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 {\Large \bf PLA-D-14-01427}

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 {\Large \sf Title: A unified scheme  of correlations using linear relative entropy}

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 {\Large \bf List of changes}

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\noindent  We would like to thank the referees for their
constructive remarks.\\
\noindent We did as best as we could to take into account all their
 comments.


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%{\bf List changes/ Editor suggestions}
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%\noindent{\Large \textcolor{red}{concerning the suggestion}}\\
% I suggest the title is amended to read\\
%"Unified scheme for correlations using linear relative entropy"\\

%\noindent\textcolor{green}{our reply}\\
%The title is changed. \\



%\noindent{\Large \textcolor{red}{concerning the suggestion}}\\
% You are requested to ask a native English speaker to correct your manuscript before resubmitting.\\

%\noindent \textcolor{green}{our reply}\\
%This is done (see the acknowledgments)\\



\begin{center}
{\bf List changes/ Reviewer 1}
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\noindent{\Large \textcolor{red}{concerning the comment}}\\
1) Page 2, Introduction, second paragraph. At the end of the first sentence of the second paragraph
"Therefore, quantification [...] last two decades [Refs]",
the authors should add some relevant and general references about the characterization of
 quantum correlations both theoretical and experimental, in order to make their reference list more up-to-date. For instance, they should cite:\\
- J.-S. Xu, K. Sun, C.-F. Li, X.-Y. Xu, G.-C. Guo, E. Andersson, R. Lo Franco and G. Compagno, Nat. Commun. 4, 2851 (2013)\\
- R. Lo Franco, B. Bellomo, S. Maniscalco, and G. Compagno, Int. J. Mod. Phys. B 27, 1345053 (2013)\\
- I. A. Silva, D. Girolami, R. Auccaise, R. S. Sarthour, I. S. Oliveira, T. J. Bonagamba, E. R. deAzevedo, D. O. Soares-Pinto, and G. Adesso, Phys. Rev. Lett. 110, 140501 (2013)\\

\noindent\textcolor{green}{our reply}\\
We added the following sentence:\\
"The
experimental and theoretical efforts, deployed in this
context, are essential to develop the appropriate strategies to
prevent against the de-coherence effects induced by the system-environment
coupling (see for instance the recent works [13,14,15] and  references
therein."\\
quoting the suggested references.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
2) Page 2, Introduction, third line of second paragraph. Besides references [13-16],
the authors should add some other relevant papers regarding the different measures of quantum correlations.
For instance, they could cite:\\
- B. Aaronson, R. Lo Franco, and G. Adesso, Phys. Rev. A 88, 012120 (2013)\\
- Paula F M, Montealegre J D, Saguia A, de Oliveira T R and Sarandy M S (2013) Europhys. Lett. 103, 50008\\
- T. R. Bromley, M. Cianciaruso, R. Lo Franco, and Gerardo Adesso, arXiv:1404.1409.\\

\noindent\textcolor{green}{our reply}\\
We added the suggested references.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
3) Page 2, Introduction, line 3 of the third paragraph.
 The adjective "semi-classical" is not a very usual term for the correlations
 in the community. I believe they mean the kind of correlations that are quantum only with respect
 to one part of the system (connected to the quantum-classical states). Therefore, I suggest
 the authors to make citations here to papers where this term is explicitly used or to briefly
 explain this expression, maybe inside brackets.\\

\noindent \textcolor{green}{our reply}\\
 Clearly the term semi-classical means correlations that are quantum only
 with respect to one component of the system.\\
 We added the following comment :\\
 ( related to quantum-classical states) after the term semi-classical.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
4) Page 3, Introduction, line 4 from the top.
The authors should change "mutual" with "total",
since the correlations are "total" not "mutual". The mutual information characterize the total correlations.\\

\noindent\textcolor{green}{our reply}\\
This is corrected. \\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
5) Page 3, line 4 of the last paragraph of Introduction. At the end of the sentence:
 "On the other hand, the symmetric part is exactly the Hilbert-Schmidt distance",
 the authors should cite Ref. [35], where this fact has been first derived.\\

