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Certifying Quantum Separability with Adaptive Polytopes
by Ties-A. Ohst, Xiao-Dong Yu, Otfried Gühne, H. Chau Nguyen
This is not the latest submitted version.
|Authors (as registered SciPost users):
|Chau Nguyen · Ties-Albrecht Ohst
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.
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
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.
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.
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 semi-definite 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 custom-tailored 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 two-qudit 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.
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 custom-tailored entanglement witnesses can be updated.