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Certifying Quantum Separability with Adaptive Polytopes
by TiesA. Ohst, XiaoDong Yu, Otfried Gühne, H. Chau Nguyen
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users):  Chau Nguyen · TiesAlbrecht Ohst 
Submission information  

Preprint Link:  https://arxiv.org/abs/2210.10054v3 (pdf) 
Code repository:  https://gitlab.com/tqo/quantumcorrelations 
Date submitted:  20231026 12:18 
Submitted by:  Ohst, TiesAlbrecht 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

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 twoparticle separability for small and mediumsize 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:
Author comments upon resubmission
Specific replies to the comments from the Reviwers were also made.
List of changes
The revision includes:
 changed to the SciPost LaTeX template
 included more references on entanglement detection in the introduction
 more detailed description of the polytope adaption algorithm (Ch. 2.2).
 added short description of the complexity of the algorithm (Ch. 2.2).
 extension of the caption of Figure 2
 merged and restructured Figure 7 and 8 in Appendix F.
Submission & Refereeing History
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Reports on this Submission
Strengths
The paper presents an algorithm intended to certify separability of a given state. The algorimth seems to be very efficient computationally, but this needs more clarifications. Certainly, the Authors achieve rather precise estimates.
Weaknesses
Some details still require clarification. In particular I expect improvement on describing the assymmetry of the system. With the new version I am somewhat confused if the algorithm also certifies entanglement.
Report
I thank the Authors for replying to my previous report. However, I am not sure if I can find the amendmments satisfactory enough at this stage. The Authors claim that the Algorithm certifies separability and such a message can be read from the description of the algorithm. Moreover, Figure 5 suggests that convergence may be expected even for 2 iterations. However, I, and I believve the Reader, would like to know about the omplexity of the algorithm. In each step, a semidefinite program must be run, and we do not know much about optimizing $tau$ operators. I think it would be beneficial to give runtimes for a few examples discusssed in the paper in context of the specs of a computer used.
Another question that would need to be discussed is the symmetry under the swap of particles. The algorithm is presented as for a system of two identical subsystems. Shall the Reader assume that in case of, say, 2x4 we simply expand the space to 4x4? But what about taus being now sigmas?
In the newest version, the Author now claim that calculating both P_in and P_out is efficient. One of the versions of the dual problem we optimize Tr rho W, suggesting an entanglement witness. Table 5 quotes both upper and lower bounds of white noise robustness. Thus I am surprised that the Authors clarified in their response that they certify separability. The above hints me towards believing that the Algorithm is indeed able to provide an entanglement witness. I ask the Authors to confirm or deny. Also, If witnesses are provided, then despite the recent literature was updated, it still lack some references relevant to finding customtailored entanglement witnesses. For example, it was Brandao to first notice the relation between an entanglement witness and a separble approximation of a state.
Finally, as a remark, I would point out that the stop criterion in the caption of Figure 5 is not informative, as it ignores the dynamics of the subsequnt iterations. If many rounds of the procedure offer the correction of 10^4 each, we are still far away. On the other hand, a lot is known about robustness of twoqudit Werner states, and it is difficult to infer how close the Algorithm has got. Only from other data it seems that it is rapidly convergent.
I am still in favour of publishing the work in SciPost, but basing on the above comments, I think there is still some space to for improvements to make it more useful to the Reader.
Requested changes
1. Explain the computational cost of the algorithm and give runtimes for few examples discussed in the paper.
2. Explain the procedure under asmymetric systems and states.
3. Explain if the newest version applies to both the inner and the outer polytope, providing both the upper and the lower limits of robustness.
