vert {\bf 1}\rangle \otimes
    \vert {\bf 1}\rangle \otimes \cdots \otimes
\vert {\bf 1}\rangle).\label{GHZ}
\end{equation}
Also, in the special situation where the overlap $p$ tends to unity
($p \rightarrow 1$ or $ \eta \rightarrow 0$ ), the state $\vert \eta
, n, m = 0 ~({\rm mod}~ 2) \rangle$ (\ref{ncs}) reduces to ground
state of a collection of $n$ qubits
\begin{equation}
\vert 0,  n , 0 ~({\rm mod}~ 2) \rangle \sim  \vert 0
\rangle \otimes\vert 0 \rangle \otimes \cdots \otimes \vert
0 \rangle,
\end{equation}
and  it is simple to check that the state $\vert \eta , 0 , 1 ~({\rm mod}~ 2)
\rangle$  becomes
a multipartite state of $W$ type~\cite{Dur00}
\begin{equation}
\vert 0,  n ,  1 ~({\rm mod}~ 2) \rangle \sim \vert\text{\rm
W}\rangle_{n}
    = \frac{1}{\sqrt{n}}(\vert 1 \rangle \otimes\vert 0\rangle \otimes \cdots\otimes
       \vert 0 \rangle  +\vert 0 \rangle \otimes\vert 1 \rangle \otimes\ldots\otimes
\vert 0 \rangle
      +\cdots
   + \vert  0 \rangle \otimes\vert 0 \rangle  \otimes \cdots\otimes \vert 1 \rangle)~.
\label{Wstate}
\end{equation}
Hence, the Shr\"odinger cat  states  $\vert \eta, n , m =
0 ~({\rm mod}~2) \rangle$  include the  $\vert\text{\rm
GHZ}\rangle_{n}$ states $(p
\rightarrow 0)$. In other hand, the states
 $\vert \eta, n , m = 1 ~({\rm mod}~2) \rangle$, constitute an interpolation between
two special classes of multi-qubits states: $\vert\text{\rm
GHZ}\rangle_{n}$ type corresponding to  $p \rightarrow 0$ and states
of $\vert\text{\rm W}\rangle_{n}$ type obtained in the special case
where $p \rightarrow 1$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ Quantum correlations in multi-partite Shr\"odinger cat states.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The structure of multipartite correlations within multi-qubit
quantum systems is a challenging and daunting  task. With the growth of number of
qubits, there are numerous ways in splitting the entire
 system to characterize how the particles are correlated. Obviously,
the bipartite splitting of the whole system is not sufficient to
capture the essential of quantum correlation existing in a multi-qubit system.
However, it must be stressed that the pairwise decomposition of total
correlation is unavoidable and  offers a good alternative
to evaluate the amount of all correlations existing in a
multipartite system. In this paper, we approach the problem of
analyzing $n$-qubit correlation using only bipartite measures.
Toward this end, we consider first the correlation between one qubit
with the remaining $(n-1)$ qubits in the state (\ref{ncs}). Thus,
the pure density matrix of the symmetric $n$-qubit system writes
$$ \rho_{n}\equiv \vert \eta, n , m \rangle\langle  \eta, n , m  \vert :=
\rho_{1\vert 23\ldots n}. $$
Furthermore, after removing $k=1, 2, \cdots, n-2$ particles from the
$n$-qubit system, the reduced density matrix $\rho_{n-k}$ can be
bi-partitioned in two subsystem, one comprises one qubit and the
remaining $(n-k-1)$ qubits are contained in the second subsystem. In this
way, after identifying the two subsystems, one can proceed with the analytical evaluation of the pairwise
quantum correlation existing among them. Also, this scheme  offers a reasonable
method to
characterize  the total amount of quantum correlation defined as the sum
of the quantum correlations for all possible bi-partitions \cite{Fanchini2}.
In this paper, we shall employ this picture to estimate the
geometric measure of quantum discord ($D_g$) in the
symmetric multi-qubit system of the form (\ref{ncs}). We give a
detailed analysis for two qubit and three qubit subsystems.  From these two specific
cases,  we give a general
algorithm to determine recursively  the pairwise quantum discord in
a reduced density describing $k$ qubit system.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Two-qubit states }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The strategy and tools we introduce for a two-qubit subsystem are
useful when extending the size of the system to encompass more
qubits. We  consider the two-qubit states extracted from the
state (\ref{ncs}) by tracing out $(n-2)$ qubits. Since the $n$
qubits are all identical, we obtain $n(n-1)/2$ identical density
matrices. They are given by
\begin{equation}\label{rho12}
\rho_{12} = {\cal N}^2 \bigg[ \vert \eta, \eta \rangle \langle \eta, \eta \vert
+ e^{im\pi} q_2 \vert -\eta, -\eta \rangle \langle \eta, \eta  \vert$$
$$+ e^{-im\pi} q_2 \vert \eta, \eta  \rangle \langle -\eta, -\eta  \vert
+  \vert -\eta, -\eta  \rangle \langle -\eta, -\eta \vert
\bigg]
\end{equation}
where   $q_2 = p^{n-2}$ . The reduced two qubit density (\ref{rho12}) can be
alternatively written as
\begin{equation}
\rho_{12} = \frac{1}{2} ~(1 + p^{n-2})~ \frac{{\cal N}^2}{{{{\cal N}_2}_+}^2} ~\vert
\eta \rangle_{2~2}\langle \eta \vert
 + \frac{1}{2} ~(1 - p^{n-2})~\frac{{\cal N}^2}{{{{\cal N}_2}_-}^2}~ Z \vert \eta
\rangle_{2~2}\langle \eta \vert Z
\end{equation}
with
$$ \vert \eta \rangle_2 = {{\cal N}_2}_+ ( \vert \eta, \eta \rangle + e^{im\pi}
\vert -\eta, -\eta \rangle) \quad  {\rm and} \quad
Z\vert \eta \rangle_2 = {{\cal N}_2}_- ( \vert \eta, \eta  \rangle - e^{im\pi} \vert
-\eta, -\eta  \rangle). $$
The normalization factors  are defined by
$$ {{\cal N}_2}^{-2}_{\pm} = 2(1 \pm p^2\cos m\pi).$$
In the computational base $\{ \vert 00 \rangle, \vert 01 \rangle, \vert 10 \rangle,
\vert 11 \rangle$,
the density matrix $\rho_{12}$ has the form of the alphabet $X$. Indeed, it is
represented by
\begin{eqnarray}
\rho_{12} = 2{\cal N}^2 \left(
\begin{array}{cccc}
q_{2+}a^4_{+}
& 0 & 0 &  q_{2+}a^2_{+}a^2_{-} \\
0 & q_{2-}a^2_{+}a^2_{-} & q_{2-}a^2_{+}a^2_{-} & 0 \\
0 & q_{2-}a^2_{+}a^2_{-} & q_{2-}a^2_{+}a^2_{-}
& 0 \\
q_{2+}a^2_{+}a^2_{-} & 0 & 0 & q_{2+} a^4_{-}
\end{array}
\right) \,
\end{eqnarray}
where
$$ a_{\pm} = \frac{\sqrt{1\pm p}}{\sqrt{2}} \qquad {\rm and} \qquad q_{2\pm} = 1 \pm
q_2 \cos m\pi.$$
The state $\rho_{12}$ can be written also as
\begin{equation}\label{12-fano}
\rho_{12}= \sum_{k,l = 0,1} \rho^{kl} \otimes \vert k \rangle \langle l \vert.
\end{equation}
This form is appropriate  to establish a relation between the Bloch components of
the $2\times 2$ matrices $\rho^{kl}$ and the correlation matrix elements associated
with the two-qubit state $\rho_{12}$. In equation (\ref{12-fano}), the matrices
$\rho^{kl}$
writes in Bloch representation as
\begin{equation}\label{0011}
 \rho^{00} =  \frac{1}{2} ( T^{00}_{0} \sigma_0 + T^{00}_{3} \sigma_3)\qquad
\rho^{11} = \frac{1}{2}  ( T^{11}_{0} \sigma_0 + T^{11}_{3} \sigma_3)
\end{equation}
and
\begin{equation}\label{0110}
 \rho^{01} = \frac{1}{2} ( T^{01}_{1} \sigma_1 + T^{01}_{2} \sigma_2 )  \qquad
\rho^{10} = \frac{1}{2} ( T^{10}_{1} \sigma_1 + T^{10}_{2} \sigma_2)
\end{equation}
where the Bloch components $T^{kl}_{\alpha}$ ($\alpha= 0, 1,2,3$) are
$$T^{kk}_{0} = {\cal N}^2 (1 + (-)^k p) (1 + (-)^k p^{n-1}\cos m\pi), \quad
T^{kk}_{3} = {\cal N}^2 (1 + (-)^k p) (1 + (-)^k p^{n-2}\cos m\pi) $$
for $k = 0,1 $, and
$$ T^{01}_{1} = T^{10}_{1} = {\cal N}^2 (1 - p^2), \qquad T^{01}_{2} = - T^{10}_{2}
= i {\cal N}^2 (1 - p^2)p^{n-2}\cos m\pi.$$
Reporting (\ref{0011}) and (\ref{0110}) in (\ref{12-fano}), one gets
\begin{equation}\label{rho12-sum}
\rho_{12} = \sum_{\alpha \beta} T_{\alpha \beta}
\sigma_{\alpha}\otimes \sigma_{\beta}
\end{equation}
where the non vanishing matrix elements $T_{\alpha \beta}$ $(\alpha,
\beta = 0,1,2,3)$ are given by
\begin{equation}\label{recurr}
T_{\alpha 0} = T^{00}_{\alpha} + T^{11}_{\alpha} ~~ {\rm for}~ \alpha = 0,3 \qquad
T_{\alpha 1} = T^{01}_{\alpha} + T^{10}_{\alpha}  ~~ {\rm for}~ \alpha = 1 $$
$$ T_{\alpha 2} = i T^{01}_{\alpha} -i T^{10}_{\alpha}  ~~ {\rm for}~ \alpha = 2
\qquad  T_{\alpha 3} = T^{00}_{\alpha} - T^{11}_{\alpha}  ~~ {\rm for}~ \alpha =0,3
\end{equation}
which gives
\begin{equation}\label{Tij12}
 T_{00} = 1, \quad T_{11} = 2{\cal N}^2 (1- p^2), \quad T_{22} = -2{\cal N}^2 (1-
p^2)~p^{n-2}\cos
m\pi,$$ $$ T_{33} = 2{\cal N}^2 (p^2 + p^{n-2}\cos m\pi), \quad
T_{03} = T_{30} = 2{\cal N}^2 (p + p^{n-1}\cos m\pi).
\end{equation}
The expressions (\ref{recurr}) establish the relations between the  Bloch components
$T^{kl}_{\alpha}$  associated with one
qubit states (\ref{0011}) and (\ref{0110}) and the two qubit Fano-Bloch tensor
elements   occurring in the two qubit density
$\rho_{12}$ (\ref{rho12-sum}). This result is ready generalizable to subsystems