\noindent\textcolor{green}{our reply}\\
The suggested reference is cited.\\




\noindent{\Large \textcolor{red}{concerning the comment}}\\
6) Page 3, third last line of the last paragraph of the introduction. When the authors write
about "the classical state and its closest product state", it is not very clear that also the
classical state must be the closest one to the system state. I suggest the authors to change
the sentence like: "In this situation the explicit derivations of the suitable closest product and classical states is achieved".\\

\noindent\textcolor{green}{our reply}\\
This is adjusted. \\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
7)  The title of section 2 should be corrected by adding "on" as:
 "Correlation quantifiers based on symmetrized linear relative entropy".
 This must be analogously done for the other subsection titles.\\

\noindent\textcolor{green}{our reply}\\
This is added. \\




\noindent{\Large \textcolor{red}{concerning the comment}}\\
8) Page 3, first line after Eq. (1). The authors should correct "... given quantum..." with "... given quantumness...".\\

\noindent\textcolor{green}{our reply}\\
This is corrected.\\





\noindent{\Large \textcolor{red}{concerning the comment}}\\
9) Page 4, first line. Please, correct "... densities matrices...." with "... density matrices...".\\

\noindent\textcolor{green}{our reply}\\
This is corrected.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
10) Page 4, second line from the top. Please correct "... von Neumann [34]" with "... von Neumann entropies [34]".\\

\noindent\textcolor{green}{our reply}\\
This is corrected.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
11) Page 4, the line before Eq. (4). At the end of the sentence the authors should explicitly write that
 $\chi_\rho$ is the closest classical state to $\rho$, so that the subsequent quantum and classical correlations are well defined.\\

\noindent\textcolor{green}{our reply}\\
Before the equation 4, we added :  $\cdots$ the state
$\rho$ and its closest classical state\\




\noindent{\Large \textcolor{red}{concerning the comment}}\\
12) Page 4, first line after E. (4). At the end of this line the authors must correct "... distance between the closest classical state $\chi_\rho$ ...".\\

\noindent\textcolor{green}{our reply}\\
This is corrected.\\


\noindent{\Large \textcolor{red}{concerning the comment}}\\
13) Page 4, first line after Eq. (5). The authors should correct "... based relative entropy ..." with "relative entropy-based ...". They should check it everywhere in the paper.\\

\noindent\textcolor{green}{our reply}\\
This is corrected.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
14) Page 4, last line before Eq. (8). At the end of this sentence the authors should cite Ref. [35], where this linearization first appears.
Moreover, they should correct Eq. (8) by changing the first $\rho_1$ with $\rho_2$, according to Eq. (11) of Ref. [35].\\

\noindent\textcolor{green}{our reply}\\
This is corrected.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
15) Page 5, Eq. (12). In this equation $S_+$ should appear instead of $S_-$. There is confusion here, since the author make reference to Eq. (9) and then write $S_-$.\\

\noindent\textcolor{green}{our reply}\\
Indeed,  there we should make reference to $S_-$. This is corrected.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
16) Page 5, first line after Eq. (13). I am not convinced that Eq. (13), containing the linear entropy $S_2$, is a symmetrized
form of the relative entropy, that contains logarithms. Maybe the authors mean "a linearized version of relative entropy". They should clarify this point.\\

\noindent\textcolor{green}{our reply}\\
after equation (13), we added "a symmetrized form of the linear relative entropy"\\




\noindent{\Large \textcolor{red}{concerning the comment}}\\
17) Page 5, last line before Eq. (14). At the end of the sentence "... the Hilbert-Schmidt distance ", the authors should cite Ref. [35], where this aspect has been first pointed out.\\

\noindent\textcolor{green}{our reply}\\
We quote the suggested reference.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
18) Page 5, second line after Eq. (14). The authors should change "symmetrized' with "symmetric".
In the successive line they should also change "antisymmetrical" with "antisymmetric", in order to be consistent with the definitions above.\\

\noindent\textcolor{green}{our reply}\\
This is corrected.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
19) Page 6, second line above Eq. (20). As usual, the authors should change " based linear relative entropy correlations" in "liner relative entropy based correlations".\\
\textcolor{green}{our reply}\\
This is corrected.\\


