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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 {\large {\bf Pairwise local quantum uncertainty in superpositions of Dicke states }}\\
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{1.5cm} {\bf xxxxxx}$^{a,b,c}${\footnote { email: {\sf
m$_{-}$daoud@hotmail.com}}}, {\bf yyyyyy}$^{d,e}$ {\footnote {
email: {\sf ahllaamara@gmail.com}}}  and {\bf zzzzzz}$^{d}$
{\footnote { email: {\sf ????????}}} \\
\vspace{0.5cm}
$^{a}${\it Department of Physics, Faculty of Sciences-Ain Chock, University Hassan II Casablanca, Morocco}\\[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}
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\medskip


We give a closed expression of local quantum uncertainty for
two-qubit $X$ states. A detailed analysis is performed for multi-partite spin systems for which we give
the pairwise local quantum uncertainty.





\vspace{1cm}








\newpage
\section{Introduction}




Quantum correlations in multipartite systems are a fundamental
resource in various protocols of quantum information processing  \cite{NC-QIQC-2000,Vedral-RMP-2002,Horodecki-RMP-2009,Guhne}. In
this respect, the characterization the degree of  quantumness  of
correlations between the different parts of a composite system is
highly desirable. During the last two decades, several quantifiers
were investigated in the literature (for a recent review see
\cite{vedral}). The most utilized
ones to capture the main features of quantum correlations are the
concurrence, entanglement of formation, quantum discord and its
different geometric versions \cite{Rungta,Ben3,Wootters,Coffman,Vedral-et-al,Ollivier-PRL88-2001}. The interest in quantum discord lies
in the existence of nonclassical correlations even in separable
states \cite{Vedral-et-al,Ollivier-PRL88-2001}. The entanglement of formation  does
not account for all nonclassical aspects of correlations and
unentangled mixed states can possess quantum correlations. This
explains the important number of works devoted to the significance
and the computation of quantum discord in different quantum systems.
However, the derivation of the explicit expression of quantum
discord of an arbitrary quantum system is very challenging in
general. An alternative way to overcome this problem consists in
utilizing geometric methods to quantify the distance between a
bipartite state and its closed one encompassing only classical
correlations \cite{Dakic2010,Bellomo1,Bellomo2}
(see also \cite{Dajka,Paula1,Aaronson,Paula2}).


Recently, it was shown that the concept of quantum discord is deeply
related to the formalism of quantum uncertainty on local
observables. This kind of quantum correlations quantifier  uses the
skew information, introduced in Ref. \cite{wigner}, to determine the
uncertainty in the measurement of an observable. More precisely, the
local quantum uncertainty is given by the minimum of the skew
information over all possible observables. This measure leads to the
closed analytical expression characterizing the quantum correlations
for any qubit-qudit bipartite system \cite{girolami}.  It must be
noticed that this quantum correlation measure is related to quantum
Fisher information \cite{luo,petz,luo2} which is the key ingredient
in estimating precisions  in quantum metrology
protocols~\cite{girolami}. It also quantifies the speed of
evolution of a quantum system \cite{girolami}.


Taking advantages from the recent progress in understanding this
kind of quantum correlations quantifier, we give in this paper the
explicit analytical expressions of local quantum uncertainty for a
two qubit $X$ states including some special states for which the
local quantum uncertainty was recently derived
\cite{Karimipour,Sen}.  As illustration we consider
the pairwise quantum correlation in multipartite symmetric qubit
states.






The paper is structured as follows. In the following section, we introduce
the concept of local quantum uncertainty. The explicit anlaytical for of this  measure of discord-like
quantum correlations is derived for an arbitrary two-qubit $X$ states.  To illustrate our purpose, we consider collective
$n$ qubit systems  possessing
parity and exchange symmetries.  A special attention is devoted to the evaluation of pairwise
local quantum uncertainty in Dicke states, even and odd spin coherent states and finally Kitagawa-Ueda system.
 Concluding remarks close this
paper.

\section{Local quantum uncertainty in two qubit $X$ states}


\subsection{Local quantum uncertainty: Definition}

The concept of local quantum uncertainty is considered as a faithful
quantifier of quantum correlation. It quantifies the minimal quantum
uncertainty in a quantum state due to a measurement of a local
observable \cite{girolami}. For a quantum state $\rho_{AB}$, the
local quantum uncertainty is defined as
\begin{equation}
\mathcal{U}(\rho_{AB}) \equiv \min_{K} \mathcal{I}(\rho_{AB}, K \otimes
I), \label{LQU}
\end{equation}
where $K$ is some local observable on the subsystem $A$ and the quantity
\begin{equation}
\mathcal{I}(\rho_{AB},  K)=-\frac{1}{2}{\rm
Tr}([\sqrt{\rho_{AB}},K]^{2})
\end{equation}
 is the skew information \cite{wigner,luo}. The skew information
represents the non-commutativity between the state and the
observable $K$.
The analytical evaluation the local quantum
uncertainty requires a
 minimization procedure  over the set of all
observales acting on the part $A$. A closed form for qubit-qudit
systems was derived in \cite{girolami}. In particular, for
qubits (spin-$\frac{1}{2}$ particles),  it is given by  \cite{girolami}
\begin{equation}
 \mathcal{U}(\rho) = 1 - \lambda_{max}\{W\},
\end{equation}
where  $\lambda_{max}$ denotes the maximal  eigenvalue of the matrix
$W$ whose matrix elements are given by
\begin{equation}\label{matrixomega}
 (W)_{ij} \equiv  {\rm
Tr}\{\sqrt{\rho_{AB}}(\sigma_{i}\otimes
{\bf I})\sqrt{\rho_{AB}}(\sigma_{j}\otimes {\bf I})\},
\end{equation}
The local quantum uncertainty is zero when the observable and the
state of the system commute. This  indicates the absence of any
quantum correlation between the components of the bipartite system.
The local quantum uncertainty can be considered  as quantum discord
measure. In this sense, it satisfies all necessary requirements.
Indeed, local quantum uncertainty vanishes for all zero quantum discord states. It
is also invariant under local unitary operations on un-measured
qubit. In the following we derive the explicit form of local quantum
uncertainty in  two-qubit $X$ states.

