Impact of Imbalanced Modulation on Security of Continuous-Variable Measurement-Device-Independent Quantum Key Distribution

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper presents a research article where the impact of imbalanced modulation at transmitters on the security of CV-MDI-QKD system has been investigated using coherent state and squeezed state for symmetric and asymmetric distances. The Authors show that imbalanced modulation can achieve a higher secret key rate and transmission distances respect to the balanced modulation protocol.

Upon a thorough examination of the manuscript, Prior to formal acceptance, I have few suggestions for the Authors' consideration:

_ Line 51-55: the sentence is not clear to me. I recommend rephrasing the sentence.
_ Line 83: perhaps the Authors meant “different” instead of “difference”.
_ Figure 1: the figure might be more intuitive if the Authors point out the \eta parameter in the figure and not only describe it in the caption.
_ Line 91: the abbreviation PM has not been explained.
_ Line 95: “blends” might not be the appropriate word. Maybe, “mixes” might be better.
_ Line 97: “two realistic homodyne detectors”
_ Line 98-102: the sentence is not clear to me. I recommend rephrasing the sentence.
_ Line 119-122: the sentence is not clear to me. I recommend rephrasing the sentence.
_ Figure 2: from the text and the caption is not clear why the graph of R is not symmetric versus the modulation variances. Perhaps, an explanation is necessary.
_ Line 188 I recommend the following paper to cite: Pirandola, S., Laurenza, R., Ottaviani, C. and Banchi, L. Fundamental limits of repeaterless quantum communications. Nat. Commun. 8, 1–15 (2017).
_ Figure 6: the unit “(km)” is missing in the label of the x-axis.
_ Figure 7: The Authors describe the plot and the result but does not show why the maximum transmission distance is obtained for these specific squeezed parameters.

_ Investigating the case of efficiency different from 1 might improve the quality of the paper.

_ The theory Section should be improved and should report more details in order to be able to derive all the plots published in this paper.

Author Response

The paper presents a research article where the impact of imbalanced modulation at transmitters on the security of CV-MDI-QKD system has been investigated using coherent state and squeezed state for symmetric and asymmetric distances. The Authors show that imbalanced modulation can achieve a higher secret key rate and transmission distances respect to the balanced modulation protocol.

Upon a thorough examination of the manuscript, Prior to formal acceptance, I have few suggestions for the Authors' consideration:

1. Line 51-55: the sentence is not clear to me. I recommend rephrasing the sentence.

Our responses:

Thanks for the reviewer’s valuable comments. We have rephrased the relevant sentence to improve its clarity. The revised sentence is as follows: The results of the above literature indicate that when the distances from Alice to the untrusted third party is equal to the distances from Bob to the untrusted third party (the symmetric case), the performance of the protocol, in terms of both maximum secure transmission distances and secret key rate, are lower than in the asymmetric case, where these distances are different.

According to the reviewer’s suggestion, we give a rephrased sentence and highlight it in the revised manuscript.
2. Line 83: perhaps the Authors meant “different” instead of “difference”.

Our responses:

Thanks for the reviewer’s valuable comments. To clarify, the term "different" is indeed what we intended to use in the sentence. The revised sentence is as follows: We then elucidate the different performance between balanced modulation and imbalanced modulation.

According to the reviewer’s suggestion, we give a modified sentence and highlight it in the revised manuscript.

3. Figure 1: the figure might be more intuitive if the Authors point out the parameter in the figure and not only describe it in the caption.

Our responses:

Thanks for the reviewer’s valuable feedback. We agree that including the detection efficiency  parameter directly in Fig.1 would make it more intuitive. We have revised the figure to append the detection efficiency  parameter within the Fig.1, in addition to describing it in the caption. We hope this modification enhances the clarity and intuitiveness of the Fig.1.

4. Line 91: the abbreviation PM has not been explained.

Our responses:

Thanks for the reviewer’s valuable comments. We apologize for the oversight about explaining the abbreviation PM. The abbreviation "PM" in line 91 is prepare-and-measure(PM) used in this context to denote the scheme. The revised sentence is as follows: In general, either coherent or squeezed state is prepared as modulated source for the preparate and measurement (PM) scheme.

According to the reviewer’s suggestion, we give a modified sentence and highlight it in the revised manuscript. We hope this addresses reviewer’s concern.