4. If the answer to the last question is "yes", the literature on customtailored entanglement witnesses can be updated.
Author: TiesAlbrecht Ohst on 20240110 [id 4235]
(in reply to Report 1 on 20231114)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:**
I thank the Authors for replying to my previous report. However, I am not sure if I can find the amendmments satisfactory enough at this stage. The Authors claim that the Algorithm certifies separability and such a message can be read from the description of the algorithm. Moreover, Figure 5 suggests that convergence may be expected even for 2 iterations. However, I, and I believve the Reader, would like to know about the omplexity of the algorithm. In each step, a semidefinite program must be run, and we do not know much about optimizing tau operators. I think it would be beneficial to give runtimes for a few examples discusssed in the paper in context of the specs of a computer used.
**Our response:**
Thank you for your comment. We agree, and now explicitly give some examples of run times for certain states so that the readers have an idea of how long it typically takes. This is given at the end of section 2.2 for random 5x5dimensional states (order of seconds in a normal desktop) and in the Caption of Table 1 for multiparticle states (order of minutes in a normal desktop).
**The Reviewer wrote:**
Another question that would need to be discussed is the symmetry under the swap of particles. The algorithm is presented as for a system of two identical subsystems. Shall the Reader assume that in case of, say, 2x4 we simply expand the space to 4x4? But what about taus being now sigmas?
**Our response:**
No, the spaces do not have to be expanded. Our algorithm switches between optimisations on Alice and Bob's generalized Bloch spheres which may have different dimensions. Importantly, in the third step of the algorithm, in which the systems A and B are exchanged, a swap operation on the state is performed which transforms a 2x4dimensional state into an 4x2dimensional state. Hence, every odd iteration step involves an optimisation of 2x2 matrices, while every even iteration step involves an optimisation of 4x4 matrices (or the other way around). In this way, the algorithm does work normally in the asymmetric setting without any additional increase in complexity compared to the symmetric setting.
In the revision, we clarified this in section 2.2.
**The Reviewer wrote:**
In the newest version, the Author now claim that calculating both P_in and P_out is efficient. One of the versions of the dual problem we optimize Tr rho W, suggesting an entanglement witness. Table 5 quotes both upper and lower bounds of white noise robustness. Thus I am surprised that the Authors clarified in their response that they certify separability. The above hints me towards believing that the Algorithm is indeed able to provide an entanglement witness. I ask the Authors to confirm or deny. Also, If witnesses are provided, then despite the recent literature was updated, it still lack some references relevant to finding customtailored entanglement witnesses. For example, it was Brandao to first notice the relation between an entanglement witness and a separble approximation of a state.
**Our response:**
Sorry for the confusion. Inner polytopes provide an inner approximation of the separable states whereas outer polytopes give rise to an outer approximation. Hence, entanglement detection is indeed possible by using the outer polytope approximation. However, the outer polytope approximation is only practical if one of the systems is a qubit, since then its lowdimensional Bloch sphere allows for an efficient construction of an outer polytope approximation to it. Such a construction is unknown for qudits. Also, the adaption techniques we developed for the inner polytopes cannot be applied to the outer approximation. We added few sentences in the end of Section 2.1. together with references on customized entanglement witnesses to clarify this. On the other hand, the algorithm with adaptive inner polytopes allows for certification of separability also for system of qudits. As this problem has been long considered to be difficult, we mainly concentrate on this aspect in the rest of the manuscript.
**The Reviewer wrote:**
Finally, as a remark, I would point out that the stop criterion in the caption of Figure 5 is not informative, as it ignores the dynamics of the subsequnt iterations. If many rounds of the procedure offer the correction of 10^4 each, we are still far away. On the other hand, a lot is known about robustness of twoqudit Werner states, and it is difficult to infer how close the Algorithm has got. Only from other data it seems that it is rapidly convergent.
**Our response:**
Thank you for pointing this out. The convergence of the Figure 5 was in fact confirmed by comparing with upper bounds given by the PPT criterion (which is good for random states as we discussed in section 2.2.) We revise the caption to make this clear.
**The Reviewer wrote:**
I am still in favour of publishing the work in SciPost, but basing on the above comments, I think there is still some space to for improvements to make it more useful to the Reader.
**Our response:**
Thank you, your comments have indeed improved the clarity of the manuscript!