\subsection{Local quantum uncertainty for $X$ states}

We consider a two-qubit system which is described by an $X$-shaped
mixed state.  Two-qubit $X$ states include various types of quantum
states usually used in investigating entanglement and quantum
correlations in various condensed matter models such ones describing
spin collective systems. In the computational basis of a two qubit
system,  the $X$ density matrices have non-zero entries only along
the diagonal and anti-diagonal and therefore they are parameterized
by seven real parameters. The corresponding symmetry is fully
characterized by the $su(2) \times su(2) \times u(1)$ subalgebra of
the full $su(4)$ algebra describing an arbitrary two-qubit system.
Many analytical calculations of concurrence, entanglement of
formation, quantum discord  can be carried  out easily for $X$
states leading to interesting results in studying their properties
and especially their evolution under dissipative processes
\cite{ref6,ref7}. The density matrix for a two-qubit $X$ state writes as
\begin{equation}
\rho =  \left(
\begin{array}{cccc}
\rho_{11} & 0 & 0 & \rho_{14} \\
0 & \rho_{22} & \rho_{23} & 0 \\
0 & \rho_{32} & \rho_{33} & 0 \\
\rho_{41} & 0 & 0 & \rho_{44}
\end{array}
\right). \label{eqn1}
\end{equation}
in the computational basis $\{ \vert 00 \rangle, \vert 01\rangle,
\vert 10\rangle, \vert 11\rangle\}$. The entries are subjected to
the normalization property (${\rm Tr} \rho= 1$), the
positivity condition ($\rho_{11}\rho_{44} \geq \vert \rho_{14}
\vert^2$ and $\rho_{22}\rho_{33} \geq \vert \rho_{23} \vert^2$) and
 the complex conjugation requirement  ($\rho_{14} = \rho^{\star}_{14}$ and $\rho_{23} = \rho^{\star}_{32}$ ). The phase factors
$e^{i\theta_{14}}= \frac{\rho_{14}}{\vert \rho_{14}\vert }$ and
$e^{i\theta_{23}}= \frac{\rho_{23}}{\vert \rho_{23}\vert }$
 of the off diagonal elements can be removed using the local unitary
transformations
$$ \vert 0 \rangle_1 \rightarrow \exp \bigg(-\frac{i}{2} ( \theta_{14}+ \theta_{23})\bigg) \vert 0 \rangle_1 \quad  \vert 0 \rangle_2 \rightarrow \exp \bigg(-\frac{i}{2} (\theta_{14}-\theta_{23} )\bigg) \vert 0 \rangle_2$$
 and  the
rank of the density matrix $\rho$ remains unchanged. Thus, all
entries of the density matrix can be taken positive.  The
eigenvalues of the density matrix $\rho$ write
$$\lambda_{1} = \frac{1}{2} t_1 + \frac{1}{2} \sqrt{t_1^2 - 4 d_1}, \quad\lambda_{2} = \frac{1}{2} t_2 + \frac{1}{2} \sqrt{t_2^2 - 4 d_2},
\quad \lambda_{3} = \frac{1}{2} t_2 - \frac{1}{2} \sqrt{t_2^2 - 4
d_2}, \quad\lambda_{4} = \frac{1}{2} t_1 - \frac{1}{2} \sqrt{t_1^2 -
4 d_1} $$ with $ t_1 = \rho_{11}  + \rho_{44}, \quad t_2= \rho_{22}
+ \rho_{33},\quad d_1 = \rho_{11}\rho_{44} - \rho_{14}\rho_{41},
{\rm and}\quad d_2= \rho_{22}\rho_{33}- \rho_{32}\rho_{23} $. The
Fano-Bloch decomposition of the state $\rho$ writes as
\begin{equation}\label{fano-bloch-rho}
\rho = \frac{1}{4}\sum_{\alpha, \beta} {R}_{\alpha \beta} \sigma_{\alpha} \otimes \sigma_{\beta}
\end{equation}
where the correlation matrix ${R}_{\alpha \beta}$ are given by $ {
R}_{\alpha \beta} = {\rm Tr} (\sqrt{\rho}~\sigma_{\alpha} \otimes
\sigma_{\beta} )$. They write
%$$ R_{03} = 1 -2  \rho_{22} - 2\rho_{44},\quad  R_{30} =    1-  2\rho_{33} - 2\rho_{44},\quad  R_{11} = 2~{\rm Re}(\rho_{32} + \rho_{41}),\quad  R_{22} =  2~{\rm Re}(\rho_{32} - \rho_{41})$$
%$$ R_{12} = -2i~{\rm Im}(\rho_{41} - \rho_{32})\quad  R_{21} =  -2i~{\rm Im}(\rho_{41} + \rho_{32}),\quad R_{00} = \rho_{11}+\rho_{22}+\rho_{33}+\rho_{44}= 1 ,\quad  R_{33} =    1-  2\rho_{22} - 2\rho_{33}.$$
$$ R_{03} = 1 -2  \rho_{22} - 2\rho_{44},\quad  R_{30} =    1-  2\rho_{33} - 2\rho_{44},\quad  R_{11} = 2~{\rm Re}(\rho_{32} + \rho_{41}),$$
$$  R_{22} =  2~{\rm Re}(\rho_{32} - \rho_{41}) ,\quad R_{00} = \rho_{11}+\rho_{22}+\rho_{33}+\rho_{44}= 1 ,\quad  R_{33} =    1-  2\rho_{22} - 2\rho_{33}.$$
For nonzero values of $t_1$ and $t_2$, it is simple to check that
the square root of the density matrix $\rho$ writes
\begin{equation}
\sqrt{\rho} =  \left(
\begin{array}{cccc}
\frac{\rho_{11}+\sqrt{d_1}}{\sqrt{t_1 + 2\sqrt{d_1}}} & 0 & 0 & \frac{\rho_{14}}{\sqrt{t_1 + 2\sqrt{d_1}}}\\
0 & \frac{\rho_{22}+\sqrt{d_2}}{\sqrt{t_2 + 2\sqrt{d_2}}} & \frac{\rho_{23}}{\sqrt{t_2 + 2\sqrt{d_2}}} & 0 \\
0 & \frac{\rho_{32}}{\sqrt{t_2 + 2\sqrt{d_2}}} & \frac{\rho_{33}+\sqrt{d_2}}{\sqrt{t_2 + 2\sqrt{d_2}}} & 0 \\
\frac{\rho_{41}}{\sqrt{t_1 + 2\sqrt{d_1}}} & 0 & 0 &
\frac{\rho_{44}+\sqrt{d_1}}{\sqrt{t_1 + 2\sqrt{d_1}}}
\end{array}
\right). \label{sqrtrho}
\end{equation}
The eigenvalues $\sqrt{\lambda_{1}}$, $\sqrt{\lambda_{2}}$,
$\sqrt{\lambda_{3}}$ and $\sqrt{\lambda_{4}}$ of the matrix
$\sqrt{\rho} $ can be rewritten as
$$\sqrt{\lambda_{1}} = \frac{1}{2}  \sqrt{t_1 + 2 \sqrt{d_1}}   + \frac{1}{2} \sqrt{t_1 - 2 \sqrt{d_1}}, \qquad \sqrt{\lambda_{2}} = \frac{1}{2} \sqrt{t_2 + 2 \sqrt{d_2}}  + \frac{1}{2} \sqrt{t_2 - 2 \sqrt{d_2}} $$
$$\sqrt{\lambda_{3}}  = \frac{1}{2} \sqrt{t_2 + 2 \sqrt{d_2}} - \frac{1}{2} \sqrt{t_2 - 2 \sqrt{d_2}}, \qquad \sqrt{\lambda_{4}} = \frac{1}{2} \sqrt{t_1 + 2 \sqrt{d_1}} - \frac{1}{2} \sqrt{t_1 - 2 \sqrt{d_1}} $$
In Fano-Bloch representation, the matrix $\sqrt{\rho}$ writes as
$$ \sqrt{\rho} = \frac{1}{4}\sum_{\alpha, \beta} {\cal R}_{\alpha \beta} \sigma_{\alpha} \otimes \sigma_{\beta}$$
with $ {\cal R}_{\alpha \beta} = {\rm Tr}
(\sqrt{\rho}~\sigma_{\alpha} \otimes \sigma_{\beta} )$. The non
vanishing matrix correlation elements ${\cal R}_{\alpha \beta}$ are
explicitly given by
$$  {\cal R}_{00} =  \sqrt{t_1 + 2 \sqrt{d_1}}  +  \sqrt{t_2 + 2 \sqrt{d_2}}   \qquad  {\cal R}_{03} =   \frac{1}{2} \frac{R_{30}  + R_{03}  }{\sqrt{t_1 + 2\sqrt{d_1}}} -  \frac{1}{2} \frac{R_{30}- R_{03} }{\sqrt{t_2 + 2\sqrt{d_2}}}$$
$$  {\cal R}_{30} = \frac{1}{2} \frac{R_{30}  + R_{03}  }{\sqrt{t_1 + 2\sqrt{d_1}}} + \frac{1}{2} \frac{R_{30}- R_{03} }{\sqrt{t_2 + 2\sqrt{d_2}}} \qquad {\cal R}_{11} = \frac{1}{2} \frac{R_{11}  + R_{22}  }{\sqrt{t_2 + 2\sqrt{d_2}}} +  \frac{1}{2} \frac{R_{11}- R_{22} }{\sqrt{t_1 + 2\sqrt{d_1}}}$$
%$$  {\cal R}_{12} =  \frac{1}{2} \frac{R_{12}  + R_{21}  }{\sqrt{t_1 + 2\sqrt{d_1}}} +  \frac{1}{2} \frac{R_{12}- R_{21} }{\sqrt{t_2 + 2\sqrt{d_2}}} \qquad  {\cal R}_{21} = \frac{1}{2} \frac{R_{12}  + R_{21}  }{\sqrt{t_1 + 2\sqrt{d_1}}} -  \frac{1}{2} \frac{R_{12}- R_{21} }{\sqrt{t_2 + 2\sqrt{d_2}}}$$
$$  {\cal R}_{22} = \frac{1}{2} \frac{R_{11}  + R_{22}  }{\sqrt{t_2 + 2\sqrt{d_2}}} -  \frac{1}{2} \frac{R_{11}- R_{22} }{\sqrt{t_1 + 2\sqrt{d_1}}} \qquad  {\cal R}_{33} =  \sqrt{t_1 + 2 \sqrt{d_1}}  -  \sqrt{t_2 + 2 \sqrt{d_2}} $$
At this stage, we have the tools to evaluate the matrix elements
defined by
$$ \omega_{ij} = {\rm Tr} \bigg(\sqrt{\rho}~(\sigma_{i} \otimes \sigma_{0}) \sqrt{\rho}~(\sigma_{j} \otimes \sigma_{0}) \bigg)$$
where $i$ and $j$ take the values 1, 2, 3. Using the following
identities
$$\{ \sigma_i , \sigma_j\} = 2\delta_{ij}\quad  {\rm Tr}(\sigma_i\sigma_j)= 2\delta_{ij} \quad {\rm Tr}(\sigma_i\sigma_j\sigma_k\sigma_l)=
2(\delta_{ij}\delta_{kl}- \delta_{ik}\delta_{jl}+
\delta_{il}\delta_{jk}),$$ one shows that the matrix $W$ is diagonal
and the diagonal elements are
\begin{eqnarray} \label{omegaii}
\omega_{ii} = \frac{1}{4} \bigg[ \sum_{\beta}\bigg( {\cal
R}^2_{0\beta} - \sum_k {\cal R}^2_{k\beta} \bigg) \bigg] +
\frac{1}{2}\sum_{\beta} {\cal R}^2_{i\beta}
 \end{eqnarray}
where $i=1,2,3$ and $\beta=0,1,2,3$. Explicitly, we have
%elements $ \omega_{ij}$ write
%$$ \omega_{ij} = \delta_{ij} \bigg[ \frac{1}{4}\sum_{\beta}\bigg( {\cal R}^2_{0\beta} - \sum_k {\cal R}^2_{k\beta} \bigg) \bigg]
%+ \frac{1}{2}\sum_{\beta} {\cal R}_{i\beta}{\cal R}_{j\beta}$$
%where $\beta=0,1,2,3$ and $k=1,2,3$. The diagonal elements are
%$$ \omega_{ii} = \frac{1}{4} \bigg[ \sum_{\beta}\bigg( {\cal R}^2_{0\beta} - \sum_k {\cal R}^2_{k\beta} \bigg) \bigg]
%+ \frac{1}{2}\sum_{\beta} {\cal R}^2_{i\beta}$$
%and the off-diagonal elements are
%$$ \omega_{ij} = \frac{1}{2}\sum_{\beta} {\cal R}_{i\beta} {\cal R}_{j\beta} \quad i\neq j$$
%Explicitly, we have
\begin{eqnarray} \label{omega11}
 \omega_{11}=  \frac{1}{4}\bigg[ 4 \bigg(\sqrt{\lambda_1} + \sqrt{\lambda_4} \bigg)\bigg(\sqrt{\lambda_2} + \sqrt{\lambda_3} \bigg) +
\frac{(R^2_{11}-R^2_{22})+(R^2_{03}-R^2_{30})}{(\sqrt{\lambda_1} +
\sqrt{\lambda_4} )(\sqrt{\lambda_2} + \sqrt{\lambda_3} )} \bigg]
 \end{eqnarray}
%\begin{eqnarray} \label{omega11}
% \omega_{11}=  \frac{1}{4}\bigg[ 4 \bigg(\sqrt{\lambda_1} + \sqrt{\lambda_4} \bigg)\bigg(\sqrt{\lambda_2} + \sqrt{\lambda_3} \bigg) +
%\frac{(R^2_{11}-R^2_{22})+(R^2_{12}-R^2_{21})+(R^2_{03}-R^2_{30})}{(\sqrt{\lambda_1}
%+ \sqrt{\lambda_4} )(\sqrt{\lambda_2} + \sqrt{\lambda_3} )} \bigg]
% \end{eqnarray}
\begin{eqnarray}\label{omega22}
\omega_{22}=  \frac{1}{4}\bigg[ 4 \bigg(\sqrt{\lambda_1} +
\sqrt{\lambda_4} \bigg)\bigg(\sqrt{\lambda_2} + \sqrt{\lambda_3}
\bigg) +
\frac{(R^2_{22}-R^2_{11})+(R^2_{03}-R^2_{30})}{(\sqrt{\lambda_1} +
\sqrt{\lambda_4} )(\sqrt{\lambda_2} + \sqrt{\lambda_3} )} \bigg]
 \end{eqnarray}
%\begin{eqnarray}\label{omega22}
%\omega_{22}=  \frac{1}{4}\bigg[ 4 \bigg(\sqrt{\lambda_1} +
%\sqrt{\lambda_4} \bigg)\bigg(\sqrt{\lambda_2} + \sqrt{\lambda_3}
%\bigg) +
%\frac{(R^2_{22}-R^2_{11})+(R^2_{21}-R^2_{12})+(R^2_{03}-R^2_{30})}{(\sqrt{\lambda_1}
%+ \sqrt{\lambda_4} )(\sqrt{\lambda_2} + \sqrt{\lambda_3} )} \bigg]
% \end{eqnarray}
\begin{eqnarray}\label{omega33}
\omega_{33} &=& \frac{1}{2}\bigg[  \bigg(\sqrt{\lambda_1} +
\sqrt{\lambda_4} \bigg)^2 + \bigg(\sqrt{\lambda_2} +
\sqrt{\lambda_3} \bigg)^2 \bigg]
 + \frac{1}{8}\bigg[
\frac{(R_{03}+R_{30})^2-(R_{11}-R_{22})^2}{\bigg(\sqrt{\lambda_1} +
\sqrt{\lambda_4} \bigg)^2}\bigg]\nonumber
\\ && +\frac{1}{8}\bigg[
\frac{(R_{03}-R_{30})^2-(R_{11}+R_{22})^2}{\bigg(\sqrt{\lambda_2} +
\sqrt{\lambda_3} \bigg)^2 }\bigg]
 \end{eqnarray}
%\begin{eqnarray}\label{Omega33}
%\omega_{33} &=& \frac{1}{2}\bigg[  \bigg(\sqrt{\lambda_1} +
%\sqrt{\lambda_4} \bigg)^2 + \bigg(\sqrt{\lambda_2} +
%\sqrt{\lambda_3} \bigg)^2 \bigg]
% + \frac{1}{8}\bigg[
%\frac{(R_{03}+R_{30})^2-(R_{11}-R_{22})^2-(R_{12}+R_{21})^2}{\bigg(\sqrt{\lambda_1}
%+ \sqrt{\lambda_4} \bigg)^2}\bigg]\nonumber
%\\ && +\frac{1}{8}\bigg[
%\frac{(R_{03}-R_{30})^2-(R_{11}+R_{22})^2-(R_{12}-R_{21})^2}{\bigg(\sqrt{\lambda_2}
%+ \sqrt{\lambda_3} \bigg)^2 }\bigg]
% \end{eqnarray}
%\begin{eqnarray}\label{Omega12}
% \omega_{12} =  \omega_{21} =  \frac{1}{2}\bigg({\cal R}_{11} {\cal R}_{21}+ {\cal R}_{12} {\cal R}_{22} \bigg)=
%\frac{1}{2} \frac{R_{11}R_{21}+ R_{22}R_{12}}{(\sqrt{\lambda_1} +
%\sqrt{\lambda_4})(\sqrt{\lambda_2} + \sqrt{\lambda_3})}
%\end{eqnarray}
%\begin{eqnarray}\label{Omega13}
%\omega_{13} =  \omega_{31} = 0 ,\quad \omega_{23} =  \omega_{32} = 0
%\end{eqnarray}