5. Line 95: “blends” might not be the appropriate word. Maybe, “mixes” might be better.

Our responses:

Thanks for the reviewer’s valuable comments. We appreciate your suggestion regarding the word choice. We have reviewed the relevant section and agree that "mixes" is a more appropriate term than "blends". We have made the necessary revision to enhance the clarity and accuracy of the text.

The updated sentence is as follows: The third party Charlie receives two independent quantum signals and mixes each other with a beam splitter(50:50).

According to the reviewer’s suggestion, we give a modified sentence and highlight it in the revised manuscript.   

6. Line 97: “two realistic homodyne detectors”

Our responses:

Thanks for the reviewer’s valuable comments. We have reviewed the wording and ensured that it accurately describes the detectors used. We have checked throughout the manuscript and correct the similar errors. The revised sentence is as follows: The output port of the interfered mode signal is measured with  quadrature by two realistic homodyne detectors:

According to the reviewer’s suggestion, we give a modified sentence and highlight it in the revised manuscript.

7. Line 98-102: the sentence is not clear to me. I recommend rephrasing the sentence.

Our responses:

Thanks for the reviewer’s valuable comments. We appreciate reviewer’s suggestion to rephrase the sentence in lines 98-102 for clarity. We have rephrased the relevant sentence to improve its clarity. The revised sentence is as follows: The detection results are publicly announced to Alice and Bob through an authenticated classical channel. The knowledge of public outcomes can enable each party (Alice or Bob) to infer the measurement results of the other party by data post-processing. In particular, the performance of modulated-squeezed state can aggrandize the resistant capacity against a two-mode Gaussian attack when Alice and Bob generate or quadrature-squeezed-states. If we assume a  quadrature-squeezed state is prepared, the modulation variances  and   should satisfy the equation

                                                                                

Where  and  denote the squeezing and anti-squeezing variances of the states, respectively.  is the excess noise of the anti-squeezing.

According to the reviewer’s suggestion, we give a modified sentence and highlight it in the revised manuscript. We hope this addresses reviewer’s concern.

8. Line 119-122: the sentence is not clear to me. I recommend rephrasing the sentence.

Our responses:

Thanks for the reviewer’s valuable comments. We have rephrased the relevant sentence to improve its clarity. The revised sentence is as follows: The third-party Charlie perform a CV Bell state measurement. On this station, the two received imbalanced modulated modes  and  are interfered at a 50:50 beam splitter (BS), then the output two modes C and D are transformed into the modes and  to model the realistic homodyne detector with detection efficiency  and electronic noise .

According to the reviewer’s suggestion, we give a modified sentence and highlight it in the revised manuscript.

9. Figure 2: from the text and the caption is not clear why the graph of R is not symmetric versus the modulation variances. Perhaps, an explanation is necessary.

Our responses:

Thanks for the reviewer’s valuable comments. The graph of R is not symmetric versus the modulation variances. The explanation is as follows: In the course of calculating the secret key rate, the equivalent excess noise due to the introduction of thermal noise from Eve in both quantum channels is not symmetric. If Alice is encoder, Bob can modify his data while Alice keeps hers unchanged during the post-processing step. This means that the symmetric case cannot result in an optimal system performance and imbalanced variance modulation cannot result in a symmetric secret key rate topological distribution.

According to the reviewer’s suggestion, we will give a modified contents and highlight it in the revised manuscript.

10. Line 188 I recommend the following paper to cite: Pirandola, S., Laurenza, R., Ottaviani, C. and Banchi, L. Fundamental limits of repeaterless quantum communications. Nat. Commun. 8, 1–15 (2017).

Our responses:

Thanks for the reviewer’s valuable comments. According to the reviewer’s suggestion, We have cited the reference “Pirandola, S., Laurenza, R., Ottaviani, C. and Banchi, L. Fundamental limits of repeaterless quantum communications. Nat. Commun. 8, 1–15 (2017)” in the revised manuscript.
11. Figure 6: the unit “(km)” is missing in the label of the x-axis.

