Evaluation of Hamiltonians from Complex Symplectic Matrices
Round 1
Reviewer 1 Report
The authors analyze some interesting aspects about the Hamiltonians, particularly involving Gaussian unitaries which are transformations preserving Gaussian states. I have to remark that the Hamiltonians are mathematical objects governing the dynamic of any system if formulated appropriately. Then if the paper is written correctly, it could be interesting. However, before proceeding, I recommend the authors the following:
1). The abstract has English mistakes which should be addressed. For example:
"In the general N mode they may formulated by a second–order polynomial in the bosonic operators".
Instead, you should say "In the general N mode approximation ..."
2). In the Line 260: "The fist step", it should be "The first step..."
Please revise the typos and the English of the paper. Subsequently, I suggest the authors to extend the conclusions and include more references. Then for example since the authors are dealing with Hamiltonians, how could the proposed methods help in analyzing certain problems like the dynamic of condensed matter systems, phase transitions, etc. The authors should just mention a little bit about this. See for instance Symmetry 11 (2019) 803; Phys. Rev. 122, 345 (1961); as well as other similar references. After the authors revise the paper in agreement with these comments, then I would happily read the paper again.
Author Response
Please see the attachment.Author Response File: 
Author Response.pdf
Reviewer 2 Report
The article generalizes the mathematical apparatus that serves the physics of coherent states, which is important for problems, for example, quantum optics. In general, the material in the article is presented very clearly, but there are some comments that require correction of the text of the article.
Minor Remarks:
1. The file of the manuscript submitted for review does not contain Appendices 1-5, which are referenced in the text (Lines 67, 200, 201, 222, 245, 253, 266, 269, 275, 373, 381). There is no indication in the section headings that these are Appendices.
2. Lines 280, 282: The bean splitter is certainly a useful device and deserves separate consideration, but the matrix decomposition technique described in this article still applies to beam splitting.
3. Fig.6. The square of $c_2$ cannot be less than $c_0$, which is negative in $C_{--}$. Perhaps, in the domains $C_{--}$ and $C_{-+}$, in the inequalities defining the domains, one should write $|c_0|$ instead of $c_0$.
A number of errors, apparently related to problems with compiling the TeX file.
1. Line 237: `seefile(pp865.m)’.
2. Text before equation (128) and after (131): `seefile(pp852.m)’, `seefile(pp853.m)’.
3. Lines 331,332: `seefile(macro LHformula in file pp264)’.
4. Lines 383,384: `qc264ann.tex’.
Typos
1. Abstract, Lines 15,16: `modes modes’
2. Line 291: `the the’.
3. Lines 56, 71: unclosed brackets.
4. Line 233: `which can be evaluated by 95’.
5. Line 372: `wit’ -> `with’?
I recommend the article for publication in Symmetry, provided that comments are taken into account and corrections are made. The comments are not too significant and the manuscript, at the discretion of the editor, may not be re-sent for review.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Reviewer 3 Report
The present paper deals with Gaussian unitaries which have a fundamental role in the field of continuous variable. Generally, N mode they may formulated by a second–order polynomial in the bosonic operators. An other important role related to Gaussian unitaries is played by the symplectic transformations in the phase space. The paper investigates the links between the two representations: the link from Hamiltonian to symplectic, governed by an exponential, and the link symplectic to Hamiltonian, governed by a logarithm. Thus, an answer is given to the non trivial question: which Hamiltonian does produce a given symplectic representation? The complex instead of the traditional real symplectic representation is considered, with the advantage of getting compact and elegant relations. The application to the single, two and three modes modes illustrates the theory.
The manuscript is interesting and it has a potential to contribute to the theory. However, the language and the presentation of the manuscript must be improved.
Also, the following most recent papers in similar direction should be mentioned:
G. Milovanovic, Quadrature Formulas of Gaussian Type for Fast Summation of Trigonometric Series, Constr. Math. Anal., 2 (4) (2019), 168-182.
W. J. Schempp, Congruence and metaplectic covariance: rational biquadratic reciprocity and quantum entanglement, Constr. Math. Anal., 4 (1) (2021), 61-80.
Based on my above comments/remarks, the manuscript needs revisions. If the above all corrections are applied, I may reconsider my decision. I suggest revision.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
The authors have addressed partially the comments. Although I consider that mentioning some applications for the model would be relevant, still I think that the paper as a purely theoretical paper looks fine. However, I suggest an introduction without equations and then reserve the equations for a review in a subsequent section. Finally, still the conclusions are too short, please extend.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