Clearly, the above results were derived for $X$ states satisfying
$t_1\neq 0$ and $t_2\neq 0$. The situation $t_1 = 0$ occurs when
$\rho_{11} = \rho_{44} = 0$ and the positivity condition of  $\rho$ (\ref{eqn1}) implies $ \rho_{14} =
0$. In this case the correlation matrix elements of the matrix $\sqrt{\rho}$ write simply as

$$  {\cal R}_{00} =   \sqrt{t_2 + 2 \sqrt{d_2}}   \qquad  {\cal R}_{03} =   -  \frac{1}{2} \frac{R_{30}- R_{03} }{\sqrt{t_2 + 2\sqrt{d_2}}}$$
$$  {\cal R}_{30} =  \frac{1}{2} \frac{R_{30}- R_{03} }{\sqrt{t_2 + 2\sqrt{d_2}}} \qquad {\cal R}_{11} = \frac{1}{2} \frac{R_{11}  + R_{22}  }{\sqrt{t_2 + 2\sqrt{d_2}}} $$
%$$  {\cal R}_{12} =  \frac{1}{2} \frac{R_{12}  + R_{21}  }{\sqrt{t_1 + 2\sqrt{d_1}}} +  \frac{1}{2} \frac{R_{12}- R_{21} }{\sqrt{t_2 + 2\sqrt{d_2}}} \qquad  {\cal R}_{21} = \frac{1}{2} \frac{R_{12}  + R_{21}  }{\sqrt{t_1 + 2\sqrt{d_1}}} -  \frac{1}{2} \frac{R_{12}- R_{21} }{\sqrt{t_2 + 2\sqrt{d_2}}}$$
$$  {\cal R}_{22} = \frac{1}{2} \frac{R_{11}  + R_{22}  }{\sqrt{t_2 + 2\sqrt{d_2}}}  \qquad  {\cal R}_{33} =   -  \sqrt{t_2 + 2 \sqrt{d_2}} $$



Similarly, in the special case where $t_2= 0$ (or equivalently $\rho_{22} = \rho_{33} = \rho_{23} = \rho_{32} = 0$), one has
$$  {\cal R}_{00} =  \sqrt{t_1 + 2 \sqrt{d_1}}    \qquad  {\cal R}_{03} =   \frac{1}{2} \frac{R_{30}  + R_{03}  }{\sqrt{t_1 + 2\sqrt{d_1}}} $$
$$  {\cal R}_{30} = \frac{1}{2} \frac{R_{30}  + R_{03}  }{\sqrt{t_1 + 2\sqrt{d_1}}}  \qquad {\cal R}_{11} =  \frac{1}{2} \frac{R_{11}- R_{22} }{\sqrt{t_1 + 2\sqrt{d_1}}}$$
%$$  {\cal R}_{12} =  \frac{1}{2} \frac{R_{12}  + R_{21}  }{\sqrt{t_1 + 2\sqrt{d_1}}} +  \frac{1}{2} \frac{R_{12}- R_{21} }{\sqrt{t_2 + 2\sqrt{d_2}}} \qquad  {\cal R}_{21} = \frac{1}{2} \frac{R_{12}  + R_{21}  }{\sqrt{t_1 + 2\sqrt{d_1}}} -  \frac{1}{2} \frac{R_{12}- R_{21} }{\sqrt{t_2 + 2\sqrt{d_2}}}$$
$$  {\cal R}_{22} =  -  \frac{1}{2} \frac{R_{11}- R_{22} }{\sqrt{t_1 + 2\sqrt{d_1}}} \qquad  {\cal R}_{33} =  \sqrt{t_1 + 2 \sqrt{d_1}}   $$


For $t_1= 0$, the results (\ref{omega11}), (\ref{omega22}) and (\ref{omega33}) give

\begin{equation}
\omega_{11} = 0 \quad \omega_{22} = 0  \quad \omega_{33} = \frac{1}{2}\bigg[    t_2 + 2 \sqrt{d_2} \bigg]
 + \frac{1}{8}\bigg[
\frac{(R_{03}-R_{30})^2-(R_{11}+R_{22})^2}{ t_2 + 2 \sqrt{d_2}}\bigg]
 \end{equation}