Our responses:

According to the reviewer’s suggestion, the unit “(km)” have been added.
12. Figure 7: The Authors describe the plot and the result but does not show why the maximum transmission distance is obtained for these specific squeezed parameters.
Our responses:

Thanks for the reviewer’s valuable comments, which has significantly improved the clarity and completeness of our manuscript. In our study, we analyze the impact of squeezed parameters on the transmission distance of the optical signal. The maximum transmission distance is obtained for these specific squeezed parameters due to the following reasons: Under the case of extreme asymmetric distances, different combinations of squeezing parameters can achieve different amounts of Shannon’s mutual information and Holevo bound  describing Eve’s information obtained from the entangling cloner. Asymmetric combinations between squeezing parameters  and  can get relatively higher secret key rate and transmission distances comparing the commonly used symmetric modulation. Squeezing state reduces the quantum noise in one quadrature of the light field at the expense of increasing the noise in the orthogonal quadrature. By optimizing the squeezing parameters, we minimize the noise in the quadrature that is most sensitive to transmission losses. The chosen squeezed parameters represent an optimal balance between noise reduction and the loss of signal.

To provide a clearer understanding, we have included additional explains and discussions in the revised manuscript that illustrate the relationship between the squeezed parameters and the transmission distances. These additions demonstrate how variations in the squeezing affect the key performance metrics, supporting our conclusion about the optimal parameters."

We believe this added explanation provides the necessary context and justification for why the specific squeezed parameters yield the maximum transmission distance.
13. Investigating the case of efficiency different from 1 might improve the quality of the paper.
Our responses:

Thanks for the reviewer’s valuable comments, which has significantly improved the clarity and completeness of our manuscript. In addition to our original analysis, we have now investigated the impact of detection efficiencies different from 1 on the performance of our revised manuscript. In this context, refers to the transmission and detection efficiency of the optical components, which can significantly affect the overall system performance.

We considered a range of detection efficiency values from 0.9 to 1.0 to account for realistic imperfections in the optical components. This range reflects typical losses encountered in practical implementations due to factors like imperfect detection. Our analysis shows that as the detection efficiency decreases, the maximum transmission distance is reduced. This is due to increased losses, which degrade the signal quality and limit the effective range of the optical communication. The results provide practical insights for improving the robustness and performance of quantum communication schemes under non-ideal conditions.

We have included these findings in almost every section of the revised manuscript, which discusses the methodology, results, and implications of varying detection efficiency. We believe this additional analysis significantly enhances the depth and applicability of our research. Thanks again for reviewer’s constructive feedback.

14. The theory Section should be improved and should report more details in order to be able to derive all the plots published in this paper.

Our responses:

We have carefully reviewed and revised the theory section to include more comprehensive details and derivations that support the plots presented in our paper. Below are the specific improvements we have made:

We have expanded the theoretical framework to include detail derivations of the key equations and formulas used in our analysis. This includes the mathematical treatment of the coherent and squeezed states, as well as the impact of varying efficiencies on these states.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The article "The effect of unbalanced modulation on the security of the distribution of quantum keys independent of continuous variable measuring devices" is devoted to the urgent problem of increasing the transmission rate of the secret key in optical fiber links and is of interest to a wide range of scientists. The authors present numerical calculations of the dependence of the secret key rate on some physical parameters, in particular, on the distance between Alice-Charlie and Bob-Charlie, types of laser beam properties.

They use the well–known general relations (1) - (4) to estimate the secret key rate for one of the possible schemes based on entangled states. Unfortunately, the authors do not show how the main functions used depend on the L_AC and L_BC distances. In addition, the above equations do not contain a difference between the coherent and squeezed states of the laser beam. Without taking into account these dependencies, all numerical calculations and corresponding conclusions have little meaning and significance. To eliminate these shortcomings, the authors need to do the following.

1.     To show the dependence of the main functions on the distances L_AC and L_BC.

2.     To show how the difference between coherent and compressed states of the laser beam can manifest itself in the analyzed optical scheme.

3.      They also need to show whether the result depends on the location (L_AC or L_BC) of Eve's attack.

There are many typos in the manuscript. Namely, double-repeated words (for example, line 27), the absence of intervals, in Figure 1 “Charile” and so on.

Author Response

Comments and Suggestions for Authors

The article "The effect of unbalanced modulation on the security of the distribution of quantum keys independent of continuous variable measuring devices" is devoted to the urgent problem of increasing the transmission rate of the secret key in optical fiber links and is of interest to a wide range of scientists. The authors present numerical calculations of the dependence of the secret key rate on some physical parameters, in particular, on the distance between Alice-Charlie and Bob-Charlie, types of laser beam properties.