comprising three or more qubits.  This issue is discussed in what follows.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Three-qubit states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The three-qubit states is extracted from the whole state (\ref{ncs})
by  removing $(n-3)$ qubits by the usual trace procedure. In this scenario,  it is simple to check that all the
three qubit density matrices are
identical. Explicitly,  they are given by
\begin{equation}\label{rho123}
\rho_{123} = {\cal N}^2 \bigg[ \vert \eta, \eta, \eta \rangle \langle \eta, \eta,
\eta \vert
+ e^{im\pi} q_3 \vert -\eta, -\eta, -\eta \rangle \langle \eta, \eta, \eta \vert$$
$$+ e^{-im\pi} q_3 \vert \eta, \eta, \eta \rangle \langle -\eta, -\eta, -\eta \vert
+  \vert -\eta, -\eta, -\eta \rangle \langle -\eta, -\eta, -\eta \vert
\bigg]
\end{equation}
where $ q_3 = p^{n-3}$. It is simply verified that the mixed three qubit state
$\rho_{123}$ is of rank two and  writes
\begin{equation}
\rho_{123} = \frac{1}{2} ~(1 + p^{n-3})~ \frac{{\cal N}^2}{{\cal N}_{3+}^2} ~\vert
\eta \rangle_{3~3}\langle \eta \vert
 + \frac{1}{2} ~(1 - p^{n-3})~\frac{{\cal N}^2}{{\cal N}_{3-}^2}~ Z \vert \eta
\rangle_{3~3}\langle \eta \vert Z
\end{equation}
where
$$ \vert \eta \rangle_3 = {\cal N}_{3+} ( \vert \eta, \eta, \eta \rangle + e^{im\pi}
\vert -\eta, -\eta, -\eta \rangle) \qquad
Z\vert \eta \rangle_3 = {\cal N}_{3-} ( \vert \eta, \eta, \eta \rangle - e^{im\pi}
\vert -\eta, -\eta, -\eta \rangle) $$
with the normalization factors $ {\cal N}_{3\pm}$  given by
$$ {\cal N}^{-2}_{3\pm} = 2(1 \pm p^3\cos m\pi).$$
In the standard computational basis $\{ \vert 000 \rangle, \vert 010 \rangle, \vert
100 \rangle, \vert 110 \rangle,
\vert 001 \rangle, \vert 011 \rangle, \vert101 \rangle, \vert 111 \rangle \}$, the
state $\rho_{123}$
 takes the matrix form
 \begin{equation}\label{3X-class2}
\frac{\rho_{123}}{2{\cal N}^2}=\left(%
\begin{array}{cccccccc}
 q_{+3}a^6_{+} & 0 & 0 & q_{+3}a^4_{+}a^2_{-} & 0 & q_{+3}a^4_{+}a^2_{-}  &
q_{+3}a^4_{+}a^2_{-}  & 0 \\
  0 &q_{-3}a^4_{+}a^2_{-} & q_{-3}a^4_{+}a^2_{-}  & 0 & q_{-3}a^4_{+}a^2_{-}  & 0 &
0 &q_{-3}a^2_{+}a^4_{-}  \\
  0 & q_{-3}a^4_{+}a^2_{-} & q_{-3}a^4_{+}a^2_{-}  & 0 & q_{-3}a^4_{+}a^2_{-}  & 0 &
0 &  q_{-3}a^2_{+}a^4_{-} \\
  q_{+3}a^4_{+}a^2_{-}  & 0 & 0 & q_{+3}a^2_{+}a^4_{-}  & 0 & q_{+3}a^2_{+}a^4_{-} &
q_{+3}a^2_{+}a^4_{-} & 0 \\
  0 & q_{-3}a^4_{+}a^2_{-}  & q_{-3}a^4_{+}a^2_{-} & 0 &  q_{-3}a^4_{+}a^2_{-} & 0 &
0 &  q_{-3}a^2_{+}a^4_{-} \\
  q_{+3}a^4_{+}a^2_{-} & 0 & 0 & q_{+3}a^2_{+}a^4_{-} & 0 & q_{+3}a^2_{+}a^4_{-} &
q_{+3}a^2_{+}a^4_{-} & 0 \\
q_{+3}a^4_{+}a^2_{-} & 0 & 0 & q_{+3}a^2_{+}a^4_{-} & 0 & q_{+3}a^2_{+}a^4_{-} &
q_{+3}a^2_{+}a^4_{-} & 0 \\
  0 & q_{-3}a^2_{+}a^4_{-} & q_{-3}a^2_{+}a^4_{-} & 0 & q_{-3}a^2_{+}a^4_{-} & 0 & 0
& q_{-3}a^6_{-} \\
\end{array}%
\right).
\end{equation}
%{\bf Attention changer $q_{\pm}$ par $q_{\pm3}$}
The density matrix is built of four blocks. The diagonal blocks appear as $X$
alphabet with non-zero
density matrix elements only along the diagonal and anti-diagonal contrarily to  the
two off diagonal blocks which have
vanishing  elements  along the diagonal and anti-diagonal. This suggests that
$\rho_{123}$ can be viewed as a
special instance of three qubit $X$ states.  Paralleling the treatment for two
qubits,  the state (\ref{3X-class2}) is  re-written as
\begin{equation}\label{3X-fano1-calss2}
\rho_{123}= \sum_{k,l = 0,1} \rho^{kl} \otimes \vert k \rangle \langle l \vert
\end{equation}
where $\vert k \rangle , \vert l \rangle $ belong to Hilbert space of the qubit $3$.
The two qubit density matrices $\rho^{kk}$ ( $k = 0,1$), occurring in the expansion
(\ref{3X-fano1-calss2}), writes in the computational basis spanned by  $\{  |0
\rangle_1 \otimes |0\rangle_2,
|0\rangle_1 \otimes |1\rangle_2 , |1\rangle_1 \otimes
|0\rangle_2,  |1\rangle_1 \otimes |1\rangle_2\} $, as
\begin{eqnarray}
\rho^{00}= 2{\cal N}^2\left(
\begin{array}{cccc}
q_{+3}a^6_{+}
& 0 & 0 &  q_{+3}a^4_{+}a^2_{-} \\
0 & q_{-3}a^4_{+}a^2_{-} & q_{-3}a^4_{+}a^2_{-} & 0 \\
0 & q_{-3}a^4_{+}a^2_{-} & q_{-3}a^4_{+}a^2_{-}
& 0 \\
q_{+3}a^4_{+}a^2_{-} & 0 & 0 & q_{+3}a^2_{+}a^4_{-}
\end{array}
\right) \,,\label{2X-00}
\end{eqnarray}
and
\begin{eqnarray}
\rho^{11}= 2{\cal N}^2\left(
\begin{array}{cccc}
 q_{-3}a^4_{+}a^2_{-}
& 0 & 0 &  q_{-3}a^2_{+}a^4_{-} \\
0 & q_{+3}a^2_{+}a^4_{-} & q_{+3}a^2_{+}a^4_{-} & 0 \\
0 & q_{+3}a^2_{+}a^4_{-} & q_{+3}a^2_{+}a^4_{-}
& 0 \\
q_{-3}a^2_{+}a^4_{-} & 0 & 0 & q_{-3}a^6_{-}
\end{array}
\right) \,. \label{2X-11}
\end{eqnarray}
For $(k = 0 , l = 1)$ and $(k = 1 , l = 0)$ , we have respectively
\begin{eqnarray}
\rho^{01} = 2{\cal N}^2 \left(
\begin{array}{cccc}
0
& q_{+3}a^4_{+}a^2_{-} & q_{+3}a^4_{+}a^2_{-}  & 0 \\
q_{-3}a^4_{+}a^2_{-} & 0 & 0  & q_{-3}a^2_{+}a^4_{-} \\
q_{-3}a^4_{+}a^2_{-} & 0 & 0
& q_{-3}a^2_{+}a^4_{-}\\
0 & q_{+3}a^2_{+}a^4_{-} & q_{+3}a^2_{+}a^4_{-} & 0
\end{array}
\right) \,,  \label{2X-01}
\end{eqnarray}
and
\begin{eqnarray}
\rho^{10} = 2{\cal N}^2 \left(
\begin{array}{cccc}
0
& q_{-3}a^4_{+}a^2_{-} & q_{-3}a^4_{+}a^2_{-} & 0 \\
 q_{+3}a^4_{+}a^2_{-} & 0 & 0  & q_{+3}a^2_{+}a^4_{-} \\
 q_{+3}a^4_{+}a^2_{-} & 0 & 0
& q_{+3}a^2_{+}a^4_{-}\\
0 &  q_{-3}a^2_{+}a^4_{-} &  q_{-3}a^2_{+}a^4_{-} & 0
\end{array}
\right) \, . \label{2X-10}
\end{eqnarray}
The Fano-Bloch representation of the matrices $\rho^{kk}$, given by  (\ref{2X-00})
and (\ref{2X-11}), take the the form
\begin{equation}\label{2X-fano-class2-1}
\rho^{kk}=\frac{1}{4}  \sum_{\alpha \beta} T^{kk}_{\alpha \beta} ~
\sigma_{\alpha}\otimes \sigma_{\beta}
\end{equation}
where $\alpha, \beta = 0,1,2,3$ and the correlation matrix elements $T^{kk}_{\alpha
\beta}$ are given by
$$T^{kk}_{\alpha \beta} = {\rm Tr}(\rho^{kk}~\sigma_{\alpha}\otimes \sigma_{\beta}).$$
The explicit expressions of the non vanishing correlation elements, in terms of the
overlap factor $p$ defined by (\ref{overlap}),  are
\begin{equation}\label{Tii}
T^{kk}_{00}= 1$$
$$ T^{kk}_{30}= T^{kk}_{03} = \frac{p}{2}~ (1 + (-)^kp)~\frac{1 + (-)^kp^{n-3}\cos
m\pi}{1 + p^{n}\cos m\pi}$$
$$T^{kk}_{11}=\frac{1}{2}  ~(1 + (-)^kp)~\frac{1 - p^{2}}{1 + p^{n}\cos m\pi}$$
$$T^{kk}_{22}= - \frac{1}{2} ~ (1 + (-)^kp)~\frac{(1 - p^{2}) p^{n-3}\cos m\pi}{1 +
p^{n}\cos m\pi}$$
$$T^{kk}_{33}=\frac{1}{2} ~ (1 + (-)^kp)~\frac{p^2 + (-)^kp^{n-3}\cos m\pi}{1 +
p^{n}\cos m\pi}
\end{equation}
Similarly, for the two-qubit states $\rho^{kl}$ ($ k \neq l$) given by (\ref{2X-01})
and (\ref{2X-10}), the Fano-Bloch representation writes
\begin{equation}\label{2X-fano-class2-2}
\rho^{kl}=\frac{1}{4}  \sum_{\alpha \beta} T^{kl}_{\alpha \beta} ~
\sigma_{\alpha}\otimes \sigma_{\beta}
\end{equation}
where the non zero matrix elements $T^{kl}_{\alpha \beta}$ are given by
\begin{equation}\label{Tij}
T^{kl}_{01}= T^{kl}_{10}= \frac{1}{2}~ \frac{1-p^2}{1 + p^{n}\cos m\pi} $$
$$T^{kl}_{02}= T^{kl}_{20}=  (-)^k \frac{i}{2}~ \frac{p(1-p^2)}{1 + p^{n}\cos m\pi}  $$
$$T^{kl}_{13}= T^{kl}_{31}= \frac{1}{2}~ \frac{(1-p^2)p^{n-2}\cos m\pi)}{1 +
p^{n}\cos m\pi} $$
$$T^{kl}_{23}= T^{kl}_{32}= (-)^k \frac{i}{2}~ \frac{(1-p^2)p^{n-2}\cos m\pi)}{1 +
p^{n}\cos m\pi}.
\end{equation}
Using  (\ref{3X-fano1-calss2}), the three-qubit state $\rho_{123}$ expands  as
\begin{equation}\label{3X-fano1}
\rho_{123}=\frac{1}{2} \Bigg[ (\rho^{00} + \rho^{11} )\otimes \sigma_0  +
(\rho^{00} - \rho^{11} )\otimes \sigma_3 +
 (\rho^{01} + \rho^{10} )\otimes \sigma_1  + i (\rho^{01} - \rho^{10} )\otimes
\sigma_2 \Bigg].
\end{equation}
Inserting  (\ref{2X-fano-class2-1}) and  (\ref{2X-fano-class2-2}) in the expression
(\ref{3X-fano1}) and using  the results (\ref{Tii}) and (\ref{Tij}) , one gets
\begin{equation}\label{density}
\rho_{123} = \frac{1}{8} \sum_{\alpha\beta} \Bigg[T_{\alpha\beta0} ~\sigma_{\alpha}
\otimes \sigma_{\beta} \otimes \sigma_{0}+
T_{\alpha\beta1} ~\sigma_{\alpha} \otimes \sigma_{\beta} \otimes \sigma_{1}+
T_{\alpha\beta2} ~\sigma_{\alpha} \otimes \sigma_{\beta} \otimes \sigma_{2}+
T_{\alpha\beta3} ~\sigma_{\alpha} \otimes \sigma_{\beta} \otimes \sigma_{3}\Bigg]
\end{equation}
where
\begin{equation}\label{relation3ret2r-class2}
T_{\alpha\beta0}=    T^{++}_{\alpha\beta} = T^{00}_{\alpha\beta} +
T^{11}_{\alpha\beta} $$
$$ T_{\alpha\beta3}= T^{--}_{\alpha\beta} = T^{00}_{\alpha\beta} - T^{11}_{\alpha\beta}
\end{equation}
with $\alpha \beta = 00, 03, 30,  11, 22, 33 $ (cf. (\ref{Tii})),
%$$ (00), (03), (30),  (11), (22), (33)$$
and
\begin{equation}\label{relation3ret2r-class22}
T_{\alpha\beta1} = T^{+-}_{\alpha\beta} = T^{01}_{\alpha\beta} + T^{10}_{\alpha\beta}$$