The quantity $\omega_{33}$ can be written also as

\begin{equation}
 \omega_{33} = \frac{1}{2}\bigg[    1  + 2 \sqrt{\rho_{22}\rho_{33}-{\rho^2_{23}}} \bigg]
 + \frac{1}{2} \bigg[  \frac{(\rho_{22}\rho_{33})^2- 2\rho_{23}^2}{1  + 2 \sqrt{\rho_{22}\rho_{33}-\rho_{23}^2}}\bigg]
 \end{equation}


Similarly,  for $t_2= 0$ one gets


\begin{equation}
\omega_{11} = 0 \quad \omega_{22} = 0  \quad \omega_{33} = \frac{1}{2}\bigg[    t_1 + 2 \sqrt{d_1} \bigg]
 + \frac{1}{8}\bigg[
\frac{(R_{03}+R_{30})^2-(R_{11}-R_{22})^2}{ t_1 + 2 \sqrt{d_1}}\bigg]
 \end{equation}

In this special case, the quantity $\omega_{33}$ can be explicitly given by
\begin{equation}
 \omega_{33} = \frac{1}{2}\bigg[    1  + 2 \sqrt{\rho_{11}\rho_{44}-{\rho^2_{14}}} \bigg]
 + \frac{1}{2} \bigg[  \frac{(\rho_{11}\rho_{44})^2- 2\rho_{14}^2}{1  + 2 \sqrt{\rho_{11}\rho_{44}-\rho_{14}^2}}\bigg]
 \end{equation}
in terms of the non vanishing density matrix elements.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Special two qubit $X$ states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Two qubit Bell states }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The two qubit $X$ states (\ref{fano-bloch-rho}) becomes of Bell type when the Bloch correlations elements $R_{30}$ and $R_{03}$ are zero. In this case, using the
results (\ref{omega11}), (\ref{omega22}) and (\ref{omega33}), the eigenvalues of the matrix $ \omega$ (cf. equation (\ref{matrixomega})) write
\begin{equation}\label{omegabell11}
\omega_{11} = 2 \bigg(\sqrt{\lambda_1 \lambda_2} +
\sqrt{\lambda_3\lambda_4 } \bigg)
\end{equation}
\begin{equation}\label{omegabell22}
\omega_{22} = 2 \bigg(\sqrt{\lambda_1 \lambda_3} +
\sqrt{\lambda_2\lambda_4 } \bigg)
\end{equation}
\begin{equation}\label{omegabell33}
\omega_{33} = 2 \bigg(\sqrt{\lambda_1 \lambda_4} +
\sqrt{\lambda_2\lambda_3 } \bigg)
\end{equation}
which coincide with  the results derived in \cite{Karimipour}. The eigenvalues
$\omega_{11}$, $\omega_{22}$ and $\omega_{33}$ rewrites also as
\begin{equation}\label{omegabell111}
\omega_{11} = \frac{1}{2} \bigg(\sqrt{(1- R_{11})^2 - (R_{22} +
R_{33})^2} + \sqrt{(1+ R_{11})^2 - (R_{22} - R_{33})^2 } \bigg)
\end{equation}
\begin{equation}\label{omegabell222}
\omega_{22} = \frac{1}{2} \bigg(\sqrt{(1- R_{22})^2 - (R_{33} +
R_{11})^2} + \sqrt{(1+ R_{22})^2 - (R_{33} - R_{11})^2 } \bigg)
\end{equation}
\begin{equation}\label{omegabell333}
\omega_{33} = \frac{1}{2}\bigg(\sqrt{(1- R_{33})^2 - (R_{11} +
R_{22})^2} + \sqrt{(1+ R_{33})^2 - (R_{11} - R_{22})^2 } \bigg)
\end{equation}
in terms of the correlation elements $R_{11}$, $R_{22}$ and $R_{33}$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Orthogonal invariant two-qubit states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The local quantum uncertainty for orthogonal invariant two qubit states was  discussed in \cite{Sen}.
Any two qubit state invariant under the operation ${\cal
O}\otimes{\cal O}$ ( with ${\cal O}$  an arbitrary orthogonal matrix) can be
expanded in terms of the three generators $\mathcal{I}$ ,
$\mathcal{F}_1$ and $\mathcal{F}_2$ as \cite{Sen}
\begin{equation}\label{O-invariant}
\rho = a \mathcal{I} + b \mathcal{F}_1 + c \mathcal{F}_2
\end{equation}
where the real parameters $a$, $b$ and $c$ are positive and  satisfy
$4a+2b+2c=1$ (trace condition), $\mathcal{I}$ is the identity and
the operators $\mathcal{F}_1$ and   $\mathcal{F}_2$
$$\mathcal{F}_1 = \sum_{ij} \vert ij \rangle \langle ji \vert \quad  \mathcal{F}_2 = \sum_{ij}  \vert ii \rangle \langle jj \vert$$
in the computational basis. The density matrix (\ref{O-invariant}) has the form $X$
\begin{equation}
\rho =  \left(
\begin{array}{cccc}
a+b+c & 0 & 0 & c \\
0 & a & b & 0 \\
0 & b & a & 0 \\
c & 0 & 0 & a+b+c
\end{array}
\right).
\end{equation}
and using the results (\ref{omega11}), (\ref{omega22}) and (\ref{omega33}), one verifies
$$ \omega_{11} = \omega_{33} =  2 \bigg(\sqrt{(a+b)(a+b+2c)} + \sqrt{a^2-b^2 } \bigg) \quad \omega_{22} =  2 \bigg(\sqrt{(a-b)(a+b+2c)} + (a+b) \bigg)$$
and one recovers the results obtained in \cite{Sen}.























%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Quantum correlation in separable states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%In this situation the trace quantum discord is
%\begin{eqnarray}
% D_{\rm g}(\rho)= \frac{1}{2}(1 - \alpha) + \sqrt{c_1(\alpha - c_1)}.
%\end{eqnarray}



%For $ \frac{2}{3} \leq \alpha \leq 1 $,


%we have $ \vert R_{11}\vert
%\leq \vert R_{33}\vert $ when $ c_1 \in [ \alpha_- , \alpha_+] $ and
%$\vert R_{33}\vert \leq \vert R_{11}\vert $ when $c_1 \in [
%0,\alpha_-] \cup [ \alpha_+, \alpha ]$ where
%$$\alpha_{\pm} = \frac{\alpha}{2} \pm \sqrt{\frac{5}{2}\alpha^2-3\alpha +1}.$$
%Accordingly, for $ \frac{2}{3} \leq \alpha \leq 1 $, the trace discord is given by
%\begin{eqnarray}
% D_{\rm g}(\rho)= \frac{1}{2}(1 - \alpha) + \sqrt{c_1(\alpha - c_1)}.
%\end{eqnarray}
%for $ c_1 \in [ \alpha_- , \alpha_+] $ and
%\begin{eqnarray}
% D_{\rm g}(\rho)= \frac{1}{2}\bigg( 1 - (\sqrt{c_1} + \sqrt{\alpha - c_1})^2\bigg)^2 + 4 \sqrt{c_1(\alpha - c_1)}(\sqrt{c_1} - \sqrt{\alpha - c_1})^2
%\end{eqnarray}
%when $c_1 \in [
%0,\alpha_-] \cup [ \alpha_+, \alpha ]$.




%we have $ \vert R_{33}\vert
%\leq \vert R_{11}\vert $ when $ c_1 \in [ \alpha_- , \alpha_+] $ and
%$\vert R_{11}\vert \leq \vert R_{33}\vert $ when $c_1 \in [
%0,\alpha_-] \cup [ \alpha_+, \alpha ]$ where
%$$\alpha_{\pm} = \frac{\alpha}{2} \pm (1-\alpha)(2\alpha -1).$$

%Accordingly, the trace discord is given by
%\begin{eqnarray}
% D_{\rm g}(\rho)= \frac{1}{2}(1 - \alpha) + \sqrt{c_1(\alpha - c_1)}.
%\end{eqnarray}
%for $ c_1 \in [ \alpha_- , \alpha_+] $
%and
%\begin{eqnarray}
% D_{\rm g}(\rho)= \frac{1}{2}\bigg( 1 - (\sqrt{c_1} + \sqrt{\alpha - c_1})^2\bigg)^2 + 4 \sqrt{c_1(\alpha - c_1)}(\sqrt{c_1} - \sqrt{\alpha - c_1})^2
%\end{eqnarray}
%when $c_1 \in [
%0,\alpha_-] \cup [ \alpha_+, \alpha ]$