They use the well–known general relations (1) - (4) to estimate the secret key rate for one of the possible schemes based on entangled states. Unfortunately, the authors do not show how the main functions used depend on the L_AC and L_BC distances. In addition, the above equations do not contain a difference between the coherent and squeezed states of the laser beam. Without taking into account these dependencies, all numerical calculations and corresponding conclusions have little meaning and significance. To eliminate these shortcomings, the authors need to do the following.

1. To show the dependence of the main functions on the distances L_ACand L_BC.

Our responses:

Thanks for the reviewer’s valuable comments, which has significantly improved the clarity and completeness of our manuscript. By analyzing the relationship between these distances and the corresponding changes in the main functions, we aim to provide a comprehensive understanding of their interdependence of the main functions on the distances L_AC and L_BC within the scheme.

For CV-MDI-QKD protocols, it has been proved that the performance of asymmetric case is superior to the symmetric case. If Alice is the encoder of information, the transmission distance L increases significantly as L_AC decreases. If Charlie’s location is close to Alice, the total transmission distance L will increase to its maximal value with the same parameters. We also investigate the performance of the protocol in the most asymmetric case, in which two-mode Gaussian attack degenerates into two independent Gaussian attacks. Based on the above comparisons, we have optimized the imbalanced modulation variances for both the coherent-state and squeezed-state protocol. For the sake of illustrating the point more clearly, the case of different distances from Alice to Bob is demonstrated on the secret key rate and maximum transmission distances. We have found that optimal network configuration is under the case of extreme asymmetric.

We have revised the manuscript to include this clearer explanation. Specifically, we have elaborated on how variations in the distances L_AC affect the performance of the protocol within our numerical calculation setup. Figure 6 shows the secure key rate in different asymmetric case, when L_AC increases, the maximal L decrease dramatically. If Charlie’s position can be close to Alice, the total transmission distance L can be a relatively longer distance.

This analysis helps us to elucidate the critical role that spatial parameters play in optimizing the overall system. We appreciate your guidance in helping us improve the clarity and precision of our manuscript. We believe these changes will enhance the reader's understanding of the significance of our study.

2. To show how the difference between coherent and compressed states of the laser beam can manifest itself in the analyzed optical scheme.

Our responses:

Thanks for the reviewer’s valuable comments, which has significantly improved the clarity and completeness of our manuscript. We provide definitions and descriptions of both coherent and squeezed states in section 2. If we assume a  quadrature-squeezed state is prepared, the modulation variances  and   should satisfy the equation , Where  and  denote the squeezing and anti-squeezing variances of the states, respectively.  is the excess noise of the anti-squeezing. If s=1, the squeezed state degenerate to a coherent state. In experimental papers, squeezing is often measured in decibels, defined so that a squeezing degree r corresponds to 10log₁₀(e^2r)dB. Thus, 10 dB corresponds to r=1.15.

We have updated the manuscript to reflect this clearer and more detailed explanation difference between coherent and squeezed states of the laser beam. We believe this revised manuscript will help readers better understand the significance of our study and the implications of using different laser beam states within the performance of the protocol.

3. They also need to show whether the result depends on the location (L_ACor L_BC) of Eve's attack.

Our responses:

Thanks for the reviewer’s valuable comments, which has significantly improved the clarity and completeness of our manuscript.

In each use of the relay, Eve may intercept the two modes A and B, and make them interact with an ensemble of ancillary vacuum modes via a general unitary U. Among the output mode, two are sent to a simulator of the relay, where they are homodyned and the result  broadcast. The remaining modes E are stored in a quantum memory, which is measured at the end of the protocol.               

In general, the injected state of the two ancillas can be separable or entangled. The simplest case is when there are no correlations, so that injected state is a tensor product and the attack collapses into a collective one-mode attacks with two independent entangling cloners. The most realistic scenario is the Gaussian attack depicted in Fig.1. In this attack, the two travelling modes A and B are mixed with two ancillary modes, E₁ and E₂, respectively.

In conclusion, the simulation results depend on the location of third party Charlie (L_AC or L_BC). Eve’s attack including one-mode attack and two-mode attack which could be anywhere on channel, can influence the performance of the protocol.

We have updated the revised manuscript to include this clearer and more detailed explanation. We believe this revision will help readers better understand the importance of the third part Charlie’s location in our study and the implications for the attack pattern of Eve on the security and performance of the protocol.

4. There are many typos in the manuscript. Namely, double-repeated words (for example, line 27), the absence of intervals, in Figure 1 “Charile” and so on.

