Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see the attached file.  

Comments for author File: Comments.pdf

Author Response

\section*{Replies to Reviewer 1} We thank the Reviewer for the positive evaluation of our work. We also appreciate the Reviewer’s helpful suggestions for improving the presentation of the manuscript.     \begin{itemize} \item \textbf{Line 77:} Yes, we have corrected the mistake. \item \textbf{Line 99:} Thank you for your comment. We have modified the text as follows: from "Consistency therefore requires that the distance between $|\psi\rangle$ and $|\psi\rangle + |d\psi\rangle$ be identical to the distance between $|\psi^\prime\rangle =e^{i\alpha}|\psi\rangle$ and $|\psi^\prime\rangle + |d \psi^\prime\rangle$." to "Therefore, consistency requires that the distance between$\lvert \psi_1 \rangle$ and $\lvert \psi_2 \rangle$ be the same as the distance between $e^{i\alpha}\lvert \psi_1 \rangle$ and $e^{i\beta}\lvert \psi_2 \rangle$, for any real $\alpha$ and $\beta$." \item \textbf{Line 114:} We agree with the reviewer. We add the following footnote: "Note that in the literature two distinct definitions of the non-infinitesimal Fubini--Study metric can be found. On the Bloch sphere, these correspond, up to a scale factor, to {\it (i)} the straight-line (chordal) distance between two points on the sphere, and {\it (ii)} the geodesic distance along the spherical surface. In the present work, we adopt definition {\it (i)}. The two definitions coincide in the infinitesimal limit." \item \textbf{Line 142:} Thank you. The mistake has been corrected, and we now refer to Eq. (13).     \item \textbf{Line 143:} We sincerely thank the Reviewer for the thorough analysis. We agree with the observation and have modified the manuscript accordingly. We have revised text as follows:   from   "Note that the set of kets in (12), obtained by varying the operators $U^\mu$, coincides with the equivalence class $[|\psi\rangle]$; moreover, all these kets share the same degree of entanglement."   to   "Note that all kets in (13) have the same degree of entanglement."   \item \textbf{Line 265:} Thanks again. We have revised text as follows:   from   "This proves that the concurrence for pure states is a special case of ED, valid for the case $M=2$ and $d_0 = d_1 = 2$"   to   "This proves that the concurrence for pure states is a special case of ED, valid for the case of two qubits."   \item \textbf{Subsection 3.3:} Thank for the observation. We have reduced the subsection 3.3 as follows: "In the special case of pure two-qubit states, the entropy of entanglement, $E_S(|\Psi\rangle)$, can be expressed explicitly as a function of the entanglement distance $E_D(|\Psi\rangle)$.   Indeed, the relation between the two-qubit concurrence and the entanglement entropy (for pure states) is well known and monotonic. Thus, using (39), one finds \begin{equation} E_S(|\Psi\rangle) = k_B\,F\!\left(\frac{1+\sqrt{1-\frac{E_D(|\Psi\rangle)}{2}}}{2}\right), \label{eq:ES_of_ED} \end{equation} where \begin{equation} F(x) = -x\ln x - (1-x)\ln(1-x). \end{equation} This relation follows directly from (39), see, for instance, Ref. [9], Eq.~(9)."     \end{itemize}  

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The authors investigate entanglement from a geometric perspective and introduce what they call an entanglement distance. However, geometric approaches to quantifying entanglement were proposed three decades ago. In particular, the geometric measure of entanglement (GME) was introduced and systematically developed in
A. Shimony, Degree of entanglement, Ann. New York Acad. Sci. 755, 675 (1995),
and later in
T. C. Wei and P. M. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states, Phys. Rev. A 68, 042307 (2003), as well as in numerous subsequent works.

The manuscript does not cite these foundational references, nor does it provide a clear comparison between the proposed “entanglement distance” and the established geometric measure of entanglement. As a consequence, it is unclear whether the authors’ construction represents a genuinely new entanglement measure, a reformulation of existing results using different terminology, or merely a special case of previously known frameworks.

Without a precise comparison to established geometric entanglement measures and a clear explanation of the conceptual or technical novelty of the present approach, the significance of the reported results remains questionable. In its current form, I therefore cannot recommend this manuscript for publication in Entropy.