$$ T_{\alpha\beta2} = T^{-+}_{\alpha\beta} = iT^{01}_{\alpha\beta} -
iT^{10}_{\alpha\beta}.
\end{equation}
with $\alpha \beta= 01, 02, 10, 20 , 13 , 23, 31, 32$ (cf. equations (\ref{Tij})).
Reporting (\ref{Tii}) and (\ref{Tij}) in
the expressions (\ref{relation3ret2r-class2}) and (\ref{relation3ret2r-class22}),
one obtains
the 32 non vanishing correlation matrix elements  $T_{\alpha\beta\gamma}$
corresponding to the three qubit state $\rho_{123}$. Subsequently,
The recursive relations (\ref{relation3ret2r-class2}) and
(\ref{relation3ret2r-class22}) offer a nice tool
to determine the correlation elements $T_{\alpha\beta\gamma}$ in terms of those
associated with the two qubit density
matrices $\rho^{kk}$ and $\rho^{kl}$ respectively  given  by
(\ref{2X-fano-class2-1}) and (\ref{2X-fano-class2-2}).
This recursive procedure is  useful in simplifying the evaluation of pairwise correlations in
three qubit state of type $\rho_{123}$ as we shall discuss hereafter.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$k$-qubit states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The previous analysis can be extended to embrace an arbitrary mixed $k$-qubit state
$(k= 2, 3, \cdots, n)$  obtained by tracing
out $(n-k)$ qubit from the state (\ref{ncs}). The resulting reduced density is given by
\begin{equation}\label{rho123k-trace}
\rho_{123\cdots k} = {\cal N}^2 \bigg[ \vert \eta, \eta,\cdots, \eta \rangle \langle
\eta, \eta,\cdots, \eta \vert
+ e^{im\pi} q_k \vert -\eta, -\eta,\cdots, -\eta \rangle \langle \eta, \eta,\cdots,
\eta \vert$$
$$+ e^{-im\pi} q_k \vert \eta, \eta,\cdots, \eta \rangle \langle -\eta,
-\eta,\cdots, -\eta \vert
+  \vert -\eta, -\eta,\cdots, -\eta \rangle \langle -\eta, -\eta,\cdots -\eta \vert
\bigg]
\end{equation}
where $ q_k = p^{n-k}$. The reduced density matrix $\rho_{123\cdots k}$ is of rank
2. Indeed, the state
(\ref{rho123k-trace}) rewrites
\begin{equation}
\rho_{123\cdots k} = \frac{1}{2} ~(1 + p^{n-k})~ \frac{{\cal N}^2}{{\cal N}_{k+}^2}
~\vert \eta \rangle_{k~k}\langle \eta \vert
 + \frac{1}{2} ~(1 - p^{n-k})~\frac{{\cal N}^2}{{\cal N}_{k-}^2}~ Z \vert \eta
\rangle_{k~k}\langle \eta \vert Z
\end{equation}
where
$$ \vert \eta \rangle_k = {\cal N}_{k+} ( \vert \eta, \eta,\cdots, \eta \rangle +
e^{im\pi} \vert -\eta, -\eta,\cdots, -\eta \rangle) \qquad
Z\vert \eta \rangle_k = {\cal N}_{k-} ( \vert \eta, \eta,\cdots, \eta \rangle -
e^{im\pi} \vert -\eta, -\eta,\cdots, -\eta \rangle) $$
and the normalization factors $ {\cal N}_{k\pm}$ are given by
$$ {\cal N}^{-2}_{k\pm} = 2(1 \pm p^k\cos m\pi).$$
The cyclic operator $Z$ is now defined by
$$ Z \vert \eta, \eta, \cdots, \eta \rangle = \vert \eta, \eta, \cdots ,\eta \rangle
\qquad  Z \vert - \eta, - \eta, \cdots , -\eta \rangle = -  \vert - \eta, - \eta,
\cdots , -\eta \rangle.$$
Clearly, for $k=2$ and $k=3$, the state $\rho_{123\cdots k} $ (\ref{rho123k-trace})
reduces to $\rho_{12} $ (\ref{rho12}) and $\rho_{123} $ (\ref{rho123}) respectively.
Using (\ref{rho123k-trace}), it is simple to check that the $k$-qubit state
$\rho_{123\cdots k}$ can
be expressed in terms of states comprising $(k-1)$-qubits. The state
$\rho_{123\cdots k}$ (\ref{rho123k-trace})  can be written also as
\begin{equation}\label{rho123k-decompose}
\rho_{123\cdots k} = \sum_{rs=1,2} \rho^{rs}_{12\cdots (k-1)} \otimes \vert r
\rangle \langle s \vert
\end{equation}
where
\begin{equation}
\rho^{rs}_{12\cdots (k-1)} \equiv \rho^{rs} =  a_+^{2-r-s}a_-^{r+s} \Bigg[
\frac{1}{2} ~(1 + p^{n-k})~
\frac{{\cal N}^2}{{\cal N}_{(k-1)+}^2} ~ Z^r\vert \eta \rangle_{(k-1)~(k-1)}\langle
\eta \vert Z^s$$
$$ + \frac{1}{2} ~(1 - p^{n-k})~\frac{{\cal N}^2}{{\cal N}_{(k-1)-}^2}~ Z^{r+1}
\vert \eta \rangle_{(k-1)~(k-1)}\langle \eta \vert Z^{s+1}\Bigg].
\end{equation}
Explicitly, the $k$-qubit matrix  (\ref{rho123k-decompose}) writes
\begin{equation}\label{rho123k}
\rho_{123\cdots k} = \frac{1}{2} (\rho^{00} + \rho^{11}) \otimes \sigma_{0} +
\frac{1}{2} (\rho^{01}+ \rho^{10}) \otimes \sigma_{1} + \frac{i}{2} (\rho^{01} -
\rho^{10}) \otimes \sigma_{2}
+ \frac{1}{2} (\rho^{00} - \rho^{11}) \otimes \sigma_{3}
\end{equation}
and the $(k-1)$-qubit states $\rho^{rs}\equiv \rho^{rs}_{12\cdots (k-1)}$ can be expanded, in Fano-Bloch
representation,  as
\begin{equation}\label{fanorhors}
\rho^{rs}= \frac{1}{2^{k-1}} \sum_{\alpha_1,\alpha_2, \cdots,\alpha_{k-1}}
T^{rs}_{\alpha_1 \alpha_2 \cdots \alpha_{k-1}}
\sigma_{\alpha_1} \otimes \sigma_{\alpha_2} \otimes \cdots \otimes
\sigma_{\alpha_{k-1}}.
\end{equation}
Hence, reporting (\ref{fanorhors}) in (\ref{rho123k-decompose}), the $k$-qubit state
$\rho_{123\cdots k}$ takes the form
\begin{equation}\label{fanorhor123k}
\rho_{123\cdots k} = \frac{1}{2^{k}} \sum_{\alpha_1,\alpha_2,
\cdots,\alpha_{k-1},\alpha_{k}} T_{\alpha_1 \alpha_2 \cdots \alpha_{k-1}\alpha_k}
\sigma_{\alpha_1} \otimes \sigma_{\alpha_2} \otimes \cdots \otimes
\sigma_{\alpha_{k-1}}\otimes \sigma_{\alpha_{k}}.
\end{equation}
where the correlation matrix elements $T_{\alpha_1 \alpha_2 \cdots
\alpha_{k-1}\alpha_k}$ express in terms of the correlations
coefficients occurring in (\ref{fanorhors}) as
\begin{equation}\label{recur-k}
  T_{\alpha_1 \alpha_2 \cdots \alpha_{k-1}0} = T^{00}_{\alpha_1 \alpha_2 \cdots
\alpha_{k-1}} + T^{11}_{\alpha_1 \alpha_2 \cdots \alpha_{k-1}} $$
$$ T_{\alpha_1 \alpha_2 \cdots \alpha_{k-1}3}= T^{00}_{\alpha_1 \alpha_2 \cdots
\alpha_{k-1}} - T^{11}_{\alpha_1 \alpha_2 \cdots \alpha_{k-1}}$$
$$ T_{\alpha_1 \alpha_2 \cdots \alpha_{k-1}1} = T^{01}_{\alpha_1 \alpha_2 \cdots
\alpha_{k-1}} + T^{10}_{\alpha_1 \alpha_2 \cdots \alpha_{k-1}}$$
$$ T_{\alpha_1 \alpha_2 \cdots \alpha_{k-1}2} = i T^{01}_{\alpha_1 \alpha_2 \cdots
\alpha_{k-1}} -i T^{10}_{\alpha_1 \alpha_2 \cdots \alpha_{k-1}}.
\end{equation}
This establishes the relations between the correlation matrix elements of $k$ and
$(k-1)$-qubit states. In this picture the correlation matrix elements
associated with a  $k$-qubit state can be recursively expressed in terms of ones
involving two qubits. The recurrence relations (\ref{recur-k}) reduce to
(\ref{recurr}) for $k=2$ and to (\ref{relation3ret2r-class2}-\ref{relation3ret2r-class22}) for $k=3$.
To illustrate further  the algorithm in deriving relations of type (\ref{recur-k}),
 we consider the case of four qubits $(k = 4)$. In this situation, the density matrix
(\ref{rho123k-trace}) becomes
\begin{equation}\label{rho1234}
\rho_{1234} = {\cal N}^2 \bigg[ \vert \eta, \eta,\eta, \eta \rangle \langle \eta,
\eta,\eta, \eta \vert
+ e^{im\pi} q_4 \vert -\eta, -\eta,-\eta, -\eta \rangle \langle \eta, \eta,\eta,
\eta \vert$$
$$+ e^{-im\pi} q_4 \vert \eta, \eta,\eta, \eta \rangle \langle -\eta, -\eta,-\eta,
-\eta \vert
+  \vert -\eta, -\eta,-\eta, -\eta \rangle \langle -\eta, -\eta,-\eta -\eta \vert
\bigg]
\end{equation}
and the expression (\ref{rho123k-decompose}) gives
\begin{equation}\label{rho1234-bis}
\rho_{1234} = \rho_{123}^{00}\otimes \vert 0\rangle \langle 0\vert +
\rho_{123}^{01}\otimes \vert0 \rangle \langle 1\vert + \rho_{123}^{10}\otimes \vert
1\rangle \langle 0 \vert
+ \rho_{123}^{11}\otimes \vert 1\rangle \langle 1 \vert
\end{equation}
where the three-qubit states $\rho_{123}^{00}$, $\rho_{123}^{01}$, $\rho_{123}^{10}$
and $\rho_{123}^{11}$ are given in the
usual computational basis as
\begin{equation}\label{rho00-4}
\frac{\rho^{00}_{123}}{2{\cal N}^2}=\left(%
\begin{array}{cccccccc}
 q_{+4}a^8_{+} & 0 & 0 & q_{+4}a^6_{+}a^2_{-} & 0 & q_{+4}a^6_{+}a^2_{-}  &
q_{+4}a^6_{+}a^2_{-}  & 0 \\
  0 &q_{-4}a^6_{+}a^2_{-} & q_{-4}a^6_{+}a^2_{-}  & 0 & q_{-4}a^6_{+}a^2_{-}  & 0 &
0 &q_{-4}a^4_{+}a^4_{-}  \\
  0 & q_{-4}a^6_{+}a^2_{-} & q_{-4}a^6_{+}a^2_{-}  & 0 & q_{-4}a^6_{+}a^2_{-}  & 0 &
0 &  q_{-4}a^4_{+}a^4_{-} \\
  q_{+4}a^6_{+}a^2_{-}  & 0 & 0 & q_{+4}a^4_{+}a^4_{-}  & 0 & q_{+4}a^4_{+}a^4_{-} &
q_{+4}a^4_{+}a^4_{-} & 0 \\
  0 & q_{-4}a^6_{+}a^2_{-}  & q_{-4}a^6_{+}a^2_{-} & 0 &  q_{-4}a^6_{+}a^2_{-} & 0 &
0 &  q_{-4}a^4_{+}a^4_{-} \\
  q_{+4}a^6_{+}a^2_{-} & 0 & 0 & q_{+4}a^4_{+}a^4_{-} & 0 & q_{+4}a^4_{+}a^4_{-} &
q_{+4}a^4_{+}a^4_{-} & 0 \\
q_{+4}a^6_{+}a^2_{-} & 0 & 0 & q_{+4}a^4_{+}a^4_{-} & 0 & q_{+4}a^4_{+}a^4_{-} &
q_{+4}a^4_{+}a^4_{-} & 0 \\
  0 & q_{-4}a^4_{+}a^4_{-} & q_{-4}a^4_{+}a^4_{-} & 0 & q_{-4}a^4_{+}a^4_{-} & 0 & 0
& q_{-4}a^2_{+}a^6_{-} \\
\end{array}%
\right)
\end{equation}
\begin{equation}\label{rho11-4}
\frac{\rho^{11}_{123}}{2{\cal N}^2}=\left(%
\begin{array}{cccccccc}
 q_{-4}a^6_{+}a^2_{-} & 0 & 0 & q_{-4}a^4_{+}a^4_{-} & 0 & q_{-4}a^4_{+}a^4_{-}  &
q_{-4}a^4_{+}a^4_{-}  & 0 \\