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{local quantum uncertainty in 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{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 $|\eta_i\rangle \equiv (1 + \eta_i
\bar\eta_i)^{-\frac{1}{2}}(|0\rangle + \eta_i |1\rangle)~
(i=1,\ldots, n)$
%$|\eta_i\rangle \equiv \alpha_i |0\rangle + \beta_i |1\rangle$ ($i=1,\ldots, n$)
 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 {D_n(k)} \rangle = \frac{1}{\sqrt{C_n^k}} \sum_{\sigma} \vert{\underbrace{0\ldots 0}_{n-k}\underbrace{1\ldots 1}_{k}}\rangle,
%\end{equation}
\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 $i = k+1, \ldots , n$. The states
(\ref{Dicke}) are the eigenstates of the collective spin operators
$J^2$ and $J_z$ defined as
$$ J_{\alpha} = \frac{1}{2} \sum_{i=1}^n \sigma_{i\alpha} \quad \alpha = x,y,z $$
where the operators $\sigma_{i\alpha}$ stand for the spin
$\frac{1}{2}$-Pauli operators. In this respect,  the symmetric qubit
states (\ref{Dicke}) are completely determined by the quantum
angular momentum $j = 2n$ which may take integer or half integer
values ( $j = {1\over 2}, 1, \frac{3}{2}, \ldots$) specifying  the
irreducible representations classes of the group $SU (2)$. The
$(2j+1)$-dimensional Hilbert space is spanned by the irreducible
tensorial set $\{ \vert j , m \rangle, m = -j , -j+1, \cdots, j-1,j
\} \equiv \{ \vert n , k \rangle = \vert j = \frac{n}{2}, m + j
\rangle, k = 0, 1, \cdots, n\}$ characterizing the spin-$j$
representations of the group $SU(2)$.\\







\noindent As we shall consider the pairwise local quantum uncertainty in
symmetric states of type (\ref{psisymD}), we need the reduced two-qubit
density matrices extracted from the whole $n$ particles system. The general form of bipartite density matrix writes
as
\begin{equation}
\rho _{ij}=\left(
\begin{array}{llll}
\rho_{11} & \rho_{21}^{*} & \rho_{31}^{*} & \rho_{41}^{*} \\
\rho_{21} & \rho_{22} & \rho_{32}^{*} & \rho_{42}^{*} \\
\rho_{31} & \rho_{32} & \rho_{33} & \rho_{43}^{*} \\
\rho_{41} & \rho_{42} & \rho_{43} & \rho_{44}
\end{array}
\right)   \label{eq:rhogood}
\end{equation}
in the computational basis $\{|00\rangle ,|01\rangle ,|10\rangle
,|11\rangle \}$ where the matrix elements are given by
\begin{equation}
\rho_{11} =\frac 14\left( 1+ 2\langle \sigma _{i3} \otimes \sigma
_{j0} \rangle +\langle \sigma _{i3}\otimes \sigma _{j3}\rangle
\right) \qquad \rho_{44} =\frac 14\left( 1- 2\langle \sigma
_{i3}\otimes  \sigma _{j0}\rangle +\langle \sigma _{i3} \otimes
\sigma _{j3}\rangle \right)
\end{equation}
\begin{equation}
\rho_{21} = \rho_{31} = \frac 12(\langle \sigma _{i+} \otimes \sigma
_{j0}\rangle + \langle \sigma _{i+} \otimes \sigma _{j3}\rangle )
\qquad \rho_{42} = \rho_{43} = \frac 12(\langle \sigma _{i+} \otimes
\sigma _{j0}\rangle -\langle \sigma _{i+} \otimes \sigma
_{j3}\rangle )
\end{equation}
\begin{equation}
\rho_{22} = \rho_{33}=\frac 14\left( 1-\langle \sigma _{i3} \sigma
_{i+}\otimes \sigma _{j3}\rangle \right)
\end{equation}
\begin{equation}
\rho_{32}  = \langle\sigma_{i+} \otimes \sigma_{j-}\rangle
\end{equation}
\begin{equation}
\rho_{41} =\frac 14(\langle \sigma _{i1} \otimes \sigma _{j1}\rangle
-\langle \sigma _{i2} \otimes \sigma _{j2}\rangle +i2\langle \sigma
_{i1}\otimes \sigma _{j2}\rangle ). \label{eq:para1}
\end{equation}
For  collective spin models, the pairwise reduced density matrix
in the standard basis, $\{|{\downarrow}{\downarrow}{\rangle},
|{\downarrow}{\uparrow}{\rangle},|{\uparrow}{\downarrow}{\rangle},
|{\uparrow}{\uparrow}{\rangle}\}$
(with ${\sigma_z}|{\uparrow}{\rangle}=|{%
\uparrow}{\rangle}$ and ${\sigma_z}|{\downarrow}{\rangle}=-|{\downarrow}{%
\rangle}$)~\cite{wang1}, can be derived in terms of the collective
operators spin. Indeed, for  states, with symmetry exchange, we have
$$  \langle \sigma_{i\alpha} \otimes \sigma_{j0}\rangle = \frac{\langle J_{\alpha}\rangle}{n}, \qquad \langle \sigma_{i\alpha} \otimes \sigma_{j\alpha}\rangle = \frac{4 \langle J^2_{\alpha}\rangle- n}{n(n-1)} $$
$$  \langle \sigma_{i1} \otimes \sigma_{j2}\rangle = \frac{\langle J_{1} J_{2} + J_{2}J_{1}\rangle}{n(n-1)}, \qquad \langle \sigma_{i+} \otimes \sigma_{j3}\rangle = \frac{4 \langle J_{+} J_{3} + J_{3}J_{+}\rangle}{n(n-1)} $$
where  $\alpha = 1, 2, 3$. It follows that the explicit expressions
for the elements of the reduced density matrix  are given by
\begin{eqnarray}
\rho_{11} =\frac{n^2-2n+4{\langle}J^{2}_{z}{\rangle}+4(n-1){\langle}
J_{z}{\rangle}}{4n(n-1)}, \quad \rho_{44}
=\frac{n^2-2n+4{\langle}J^{2}_{z}{\rangle}-4(n-1){\langle}
J_{z}{\rangle}}{4n(n-1)}
\end{eqnarray}
\begin{eqnarray}
\rho_{21} = \rho_{31}
=\frac{(n-1){\langle}J_{+}{\rangle}+{\langle}J_{+}J_{z}+ J_{z}J_{+}{
\rangle}}{2n(n-1)},\quad \rho_{42} = \rho_{43}
=\frac{(n-1){\langle}J_{+}{\rangle}{\pm}{\langle}J_{+}J_{z}+
J_{z}J_{+}{ \rangle}}{2n(n-1)}
\end{eqnarray}
\begin{eqnarray}
\rho_{22} = \rho_{33}= \rho_{23} = \rho_{32}=
\frac{n^2-4{\langle}J^2_{z}{\rangle}}{4n(n-1)}=\frac{{\langle}
J^2_x+J^2_y{\rangle}-n/2}{n(n-1)},
\end{eqnarray}
\begin{eqnarray}
\rho_{41} = \frac{{\langle}J^2_+{\rangle}}{n(n-1)},
\end{eqnarray}
%$w=y$, for $\sum_{\alpha=x,y,z}J^2_{\alpha}=J^2=
%\frac{N}{2}(\frac{N}{2}+1)$.
%For the symmetric states with parity conservation, we find
%$x_{\pm}=0$
For  states with parity symmetry,  the
density matrix commutes with the operator $\sigma_3 \otimes
\sigma_3$. This implies $\rho_{12} = \rho_{13} = \rho_{42} =
\rho_{43}= 0$. In fact, for states with parity symmetry, we have $
\langle J_{1}\rangle = \langle J_{2} \rangle = 0$ and $ \langle
J_{1}J_{3}\rangle = \langle J_{2}J_{3} \rangle = 0$. Hence the
pairwise reduced density matrix is $X$ shaped  and  writes as

\begin{equation}
\rho _{ij}=\left(
\begin{array}{llll}
\rho_{11} & 0 & 0 & \rho_{41}^{*} \\
0 & \rho_{22} & \rho_{22} & 0 \\
0 & \rho_{22} & \rho_{22} & 0 \\
\rho_{41} & 0 & 0 & \rho_{44}
\end{array}
\right)   \label{rho:2}
\end{equation}




We note that the local unitary transformation
$$ \vert 0 \rangle_k \rightarrow \exp \bigg(\frac{i}{2} (\theta)\bigg) \vert 0 \rangle_k $$
eliminates the phase factors of the matrix element $\rho_{41}$ with
$\rho_{41} = \vert\rho_{41} \vert e^{i\theta}$ and $k=1,2$ labels
the subsystems 1 and 2. It follows that non-zero elements of the
correlation matrix take the simple form
\begin{equation}\label{matrix-R-special}
R_{03} = R_{30} = \rho_{11} - \rho_{44} \quad R_{11} = 2( \rho_{22} + \vert \rho_{41}\vert) \quad R_{22} = 2( \rho_{22} - \vert \rho_{41}\vert) \quad R_{33} = 1 - 4 \rho_{22}
\end{equation}



