Distance between Bound Entangled States from Unextendible Product Bases and Separable States

Round 1

Reviewer 1 Report

In the manuscript, entitled “Distance between bound entangled states from unextendible product bases and separable states”, Marcin Wiesniak et al employ a tailor entanglement witness method for unextendible product basis bound entangled states by using the Gilbert algorithm. They demonstrate their numerical results of their algorithm and show that their witness is more optimal than the ones given by the BGR construction.

 

This topic is interesting, but I could not see what the new significance brought by the work. I cannot support its publication unless the authors can give a convincing argument on this.

 

It seems that this work just applies the approximation method and algorithm discussed in [10] and [11] to a special case, bound entangled states. In the most of the first 4 pages (out of 7), the authors review the results in literature and another page for reference. Even the algorithm has already been presented in literature. From what I can see of Section 5 (one page of the authors work), the authors merely wrote a program according to the algorithm and apply it to bound entangled states. Such kind of implementation does not qualify as a scientific publication.

 

The meaning of different colored points in the caption of figure 2 is not consist with the one in the main text, which may lead to different conclusions of which method is more optimal.

 

There are several clerical errors and format problems in this manuscript, e.g. the parenthesis is missing by half in line.50, and figure 1 is hard to understand. These errors should be checked again and revised correctly.

Author Response

We thank the the reviewer for his time and effort put in preparing this report.
(S)he writes:

"In the manuscript, entitled “Distance between bound entangled states from unextendible product bases and separable states”, Marcin Wiesniak et al employ a tailor entanglement witness method for unextendible product basis bound entangled states by using the Gilbert algorithm. They demonstrate their numerical results of their algorithm and show that their witness is more optimal than the ones given by the BGR construction.

This topic is interesting, but I could not see what the new significance brought by the work. I cannot support its publication unless the authors can give a convincing argument on this.

It seems that this work just applies the approximation method and algorithm discussed in [10] and [11] to a special case, bound entangled states. In the most of the first 4 pages (out of 7), the authors review the results in literature and another page for reference. Even the algorithm has already been presented in literature. From what I can see of Section 5 (one page of the authors work), the authors merely wrote a program according to the algorithm and apply it to bound entangled states. Such kind of implementation does not qualify as a scientific publication."


We thank the Referee for an accurate description of our work. Naturally, we cannot agree on total insignificance of our work. First, we confirm that the main methodology was already introduced in Ref. [10]. However, it is not unusual to introduce an analytical tool in one work and apply it to more specific cases in subsequent works. The difference between this manuscript and Ref. [10] is that here we specifically focus on the final state yielded by the algorithm as an operator generating the entanglement witness rather than studying the convergence of the distance. We must agree that this is not a major improvement, but certainly it is one. We are quite successful with this analysis, finding witnesses for 133 out of 146 states. Furthermore, these witnesses are more optimal than BGR witnesses.

The improvement of these witnesses has its experimental consequences. Entangled states, regardless of their form, are useful at the very least for randomness amplification or cryptographic key extraction. In an experiment, we never perfectly realize the reference state, there is always a slight deviation. This deviation could be larger for the state still to reveal entanglement with more optimal witnesses. In other words, witnesses found in this manuscript allow for more experimental error than BGR witnesses.

We believe that the composition of our manuscript is not its disadvantage, describing the results we use on one hand, and being clear but brief on the novel contribution on the other. All the necessary references are given, where more details can be found if desired.

As a result of this remark by the Referee, we have added the following paragraph to conclusions:
'Our results therefore have multi-fold scientific aspects. First, we have partially found the order of degree of entanglement of UPBBE states, by recognizing that in the given dimension, those farthest from the set of separable states have a smaller central tile. Second, the witnesses yielded by the algorithm are (in most cases) more optimal than the BGR construction. As a consequence, our witnesses allow for larger imperfections in an experimental realizations. This could be relevant in quantum randomness amplification and cryptographic key distribution protocols.'

"The meaning of different colored points in the caption of figure 2 is not consist with the one in the main text, which may lead to different conclusions of which method is more optimal."

We thank the Referee for pointing out this mistake. It was corrected.

"There are several clerical errors and format problems in this manuscript, e.g. the parenthesis is missing by half in line.50, and figure 1 is hard to understand. These errors should be checked again and revised correctly."

