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\begin{center}

 {\Large \bf IJQI-D-14-00125}

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 {\Large \sf Title:  Pairwise quantum and classical correlations in  multi-qubits states via linear relative entropy}

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 {\Large \bf List of changes}

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\noindent  We would like to thank the referee for his
constructive suggestions and pertinent remarks.\\
\noindent We did as best as we could to take into account all his
 comments.\\

%The authors are  submitting a paper which  in the direct  continuation of their two previous papers (PLA 2012, for two of them,  and JPA 2013, for all of them). They develop  a formalism for calculating the pairwise correlations %existing in a multi-qubit system. This formalism heavily rests upon proposals given in the  highlighted papers of Modi et al (PRL 2010) and Dakic et al (PRL 2010) in view of quantify correlations through relative entropy. In order %to make computations manageable, the authors use a linearized version of the relative entropy.

%The content is informative and correct, as much I could check these routine algebraic computations.

\noindent{\Large \textcolor{red}{concerning the comment}}\\
\noindent Now, I am not convinced by the physical relevance of such explorations. Indeed, the authors fail to justify on a physical level the linear approximation $\log \rho \approx \rho - Id$. Of course, there is a technical justification, since computations become a lot easer. But what happens when one copes with the deep quantum regime?\\

\noindent\textcolor{green}{our reply}

\noindent{\bf We thank the referee for this important comment}.  The way of introducing the linear entropy  (in the first version of the paper) is
reformulated. In fact,  in the previous version, the reader can understand that  the whole analysis
are performed only  for density matrices approaching  completely mixed states.  To avoid any misunderstanding,  we
define (in the revised version) the linear relative entropy (see formula (24)) for arbitrary density matrices. After, equation (29), we mention that for density matrices  approaching the unit matrix,
the relative entropy coincides with linear relative entropy.\\




\noindent{\Large \textcolor{red}{concerning the comment}}\\
\noindent There are some strange formulations through the text. For instance, they name ``classical state'' a diagonal mixed state (e.g., Eq. 19).
For the physics community, a classical state is a point in the phase space of a mechanical system. Is it necessary to distort in such confusing way the commonly accepted terminology? \\

\noindent\textcolor{green}{our reply}

\noindent We replaced the name "classical states" by "quantum-classical states" which is more appropriate for states presenting zero quantum correlations and
containing only classical correlations(classically correlated states).
Some authors use the term "classical" to refer to classically correlated states. We agree with the referee that such distortion
in not necessary. \\






\noindent{\Large \textcolor{red}{concerning the comment}}\\
\noindent Another point is the mathematical difference between ``pure bipartite states'', which are $2n \times 2n$ complex matrices and ``mixed bipartite states'' which are $4 \times 4$ complex matrices. \\

\noindent\textcolor{green}{our reply}

\noindent In the pure bi-partitioning scheme, the whole system is mapped into a two qubit system (this mapping is defined by eqs (6) and (7)) so that
the density matrix is of size $4 \times 4$  (equation (8) gives (10)). The $ 4 \times 4$ density matrix for the second scheme is obtained by tracing out $(n-2)$ qubits and
it is different from the density matrix obtained in the first scheme. These two schemes give all possible bi-partitions of the system.
\\






\noindent{\Large \textcolor{red}{concerning the comment}}\\
\noindent Another strange sentence  is found in page 9, below Eq. (41), ``The closest classical state to $\rho_{k,n-k}$ is the eigenvector ...". Eigenvector of what?
The authors should be more careful and rigorous  in presenting  the mathematical content and notations of their work.\\

\noindent\textcolor{green}{our reply}\\
\noindent 1. This sentence is removed and replaced by :
"To find
the explicit expression of the
closest quantum-classical state to $\rho_{k,n-k}$, we follow the
 procedure reported in [12] for an arbitrary two qubit system."


\noindent 2. We added a comment concerning the Schmidt decomposition (8) concerning
 the definitions of the vectors $\vert {\bf 0} \rangle_{k}, \vert {\bf 1} \rangle_{k}$  and
 $\vert {\bf 0} \rangle_{n-k}, \vert {\bf 0} \rangle_{n-k}$ (the second line after equation (9)). This
 responds also to your pervious comment.


\noindent 3. We also replaced $a_3$ in equation (46) (and in the sequel of section 5) by $c_3$ to avoid any confusion with
 the notation $a_3$ used in section 4.


\noindent 4. We replaced the correlation matrix elements $R_{\alpha, \beta}$ by  ${\cal R}_{\alpha, \beta}$ for mixed two qubit states $\rho_{12}$ to
avoid any possible confusion with those associated with the pure case.\\





\noindent{\Large \textcolor{red}{concerning the comment}}\\
\noindent Also the authors should give a clear account of what is new in their submission with regard to their recent papers that I mentioned above.\\


\noindent\textcolor{green}{our reply}\\
\noindent We agree with the referee that this clarification is of paramount  importance for the readers.\\
Two paragraphs are added to make clear two essentials points (one in the introduction and the second in the beginning
of section 3). \\
In the fourth paragraph in the introduction, we added "In this picture,
the pairwise quantum correlations $\cdots$ existing in  $n$-qubit systems
 with parity and exchange invariance." \\
 In particular, we give a clear account of the novelties of our work which can be summarized as follows:\\
 1. We show
 that the quantum discord $D_2$ (with linear relative entropy) is exactly the quantum discord $D_g$ (based on Hilbert-Schmidt distance) derived
 in the paper [20] (by two authors of us)\\
2. In the other hand, we investigate the classification of the states (quantum, quantum-classical, product...), of the system under consideration, according their nature. This
 question was not considered in the papers [20,21].\\
3. Finally, we define a linearized version of the relative entropy that is tractable  from an analytical point of view in comparison
 with the unified view proposed in [19].\\
 Similarly, in the beginning
of section 3, we stress  that the present work is a continuation of the paper [20].\\







\noindent{\Large \textcolor{red}{concerning the comment}}\\
\noindent There are misprints which should be corrected.\\

\noindent\textcolor{green}{our reply}\\
\noindent They are corrected.



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%\begin{enumerate}
%  \item \emph{an unified }should be \emph{a unified } (page 6, line 6)
%  \item \emph{as excepted} should be \emph{as expected} (page 10, line 12)
%  \item \emph{it simple} should be \emph{it is simple} (page 10, line -9 from the bottom)
% \item  \emph{zero} should be \emph{at zero} (page 10, line -2 from the bottom)
% \item \emph{in} should be \emph{is} (page 12, line 1)
%  \item \emph{cliasscal} should be \emph{classical} (page 12, line -7 from the bottom)
%   \item \emph{simple} should be \emph{is simple} (page 12, line -4 from the bottom)
%    \item \emph{main}  what? (page 13, line -11 from the bottom)
%     \item \emph{partioning} should be \emph{partitioning} (page 14, line 11)
% \item \emph{reflects} should be \emph{reflects the fact} (page 15, line -2 from the bottom)
% \item \emph{a pure two qubit sub-systems} in page 17: please correct.
%\end{enumerate} 