\subsection{Pairwise Local quantum uncertainty in Dicke
states}


The Dicke states are defined as the equal superposition of all basis
states of $n$ qubits having exactly $k$ excitations (\ref{Dicke}). Nowadays it is
commonly accepted that this family of symmetric states can be an
useful resource in various quantum protocols for two main reasons.
First, they can be generated experimentally. indeed, the generation
of Dicke states with trapped-ion qubits have been proposed \cite{Retzker}. On the other hand,  quantum correlations  in Dicke states
are highly robust in presence of  external decoherence effects and
especially measurements on individual qubits \cite{Stockton}.  From the equation (\ref{eq:bbbb}), it is simply verified that the
reduced density matrix $\rho _{12}$, describing  two qubits extracted from the state (\ref{Dicke}), is given by
\begin{equation}
\rho _{12}=\left(
\begin{array}{llll}
\rho_{11} & 0 & 0 & \rho_{14}\\
0 & \rho_{22} & \rho_{23} & 0 \\
0 & \rho_{32} & \rho_{33} & 0 \\
\rho_{41} & 0 & 0 & \rho_{44}
\end{array}
\right)   \label{eq:rho}
\end{equation}
with $\rho_{14}= \rho_{41}=0$ and the non vanishing elements are
\begin{eqnarray}
\rho_{11} &=&\frac{k(k-1))}{n(n-1)},  \nonumber \\
\rho_{44} &=&\frac{(n-k)(n-k-1)}{n(n-1)},  \nonumber \\
 \rho_{22} &=& \rho_{23} ~=~ \rho_{32}~ = ~\rho_{33} ~=~ \frac{k(n-k)}{n(n-1)}.  \label{eq:elements}
\end{eqnarray}
From the correlation matrix elements (\ref{matrix-R-special}), one obtains
\begin{eqnarray}
R_{11} &=& R_{22} ~=~  \frac{2k(n-k))}{n(n-1)},  \nonumber \\
R_{33} &=& 1 - \frac{4k(n-k)}{n(n-1)},  \nonumber \\
R_{03} &=& R_{30} ~=~ \frac{2k-n}{n}.  \label{eq:elements}
\end{eqnarray}
Using the expressions (\ref{omega11}), (\ref{omega22}) and
(\ref{omega33}), one finds
\begin{equation}
\omega_{11} = \omega_{22} = \sqrt{\frac{2k(n-k)}{n(n-1)}} ~ \bigg(
\sqrt{k(k-1)} + \sqrt{(n-k)(n-k-1)}\bigg)
\end{equation}
\begin{equation}
\omega_{33} = 1 + 2 ~\frac{k(k-n)}{n(n-1)}
\end{equation}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Local quantum uncertainty in spin coherent states}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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 $c_k$ ($k=0,\ldots,n$) stand for the complex expansion
coefficients. In particular, when the qubit are all identical
($\eta_i = \eta$ for all qubits), it is simply verified 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 (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 $|1\rangle \equiv
|\frac{1}{2} ,+ \frac{1}{2}\rangle$). The standard $ SU(2)$ coherent
states are obtained by the action of an element of the coset space
$SU(2)/ U(1)$
\begin{equation}
 D_j(\xi) = \exp (\xi J_{+} - \xi^{\ast} J_{-}) \ ,
\end{equation}
on the extremal state $|j,-j\rangle$. This action  gives  the states
\begin{equation}
|j,\eta\rangle = D_j(\xi) |j,-j\rangle = \exp(\xi J_{+} - \xi^{\ast}
J_{-}) |j,-j\rangle = (1+|\eta|^{2})^{-j} \exp(\eta J_{+})
|j,-j\rangle \ ,   \label{4.7}
\end{equation}
where $\eta = (\xi/|\xi|)\tan |\xi|$. In the standard angular momentum basis $\{ \vert j , m
\rangle\}$, they write
\begin{equation}\label{cs-spinj}
|j,\eta\rangle = (1+|\eta|^{2})^{-j} \sum_{m=-j}^{j} \left[
\frac{(2j)!}{(j+m)!(j-m)!} \right]^{1/2} \eta^{j+m} |j,m\rangle \ .
\end{equation}
They satisfy the resolution to identity property
\begin{equation}
\int d\mu(j,\eta) |j,\eta\rangle \langle j,\eta| = I \ ,
\mbox{\hspace{1.0cm}} d\mu(j,\eta) = \frac{2j+1}{\pi}
\frac{d^{2}\!\eta}{(1+|\eta|^{2})^{2}} \ .  \label{4.10}
\end{equation}
The spin coherent states are not orthogonal to each other:
\begin{equation}
\langle j,\eta_{1}|j,\eta_{2}\rangle = (1+|\eta_{1}|^{2})^{-j}
(1+|\eta_{2}|^{2})^{-j} (1 + \eta_{1}^{\ast} \eta_{2})^{2j} \ .
\label{4.9}
\end{equation}
The resolution to identity makes possible to expand an arbitrary
state in terms of the coherent states $|j,\eta\rangle$. In the
special case $j = \frac{1}{2}$, the spin coherent states
(\ref{cs-spinj}) reduce to
\begin{equation}
|  \eta \rangle = \frac{1}{\sqrt{1 + \bar\eta \eta}}
|\downarrow\rangle
           + \frac{\eta}{\sqrt{1 + \bar\eta \eta}} |\uparrow\rangle . \label{coh}
\end{equation}
Here and in the following $| \eta \rangle$ is short for the
spin-$\frac{1}{2}$ coherent state $|{\frac{1}{2}}, \eta\rangle$ with
$|\uparrow\rangle \equiv |\frac{1}{2},\frac{1}{2}\rangle$ and
$|\downarrow \rangle \equiv |\frac{1}{2} ,- \frac{1}{2}\rangle$). It
is important to notice that the  tensorial product of two  $SU(2)$
coherent states $ |j_1, \eta \rangle$ and $ |j_2, \eta \rangle$
produces a spin-$(j_1+j_2)$ coherent state labeled by the same
variable:
\begin{equation}\label{split}
 |j_1, \eta \rangle \otimes | j_2 , \eta \rangle
  =  (D_{j_1} \otimes D_{j_2})\, (|j_1,j_1\rangle \otimes |j_2,j_2\rangle)
  =  D_{j_1+j_2}\, |j_1+j_2,j_1+j_2\rangle
    \; = \; |j_1 + j_2, \eta \rangle .
\end{equation}
Only coherent states possess this remarkable property. It allows  to
write any spin-$j$ coherent states as a $2j$ tensorial product of
spin-$\frac{1}{2}$
 coherent states:
\begin{equation}
|j, \eta \rangle =  \left(|\eta\rangle \right)^{\otimes 2j}
            =   \left( \frac{1}{\sqrt{1 + \bar\eta \eta}}
|\downarrow\rangle
           + \frac{\eta}{\sqrt{1 + \bar\eta \eta}} |\uparrow\rangle \right)^{\otimes 2j} \nonumber
          =  (1 + \bar\eta \eta)^{-j}\sum_{m=-j}^{+j} {2 j \choose j + m}^{\frac{1}{2}}
         \eta^{j+m}
          |j,m\rangle,  \label{Coh}
\end{equation}
reflecting that a spin-$j$ coherent state may be viewed as a
multipartite state containing  $2j$ qubits.

The even and odd spin coherent states are defined by
\begin{equation}\label{ncs}
 \vert j, \eta , m \rangle  =  {\cal N}_m ( \vert j, \eta \rangle + e^{im\pi} \vert j, - \eta
 \rangle)
\end{equation}
where the integer $m \in \mathbb{Z}$ takes the values $m = 0 ~({\rm
mod}~2)$ and $m = 1~ ({\rm mod}~2)$. The normalization factor ${\cal
N}_m$ is
$$ {\cal N}_m = \big[ 2 + 2 p^{2j} \cos m \pi\big]^{-1/2}$$
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}
For  $ j = \frac{1}{2}$, the even and odd coherent states coincide
with $\vert \uparrow \rangle $ and $ \vert \downarrow \rangle$. They
can be identified with basis states for a logical qubit as $ \vert
0\rangle \rightarrow \vert \uparrow \rangle$ and $ \vert 1 \rangle
\rightarrow \vert \downarrow \rangle$. In this manner,  the states $
\vert j, \eta , m \rangle$ can be viewed as multipartite fermionic
coherent states:
\begin{equation}\label{cs-2j}
 \vert j, \eta , m \rangle  =  {\cal N}_m ( \left(|\eta \rangle \right)^{\otimes 2j} + e^{im\pi} \left(|-\eta \rangle \right)^{\otimes
 2j}).
\end{equation}



\noindent The decomposition property (\ref{split}) provides us with
a picture where even and odd spin coherent states can be considered
as comprising multipartite spin subsystems. This is our main
motivation to investigate the quantum correlations present in a
single spin coherent state. This issue is discussed in what follows.