\noindent{\Large \textcolor{red}{concerning the comment}}\\
20) Page 7, first line after Eq. (26). The condition on the coefficients $c_1$, $c_2$ to be comprised between 0 and 1 should
be anticipated when these coefficients appear for the first time,
that is together with the condition $c_1+c_2 <= 1$ in the second line after Eq. (24). In fact, the state must be well-defined in both the representations (matrix and Fano-Bloch).\\

\noindent\textcolor{green}{our reply}\\
The condition $c_1+c_2 \leq 1$ is now given after equation (26).\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
21) Page 8, Eq. (28). To be complete, the authors should also write that the $\sigma_i (i=1,2,3)$ are the Pauli matrices and $\sigma_0$ is the identity matrix.\\

\noindent\textcolor{green}{our reply}\\
We defined $\sigma_i (i=1,2,3)$ and $\sigma_0$  (in this revised version) after equation (15). \\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
22) Page 8, Eq. (29). The authors should define this distance, because it never appears before. I believe it is defined by Eq. (17) plus (10)
for consistency, but the authors must define it here or before. Moreover, this definition will be useful for giving the explicit expressions
of the quantifiers of correlations, $T_2$ and the others, just in terms of this distance.\\

\noindent\textcolor{green}{our reply}\\
In the line before equation (29), we added: "Using the definition (16)"\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
23) Page 8, Eq. (32). Due to the known fact about the closest product state for the Hilbert-Schmidt
distance and relative entropy, for Bell-diagonal states the closest product state is expected to
 be the maximally mixed state, product of the marginals. This should be valid also for
 the $T_2$ based on linear relative entropy.
 The authors should check when their density matrix of Eq. (24) can reduce to
 a Bell-diagonal state and verify this fact. For instance, it is seen later in Eq. (37) that for $c_1=c_2$ the density matrix takes the form of a Bell-diagonal state.
 For this condition, $R_{30}=R_{03}=0 $ and $a_3=0$. This means that the closest product state of Eq. (32) is just the maximally mixed density matrix, as expected.
 The authors should comment on this point.\\

\noindent\textcolor{green}{our reply}\\
After equation (32), we added the following comment: \\
It is interesting to note that for $c_1=c_2$, the density matrix  $\rho$ (24) becomes a Bell-diagonal state.  In this special case, the matrix elements $R_{30}$
and $R_{03}$, given by (25), vanish and from (31) we have $a_3 = 0$. This implies that the closest product state (32) is just the product
of the marginals $\rho_1 = \frac{1}{2}\sigma_0$ and $\rho_2 = \frac{1}{2}\sigma_0$ of $\rho$.\\





\noindent{\Large \textcolor{red}{concerning the comment}}\\
24) Page 9, Eq. (33). I do believe that the authors use Eq. (29) to obtain the expression of $T_2$, together with $a_3$ given by Eq. (31).
They should write like this in the manuscript. Moreover, there is a difference between Eq. (29) and Eq. (33) and it is not clear how the two expressions can be equal.
The authors should check this point carefully, since it could be occurred a confusion between the geometric quantifier $T_g$ and the linear entropy based $T_2$.\\

\noindent\textcolor{green}{our reply}\\
We have checked this point. $T_2$, given by (33), is obtained by using the equation (17) and (10). $T_g$, given by (63), is exactly
the Hilbert-Schmidt distance (29). We believe that there is no risk of confusion.\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
25) Page 10, Figure 1 and the other figures as well.
This plot, as well as all the other plots present in the manuscript, is not clear.
The authors should indicate in the caption of any figure the values of $\alpha$ corresponding to the
solid colored curves. The displayed number size in the axes and the axes labels should be increased.
Moreover, in order to make the figures clear also in gray scale, the authors should use different dashing for the curves.\\

\noindent\textcolor{green}{our reply}\\
This is addressed.\\


\noindent{\Large \textcolor{red}{concerning the comment}}\\
26) Page 15, line 6 from the top. I suggest the authors to refer to $L_2^+$ as the "quantity" and not as the "linear relative entropy" since it is confusing.
Moreover, the authors should comment here about the fact that, when $c_1=c_2$, that is when the state is reduced to a Bell-diagonal state form, this quantity is zero and the total correlations
 are equal to the sum $ D_2 + C_2 = T_2$, as expected for Bell-diagonal states.\\