  0 &q_{+4}a^4_{+}a^4_{-} & q_{+4}a^4_{+}a^4_{-}  & 0 & q_{+4}a^4_{+}a^4_{-}  & 0 &
0 &q_{+4}a^2_{+}a^6_{-}  \\
  0 & q_{+4}a^4_{+}a^4_{-} & q_{+4}a^4_{+}a^4_{-}  & 0 & q_{+4}a^4_{+}a^4_{-}  & 0 &
0 &  q_{+4}a^2_{+}a^6_{-} \\
  q_{-4}a^4_{+}a^4_{-}  & 0 & 0 & q_{-4}a^2_{+}a^6_{-}  & 0 & q_{-4}a^2_{+}a^6_{-} &
q_{-4}a^2_{+}a^6_{-} & 0 \\
  0 & q_{+4}a^4_{+}a^4_{-}  & q_{+4}a^4_{+}a^4_{-} & 0 &  q_{+4}a^4_{+}a^4_{-} & 0 &
0 &  q_{+4}a^2_{+}a^6_{-} \\
  q_{-4}a^4_{+}a^4_{-} & 0 & 0 & q_{-4}a^2_{+}a^6_{-} & 0 & q_{-4}a^2_{+}a^6_{-} &
q_{-4}a^2_{+}a^6_{-} & 0 \\
q_{-4}a^4_{+}a^4_{-} & 0 & 0 & q_{-4}a^2_{+}a^6_{-} & 0 & q_{-4}a^2_{+}a^6_{-} &
q_{-4}a^2_{+}a^6_{-} & 0 \\
  0 & q_{+4}a^2_{+}a^6_{-} & q_{+4}a^2_{+}a^6_{-} & 0 & q_{+4}a^2_{+}a^6_{-} & 0 & 0
& q_{+4} a^8_{-} \\
\end{array}%
\right),
\end{equation}
\begin{equation}\label{rho01-4}
\frac{\rho^{01}_{123}}{2{\cal N}^2}=\left(%
\begin{array}{cccccccc}
 0 & q_{+4}a^6_{+}a^2_{-} &  q_{+4}a^6_{+}a^2_{-} & 0 &  q_{-4}a^6_{+}a^2_{-} & 0  &
0  & q_{-4}a^4_{+}a^4_{-} \\
q_{-4}a^6_{+}a^2_{-} & 0 & 0  & q_{-4}a^4_{+}a^4_{-} & 0   & q_{+4}a^4_{+}a^4_{-}  &
q_{+4}a^4_{+}a^4_{-} & 0  \\
q_{-4}a^6_{+}a^2_{-} & 0 & 0 & q_{-4}a^4_{+}a^4_{-} & 0   &  q_{+4}a^4_{+}a^4_{-} &
q_{+4}a^4_{+}a^4_{-} & 0 \\
0 & q_{+4}a^4_{+}a^4_{-} & q_{+4}a^4_{+}a^4_{-} & 0  &  q_{-4}a^4_{+}a^4_{-} & 0 & 0
& q_{-4}a^2_{+}a^6_{-} \\
q_{-4}a^6_{+}a^2_{-} & 0  & 0 &  q_{-4}a^4_{+}a^4_{-} & 0 &   q_{-4}a^4_{+}a^4_{-} &
 q_{-4}a^4_{+}a^4_{-} &  0 \\
0 &  q_{+4}a^4_{+}a^4_{-} &  q_{+4}a^4_{+}a^4_{-} & 0 &  q_{+4}a^4_{+}a^4_{-} & 0 &
0 &  q_{+4}a^2_{+}a^6_{-}\\
0 &  q_{+4}a^4_{+}a^4_{-} & q_{+4}a^4_{+}a^4_{-} & 0 & q_{+4}a^4_{+}a^4_{-} & 0 & 0
& q_{+4}a^2_{+}a^6_{-}\\
q_{-4}a^4_{+}a^4_{-} & 0 & 0 & q_{-4}a^2_{+}a^6_{-} & 0 & q_{-4}a^2_{+}a^6_{-} &
q_{-4}a^2_{+}a^6_{-}& 0 \\
\end{array}%
\right),
\end{equation}
and
\begin{equation}\label{rho10-4}
\rho^{10}_{123} = \big( \rho^{01}_{123} \big)^t.
\end{equation}
Obviously, with increasing number of qubits, the size of density matrices is
growing. This induces inevitably complicated algebraic manipulation  in deriving computable expressions of
quantum correlations. However, the recursive algorithm presented
above, offers an alternative way for symmetric multi-qubit (\ref{ncs}), to reduce the complexity
in determining analytical evaluation of geometric discord.  The expression (\ref{rho1234-bis})
allows us to write the correlations
factors $T_{\alpha_1 \alpha_2 \alpha_3 \alpha_4}$ in terms
of those corresponding to three-qubit density matrices $\rho^{00}_{123}$,
$\rho^{01}_{123}$, $\rho^{10}_{123}$
and $\rho^{11}_{123}$ (cf. matrices (\ref{rho00-4})-(\ref{rho10-4})). Indeed, the state $\rho_{1234}$ expands in the Fano-Bloch
representation as
\begin{equation}\label{rho123k-bis}
\rho_{1234} = \frac{1}{2^4}\sum_{\alpha_1, \alpha_2, \alpha_3, \alpha_4} T_{\alpha_1
\alpha_2 \alpha_3 \alpha_4} \sigma_{\alpha_1}\otimes \sigma_{\alpha_2}
\otimes \sigma_{\alpha_3} \otimes \sigma_{\alpha_4}
\end{equation}
and by re-equating (\ref{rho1234-bis}) as
\begin{equation}\label{rho1234-bis-bis}
\rho_{1234} = \frac{1}{2} (\rho_{123}^{00}+ \rho_{123}^{11}) \otimes \sigma_0 +
\frac{1}{2} (\rho_{123}^{01}+ \rho_{123}^{10})\otimes \sigma_1
+ \frac{i}{2} (\rho_{123}^{01} - \rho_{123}^{10}) \otimes \sigma_2
+ \frac{1}{2} (\rho_{123}^{00} - \rho_{123}^{00}) \otimes \sigma_3,
\end{equation}
it is simple to see that
\begin{equation}\label{recu-4}
T_{\alpha_1 \alpha_2 \alpha_3 0}  = T^{00}_{\alpha_1 \alpha_2 \alpha_3} +
T^{11}_{\alpha_1 \alpha_2 \alpha_3} $$
$$ T_{\alpha_1 \alpha_2 \alpha_3 1} = T^{01}_{\alpha_1 \alpha_2 \alpha_3} +
T^{10}_{\alpha_1 \alpha_2 \alpha_3}$$
$$ T_{\alpha_1 \alpha_2 \alpha_3 2} = iT^{01}_{\alpha_1 \alpha_2 \alpha_3} - i
T^{10}_{\alpha_1 \alpha_2 \alpha_3}$$
$$ T_{\alpha_1 \alpha_2 \alpha_3 3}= T^{00}_{\alpha_1 \alpha_2 \alpha_3} -
T^{11}_{\alpha_1 \alpha_2 \alpha_3}
\end{equation}
where the quantities $T^{kl}_{\alpha_1, \alpha_2, \alpha_3}$, defined so that
\begin{equation}\label{rhokl123}
\rho^{kl}_{123} = \frac{1}{2^3}\sum_{\alpha_1, \alpha_2, \alpha_3} T^{kl}_{\alpha_1
\alpha_2 \alpha_3} \sigma_{\alpha_1}\otimes \sigma_{\alpha_2}
\otimes \sigma_{\alpha_3},
\end{equation}
can be obtained easily following the method developed above for three and two qubit
states (modulo some obvious substitutions). In this manner, one verifies
 that the only non vanishing elements $T_{\alpha_1 \alpha_2 \alpha_3 \alpha_4}$  are
those with indices  $({\alpha_1, \alpha_2, \alpha_3, \alpha_4})$ belonging to the
following  set of quadruples
$$\{ \{ 00, 11, 22, 33, 03, 30\} \times \{0,3\} \times \{ 0,3\},$$
$$ \{ 00, 11, 22, 33, 03, 30\} \times \{1,2\} \times \{ 1,2\},$$
$$ \{ 01, 10, 20, 02, 13, 31, 23, 32\} \times \{1,2\} \times \{ 0,3\},$$
$$ \{ 01, 10, 20, 02, 13, 31, 23, 32\} \times \{0,3\} \times \{ 1,2\} \}.$$
Finally, we stress that  the recursive approach giving  the Fano-Bloch
components
for an arbitrary $k$-qubit state in terms of those involving $(k-1)$-qubits is of central importance for our
purpose. This
gives a simple way to specify the correlation matrix elements for
$k$-qubits state in  terms of ones associated with its sub-components which render  more tractable the computation of
all pairwise geometric quantum discord  in the multi-qubit Shro\"odinger cat states
(\ref{ncs}).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Geometric measure of quantum discord and classical states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We now face the question of determining the explicit form of pairwise geometric discord in
the $k$-qubit mixed state (\ref{rho123k-trace}) when partitioned   into two subsystems: one containing a single
qubit and the second part is grouping the rest of the $(k-1)$ qubits living in $2^{k-1}$ dimensional Hilbert space.
For this end, it is necessary to find the
expression of closest classical states to the states of type (\ref{rho123k-trace})
when the distance is measured by Hilbert-Schmidt
norm defined by
$$ || \rho - \rho' ||^2 = {\rm Tr}\big[(\rho - \rho')^2\big]$$
for all density operators acting on the Hilbert space ${\cal H}_n$.
To extend the procedure developed in \cite{Dakic2010}, we shall begin with the simplest case $k=2$. A special focus
concerns the derivation of the closest classical states to the two-qubit state (\ref{rho12-sum}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Two-qubit states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the density matrix (\ref{rho12-sum}), which reads also as
\begin{equation}\label{rho12sum}
\rho_{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_{11}~
\sigma_{1}\otimes \sigma_{1} + T_{22}~
\sigma_{2}\otimes \sigma_{2}+T_{33}~
\sigma_{3}\otimes \sigma_{3}\Bigg],
\end{equation}
 the zero-discord or classical states are given by
\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 $1$ and $\rho_i^{2}$ $(i = 1, 2)$ are
reduced density matrices attached  the second qubit. It can be
rewritten  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
$$ t = p_1-p_2 \qquad e_i = \langle \psi_1 \vert \sigma_i \vert \psi_1 \rangle  \qquad
(s_{\pm})_{j} = {\rm Tr}\big((p_1\rho_1^{2} \pm p_2\rho_2^{2}) \sigma_{j} \big).$$
The distance between the density matrix $\rho_{12}$ (\ref{rho12sum}) and the
classical state $\chi_{12}$ (\ref{xi12}), as
measured by Hilbert-Schmidt norm, is
\begin{equation}\label{HS12}
|| \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}
%To get the closest classical state to the quantum state $\rho_{12}$, one performs the
The minimal distance is obtained by minimizing  the the Hilbert-Schmidt norm
(\ref{HS12}) with respect to the parameters $t$, $(s_+)_{i}$ and $(s_-)_{i}$. This
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  the solutions (\ref{minHS12}) in (\ref{HS12}), one gets
\begin{equation}\label{HS12-compact}
|| \rho_{12} - \chi_{12}||^2 = \frac{1}{4} \bigg[  {\rm Tr}K - \vec{e}^t ~K~ \vec{e}
\bigg]
\end{equation}
where $\vec{e}^t = (e_1, e_2, e_3)$ and the matrix $K$ is defined by
\begin{equation}\label{matrixK12}
K = {\rm diag}(T_{11}^2, T_{22}^2, T_{30}^2+T_{33}^2).
\end{equation}
From  the result (\ref{Tij12}), the eigenvalues of the  matrix $K$ (\ref{matrixK12}) read
\begin{equation}\label{GD12lambda1}
k_1 \equiv T_{11}^2 = \frac{(1-p^2)^2}{(1+p^n\cos m\pi)^2},
\end{equation}
\begin{equation}\label{GD12lambda2}
k_2 \equiv T_{22}^2 = \frac{(1-p^2)^2 p^{2(n-2)}}{(1+p^n\cos m\pi)^2},
\end{equation}
\begin{equation}\label{GD12lambda3}