Our responses:

Thank you for your thorough review of our manuscript and for your valuable feedback. We have carefully addressed the typographical errors you identified. Below is a detailed response to your comments: The double-repeated word on line 27 has been corrected. We have thoroughly reviewed the entire manuscript to ensure all instances of this issue have been resolved. We have corrected the missing spaces between words throughout the manuscript to ensure proper spacing. The spelling error "Charile" in Figure 1 has been corrected to the intended term.

We greatly appreciate you pointing out these issues. We have performed a comprehensive proofreading of the revised manuscript to ensure all typographical and formatting errors have been corrected.

We have uploaded the revised version of the manuscript for your review. Thank you again for your constructive comments and your assistance in improving the quality of our work.

 

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Considering the fact that quantum key distribution (QKD) is attracting great interest in the field of secure optical communications, the work presented here on a continuously variable QKD protocol based on the measurement-device-independent quantum cryptography using unbalanced modulation may be significant in this field. I recommend the publication of this work if it contains some clarifications and improvements:

 

1) In line 27, in the sentence “Owing to the the advantage...” delete one “the”.

2) In the introduction, the authors define the quantum key distribution as unconditionally secure, which is guaranteed by the primary principles of quantum mechanics, but later they discuss all possible attacks. There is an inconsistency here. If quantum cryptography based on physical principles were really unconditionally secure, why do we know the different attack possibilities. The authors should clarify or resolve this inconsistency.

3) The authors should explain the structure and purpose of the EPR sources shown in Figure 1.

4) Authors should ensure that the X and Y axes of all graphs are always labeled and the units are clearly stated. For charts with multiple data series, legends should be provided and the series should be sufficiently distinguishable.

5) All acronyms (standard acronyms and author-defined acronyms) should be defined the first time they are mentioned in the paper. For example, PSA is not defined as “phase-sensitive amplifier” but is used in the conclusion. Similarly, BS is not defined as a “beam splitter” and is not labeled in Figure 1. Furthermore, the PLOB is never defined in the paper.

After my main objections to the manuscript have been answered, corrected and added as further explanations, the paper can be accepted.

Author Response

Comments and Suggestions for Authors

Considering the fact that quantum key distribution (QKD) is attracting great interest in the field of secure optical communications, the work presented here on a continuously variable QKD protocol based on the measurement-device-independent quantum cryptography using unbalanced modulation may be significant in this field. I recommend the publication of this work if it contains some clarifications and improvements:

1. In line 27, in the sentence “Owing to the the advantage...” delete one “the”.

Our responses:

Thanks for the reviewer’s valuable comments. We have corrected this typo by deleting the extra "the" in line 27. We appreciate your attention to detail and believe this correction improves the clarity of our manuscript.

2. In the introduction, the authors define the quantum key distribution as unconditionally secure, which is guaranteed by the primary principles of quantum mechanics, but later they discuss all possible attacks. There is an inconsistency here. If quantum cryptography based on physical principles were really unconditionally secure, why do we know the different attack possibilities. The authors should clarify or resolve this inconsistency.

Our responses:

Thanks for the reviewer’s valuable comments. We appreciate your comment and explain the introduction to clarify this point. Quantum key distribution (QKD) is often described as unconditionally secure based on the fundamental principles of quantum mechanics. The security of quantum key distribution is guaranteed by the basic principles of quantum mechanics, and any eavesdropping behaviors are detectable due to the inevitable perturbation of the quantum states on which the key information is encoded. Under some ideal assumptions, the CV-QKD protocols can be strictly proved to be unconditionally secure. However, in realistic implementations, the discrepancy between practical devices and their ideal models may lead to some potential security loopholes. So the eavesdropper can exploit these loopholes to carry out different attack and acquire secret key information without being noticed. Therefore, while the theoretical foundation of QKD promises unconditional security, practical considerations necessitate the study and mitigation of these possible attacks to ensure the robustness of QKD systems. This revised manuscript aims to reconcile the theoretical security of QKD with the practical challenges and potential vulnerabilities that need to be addressed in real-world implementations.

We have updated the manuscript accordingly and believe this clarification resolves the inconsistency. Thank you for your constructive feedback.

3) The authors should explain the structure and purpose of the EPR sources shown in Figure 1.

