Implementation of Two-Mode Gaussian States Whose Covariance Matrix Has the Standard Form

Round 1

Reviewer 1 Report

This manuscript deals with the covariance matrix (CM) of two-mode Gaussian states, which, together with the mean vector, fully describes these states. The manuscript moves from the standard form (SF) introducing an architecture that implements with primitive components the given two-mode Gaussian state having the CM with the SF. Indeed, this structure uses a minimal number of components, namely: a beam-splitter, followed by two local squeezing operators on each mode, and another final beam-splitter. As far as I known, both the architecture and the implementation are correct. The advantage of this architecture is that it gives a precise non-redundant physical meaning to the generation of the Gaussian state. The manuscript is fairly well-written, all the work seems to be correct, and the results should be of interest to a reasonable number of researchers. Thus, I can recommend this manuscript for publication in Symmetry.

Author Response

Thank you very much for your comments. Actually the paper has undergone a review to meet the requirements of the other reviewers, in particular in the introduction we detailed better the contribution and the possible applications, the conclusions have been re-written completely and the first part slightly reduced.

Reviewer 2 Report

Report on symmetry-1773894:

The authors investigated the implementation of two–mode Gaussian states whose covariance matrix. In the two–mode the (ordinary) covariance matrix is a real symmetric matrix of order 4, therefore it depends on 10 real variables. However, there is a very efficient representation, called the standard form of the covariance matrix, where the degree of freedom is reduced to 4 real variables, while preserving all the relevant information on the state. The standard form can be easily evaluated using a set of symplectic invariants. Such results may have potential applications in quantum control devices and quantum networks.

Entanglement plays an important role in quantum communication and mechanics, the multi-mode entanglement works have been developed experimentally and theoretically. The authors deal with the covariance matrix of two–mode Gaussian states, which, together with the mean vector, fully describes these states. The paper moves from the standard form introducing an architecture that implements with primitive components the given two–mode Gaussian state having the covariance matrix with the standard form. The architecture consists of a beam splitter, followed by the parallel set of two single–mode real squeezers, followed by another beam splitter.

The manuscript reports a significant advance and offers incremental improvement to existing work in the references. The similar enhanced intensity-difference squeezing, multimode quantum correlation, non-Gaussian nature and entanglement [PRA 96, 043847 (2017); Ann. Phys. 422, 168316(2020); PRA 103, 013704 (2021)] are theoretically and experimentally studied in atomic and atomic-like ensemble. The authors should clarify the entanglement origin result from linear path indistinguishable superposition, or from nonlinearity multi-wave mixing processes? The English throughout this manuscript might be further revised to meet the standard of Symmetry. The authors should give a brief discussion on this issue in the INTRODUCTION section to highlight the novelty of the current work. In addition, the possible experimental realization techniques should be discussed ?

The author’s work of the implementation of two–mode Gaussian states whose covariance matrix is useful and interesting. The advantage of this mean vector architecture is that it gives a precise non-redundant physical meaning of the generation of the Gaussian state. Essentially, all the relevant information is contained in this simple vector architecture. After above improvement, I could recommend publication of the manuscript in Symmetry.

Author Response

Thank you very much for your comments.

Actually we revised all the paper trying to answer to all the isses raised by the reviewers. The main changes are outlined in blue in the new text. 

> The authors should clarify the entanglement origin result from linear path

> indistinguishable superposition, or from nonlinearity multi-wave mixing

> processes?

We specify in the revised version that entanglement in this case arises basically from the beam-splitter, therefore it can arise from  the non-classical nature of the input  fields for pure states, deriving for example by squeezing, or by multi-wave mixing again in the beam-splitter, due to the mixed thermal states that are taken as the most general input sources. 

We remarked that our main objective is to obtain a desired covariance matrix of the Gaussian state, therefore the actual source of the entanglement is not very relevant for the analysis of the covariance matrix.

together with the comments, we added also a couple of new references on that.

> The English throughout this manuscript might be further revised to meet the

> standard of Symmetry.

We revised the English with the help of a native speaker and we hope that the version meets the standard of Symmetry. 

> The authors should give a brief discussion on this issue in the INTRODUCTION

> section to highlight the novelty of the current work. In addition, the possible

> experimental realization techniques should be discussed ?

We have rewritten completely the introduction, point out applications of the theory and experiment with Gaussian states.

Reviewer 3 Report

The paper is very well written and almost belongs to the class of the seminal paper.

I have no significant comments regarding science, i.e., the physics of Gaussian states. However, it's still possible to improve the text. 

Comments:

For a broad readership of  The  Symmetry journal, its essential to provide a detailed explanation /definition of thermal and vacuum states and link them with von Neumann entropy and energy

The conclusion part should be rewritten entirely. The conclusion should be clear, concise, and unambiguous.

The part under the titles reader 1, 2, and 3 should be transferred to Appendix and extended if the authors want.

Also, the possible applications of the presented research in materials science, optics, structure light, quantum optics, and technology are missing. It's important to highlight the potential application of Gaussian states' fundamental and foundational study.

Author Response

Thank you for your review. Please check the new version , with the main changes outlined in blue.

> For a broad readership of  The  Symmetry journal, its essential to provide a

> detailed explanation /definition of thermal and vacuum states and link them

> with von Neumann entropy and energy

We added a description of thermal and vacuum states, with new references to 

> The conclusion part should be rewritten entirely. The conclusion should be

> clear, concise, and unambiguous.

> The part under the titles reader 1, 2, and 3 should be transferred to Appendix

> and extended if the authors want.

We have rewritten completely the conclusions, moving the suggested reading tracks to a new Appendix.

> Also, the possible applications of the presented research in materials science,

> optics, structure light, quantum optics, and technology are missing. It's

> important to highlight the potential application of Gaussian states'

> fundamental and foundational study.

We have changed in a very substantial way the introduction, pointing out several applications.