We have performed a spellcheck. The figure was changed and we hope that the Referee will find it more helpful.

In conclusion, we hope that we have presented the relevance of our results in a way convincing to the Referee. We have followed his suggestions concerning improvement of our work.

Reviewer 2 Report

The authors study application of own developed (numerical) method, based on Gibert algorithm, of tailoring an approximate entanglement witness (W) for UPB BE states. The approximate method, based on fact that  W lies in the plane perpendicular to the line between the reference state and its CSS, treats the tilt in this plane due to approximation on CSS (in the sense of Hilbert-Schmidt measure). The authors argue that their method provides „more optimal“ W (i.e. the better approximation to) than the method suggested by Bandyopadhyay, Ghosh and Roychowdhury (BGR). Their results demonstrate that indeed the proposed method, based on Gilbert algortihm, in considered Hilbert d x d space, d=3,4,5,6, finds in most cases (although not all!) the   better approximation to the witness to UPB BE states than the BGR construction. Moreover, they find entanglement correlations in several states that were not recognized as such by the BGR construction.

 

In my point of view, the result is based on previously well established numerical methods by the authors and I have no reason to doubt the correctness of the presented calculations. The presented method might be a valuable tool for constructing entanglement witness and thus studying entangled states in the field of quantum communications.

 

I recommend the paper for publication, but some issues in the text need to be sorted out before:

 

All abbreviations and symbols used in the text, despite how common they are within the field, need to be defined, namely, LOCC (l. 21), \rho and \sigma (l. 50), PPT (l. 84), <W>_SEP (l. 128). Also authors should choose between “UPB BE” or “UPBBE” because both versions are used along the text. 1 is not clear in the sense that indices in the figure do not reflect the description, e.g. 1…m should be the “height” of right upper block, while m in the picture is a part of the tile, neighbouring o with no “…” (indices) between them. Symbols in Eq. (1) should be defined (first to left in the bracket) In l. 92 the ambiguity between “k” and “K” related to Eq. (1) should be resolved. I do not seem to get the relation between expression in l. 139 with the examples counted in the sentence before that one. At the end of the sentence in l. 96-98, briefly mentioning the possible experimental setup, it would be good to give some references to it, or some of them. In the sentence l. 163-165, counting the number of corrections calculated to find a witness for each particular case, it is not clear if these numbers are because the HALT criterion was met or due to some other reason (e.g. inability to calculate further, insufficient precision etc). Some punctuation/capitalization should be dome in the text (l. 163 “We” -> “we”; section 182-183 “this state” -> “This state”; l. 238 space missing; l. 239 full stop missing )

 

 

 

Capitalisation/punctuation

Author Response

We thank the Reviewer for his time spent on preparing his report. We have revised typos pointed out by him. We have revised the formula in line 139. We thank you for noticing it. We also explained more on HALT conditions. As the algorithm is exponentially slower with the number of corrections already made (see Ref. [10]), sometimes it is difficult to distinguish between the HALT based on the final number of corrections and feasibility to go further. This was the case for 3x3 and 6x6 cases. Unfortunately, we were not able to find suitable references mentioned by the Reviewer concerning emulation of the states, we will happily accept any detailed suggestion.

Round 2

Reviewer 1 Report

I have read the authors’ reply and check the revisions. The mistakes I pointed before have been corrected and figure 1 has been clarified. I agree that this work has implemented some witness and numerically show it is better than the BGR witnesses for the studied cases, as shown in figure 2. I am still not sure about the significance on the work, but I would like to leave the decision to the editor.

Minor comment, I do not think the authors should claim “our witnesses” in the abstract. The witness has been proposed in literature.

Author Response

We thank the Referee for reviewing our manuscript once more. We regret our argument were not convincing for him, but we can at least thank him for leaving the final decision to the editor.

Probably the  Referee means that we obtain (approximations of) Bertlmann-Durstberg-Krammer-Heismayer witnesses. To avoid any future confusion we have replaced "our witnesses" the abstract with "witnesses found with the Gilbert algorithm in this work"

Sincerely Yours,

Marcin Wiesniak,

Palash Pandya,

Omer Sakarya,

Bianka Woloncewicz,

The Authors