Author Response

\section*{Replies to Reviewer 2}     We thank the Reviewer for bringing this important point to our attention, and we apologize for this significant omission. We have added a discussion of the Geometric Measure of Entanglement (GME), originally proposed several decades ago, as emphasized in the Reviewer’s report. In addition, we have included a comparison between the Entanglement Distance and the GME, and we have added the appropriate references to the relevant works.   We remain fully available to provide further details or clarifications regarding the GME, should the Reviewer require them.   \textbf{Line 49} We have added the following text to the Introduction:   "The Entanglement Distance is a measure derived from a geometric approach to quantum entanglement [6,42,43]. It is important to emphasize that, already around three decades ago, entanglement was investigated from a geometric perspective. In particular, in a pioneering work, Shimony introduced one of the earliest geometric definitions of the degree of entanglement for pure quantum states: the Geometric Entanglement Measure (GEM). In this framework, entanglement is interpreted as the squared Hilbert-space distance to the nearest separable state. Shortly thereafter, T. C. Wei and P. M. Goldbart reformulated and generalized Shimony’s idea in terms of overlaps rather than Euclidean distances, the measure was extended to mixed states via the convex roof construction and was proven to be an entanglement monotone.   The Entanglement Distance arises from a different geometric principle. Instead of measuring the distance to the set of separable states, ED is derived from the intrinsic Riemannian geometry of projective Hilbert space equipped with the Fubini–Study metric. Local unitary transformations generate orbits onto the projective Hilbert space. The elements of each orbit share the same degree of entanglement. Thus, ED is defined through the pullback of the Fubini–Study metric onto local unitary orbits."    

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors The paper is about the geometric aspects of entanglement, where the authors
have derived an entanglement distance which has the interpretation of a
quantum correlation measure. The paper is well written, clear, and consistent.
For the sake of improvement, the following are requested to be included: 1) The advantages of this entanglement distance over other geometrical correlation measures,
for instance, the Bures distance, and the Hellinger geometric discord. 2) Since this entanglement distance has the meaning of the quantum correlation measure,
it is good to add an illustration using an example of an entangled state
and calculating its entanglement distance. 3) A comparison between the concurrence and the entanglement distance
has been established in section 3.2, which drew our attention in the paper: “The Separability Problem in Two Qubits Revised”, Symmetry 2023, 15, 2089. https://doi.org/10.3390/sym15112089,
where the concurrence has been highlighted as a separability criterion.

 

Comments for author File: Comments.pdf

Author Response

\section*{Replies to Reviewer 3}   We thank the Reviewer for the positive evaluation of our work. We also sincerely appreciate the Reviewer’s valuable suggestions to improve the clarity of the manuscript.     \begin{enumerate}[label=\arabic*)] \item We have added the following comment to the Introduction: "Note that several other entanglement measures based on geometric correlations have been proposed in the literature, such as those based on the Bures distance [34] or on the Hellinger geometric discord [35]. However, ED has the advantage of admitting a closed, explicit mathematical expression that does not require any minimization procedure, unlike these latter measures."       \item We have added an explicit illustration of the calculation of the Entanglement Distance for a three-qubit entangled state depending on two parameters. In addition, we have included a three-dimensional plot of the ED as a function of the two parameters characterizing the state.   The results are reported in the new Section "Example: calculation of the Entanglement Distance"         \item We thank the Reviewer for the suggestion. We have added the reference and cited it in Sec. 3.2. as Ref. [50].     \end{enumerate}

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have addressed the points I made in my first report.  I recommend that the current version be published after two small changes are made:

1.  In the abstract, the word "independently" should be deleted, since concurrence and entanglement of formation for a pair of qubits are not independent.  
2. In the version I downloaded, Eq. (11) includes some nonsense characters.  I suppose this is  a simple typographical conversion problem, easily fixed.

Author Response

We thank the reviewer.

1) We agree with the comment, we have removed the word "independently" in the abstract.

2) We have fixed the error.

Thank you again.

Reviewer 3 Report

Comments and Suggestions for Authors

The additional details provide a better version, and the paper is recommended for publication.

Author Response

We thank the reviewer.