\subsubsection{Pure bipartite spin coherent states}


Let us first  consider the following balanced superposition of spin coherent
states
\begin{equation}\label{cs-theta}
 \vert j, \eta , \theta \rangle  =  {\cal N}_{\theta} ( \vert j, \eta \rangle +
 e^{i\theta}  \vert j, -\eta \rangle)
\end{equation}
where the normalization factor is given by $|{\cal N}_{\theta}|^{-2}  = 2 + 2
p^{2j}\cos\theta $. Using the factorization or the splitting
property of spin coherent states (\ref{split}), the states
(\ref{cs-theta}) can be also expressed as
\begin{equation}\label{cs-2q}
 \vert j, \eta , \theta \rangle  =  {\cal N}_{\theta} ( \vert j_1, \eta \rangle\otimes\vert j_2, \eta \rangle +
 e^{i\theta} \vert j_1, -\eta \rangle\otimes\vert j_2, -\eta \rangle)
\end{equation}
with $j = j_1+j_2$. They can rewritten as a two qubit states in the
basis
$$ \vert
j_i, \eta , 0 \rangle \longrightarrow \vert  0 \rangle_{j_i} \qquad
\vert j_i, \eta , \pi\rangle \longrightarrow \vert  1 \rangle_{j_i},
\quad i=1,2.
$$
defined by means of  odd and even spin coherent associated with the
angular momenta $j_1$ and $j_2$. Indeed, for each subsystem, an
orthogonal basis $\{ \vert 0 \rangle_l , \vert 1 \rangle_l\}$, with
$ l = j_1$ or $j_2$, can be defined as
\begin{equation}\label{base0}
\vert 0 \rangle_l = \frac{ \vert l , \eta  \rangle +  \vert l ,
-\eta \rangle}{\sqrt{2(1 + p^{2l})}}
   \qquad \vert 1 \rangle_l = \frac{\vert l , \eta \rangle -  \vert l , -\eta
\rangle}{{\sqrt{2(1- p^{2l})}}}.
\end{equation}
The bipartite density state $\rho = \vert j, \eta , m\rangle \langle
j, \eta , m \vert$ is pure. The concurrence in this pure bipartite
system writes
\begin{equation}
{\cal C}_{j_1,j_2}(\theta) =
\frac{\sqrt{1-p^{4j_1}}\sqrt{1-p^{4j_2}}}{1+p^{2j}\cos
\theta}.\label{concurence1}
\end{equation}
Using the Schmidt decomposition, the state (\ref{cs-2q}) can be
written as
\begin{equation}\label{cs-shmidt}
 \vert j, \eta , \theta \rangle  =  \sqrt{\lambda_+} \vert + \rangle_1\otimes\vert + \rangle_2
 + \sqrt{\lambda_-}
 \vert - \rangle_1\otimes\vert - \rangle_2
\end{equation}
where $\lambda_{\pm}$ denote the eigenvalues of the reduced density
of the first subsystem $\rho_{j_1} = {\rm Tr}_{j_2} (\rho)$ obtained
by tracing out the spin $j_2$ . They write as
\begin{equation}\label{lambda}
\lambda_{\pm}= \frac{1}{2}\bigg( 1 \pm \sqrt{1 - {\cal C}^2} \bigg).
\end{equation}
in terms of the concurrence ${\cal C}\equiv{\cal
C}_{j_1,j_2}(\theta)$ given by (\ref{concurence1}). In the basis $\{
\vert + \rangle_1\otimes\vert + \rangle_2, \vert +
\rangle_1\otimes\vert - \rangle_2, \vert - \rangle_1\otimes\vert +
\rangle_2, \vert - \rangle_1\otimes\vert - \rangle_2 \}$, the
density matrix $ \rho_{j_1,j_2} (\theta) = \vert j, \eta , \theta
\rangle \langle j, \eta , \theta \vert $ takes the form
\begin{equation}
 \rho_{j_1,j_2} (\theta)=\left(
\begin{array}{llll}
\lambda_+ & 0 & 0 & \sqrt{\lambda_+\lambda_-}\\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\sqrt{\lambda_+\lambda_-} & 0 & 0 & \lambda_-
\end{array}
\right)   \label{eq:rhoj1j2}
\end{equation}
Using the result (\ref{omegaii}), one verifies that $\omega_{11} =
0$ , $\omega_{22} = 0$ and $\omega_{33} = 1 - 4\lambda_+\lambda_-$.
It follows that the local quantum uncertainty coincides with the squared
concurrence (\ref{concurence1})
\begin{equation}\label{LQU-pure}
 \mathcal{U}( \rho_{j_1,j_2} (\theta)) = {\cal C}^2_{j_1,j_2}(\theta).
\end{equation}
For $\theta = m \pi$ $(m \in \mathbb{Z})$, the logical qubits $
\vert j, \eta , m=0 \rangle$ and $ \vert j, \eta , m=1 \rangle$
coincide with even and odd spin coherent states. They behave like a
multipartite state of Greenberger-Horne-Zeilinger (${\rm GHZ}$) type
\cite{GHZ} in the asymptotic limit $p \rightarrow 0 $. Indeed, in
this limit, 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 $\vert j ,
\eta, m \rangle$ becomes of ${\rm GHZ}$-type
\begin{equation}
\vert j , \eta, m \rangle \sim \vert {\rm GHZ}\rangle_{2j} = \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}
which is maximally entangled and the bipartite local quantum
uncertainty is $\mathcal{U}( \rho_{j_1,j_2} (\theta=m\pi))= 1$.



Another interesting limiting case concerns the situation where $p^2
\rightarrow 1$ (or $ \eta \rightarrow 0$ ). In this case  the state $\vert j , \eta, m = 0 ~({\rm mod}~
2) \rangle$ (\ref{cs-2j}) reduces to ground state of a collection of
$2j$ fermions
\begin{equation}
\vert j,  0 , 0 ~({\rm mod}~ 2) \rangle \sim  \vert \downarrow
\rangle \otimes\vert \downarrow \rangle \otimes \cdots \otimes \vert
\downarrow \rangle,
\end{equation}
which is completely separable and
$$\mathcal{U}( \rho_{j_1,j_2} (\theta= m\pi)= 0.$$
The odd spin coherent state $\vert j , \eta , 1
~({\rm mod}~ 2) \rangle$  becomes a multipartite state of W
type~\cite{Dur00}
\begin{equation}
\vert j,  0 ,  1 ~({\rm mod}~ 2) \rangle \sim \vert\text{\rm
W}\rangle_{2j}
    = \frac{1}{\sqrt{2j}}(\vert \uparrow \rangle \otimes\vert \downarrow \rangle \otimes \cdots\otimes
       \vert \downarrow \rangle  +\vert \downarrow \rangle \otimes\vert \uparrow\rangle \otimes\ldots\otimes \vert \downarrow\rangle
      +\cdots
   + \vert  \downarrow \rangle \otimes\vert \downarrow \rangle  \otimes \cdots\otimes \vert \uparrow\rangle)~.
\label{Wstate}
\end{equation}
and, in this limiting situation, the local quantum uncertainty is
given by
$$ \mathcal{U}( \rho_{j_1,j_2} (\theta = \pi)) = 4 \frac{j_1j_2}{j_1+j_2}.$$
%The even spin coherent states  $\vert j, \eta , m = 0 ~({\rm mod}~2)
%\rangle$ interpolate continuously between ${\rm GHZ}_{2j}$ states
%$(p \rightarrow 0)$ and the completely separable state $\vert
%\downarrow \rangle \otimes\vert \downarrow \rangle \otimes \cdots
%\otimes \vert \downarrow \rangle$ $(p \rightarrow 1)$. In the odd
%case, corresponding  to
% $\vert j, \eta , m = 1 ~({\rm mod}~2) \rangle$, we obtain states
%interpolating between states of ${\rm GHZ}_{2j}$ type $(p
%\rightarrow 0)$ and states of Werner ${\rm W}_{2j}$ type $(p
%\rightarrow 1)$.
\subsubsection{Mixed bipartite states}
Now, we consider bipartite mixed density matrices  $\rho_{ij}$ obtained by a trace
procedure consisting in removing  all degrees of freedom of all qubits except the two qubits  $i$ and
$j$.  Since we consider quantum systems possessing the exchange symmetry, the trace procedure leads to  $2j(2j-1)/2$ identical density matrices $\rho_{12}$.
After some algebra, one gets
\begin{eqnarray}
\rho_{12} = {\cal N}^2(\vert \eta , \eta \rangle \langle \eta , \eta
\vert +\vert -\eta , -\eta \rangle \langle -\eta , -\eta | +
e^{im\pi } q |-\eta , -\eta \rangle \langle \eta , \eta \vert +e^{-i
m \pi }q \vert \eta , \eta \rangle \langle -\eta , -\eta \vert ).
\label{rhoij-vect}
\end{eqnarray}
The quantity $q$  occurring in (\ref{rhoij-vect}) is defined by
$$q = p^{2j-2}.$$

Setting $\eta = e^{i\phi}\sqrt{\frac{1-p}{1+p}}$, the density matrix
takes the form
\begin{equation}
\rho_{12} = \frac{1}{4(1+ p^{2j}\cos m\pi)} \left(
\begin{smallmatrix} (1+p)^2(1+q\cos m\pi)& 0
    & 0& e^{-2i\phi}(1-p^2)(1+q\cos m\pi
)\\
0  & (1-p^2)(1-q\cos m\pi ) & (1-p^2)(1-q\cos m\pi
) & 0 \\
0  & (1-p^2)(1-q\cos m\pi ) & (1-p^2)(1-q\cos m\pi
) & 0 \\
e^{2i\phi}(1-p^2)(1+q\cos m\pi ) & 0 & 0 & (1-p)^2(1+q\cos m\pi)
\end{smallmatrix}
\right) \label{rhoij}
\end{equation}
in the computational basis. The phase factor $\phi$ will be taken equal to zero. In other words,
the phase factor can be removed by a loacal transformation and the local quantum uncertainty remains unchanged as we discussed here above.
The bipartite mixed density $\rho_{12}$ (\ref{rhoij}) writes in
Fano-Bloch representation as
\begin{equation}
\rho_{12} = \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
$$ R_{00} = 1, \quad R_{11} =  \frac{1- p^2}{1+ p^{2j}\cos m\pi}, \quad R_{22} = \frac{(p^2-1)~p^{2j-2}\cos
m\pi}{1+ p^{2j}\cos m\pi},$$ $$ R_{33} = \frac{p^2 + p^{2j-2}\cos
m\pi}{1+ p^{2j}\cos m\pi}, \quad R_{03} =  R_{30} = \frac{p +
p^{2j-1}\cos m\pi}{1+ p^{2j}\cos m\pi}.$$