k_3 \equiv T_{30}^2+T_{33}^2 = \frac{(p^2+ p^{2(n-2)})(1+p^2) + 4 p^n\cos m\pi}{(1+p^n\cos m\pi)^2}.
\end{equation}
Using (\ref{HS12-compact}), it is  easily seen  that  the minimal Hilbert-Schmidt distance
is obtained for the vector $\vec{e}$ associated with the maximal eigenvalue
$k_{\rm max} = {\rm max}(k_1,k_2,k_3) $. Thus, the geometric measure of quantum discord in the state
$\rho_{12}$ is  given by
\begin{equation}\label{GD12}
D_{\rm g}(\rho_{12}) = \frac{1}{4} (k_{1}+k_{2}+k_{3}-k_{\rm
max}).
\end{equation}
Comparing (\ref{GD12lambda1}) and (\ref{GD12lambda2}),  we have
$k_2 < k_1$. This implies that $k_{\rm max}$ can be either
$k_1$ or $k_3$. In this respect, to find  the closest classical states,  two situations must be
considered separately.  We begin first with the  case where $k_{\rm max} =
k_3$. The eigenvector associated with this maximal eigenvalue  is
$\vec{e}=(e_1=0,e_2=0,e_3=1)^t$. Reporting this result in (\ref{minHS12}),the closest classical state (\ref{xi12}) is
simply given by
\begin{equation}\label{xi12-2bits+}
\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}
Similarly, the eigenvector associated to $k_{\rm max} = k_1$ is
$\vec{e}=(e_1=1,e_2=0,e_3=0)^t$ and
from (\ref{minHS12}) one gets
\begin{equation}\label{xi12-2bits-}
\chi^{(1)}_{12} = \frac{1}{4}\Bigg[ \sigma_{0}\otimes \sigma_{0}
 + T_{03} ~\sigma_0 \otimes \sigma_{3}
+ T_{11}~\sigma_1 \otimes \sigma_{1}\Bigg].
\end{equation}
Beside the explicit derivation of closest classical states (\ref{xi12-2bits+}) and
(\ref{xi12-2bits-}), another important point to be emphasized
is the relation between the matrix $K$ (\ref{matrixK12}), which encodes the
geometric measure
quantum correlations in the state $\rho_{12}$, and the Bloch components of the
one-qubit density matrices $\rho^{ii}$ $(i=1,2)$ and
$\rho^{ij}$ $(i\neq j)$ given respectively by (\ref{0011}) and (\ref{0110}). For this purpose, we  use  the relations (\ref{recurr}) to
rewrite the matrix $K$ (\ref{matrixK12}) as
\begin{equation}
K = {\rm diag}( 2 (T^{01}_1)^2 , - 2 (T^{01}_2)^2,  (T^{00}_3)^2  +  (T^{11}_3)^2 ).
\end{equation}
Furthermore, for one-qubit states $\rho^{00}$, $\rho^{01}$, $\rho^{10}$ and
$\rho^{11}$, we introduce the  analogues
of the matrix $K$ (\ref{matrixK12}). For the states $\rho^{00}$ and
$\rho^{11}$ (\ref{0011}), we define the $3\times3$ matrix
$$ K^{kk} = (0, 0, T^{kk}_3)^t ~(0, 0, T^{kk}_3), \qquad k = 0,1, $$
and similarly, we introduce the matrices
$$K^{kl} = (T^{kl}_1, iT^{kl}_2, 0)^t ~(T^{kl}_1, iT^{kl}_2, 0) \quad {\rm for}
\quad (k,l) = (0,1)~{\rm or}~(1,0).$$
for the  states  $\rho^{01}$ and $\rho^{10}$ (\ref{0110}). They satisfied the remarkable additivity relation
\begin{equation}\label{additivitydesK}
K = 2 (K^{00} + K^{01} +  K^{10} +  K^{11} ).
\end{equation}
This relation holds also for the states containing three or more qubits
as a consequence  of the symmetry invariance of the
multi-qubit system under consideration.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Three-qubit states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We now extend the previous results to  the three qubit
states of the form (\ref{rho123}).  More especially, we analytically determine the  pairwise
quantum discord between the qubit $1$ and
the two qubit subsystem $(23)$.  We also  give the explicit forms of the closest
classical  tripartite states. To begin, we write the  density matrix (\ref{density})
as follows
\begin{equation}\label{3X-fano}
\rho_{123}=\frac{1}{8}\bigg[T_{000} ~\sigma_0\otimes \sigma_0\otimes \sigma_0 +
T_{300}~\sigma_3\otimes
\sigma_0\otimes \sigma_0 +  \sum_{(\beta,\gamma)\neq (0,0)} T_{0 \beta \gamma}~
\sigma_{0}\otimes \sigma_{\beta}\otimes
\sigma_{\gamma} +  \sum_i \sum _{(\beta,\gamma)\neq (0,0)} T_{i \beta \gamma}
~\sigma_{i}\otimes \sigma_{\beta}\otimes
\sigma_{\gamma}\bigg].
\end{equation}
The classical states  (i.e., states presenting zero discord between the qubit $1$
and the subsystem $23$) are of the form
\begin{equation}\label{chi}
\chi_{1|23} = p_1 \vert \psi_1 \rangle \langle \psi_1 \vert \otimes \rho_1^{23} +
p_2 \vert \psi_2 \rangle \langle \psi_2 \vert \otimes \rho_2^{23}
\end{equation}
where $\{ \vert \psi_1 \rangle , \vert \psi_2 \rangle \}$ is an orthonormal basis
related to the qubit $1$. The density matrices  $\rho_i^{23}$ ($i=1,2$)
corresponding to the subsystem $23$ write as
$$ \rho_i^{23} = \frac{1}{4} \bigg[  \sum_{\alpha,\beta} {\rm Tr}(\rho_i^{23}
\sigma_{\alpha}\otimes \sigma_{\beta})\sigma_{\alpha}\otimes \sigma_{\beta}\bigg].$$
The Fano-Bloch form of the  tripartite classical state (\ref{chi}) is given by
\begin{equation}\label{chi-matrix}
\chi_{1|23} = \frac{1}{8}\Bigg[ \sigma_{0}\otimes \sigma_{0} \otimes \sigma_{0}
+ \sum_{i=1}^3 te_i ~~\sigma_i \otimes \sigma_{0} \otimes \sigma_{0} $$
$$+ \sum_{(\alpha,\beta)\neq (0,0)} (s_+)_{\alpha,\beta} ~\sigma_0 \otimes
\sigma_{\alpha}\otimes \sigma_{\beta}
+ \sum_{i=1}^3  \sum_{(\alpha,\beta)\neq (0,0)}  e_i (s_-)_{\alpha,\beta} ~\sigma_i
\otimes \sigma_{\alpha}\otimes \sigma_{\beta}\Bigg]
\end{equation}
where
$$ t = p_1-p_2 \qquad e_i = \langle \psi_1 \vert \sigma_i \vert \psi_1 \rangle  \qquad
(s_{\pm})_{\alpha,\beta} = {\rm Tr}\big((p_1\rho_1^{23} \pm p_2\rho_2^{23})
\sigma_{\alpha}\otimes \sigma_{\beta}\big).$$
The Hilbert-Schmidt distance between the three qubit state $\rho_{123}$
(\ref{3X-fano}) and a classical state, having the form  (\ref{chi-matrix}), is
\begin{equation}\label{HS}
|| \rho_{1|23} - \chi_{1|23}||^2 = \frac{1}{8} \bigg[  (t^2 -2te_3T_{300} +
T^2_{300}) +  \sum_{(\alpha,\beta)\neq (0,0)}
(T_{0\alpha\beta} -  (s_+)_{\alpha,\beta})^2 + \sum_{i=1}^3
\sum_{(\alpha,\beta)\neq (0,0)} (T_{i\alpha\beta} -  e_i(s_-)_{\alpha,\beta})^2
\bigg].
\end{equation}
To derive the closest classical state as measured by Hilbert-Schmidt, an
optimization with  respect to the parameters $t$, $e_i$ ($i=1,2,3$) and
$(s_{\pm})_{\alpha,\beta}$
is performed. The minimal distance is attainable by setting zero the partial
derivatives of the
Hilbert-Schmidt distance (\ref{HS})  with respect to $t$ and
$(s_{\pm})_{\alpha,\beta}$.  This gives
\begin{equation}\label{solu}
t = e_3 T_{300}  \qquad (s_{+})_{\alpha,\beta} = T_{0\alpha\beta}   \qquad
(s_{-})_{\alpha,\beta} = \sum_{i=1}^3 e_i T_{i\alpha\beta}.
\end{equation}
Reporting the solutions (\ref{solu}) in (\ref{HS}), one obtains
\begin{equation}\label{HS123}
|| \rho_{1|23} - \chi_{1|23}||^2 = \frac{1}{8} \bigg[  T^2_{300} - e^2_3T^2_{300}
+ \sum_{i=1}^3  \sum_{(\alpha,\beta)\neq (0,0)} T^2_{i\alpha\beta} - \sum_{i,j=1}^3
\sum_{(\alpha,\beta)\neq (0,0)} e_ie_jT_{i\alpha\beta}T_{j\alpha\beta}\Bigg]
\end{equation}
which has to  be optimized with respect to the three components of the unit vector $\vec{e} =
(e_1, e_2, e_3)$. Inspired by the two qubit example, a more convenient form of the distance
(\ref{HS123}) is
\begin{equation}\label{HS1}
|| \rho_{1|23} - \chi_{1|23}||^2 = \frac{1}{8} \big[  ||x||^2 + ||T||^2
-\vec{e}(xx^t + TT^t)\vec{e}~^t\big]
\end{equation}
in terms of the   $1\times 3$ matrix defined by
\begin{equation}\label{x123}
x^t:=(0,0,T_{300})
\end{equation}
and the $3\times 15$ matrix given by
\begin{equation}\label{T123}
T := (T_{i\alpha  \beta}) ~~~ {\rm with}~~~~ (\alpha,\beta) \neq (0,0).
\end{equation}
Setting
\begin{equation}\label{matrixK-class2}
K = xx^{t}+TT^{t},
\end{equation}
and reporting (\ref{x123}) and (\ref{T123}) in (\ref{matrixK-class2}), one obtains
after some  tedious calculations
\begin{equation}\label{matrixK1}
K= {\rm diag}(k_1, k_2, k_3)
\end{equation}
where  $k_1$, $k_2$  and $k_3$ are given by
\begin{equation}\label{k1k2k3}
k_1 = \sum_{i=1,2} \sum_{j=0,3} T^2_{1ij}+ T^2_{1ji}, \quad k_2 = \sum_{i=1,2}
\sum_{j=0,3} T^2_{2ij}+ T^2_{2ji},
\quad k_3 = \sum_{i=0,3} \sum_{j=0,3} T^2_{3ij} +  \sum_{i=1,2} \sum_{j=1,2}
T^2_{3ij}.
\end{equation}
Using the relations (\ref{relation3ret2r-class2}) and
(\ref{relation3ret2r-class22}), the eigenvalues of the matrix $K$ can be
re-expressed in terms of the bipartite correlations elements
$T_{\alpha \beta}$ associated with the the qubit density matrices $\rho^{00}$,
$\rho^{01}$, $\rho^{10}$
and $\rho^{11}$ (cf. (\ref{2X-fano-class2-1}) and (\ref{2X-fano-class2-2})).
Therefore, one has
\begin{equation}
k_1=2[(T_{11}^{00})^2+(T_{11}^{11})^2]+4|
T^{01}_{10}|^2+4 |T^{01}_{13}|^2,
\end{equation}
\begin{equation}
k_{2}= 2[(T_{22}^{00})^2+(T_{22}^{11})^2]+4|
T^{01}_{20}|^2+4 |T^{01}_{23}|^2,
\end{equation}
\begin{equation}
k_{3}= 2[(T_{30}^{00})^2+(T_{30}^{11})^2] +2[(T_{33}^{00})^2+(T_{33}^{11})^2]+4|
