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Local Transformations of Multiple Multipartite States
by Antoine Neven, David Gunn, Martin Hebenstreit, Barbara Kraus
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Submission summary
| Authors (as registered SciPost users): | David Kenworthy Gunn · Antoine Neven |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2007.06256v2 (pdf) |
| Date submitted: | 2021-02-15 17:46 |
| Submitted by: | Gunn, David Kenworthy |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.
List of changes
- The introduction was modified in response to the referee’s comments about the reason for studying pure state transformations. Additionally, we made small changes to make the text more readable and precise.
- In Section V, we corrected the typo in Fig.3 and added a sentence discussing permutations of Schmidt coefficients, providing a reference (Ref. [57]) to related work and an upper bound on the number of possible permutations (Eq. 57).
- We modified the conclusion to clarify the outlook of our work as suggested by the referee.
- We updated the Acknowledgements to include Ref.[55].
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2021-5-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2007.06256v2, delivered 2021-05-14, doi: 10.21468/SciPost.Report.2913
Strengths
1-The manuscript opens a new window on multi-partite entanglement by using the notion of multi-state LOCC.
2-The authors develop a solid intuition how this notion behaves and answer a couple of key questions that clarify what multi-state LOCC can achieve.
Weaknesses
1-The manuscript only focuses on pure states (rather than also considering mixed states), but the authors make a compelling case that this already represents a meaningful first step.
2-The manuscript is non-exhaustive, as it rather explores a number of natural questions (in the context of multi-state LOCC) without the claim of being complete, as many questions are notoriously difficult (such as a full classification multi-partite multi-state transformations).
Report
The authors explore the notion of multi-state LOCC to study multi-partite entanglement. More precisely, they ask the question what types of pure states can be transformed into each other under regular LOCC if one is allowed to tensor an additional auxiliary state (in the same Hilbert space with the same multi-partite splitting) to the original state, such that the auxiliary state may transform into a different state (though still in a tensor product with the rest).
The main results are discussed in section IV and V. While section IV focuses on the multi-partite case, section V restricts to the bi-partite case (where the authors restrict to LU transformations). In particular, the authors show that multi-state LOCC can change the SLOCC class of a target state by only applying multi-state local unitaries and they identify several interesting multi-state local unitaries in the bi-partite setting. Overall, they make a compelling case that multi-state LOCC gives rise to rich structure that should be further explored.
The manuscript is well-written and makes interesting progress in the challenging field of multi-partite entanglement. The article thereby meets the SciPost selection criterion of "opening a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work". I therefore recommend publication in SciPost in the current version, but added a list of suggested changes, which are mostly cosmetic. However, I also tried to highlight a few spots where small modifications would potentially increase clarity - in particular, for some of the figures.
Requested changes
1-Shouldn't we have in equation (1) $S\in\mathrm{GL}(d,\mathbb{C})^{\otimes d}$? Or am I missing something?
2-There is a new paragraph after equation (3), i.e., see indent, which should probably be removed.
3-In equation (7) max is upright, while min is italic (variables).
4-It took me some time to fully understand the example illustrated in figure 2. In particular, I first thought that the illustration with the beige boxes has something to do with the three parties, before realizing that you (probably) just meant to indicate where the LU acts to swap the respective basis vectors. This could be just said more explicitly in the caption. It would have also helped me if you had maybe illustrated by arrows which respective basis states are permuted/swapped.
5-In lemma 3, there seem to be again unwanted indents / new paragraphs and in equation (20) there is an additional closing parenthesis in $\gamma^{(i))}_j$.
6-Figure 4 and the following figures look a bit pixeled and it would be good to convert it to vector graphics.
7-The authors could give a broader outlook on which questions remain open. They already did this in response to the first referee report (asymptotic case, restriction to physically relevant states), but it would also be interesting in regards to their results on the bi-partite case.
8-The label LU for the left-right arrows in the left figure are difficult to read. I would probably just remove them (and only have the arrows) and potentially make the state illustrations a bit larger
Author: David Kenworthy Gunn on 2021-06-11 [id 1500]
(in reply to Report 2 on 2021-05-14)We would like to thank the referee for their time and for their constructive feedback. Regarding the revisions suggested, we have the following responses:
It should be S∈GL(d,C)⊗n. We have also corrected a similar typographical error in Eq.2
We corrected this typographical error.
We corrected this typographical error and ensured all subsequent min/max are consistent.
We opted to add a further explanation in the caption. Hopefully, it is now clear the beige boxes indicate the local unitaries which are swapping the internal GHZ and W states.
We corrected this typographical error.
We followed this suggestion. Now, Fig. 4 and all subsequent images are vector graphics.
We took this suggestion on board. As a result, we modified our conclusion section. Firstly, we restructured the section so that the final paragraph is focused on the overall outlook of multi-state LOCC. As a result, in the final paragraph, we first discuss what we believe the focus of further investigations should be in the multipartite case (this discussion was moved from an earlier part of the conclusion). We then also provided further outlook with regards to the bipartite case (being mindful of the extensive research that has already been conducted, for example in entanglement catalysis).
In order to increase the readability of the label "LU" within Figure 8, we have increased the font size and we have chosen a different colour. Moreover, we have increased the size of the state illustrations a little bit, as suggested by the referee.
We would once again like to thank the referee for the detailed, constructive comments, and believe our paper has been improved because of them.
Sincerely, The authors.