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Certifying Quantum Separability with Adaptive Polytopes
by Ties-A. Ohst, Xiao-Dong Yu, Otfried Gühne, H. Chau Nguyen
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Submission summary
| Authors (as registered SciPost users): | Chau Nguyen · Ties-Albrecht Ohst |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2210.10054v2 (pdf) |
| Code repository: | https://gitlab.com/tqo/quantum-correlations |
| Date submitted: | 2023-04-05 10:45 |
| Submitted by: | Ohst, Ties-Albrecht |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
The concept of entanglement and separability of quantum states is relevant for several fields in physics. Still, there is a lack of effective operational methods to characterise these features. We propose a method to certify quantum separability of two- and multiparticle quantum systems based on an adaptive polytope approximation. This leads to an algorithm which, for practical purposes, conclusively recognises two-particle separability for small and medium-size dimensions. For multiparticle systems, the approach allows to characterise full separability for up to five qubits or three qutrits; in addition, different classes of entanglement can be distinguished. Finally, our methods allow to identify systematically quantum states with interesting entanglement properties, such as maximally robust states which are separable for all bipartitions, but not fully separable.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-10-3 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2210.10054v2, delivered 2023-10-03, doi: 10.21468/SciPost.Report.7888
Report
The manuscript addresses the significant issue of characterizing entanglement and separability in quantum states, a topic of great relevance in various fields of physics. The authors introduce a novel method based on adaptive polytope approximation to certify quantum separability in two- and multiparticle quantum systems. Their approach presents a practical algorithm that can conclusively identify two-particle separability for small and medium-sized dimensions and characterize full separability for multiparticle systems, such as up to five qubits or three qutrits. Furthermore, it allows for the distinction of different classes of entanglement and the identification of quantum states with intriguing entanglement properties, including maximally robust states.
The manuscript presents a valuable contribution to the field of quantum entanglement and separability certification, addressing a gap left by previous computationally demanding methods. To me, it seems they are using See-saw type coupled SDPs. The proposed algorithm demonstrates efficiency and versatility, making it applicable to quantum systems with relatively high dimensions and many particles. It offers the ability to certify a range of states and investigate various entanglement robustnesses. I recommend accepting the manuscript in its current form, as the presented approach is straightforward, useful, and appears to be highly effective.
Report #1 by Anonymous (Referee 1) on 2023-8-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2210.10054v2, delivered 2023-08-17, doi: 10.21468/SciPost.Report.7668
Report
The present manuscript describes an approximate method to determine nonseparability of a given state. This is done by approximating local sets of states by polytopes. Subsequently, it is tested whrether the state does belong into the convex hull of the product of such polytopes.
I think the manuscrip is presented in a rather clear manner, but there are a few points that definetely need an improvement. For example, the description of the algorithm of the adaptive plytopes is unclear. The question is what is changed between the rounds. Is it only the direction of the $tau$s, or is it also their number. Figure 2 is not really informative in this respect, it just shows switching sides. The next question is how exactly the state is determined to be in or outside the large polytope, I expect it to be by the wonder of semidefinite programming, but if there more details, I would be happy to learn them and the Readers with me. After all, this is the whole point of this game. Also, does tis procedure has to be repeated for every new state tested, or it constitues a "database" for future states. Then, what is the actual computational complexity of the protocol and how does it scale (roughly) with . Another point is that Figure 4 utilizes mixtures of random, unknown states, so their information content is very limited. Finally, the review on recent advances on entanglement detection is rather selective and not quite close to complete. The Authors should update the literature. At this moment, I am weakly not in favour of accepting the manuscript to SciPost Physics, but a revision is definetely needed and i would rather postpone my recommendation until then
Author: Ties-Albrecht Ohst on 2023-10-26 [id 4067]
(in reply to Report 1 on 2023-08-17)
Thank you for your effort in reviewing our manuscript. We have revised the manuscript taking your comments into account. Below we discuss the points you have raised.
**The Reviewer wrote:**
>For example, the description of the algorithm of the adaptive plytopes is unclear. The question is what is changed between the rounds. Is it only the direction of the taus, or is it also their number. Figure 2 is not really informative in this respect, it just shows switching sides.
**Our response:**
In consideration that the algorithm might not be as clear as we intended to, we have now added several paragraphs in Section 2.2.
In addition, the caption Figure 2 is extended to accommodate more information about the algorithm.
We hope that with this, the algorithm becomes clearer to the readers.
In short, to clarify your question, the party whose set of states is replaced by a polytope is changed in every round of the iteration.
We also would like to keep Figure 2, which should become clearer with the new text added.
**The Reviewer wrote:**
>The next question is how exactly the state is determined to be in or outside the large polytope, I expect it to be by the wonder of semidefinite programming, but if there more details, I would be happy to learn them and the Readers with me. After all, this is the whole point of this game. Also, does tis procedure has to be repeated for every new state tested, or it constitues a "database" for future states. Then, what is the actual computational complexity of the protocol and how does it scale (roughly) with.
**Our response:**
We have the impression that there might be a misunderstanding, that the polytopes used in our algorithm approximate the whole set of separable states.
This is not the case.
It is important to emphasize that the polytopes used in our algorithm are intended to approximate one of the local state spaces, either the one of Alice or Bob.
They are not an approximation for the global set of separable states shared by Alice and Bob.
The generated approximation of the global set of separable states is not a polytope.
Regarding the question whether the procedure has to be repeated for every new input state, the answer is `yes'. As long as there is no knowledge what the separable decomposition of a state could be, initialising a new random polytope is a fairly good option. However, this has no significant effects on the complexity of the algorithm.
The complexity of the algorithm scales linearly in the size of the polytope and quadratically in the local dimension on one side. A paragraph in the Section 2.2. is added to explain this aspect.
**The Reviewer wrote:**
>Another point is that Figure 4 utilizes mixtures of random, unknown states, so their information content is very limited.
**Our response:**
The purpose of Figure 4 is to emphasize on the generic nature of the algorithm, which can be applied to a general quantum state without any special structure. Once the state has more structure or symmetry (such as a GHZ state or the W state alone), determining their separability could be much more easier as known in the literature. With that in mind, we still want to keep this figure in this resubmission.
**The Reviewer wrote:**
>Finally, the review on recent advances on entanglement detection is rather selective and not quite close to complete. The Authors should update the literature.
**Our response:**
Notice that we are concentrating on the problem of verification of separability. This is complementary to the problem of entanglement detection. As we emphasized in the introduction, there are many methods for entanglement detection and the corresponding literature is indeed large. The verification of separability has been long considered to be a difficult problem, for which there are only handful of methods, which are mentioned in the introduction to our best knowledge.
However, we do agree that adding more references on entanglement detection would put the work into a larger context. We have added five recent works to our references (see Ref.[9-13]) regarding entanglement detection in this resubmission.
**The Reviewer wrote:**
>At this moment, I am weakly not in favour of accepting the manuscript to SciPost Physics, but a revision is definetely needed and i would rather postpone my recommendation until then.
**Our response:**
With our clarification, we hope that it would meet the standard of SciPost Physics.
Author: Ties-Albrecht Ohst on 2023-10-26 [id 4066]
(in reply to Report 2 on 2023-10-03)Thank you for your effort in reviewing our manuscript.
We do hope that our results contribute to the better understanding of the problem of certifying separability in entanglement theory.