T^{01}_{31}|^2+4 |T^{01}_{32}|^2.
\end{equation}
Finally, using  (\ref{Tii}) and (\ref{Tij}), we obtain
\begin{equation}\label{L11}
k_{1} = 2 ~ \frac{(1-p^2)^2(1+p^2)}{(1+p^n\cos m\pi)^2},
\end{equation}
\begin{equation}\label{L22}
k_{2} = 2 ~ \frac{(1-p^2)^2(1+p^2)p^{2(n-3)}}{(1+p^n\cos m\pi)^2},
\end{equation}
\begin{equation}\label{L33}
k_{3} = 2 ~ \frac{(p^2+ p^{2(n-3)})(1+p^4) + 4 p^n\cos m\pi}{(1+p^n\cos m\pi)^2}.
\end{equation}
The minimal value of the Hilbert-Schmidt distance (\ref{HS1}) is reached when
$\vec{e}$ is the
eigenvector associated to  the largest eigenvalue of the matrix defined by
(\ref{matrixK-class2}). We denote by $k_{\rm max}$ the largest eigenvalue
among $k_1$, $k_2$ and $k_3$. Since $k_1 \geq k_2$, $k_{\rm max}$ is
$k_2$ or $k_3$ depending on the number of qubits $n$ and the overlap $p$. Notice
that the sum of the eigenvalues $k_1$, $k_2$ and $k_2$ of the matrix $K$ is
exactly the sum of the Hilbert-Schmidt norm of the matrices $x$ (\ref{x123}) and $T$
(\ref{T123})  $(i.e. ~k_1 + k_2 + k_3  = ||x||^2 + ||T||^2)$. Accordingly,
the minimal Hilbert-Schmidt distance (\ref{HS1}) writes as
\begin{equation}\label{Dg}
D_{\rm g} (\rho_{1|23}) = \frac{1}{8} (k_1 + k_2 + k_3 - k_{\rm max})
\end{equation}
and gives the geometric measure of the pairwise quantum discord in the state
$\rho_{123}$ partitioned in
the subsystems $1$ and $23$. When the matrix elements of the density matrix
$\rho_{123}$ (\ref{3X-fano1}) are such that
$k_{\rm max} = k_1$, the closest classical state is given by
\begin{equation}\label{chi-matrix-i}
\chi^{(1)}_{1|23} = \frac{1}{8}\Bigg[ \sigma_{0}\otimes \sigma_{0} \otimes \sigma_{0}
+ \sum_{(\alpha,\beta)\neq (0,0)} T_{0\alpha\beta} ~\sigma_0 \otimes
\sigma_{\alpha}\otimes \sigma_{\beta}+
\sum_{(\alpha,\beta)\neq (0,0)} T_{1\alpha\beta}   ~\sigma_1 \otimes
\sigma_{\alpha}\otimes \sigma_{\beta}\Bigg].
\end{equation}
Conversely, in the situation where $k_{\rm max} = k_3$,  one finds
\begin{equation}\label{chi-matrix-i}
\chi^{(3)}_{1|23} = \frac{1}{8}\Bigg[ \sigma_{0}\otimes \sigma_{0} \otimes \sigma_{0}
+  T_{300} ~\sigma_3 \otimes \sigma_{0}\otimes \sigma_{0}$$
$$+ \sum_{(\alpha,\beta)\neq (0,0)} T_{0\alpha\beta}   ~\sigma_0 \otimes
\sigma_{\alpha}\otimes \sigma_{\beta}
+ \sum_{(\alpha,\beta)\neq (0,0)} T_{3\alpha\beta}   ~\sigma_3 \otimes
\sigma_{\alpha}\otimes \sigma_{\beta}\Bigg].
\end{equation}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$k$-qubit states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Based on the previous analysis, we now
 determine  the analytic  expression of the geometric
 discord in the $k$-qubit state (\ref{rho123k-trace}) when a bipartite splitting of
type  $1\vert 23\cdots k$ is considered. In this respect,
 we expand the density matrix $\rho_{12\cdots k}$ (\ref{fanorhor123k}) as
\begin{equation}\label{3X-fanok}
\rho_{12\cdots k}=\frac{1}{2^k}\bigg[T_{00 \cdots 0} ~\sigma_0\otimes \sigma_0
\cdots \otimes \sigma_0 + T_{30 \cdots 0}~\sigma_3\otimes
\sigma_0\otimes \cdots \sigma_0 $$
$$+  \sum_{(\alpha_2, \cdots \alpha_k)\neq (0,\cdots, 0)} T_{0 \alpha_2 \cdots
\alpha_k}~ \sigma_{0}\otimes \sigma_{\alpha_2}\otimes \cdots
\sigma_{\alpha_k} +  \sum_i \sum _{(\alpha_2, \cdots \alpha_k)\neq (0,\cdots, 0)}
T_{i \alpha_2 \cdots \alpha_k} ~\sigma_{i}\otimes \sigma_{\alpha_2}\otimes \cdots
\sigma_{\alpha_k}\bigg]
\end{equation}
in terms of the non vanishing correlations coefficients $T_{\alpha_1 \alpha_2 \cdots \alpha_k}$. Any  $k$-qubit state
having zero discord is necessarily of the form
\begin{equation}\label{chik}
\chi_{1|23\cdots k} = p_1 \vert \psi_1 \rangle \langle \psi_1 \vert \otimes
\rho_1^{23\cdots k} + p_2 \vert \psi_2 \rangle \langle \psi_2 \vert \otimes
\rho_2^{23\cdots k}
\end{equation}
where $\{ \vert \psi_1 \rangle , \vert \psi_2 \rangle \}$ is an orthonormal basis
related to the qubit $1$. The density matrices  $\rho_i^{23\cdots k}$ ($i=1,2$),
corresponding to the subsystem $(23\cdots k)$ composed of $(k-1)$ qubits, write as
$$ \rho_i^{23\cdots k } = \frac{1}{2^{k-1}} \bigg[  \sum_{\alpha_2, \cdots \alpha_k}
{\rm Tr}(\rho_i^{23\cdots k}
\sigma_{\alpha_2}\otimes \cdots \sigma_{\alpha_k})\sigma_{\alpha_2}\otimes \cdots
\sigma_{\alpha_k}\bigg].$$
To examine the pairwise quantum correlations in the states (\ref{rho123k-trace}),
the appropriate form for  the classical state (\ref{chik}) is
\begin{equation}\label{chi-matrixk}
\chi_{1|23\cdots k} = \frac{1}{2^k}\Bigg[ \sigma_{0}\otimes \sigma_{0} \cdots
\otimes \sigma_{0}
+ \sum_{i=1}^3 te_i ~~\sigma_i \otimes \sigma_{0}\cdots \otimes \sigma_{0} $$
$$+ \sum_{(\alpha_2,\cdots \alpha_k)\neq (0,\cdots, 0)} (s_+)_{\alpha_2,\cdots,
\alpha_k} ~\sigma_0 \otimes \sigma_{\alpha_2}\otimes \cdots \sigma_{\alpha_k}
+ \sum_{i=1}^3  \sum_{(\alpha_2,\cdots \alpha_k)\neq (0,\cdots, 0)}  e_i
(s_-)_{\alpha_2,\cdots, \alpha_k} ~\sigma_i \otimes \sigma_{\alpha_2}\otimes \cdots
\sigma_{\alpha_k}\Bigg]
\end{equation}
where
$$ t = p_1-p_2 \qquad e_i = \langle \psi_1 \vert \sigma_i \vert \psi_1 \rangle  \qquad
(s_{\pm})_{\alpha_2,\cdots, \alpha_k} = {\rm Tr}\big((p_1\rho_1^{23\cdots k} \pm
p_2\rho_2^{23\cdots k}) \sigma_{\alpha_2}\otimes \cdots \sigma_{\alpha_k}\big).$$
The Hilbert-Schmidt distance between the state $\rho_{123\cdots k}$
(\ref{3X-fanok}) and a classical state of the from  (\ref{chi-matrixk}) is given by
the following expression
\begin{equation}\label{HSk}
|| \rho_{1|23\cdots k} - \chi_{1|23\cdots k}||^2 = \frac{1}{2^k} \bigg[  (t^2
-2te_3T_{30 \cdots 0} + T^2_{30\cdots 0}) +  \sum_{(\alpha_2,\cdots \alpha_k)\neq
(0,\cdots, 0)}
(T_{0\alpha_2 \cdots \alpha_k} -  (s_+)_{\alpha_2,\cdots, \alpha_k})^2$$
$$ + \sum_{i=1}^3  \sum_{(\alpha_2,\cdots \alpha_k)\neq (0,\cdots, 0)} (T_{i\alpha_2
\cdots \alpha_k} -  e_i(s_-)_{\alpha_2,\cdots, \alpha_k})^2  \bigg]
\end{equation}
which must be minimized with  respect to the parameters $t$, $e_i$ ($i=1,2,3$) and
$(s_{\pm})_{\alpha_2,\cdots, \alpha_k}$. Setting the partial derivatives of (\ref{HSk}), with
respect to the parameters $t$ and $(s_{\pm})_{\alpha_2,\cdots, \alpha_k}$,  equal to
zero, we get
\begin{equation}\label{soluk}
t = e_3 T_{30\cdots 0}  \qquad (s_{+})_{\alpha_2,\cdots, \alpha_k} =
T_{0\alpha_2\cdots \alpha_k}   \qquad
(s_{-})_{\alpha_2,\cdots, \alpha_k} = \sum_{i=1}^3 e_i T_{i\alpha_2 \cdots \alpha_k}.
\end{equation}
In particular, these solutions reduce to equations (\ref{minHS12}) and (\ref{solu}) for  $k=2$ and $k=3$ respectively.
Reporting the conditions (\ref{soluk}) in the expression (\ref{HSk}), one obtains
\begin{equation}\label{HSkk}
|| \rho_{1|23\cdots k} - \chi_{1|23\cdots k}||^2 = \frac{1}{2^k} \bigg[
T^2_{30\cdots 0} - e^2_3T^2_{30\cdots 0}
+ \sum_{i=1}^3  \sum_{(\alpha_2,\cdots, \alpha_k)\neq (0,0)} T^2_{i\alpha_2 \cdots
\alpha_k}$$
$$ - \sum_{i,j=1}^3  \sum_{(\alpha_2,\cdots, \alpha_k)\neq (0,\cdots, 0)}
e_ie_jT_{i\alpha_2 \cdots \alpha_k}T_{j\alpha_2 \cdots \alpha_k}\Bigg].
\end{equation}
 After some algebra, the
distance  (\ref{HSkk}) takes the following compact form
\begin{equation}\label{HSkkk}
|| \rho_{1|23\cdots k} - \chi_{1|23\cdots k}||^2 = \frac{1}{2^k} \big[  ||x||^2 +
||T||^2 -\vec{e}(xx^t + TT^t)\vec{e}~^t\big]
\end{equation}
in terms of the   $1\times 3$ matrix defined by
\begin{equation}\label{x123k}
x^t=(0,0,T_{30\cdots 0})
\end{equation}
and the $3\times (4^{k-1}-1)$ matrix given by
\begin{equation}\label{T123k}
T = (T_{i\alpha_2 \cdots  \alpha_k}) ~~~ {\rm with}~~~~ (\alpha_2,\cdots, \alpha_k)
\neq (0,\cdots, 0)
\end{equation}
which are the extended versions of the matrices (\ref{x123}) and (\ref{T123})
introduced for $k=3$. The equation (\ref{HSkkk}) suggests that the pairwise quantum correlation is completely characterized  by
the eigenvalues of the matrix
\begin{equation}\label{MatrixKk}
K = xx^t + TT^t.
\end{equation}
The analytical expression  of these eigenvalues for an arbitrary multi-qubit
state constitutes a very complex task. However, this complexity is considerably
reduced for  the states $\rho_{12\cdots k}$ by exploiting their
parity symmetry (i.e. commutes with $\sigma_3 \otimes \sigma_3 \cdots
\sigma_3$). This implies that the matrix $T$ (\ref{T123k})  writes formally  as
$$ T = \sum_{\alpha_2,\cdots, \alpha_k} ( T_{ 1 \alpha_2 \cdots \alpha_k} , T_{ 2
\alpha_2 \cdots \alpha_k} , 0 )^t +
\sum_{\alpha_2,\cdots, \alpha_k} ( 0, 0 , T_{ 3 \alpha_2 \cdots \alpha_k} )^t.$$
This form is more appropriate to show that the product $TT^t$ is  diagonal. The density matrix