\noindent\textcolor{green}{our reply}\\
 Before figure 4, we added the following comment:\\
 "Remark that for
 $c_1=c_2$, the density matrix $\rho$ (24) has the form of a Bell-diagonal state.
 In this case,  we have $R_{03}=0$ and $a_3 = 0$ which implies that the quantity $L_2^+$ is zero. This
 shows that in Bell-diagonal states, the  total correlation $T_2$ is
 exactly the sum of quantum discord $D_2$ and classical
 correlation $C_2$."\\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
27) Page 17, Section 4. At the end of the concluding remarks, the authors should briefly comment on the possibility of
ambiguity in the definition of quantum and classical correlations due to degeneracies arising from the optimization procedure
over the state space, as pointed out in a very recent paper (F. M. Paula et al., arXiv:1408.1562).
 For instance, they could write that the possible redefinitions of the quantifiers based on linear relative entropy in order to remove this problem will be treated elsewhere.\\

\noindent\textcolor{green}{our reply}\\
We thank the referee for this interesting  reference and suggestion. We added in the conclusion the
following comment:\\
"Finally, we mention the general approach, recently proposed in [50],
to remove the unexpected ambiguities leading to multi-valued quantum and classical correlations. Such
ambiguities are essentially due  to degeneracies arising from
the optimization procedures of  the distance functions serving as correlations
measures. In this context, we believe that it is worthwhile to examine
 the possible redefinitions of linear
relative entropy quantifiers to avoid such problems.
We hope to treat this issue in
a forthcoming work."




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{\bf List changes/ Reviewer 2}
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\noindent{\Large \textcolor{red}{concerning the comment}}\\
In Eq. (17), all the relevant quantifiers are introduced. However, it is not clear enough
(at least it is not clearly explained) how difficult is the calculation of the closest states in general,
given that in the rest of the manuscript only a very simple case is treated explicitly
(for which the original discord could be calculated easily).
This point is in my opinion crucial, as the only advantage with respect to the full relative entropy is probably the analytical simplicity of the measure.\\


\noindent\textcolor{green}{our reply}\\
Indeed equation (17) gives all the relevant quantifiers expressed in terms of linear relative entropy. In this equation $T_2$, $D_2$ , $C_2$ and $L_2$
are the linearized version of relative entropy-based quantifiers. The explicit derivation of their
expressions requires minimization procedures that are in general very complicated to achieve. In this sense, the linear relative
offers an alternative way to tackle this issue. On other hand, the states of type  (24) are chosen such that solution of the equation (30)
can be explicitly derived (we note that for a general $X$ states such solution are extremely difficult to get (see
the equation (13) in the paper by Bellomo et al : arXiv:1112.6370 [quant-ph] ))\\
{\bf According to the reviewer's comment, we added the following comment (after Eq.(30))}:\\
"This cubic equation can be solved using Cardano's formula. It
is important to note  that for $X$ states without exchange symmetry, the explicit
determination of the Bloch coefficients $a_i$ and $b_i$  in (28) is complicated (see [41]).
This
explains why we deliberately chose to consider only  two qubit states of type (24)
 which are invariant under permutation and parity
transformations" \\



\noindent{\Large \textcolor{red}{concerning the comment}}\\
Another point I'd like to bring to the authors' attention concern Sec. 3.3.1,
that is, the calculation of quantum correlations for the $X$ state under study. As stated after Eq. (22)
$D_g=D_2$. In this case, I do not see any need to calculate the discord in such a way,
 as the result should have already be obtained in the literature (probably in Refs. [35,36]). Then, such a detailed section is in my opinion redundant.\\


\noindent\textcolor{green}{our reply}\\
In this subsection, we recall only the main results of geometric discord that are needed for our purpose. In particular,
we discuss
the conditions on the parameters $c_1$ and $c_2$ such that $\lambda_1 \geq \lambda_3$ or  $\lambda_3 \geq \lambda_1$. These
conditions are important to determine the amount of the other correlations.
In this respect, we think that there is no risk of redundancy.\\








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