Our responses:

Thanks for the reviewer’s valuable comments. In our paper, we focus on the equivalent entanglement-based(EB) model of the protocol (see reference [F. Grosshans, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, Quantum Inf. Comput. 3, 535 (2003)] for more details) instead of the prepare and measure(PM) version, for the former scheme is often using in the security analysis of QKD, while the latter is easy to implement. Once the security of the EB scheme is proved, the security of the PM version is easily obtained because of the equivalence between two schemes (see reference [Z. Li, Y. Zhang, F. H. Xu, X. Peng, and H. Guo, Phys. Rev. A 89, 8, 052301 (2014)] for more details), EPR denote two-mode squeezed state. Alice and Bob keep mode A and B on each side, and the send the other modes A₀ and B₀ to the untrusted third party (Charlie) through two insecure quantum channels. By measuring the mode A and B using different detection mode(homodyne or heterodyne ), the modes A₀ and B₀ are projected to coherent or squeezed state as the source of the protocol.

We have updated the manuscript to include this detailed explanation, providing a clear understanding of the structure and purpose of the EPR sources in our optical scheme.

4) Authors should ensure that the X and Y axes of all graphs are always labeled and the units are clearly stated. For charts with multiple data series, legends should be provided and the series should be sufficiently distinguishable.

Our responses:

Thanks for the reviewer’s valuable comments. We appreciate your attention to detail and agree that clear labeling and distinction in graphs are essential for effective communication of our results. We have carefully reviewed each figure in the manuscript to ensure these standards are consistently applied. We believe these changes enhance the clarity and readability of our graphical data presentations. Thank you again for your constructive comments.

5) All acronyms (standard acronyms and author-defined acronyms) should be defined the first time they are mentioned in the paper. For example, PSA is not defined as “phase-sensitive amplifier” but is used in the conclusion. Similarly, BS is not defined as a “beam splitter” and is not labeled in Figure 1. Furthermore, the PLOB is never defined in the paper.

Our responses:

Thanks for the reviewer’s valuable comments, which has significantly improved the clarity and completeness of our manuscript. We appreciate your attention to detail and agree that all acronyms should be clearly defined to ensure readers can easily follow the content. We have made the following changes to address these issues:

We have defined "PSA" as "phase-sensitive amplifier" the first time it is mentioned in the manuscript, "BS" is now defined as "beam splitter" upon its first mention and is also labeled in Figure 1. "PLOB" has been defined as "Pirandola-Laurenza-Ottaviani-Banchi bound" at its first mention in the paper and cite the related references.

We have reviewed the entire manuscript to ensure all acronyms are defined at their first occurrence. We have also ensured that all relevant figures, including Figure 1, are appropriately labeled with defined acronyms.

We believe these revisions will enhance the clarity and readability of our manuscript, making it more accessible to readers.

After my main objections to the manuscript have been answered, corrected and added as further explanations, the paper can be accepted.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I thank the Authors for implementing and taking into consideration my observations and suggestions to the manuscript. I found an imprecision:

line 159-161: Spaces between words are needed.

Author Response

Comments and Suggestions for Authors

1. I thank the Authors for implementing and taking into consideration my observations and suggestions to the manuscript. I found an imprecision:
   line 159-161: Spaces between words are needed.

Our responses:

Thank you for your thorough review of our manuscript and valuable feedback. There are no mistakes in the Latex files, but there are the errors in the pdf files due to our negligence. We have carefully addressed the typographical errors you identified. We have performed a comprehensive proofreading of the revised manuscript to ensure all typographical and formatting errors have been corrected. Thank you again for your constructive comments and your assistance in improving the quality of our work.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The authors took into account all the comments and gave the necessary answers.

Unfortunately, there are many typos yet. See, for example, lines 159- 161. The intervals between words or characters are often skipped.

Author Response

Comments and Suggestions for Authors

1. The authors took into account all the comments and gave the necessary answers.

Unfortunately, there are many typos yet. See, for example, lines 159- 161. The intervals between words or characters are often skipped.

Our responses:

Thank you for your thorough review of our manuscript and valuable feedback. There are no mistakes in the Latex files, but there are the errors in the pdf files due to our negligence. We have carefully addressed the typographical errors you identified. We have performed a comprehensive proofreading of the revised manuscript to ensure all typographical and formatting errors have been corrected. Thank you again for your constructive comments and your assistance in improving the quality of our work.

Author Response File: Author Response.docx