$\rho_{12\cdots k}$ possesses also exchange symmetry.  Subsequently, since its elements  are all reals and in view of the recurrence relation
(\ref{recur-k}), the off-diagonal entries of the matrix $TT^t$ vanish. This result has been shown
already  for $k=2$ and $k=3$ and will be explicitly
proved hereafter for $k=4$.  It follows that  the matrix $K$ (\ref{MatrixKk})  is
diagonal
$$ K = {\rm diag} (k_1 , k_2 , k_3)$$
and the corresponding  eigenvalues are
$$ k_1 = \sum_{\alpha_2,\cdots, \alpha_k} T^2_{ 1 \alpha_2 \cdots \alpha_k}$$
$$  k_2 = \sum_{\alpha_2,\cdots, \alpha_k} T^2_{ 2 \alpha_2 \cdots \alpha_k} $$
$$ k_3 = T^2_{30\cdots 0} + \sum_{\alpha_2,\cdots, \alpha_k\neq 0} T^2_{ 3 \alpha_2
\cdots  \alpha_k}.$$
They can be explicitly expressed in terms of the overlap $p$ (\ref{overlap}) like the special cases $k=2$
and $k=3$. This requires the expressions of $k$-components tensor correlations $T_{ \alpha_1 \alpha_2 \cdots \alpha_k}$ in terms of those involving two
qubits. This recursive procedure requires  tedious  but feasible calculations.
To exemplify this method, we consider the situation where $k=4$. In this case,
the $3\times 63$ matrix elements
of $T$ defined by (\ref{T123k}) can be explicitly derived  using the equation
(\ref{recu-4}). A straightforward but lengthy computation  shows that the $3 \times
3$ matrix $K$ is diagonal
and the corresponding  eigenvalues are
\begin{equation}\label{k1-4}
k_1 =  \sum_{k = 0,3 }\sum_{i=1,2} \sum_{j=0,3} T^2_{1kji} + \sum_{k =
1,2}\sum_{j=1,2} \sum_{i=1,2} T^2_{1kji}$$
$$ +\sum_{k = 0,3 }\sum_{j=1,2} \sum_{i=0,3} T^2_{1kji} + \sum_{k = 1,2}\sum_{j=0,3}
\sum_{i=0,3} T^2_{1kji},
\end{equation}
\begin{equation}\label{k2-4}
k_2 =  \sum_{k = 0,3 }\sum_{i=1,2} \sum_{j=0,3} T^2_{2kji} + \sum_{k =
1,2}\sum_{j=1,2} \sum_{i=1,2} T^2_{2kji}$$
$$ + \sum_{k = 0,3 }\sum_{j=1,2} \sum_{i=0,3} T^2_{2kji} + \sum_{k =
1,2}\sum_{j=0,3} \sum_{i=0,3} T^2_{2kji},
\end{equation}
\begin{equation}\label{k3-4}
k_3 =   \sum_{k = 0,3 }\sum_{i=0,3} \sum_{j=0,3} T^2_{3kji} + \sum_{k =
1,2}\sum_{j=1,2} \sum_{i=0,3} T^2_{3kji}$$
$$ + \sum_{k = 0,3 }\sum_{j=1,2} \sum_{i=1,2} T^2_{3kji} + \sum_{k =1,2}\sum_{j=0,3}
\sum_{i=1,2} T^2_{3kji}.
\end{equation}
The expressions (\ref{k1-4}), (\ref{k2-4}) and (\ref{k3-4})  can be simplified
further. Indeed,  from the relations
(\ref{recur-k}),  which reproduce the expressions (\ref{recu-4}) for $k=4$, one obtains
\begin{equation}\label{k11-4}
k_1 = 2\bigg[ \sum_{k = 0,3 }\sum_{j=1,2} ( (T^{00}_{1kj})^2 + (T^{11}_{1jk})^2 ) +
\sum_{k =  1,2} \sum_{j=0,3} ( (T^{00}_{1kj})^2 + (T^{11}_{1jk})^2 )\bigg]$$
$$ +4 \bigg[\sum_{k = 0,3 } \sum_{j=0,3} T^{01}_{1kj}T^{10}_{1kj} + \sum_{k =
1,2}\sum_{j=1,2} T^{01}_{1kj}T^{10}_{1kj}\bigg],
\end{equation}
\begin{equation}\label{k21-4}
k_2 = 2\bigg[ \sum_{k = 0,3 }\sum_{j=1,2} ( (T^{00}_{2kj})^2 + (T^{11}_{2jk})^2 ) +
\sum_{k =  1,2} \sum_{j=0,3} ( (T^{00}_{2kj})^2 + (T^{11}_{2jk})^2 )\bigg]$$
$$ +4 \bigg[\sum_{k = 0,3 } \sum_{j=0,3} T^{01}_{2kj}T^{10}_{2kj} + \sum_{k =
1,2}\sum_{j=1,2} T^{01}_{2kj}T^{10}_{1kj}\bigg],
\end{equation}
\begin{equation}\label{k31-4}
k_3 = 2\bigg[ \sum_{k = 0,3 }\sum_{j=0,3} ( (T^{00}_{3kj})^2 + (T^{11}_{3jk})^2 ) +
\sum_{k = 1,2} \sum_{j=1,2} ( (T^{00}_{3kj})^2 + (T^{11}_{3jk})^2 )\bigg]$$
$$ +4 \bigg[\sum_{k = 0,3 } \sum_{j=1,2} T^{01}_{3kj}T^{10}_{3kj} + \sum_{k =
1,2}\sum_{j=0,3} T^{01}_{3kj}T^{10}_{3kj}\bigg],
\end{equation}
in terms of  the three qubit correlation elements $T^{kl}_{\alpha\beta\gamma}$
associated with the  density matrices $\rho^{kl}_{123}$ (\ref{rhokl123}). The second step consists in
expressing the 3-qubit correlation matrix elements $T^{kl}_{\alpha\beta\gamma}$ as expansion of
bipartite correlations
associated with two qubit subsystems $T^{kl}_{\alpha\beta}$. This is simply achieved
by  using the recurrence relations of type (\ref{relation3ret2r-class2}) and
(\ref{relation3ret2r-class22}) (modulo some obvious substitution). Finally, one has
\begin{equation}
k_1 = 16 {\cal N}^4 (1 - p^2)(1 - p^{6}),\label{lambda14}
\end{equation}
\begin{equation}
k_2 = 16 {\cal N}^4 (1 - p^2)(1 -
p^{6})p^{2(n-4)},\label{lambda24}
\end{equation}
\begin{equation}
k_3 = 16 {\cal N}^4 \bigg[(1 + p^{6})(p^2 + p^{2(n-4)}) + 4
 p^n \cos m\pi\bigg],\label{lambda34}
\end{equation}
and the pairwise quantum discord is
\begin{equation}
D_g(\rho_{1|234} ) = \frac{1}{4}~ {\rm min}\{ k_1 + k_2 ,
k_1 + k_3 , k_2 + k_3\}.
\end{equation}
This example illustrates that this hierarchical or recursive method, when carried
out properly, leads to the pairwise
quantum correlation between a single qubit and a $(k-1)$-qubits cluster  in the
$k$-qubit mixed states $\rho_{1\vert23\cdots k}$. But
one should recognize that it involves lengthy expressions and tedious calculations.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Pairwise encoding}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Another adequate approach to tackle the pairwise correlation in states of type
$\rho_{1\vert23\cdots k}$, consists in mapping
the physical states of the subsystem containing $(k-1)$-qubits  onto  two logical
qubits.  This idea was recently
considered in \cite{daoud2,DaoudPLA,DaoudIJQI} to examine the pairwise quantum
correlations in multi-qubit systems. So in this section, we shall consider the
scenario where the
information contained in  the  cluster of $(23\cdots k)$ is encoded in two logical
qubits
$\{\vert  0\rangle_{23\cdots k} ,\vert 1\rangle_{23\cdots k} \}$ defined by
\begin{equation}
\vert\eta, \eta, \cdots \eta \rangle \equiv  b_+ \vert  0 \rangle_{23\cdots k} + b_-
\vert 1 \rangle_{23\cdots k}
\qquad \vert -\eta, -\eta, \cdots, -\eta \rangle \equiv b_+ \vert 0
\rangle_{23\cdots k} - b_-
 \vert  1 \rangle_{23\cdots k} ~,\label{base23k}
\end{equation}
where $$b_{\pm} = \sqrt{\frac{1 \pm p^{k-1}}{2}} .$$
According to this splitting scheme, the density matrix  (\ref{rho123k-trace})
rewrites, in the basis
$\{ \vert 0 \rangle \otimes \vert  0\rangle_{23\cdots k}, \vert 0 \rangle \otimes
\vert  1\rangle_{23\cdots k}
,\vert 1 \rangle \otimes \vert  0\rangle_{23\cdots k},\vert 1 \rangle \otimes \vert
1\rangle_{23\cdots k} \}$, as
\begin{equation}
\rho_{1(23\cdots k)} = 2{\cal N}^2 \left( \begin{smallmatrix}
a_+^2b_+^2(1+q_k \cos m\pi)& 0
    & 0& a_+a_-b_+b_-(1+q_k\cos m\pi
)\\
0  & a_+^2b_-^2(1-q_k\cos m\pi ) & a_+a_-b_+b_-(1-q_k\cos
m\pi
) & 0 \\
0  & a_+a_-b_+b_-(1-q_k\cos m\pi ) & a_-^2b_+^2(1-q_k \cos
m\pi
) & 0 \\
a_+a_-b_+b_-(1+q_k \cos m\pi ) & 0 & 0 & a_-^2b_-^2(1+q_k \cos
m\pi)
\end{smallmatrix}
\right), \label{rho23k}
\end{equation}
or equivalently, in the Fano-Bloch representation,  as
\begin{equation}
\rho_{1(23\cdots k)} = \frac{1}{4}\sum_{\alpha \beta} R_{\alpha \beta}
\sigma_{\alpha}\otimes \sigma_{\beta}
\end{equation}
where the non vanishing matrix elements $R_{\alpha \beta}$ $(\alpha,
\beta = 0,1,2,3)$ are given by
$$ R_{00} = 1, \quad R_{11} = 2{\cal N}^2 \sqrt{(1- p^2)(1- p^{2(k-1)})}, \quad
R_{22} = -2{\cal N}^2 \sqrt{(1- p^2)(1- p^{2(k-1)})}~p^{n-k}\cos
m\pi,$$ $$ R_{33} = 2{\cal N}^2 (p^k + p^{n-k}\cos m\pi), \quad
R_{03} = 2{\cal N}^2 (p^{k-1} + p^{n-k+1}\cos m\pi), \quad R_{30} =
2{\cal N}^2 (p + p^{n-1}\cos m\pi).$$
Following the standard procedure, presented here above for a two qubit system,  the
geometric discord
is
\begin{equation}
D_{\rm g}(\rho_{1(23\cdots k)}) = \frac{1}{4}~ {\rm min}\{ l_1 + l_2 ,
l_1 + l_3 , l_2 + l_3\}.
\end{equation}
where
$$l_1 = R_{11}^2, \quad  l_2 = R_{22}^2, \quad l_3 = R_{30}^2 + R_{33}^2.$$
Explicitly, the quantities $l_1$, $l_2$ and $l_3$
 are given by
\begin{equation}
l_1 = 4 {\cal N}^4 (1 - p^2)(1 - p^{2(k-1)})\label{lambda1}
\end{equation}
\begin{equation}
l_2 = 4 {\cal N}^4 (1 - p^2)(1 -
p^{2(k-1)})p^{2(n-k)}\label{lambda2}
\end{equation}
\begin{equation}
l_3 = 4 {\cal N}^4 \bigg[(1 + p^{2(k-1)})(p^2 + p^{2(n-k)}) + 4
 p^n \cos m\pi\bigg]\label{lambda3}.
\end{equation}
Remarkably, the quantities  $(l_1, l_2, l_3)$ reduces (up to the overall
multiplicative factor $2^{k-2}$) to
\big((\ref{GD12lambda1}),(\ref{GD12lambda2}),(\ref{GD12lambda3})\big)
for $k=2$, \big((\ref{L11}),(\ref{L22}),(\ref{L33})\big) for $k=3$ and
\big((\ref{lambda14}),(\ref{lambda24}),(\ref{lambda34})\big)  for $k=4$. Moreover, we
have
\begin{equation}
D_{\rm g}(\rho_{1(23\cdots k)}) =  \frac{1}{2^{k-2}} D_{\rm g}(\rho_{1\vert 23\cdots
k}).
\end{equation}
It turns out that the two bi-partitioning schemes, discussed in this paper, characterize equivalently the
pairwise geometric quantum discord in multi-partite Shr\"odinger cat states (\ref{ncs}).