From the equations (\ref{omega11}), (\ref{omega22}) and
(\ref{omega33}), one obtains

\begin{equation}\label{om11}
\omega_{11} = \sqrt{\frac{1-p^2}{1+p^2}} ~\frac{\sqrt{1-
p^{4j-4}}}{1+ p^{2j}\cos m\pi}
\end{equation}
\begin{equation}\label{om22}
\omega_{22} =  p^2 \sqrt{\frac{1-p^2}{1+p^2}} ~\frac{\sqrt{1-
p^{4j-4}}}{1+ p^{2j}\cos m\pi}
\end{equation}
\begin{equation}\label{om33}
\omega_{33} = \frac{2p^2}{1+p^2} \frac{1+p^{2j-2}\cos m\pi}{1+
p^{2j}\cos m\pi}
\end{equation}

Since $ \omega_{22} \leq \omega_{11}$, we have $\omega_{\rm max} =
{\rm max}(\omega_{11},\omega_{22})$ and one verifies that

$${\rm sign} (\omega_{11}-\omega_{33}) = {\rm sign}\bigg(2(1-p^4)- (1+3p^4)(1+p^{2(j-1)}\cos m\pi)\bigg)$$

\textcolor[rgb]{1.00,0.00,0.00}{{\bf Etudier le signe dans les $j = 1, 3/2, 2, 5/2,  ....$ pour
$m=0$ et $m=1$}}

\textcolor[rgb]{1.00,0.00,0.00}{Tracer LQU en fonction de $p^2$ pour $j = 1, 3/2, 2, 5/2,  ....$}\\

We note that for the special case $j = 1$, the state
(\ref{cs-theta}) is a pure state with $j_1= j_2= 1/2$. In this case,
it is simple to verify that $\omega_{11} = \omega_{22} = 0$ and
$\omega_{33} = \frac{4p^2}{(1+p^2)^2}$ for $m=0$ and $\omega_{33} = 0 $
for $m=1$. Therefore, in this case the local quantum uncertainty writes
$$\mathcal{U}( \rho_{1/2,1/2}) = \frac{(1-p^2)^2}{(1+p^2)^2} $$ for $m=0$ and
$$\mathcal{U}( \rho_{1/2,1/2}) = 1 $$ for $m = 1$ in agreement with the result (\ref{LQU-pure}).

For $j> 1$ and $p \longrightarrow 0$, one obtains
$$ \omega_{11} = 1,  \omega_{22} = 0,  \omega_{33} = 0$$
and the local quantum uncertainty is zero.\\


Similarly, for $j> 1$ and $p^2 \longrightarrow 1$, it is simple to
check that the local quantum uncertainty vanishes for even spin
coherent states $(m=0)$. In this limiting situation, the matrix
elements (\ref{om11}), (\ref{om22}) and (\ref{om33}) become
$$ \omega_{11} = \frac{j-2}{\sqrt{2} j},  \omega_{22} = \frac{j-2}{\sqrt{2} j},  \omega_{33} = \frac{j-1}{ j}$$
for odd spin coherent states $(m=1)$. The local quantum uncertainty
is then given by
$$ \mathcal{U}( \rho_{12} (m=1)) \longrightarrow \frac{1}{j}.$$

\begin{center}
\includegraphics[width=4in]{fig1.eps}\\
{\bf Figure 1.}  {\sf The  local quantum uncertainty  
in symmetric states $(m=0)$ for different values of $j$.}
\end{center}
\begin{center}
\includegraphics[width=4in]{fig2.eps}\\
{\bf Figure 2.}  {\sf The local quantum uncertainty
in antisymmetric states $(m=1)$ for different values of $j$.}
\end{center}





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



In conclusion, we have derived the analytical expression of local quantum uncertainty for two-qubit
in $X$-states. This quantum correlations quantifier provides an efficient and computable way to characterize the nature
of correlations present in a multi-partite quantum system. Moreover, the analytical results obtained in this paper covers
some special class of two-qubit states recently investigated in the literature. We quote  Bell states and orthogonally invariant two-qubit
system examined in \cite{Karimipour} and  \cite{Sen}, respectively. We also evaluated the pairwise local quantum uncertainty in
multi-partite systems with exchange and parity symmetries.
As illustration, we quantified the pairwise quantum correlations in  balanced superpositions of Dicke states. Our interest in such
states is mainly motivated by their relevance in various collective spin systems such as Dicke model \cite{Dicke} and Lipkin-Meshkov-Glick \cite{LMG} model exhibiting
quantum phase transition.





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\subsection{Kitagawa-Ueda state}

We discuss an ensemble of $n$ spin-$\frac 12$ with exchange
symmetry whose dynamics is governed by the nonlinear Hamiltonian given by
$$H=\chi J_x^2 = \frac \chi 4 \sum_{i,j=1}^{n} \sigma_{i1}\otimes \sigma_{j1}$$
where $\chi$ is the coupling constant. This Hamiltonian involves all
pairwise interactions.  We assume that  the system is initially
 prepared in the product state $|n, 0\rangle=|111,...,1\rangle $. Considering its dynamic evolution,
 the evolved state at time $t$ writes
\begin{equation}
|\Psi (t)\rangle = e^{-i\chi tJ_x^2}|n, 0\rangle.
\end{equation}
Using the results obtained in \cite{Kitagawa} the following
expectation values are obtained ($\theta =2\chi t$)

\begin{eqnarray}
\langle J_x\rangle  &=&\langle J_y\rangle =0,  \label{eq:aaaa} \nonumber \\
\langle J_z\rangle  &=&-\frac n2\cos ^{n-1}\left( \frac \theta 2\right)  \nonumber  \\
\langle J_x^2\rangle  &=&n/4 \nonumber  \\
\langle J_y^2\rangle  &=&\frac 18\left( n^2+n-n(n-1)\cos
^{n-2}\theta \right)
\nonumber \\
\langle J_z^2\rangle  &=&\frac 18\left( n^2+n+n(n-1)\cos
^{n-2}\theta \right)
\nonumber \\
\langle [J_{+},J_z]_{+}\rangle  &=&0 \nonumber \\
\langle [J_x,J_y]_{+}\rangle  &=&\frac 12n(n-1)\cos
^{n-2}\left(\frac \theta 2\right)\sin \left(\frac \theta 2\right).
\label{eq:bbbb}
\end{eqnarray}




The expectation values of local spin operators are given by



$$  \langle \sigma_{i3} \otimes \sigma_{j0}\rangle = -  \cos ^{n-1}\left( \frac \theta 2\right)  , \qquad \langle \sigma_{i3} \otimes \sigma_{j3}\rangle = \frac 12 \bigg(1
+\cos ^{n-2}\theta  \bigg)$$
$$  \langle \sigma_{i+} \otimes \sigma_{j-}\rangle =  \frac 18 \bigg(1
-\cos ^{n-2}\theta  \bigg), \qquad \langle \sigma_{i-} \otimes
\sigma_{j-}\rangle =  \frac 18 \bigg(\cos ^{n-2}\theta -1
\bigg)-\frac i2 \cos ^{n-2}\left( \frac \theta 2\right) \sin \left(
\frac \theta 2\right)$$



The elements of the two-qubit reduced density matrix write


\begin{eqnarray}
\rho_{11}  &=& \frac 18 \bigg( 3 + \cos ^{n-2}\theta - 4 \cos ^{n-1}\left( \frac \theta 2\right) \bigg)  \nonumber \\
\rho_{44}   &=& \frac 18 \bigg( 3 + \cos ^{n-2}\theta + 4 \cos ^{n-1}\left( \frac \theta 2\right) \bigg)   \nonumber  \\
\rho_{41}   &=& \frac 18 \bigg(\cos ^{n-2}\theta -1 \bigg)+\frac i2 \cos ^{n-2}\left( \frac \theta 2\right) \sin \left( \frac \theta 2\right)\nonumber  \\
\rho_{22} &=& \rho_{23} = \rho_{32}= \rho_{33} = \frac 18 \bigg(1
-\cos ^{n-2}\theta  \bigg)
\end{eqnarray}



The equations (\ref{omega11}), (\ref{omega22}) and (\ref{Omega33})
lead to

\begin{equation}
\omega_{11} = \sqrt{2\rho_{22}} ~\frac{ \rho_{11} + \rho_{44}+ 2
\vert \rho_{14}\vert + 2 \sqrt{\rho_{11}\rho_{44}- \vert
\rho_{14}\vert^2}}{\sqrt{\rho_{11} + \rho_{44} + 2
\sqrt{\rho_{11}\rho_{44}- \vert \rho_{14}\vert^2}}}
\end{equation}
\begin{equation}
\omega_{22} = \sqrt{2\rho_{22}} ~\frac{ \rho_{11} + \rho_{44}- 2
\vert \rho_{14}\vert + 2 \sqrt{\rho_{11}\rho_{44}- \vert
\rho_{14}\vert^2}}{\sqrt{\rho_{11} + \rho_{44} + 2
\sqrt{\rho_{11}\rho_{44}- \vert \rho_{14}\vert^2}}}
\end{equation}
\begin{equation}
\omega_{33} = \rho_{11} + \rho_{44} - 4 \frac{\vert
\rho_{14}\vert^2}{\rho_{11} + \rho_{44} + 2
\sqrt{\rho_{11}\rho_{44}- \vert \rho_{14}\vert^2}}
\end{equation}


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