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

In this paper, we developed a general method to evaluate
the pairwise geometric discord
in a mixed multi-qubit states. We especially focused on reduced density matrices $\rho_{123\cdots k}$
obtained by a trace procedure from a balanced  superpositions of
 $n$-qubit states possessing
parity invariance and exchange symmetry. Closed analytical expressions for the geometric quantum discord based
on Hilbert-Schmidt distance were derived. Two  splitting scenarios were discussed. In the first
one,  a recursive algorithm is proposed  to determine
explicitly the pairwise geometric discord between the first qubit and
 the remaining $(k-1)$ qubits.  For the Shr\"odinger cat sates considered in this work, the
 parity and exchange symmetries offer enormous advantages
in determining the geometric measure of quantum discord. In particular, a recursive prescription,
 expressing the multi-component correlations tensors in terms of two-qubit correlation matrices elements, is obtained.
 This constitutes the key ingredient
in deriving the pairwise geometric quantum discord. Another important issue we examined in this
 work concerns the explicit derivation  of classical (zero discord) states. We also
 presented a second splitting scheme according to which a $(k-1)$-qubit cluster is viewed as single qubit
 so that the whole  $k$-qubit system is mapped into a two-qubit system. We have shown that in both bi-partitioning
 schemes, the Hilbert-Schmidt distance characterize  equivalently the bipartite quantum correlations.
  The second encoding scheme appears to be particularly convenient and more tractable
  from a computational point of view, contrarily to
  the first one which requires lengthy calculations. We believe that the results obtained in this work can be extended to other classes of
multi-qubit states. Also, they can used in evaluating
multipartite geometric quantum discord
 in the spirit of the results recently obtained in
\cite{Rulli,Ma}. Finally, another interesting issue,
that deserves a special attention, concerns the distribution and the so-called monogamy property  of geometric quantum
discord between the different components in multi-partite Shr\"odinger cat states. In this sense,  we believe that the results obtained in  this
work can be useful.





















\begin{thebibliography}{99}
\bibitem{Horodecki-RMP-2009} R. Horodecki, P. Horodecki, M. Horodecki and K.
Horodecki, Rev. Mod. Phys. {\bf 81}(2009) 865.

\bibitem{Guhne} O. G\"uhne and G. T\'oth, Phys. Rep. {\bf 474} (2009) 1.

\bibitem{vedral} K. Modi, A. Brodutch, H. Cable, T. Paterek and V. Vedral,
 Rev. Mod. Phys. {\bf 84} (2012) 1655.

\bibitem{NC-QIQC-2000} M.A. Nielsen and I.L. Chuang, {\it Quantum Computation and
Quantum Information} (Cambridge Univ. Press, Cambridge,
2000).

\bibitem{Vedral-RMP-2002} V. Vedral, Rev. Mod. Phys. {\bf 74} (2002)
197.

\bibitem{Bennett1} C.H. Bennett, G. Brassard, C. Cr\'epeau, R. Jozsa, A. Peres and
W.K. Wootters, Phys. Rev. Lett. {\bf 70} (1993) 1895.

\bibitem{Bennett2}  C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett. {\bf 69} (1992)
2881.

\bibitem{Ekert}   A.K. Ekert, Phys. Rev. Lett.
{\bf 67} (1991) 661.

\bibitem{Rungta}  P. Rungta, V. Buzek, C.M. Caves, M. Hillery and G.J. Milburn,
Phys. Rev. A {\bf 64} (2001) 042315.

\bibitem{Ben3}  C.H. Bennett, D.P. DiVincenzo, J. Smolin and W.K.
Wootters, Phys. Rev. {\bf A 54} (1996) 3824.

\bibitem{Wootters}  W.K. Wootters, Phys. Rev. Lett. {\bf 80} (1998) 2245.

\bibitem{Coffman}  V. Coffman, J. Kundu and W.K. Wootters, Phys. Rev.
A {\bf 61} (2000) 052306.

\bibitem{Vedral-et-al} L. Henderson and V. Vedral, J. Phys. A {\bf 34}(2001)  6899;
V. Vedral, Phys. Rev. Lett. {\bf 90}  (2003) 050401; J. Maziero, L.
C. Cel\'eri, R.M. Serra and V. Vedral, Phys. Rev A {\bf 80} (2009)
044102.

\bibitem{Ollivier-PRL88-2001} H. Ollivier and W.H. Zurek, Phys. Rev. Lett. {\bf 88}
(2001) 017901.

\bibitem{Dakic2010} B. Dakic, V. Vedral and C. Brukner, Phys. Rev. Lett. {\bf 105}
(2010) 190502.

\bibitem{Bellomo1}
B. Bellomo,  R. Lo Franco and G. Compagno, Phys. Rev. A 86, 012312 (2012).

\bibitem{Bellomo2}
B. Bellomo, G.L. Giorgi, F. Galve, R. Lo Franco, G. Compagno and R. Zambrini,
Phys. Rev. A {\bf 85} (2012) 032104.

\bibitem{Dajka} J. Dajka, J. Luczka and P. H\"anggi,
Phys. Rev. A {\bf 84} (2011) 032120.

\bibitem{Paula1} J.D. Montealegre, F.M. Paula, A. Saguia and M.S. Sarandy,
Phys. Rev. A {\bf 87} (2013) 042115.

\bibitem{Aaronson} B. Aaronson, R. Lo Franco, G. Compagno and G. Adesso,
New J. of Physics {\bf 15} (2013) 093022.

\bibitem{Paula2} F.M. Paula, T.R. de Oliveira and M.S. Sarandy,
Phys. Rev. A {\bf 87} (2013) 064101.

\bibitem{Rana} S. Rana and P. Parashar, Phys. Rev. A {\bf 85} (2012)  024102

\bibitem{Hassan} A.S.M. Hassan, B. Lari  and P. S. Joag, Phys. Rev. A {\bf 85}
(2012) 024302.

\bibitem{Rau} S.Vinjanampathy and A.R.P Rau, J. Phys. A: Math. Theor. {\bf 45}
(2012) 095303.

\bibitem{Zhou} J. Zhou and H. Guo, Phys. Rev. A {\bf 87} (2013) 062315.

\bibitem{solano} T. Bastin, S. Krins, P. Mathonet, M.  Godefroid, L.  Lamata and E.
Solano, Phys. Rev. Lett. {\bf 103} (2009) 070503.

\bibitem{mixed} T. Bastin, P.  Mathonet and E. Solano, Operational Entanglement
Families of Symmetric Mixed N-qubit states, quant-ph/arxiv:1011.1243

\bibitem{usa1}  A.R. Usha Devi, Sudha and A.K. Rajagopal,   Determining the whole
pure symmetric N-qubit state from its parts,  quant-ph/arXiv:1003.2450.

\bibitem{usa2} A.R. Usha Devi, Sudha and A.K. Rajagopal, Quantum Information
Processing {\bf 11} ( 2012 ) 685.

\bibitem{markham1} D. Markham, Phys. Rev. A {\bf 83} (2011)  042332.

\bibitem{markham2} M. Aulbach, D. Markham and M.  Murao, New. J. Phys., {\bf 12}
(2010) 073025.

\bibitem{gebastin} J. Martin, O. Giraud, P.A. Braun, D. Braun and T.  Bastin, Phys.
Rev. A {\bf 81}  (2010) 062347.

\bibitem{markham3} M. Aulbach, D. Markham and M.  Murao,  Lecture Notes in Computer
Science {\bf 6519}  (2011)  141.

\bibitem{Maj32}  E. Majorana, Nuovo Cimento {\bf 9} (1932) 43.

\bibitem{Dic54}  R.H. Dicke, Phys. Rev. {\bf 93} (1954) 99.

\bibitem{daoud2} M. Daoud, R. Ahl Laamara and W. Kaydi,
J. Phys. A: Math. Theor. {\bf 46} (2013) 395302.

\bibitem{Wineland} D. Leibfried, E. Knill, S. Seidelin, J. Britton, R.B. Blakestad,
J. Chiaverini, D.B. Hume, W.M. Itano, J.D. Jost, C. Langer, R.
Ozeri, R. Reichle and D.J. Wineland, Nature {\bf 438} (2005) 639.

\bibitem{Haroche} M. Brune, S. Haroche, J.M. Raimond, L. Davidovich and N. Zagury,
Phys. Rev. A {\bf 45} (1992) 5193

\bibitem{ghz} D. M. Greenberger, M.A.  Horne, A. Shimony and  A. Zeilinger, Am. J.
Phys. {\bf 58} (1990) 1131.

\bibitem{Dur00} W. D\"ur, G. Vidal and J.I. Cirac, Phys. Rev. A
{\bf 62} (2000) 062314.

\bibitem{Fanchini2}  F.F. Fanchini, M.C. de Oliveira, L.K. Castelano  and M.F.
Cornelio, Phys. Rev. A {\bf 87} (2013) 032317.

\bibitem{Modi} K. Modi, T. Paterek, W. Son, V. Vedral and M. Williamson,
Phys. Rev. Lett. {\bf 104} ( 2010 ) 080501.

\bibitem{DaoudPLA} M. Daoud and R. Ahl Laamara, Phys. Lett. A {\bf 376} (2012) 2361.

\bibitem{DaoudIJQI} M. Daoud and R. Ahl Laamara, International Journal of Quantum
Information {\bf 10} (2012) 1250060.

\bibitem{Rulli} C.C. Rulli and M.S. Sarandy, Phys. Rev. A {\bf 84} (2011) 042109.

\bibitem{Ma} Z-H Ma, Z-H Chen and F.F. Fanchini, New J. Phys. {\bf 15} (2013) 043023.

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



\end{thebibliography}

\end{document}










