SciPost Submission Page
Certifying Temporal Correlations
by Harshank Shrotriya, Leong-Chuan Kwek, Kishor Bharti
Submission summary
| Authors (as registered SciPost users): | Harshank Shrotriya |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2206.06092v2 (pdf) |
| Date submitted: | 2023-01-04 05:12 |
| Submitted by: | Shrotriya, Harshank |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Self-testing has been established as a major approach for quantum device certification based on experimental statistics under minimal assumptions. However, despite more than 20 years of research effort most of the self-testing protocols are restricted to spatial scenarios (Bell scenarios), without any temporal generalisations known. Under the scenario of sequential measurements performed on a single quantum system, we build upon previous works which used semi-definite programming (SDP) based methods to bound sequential measurement inequalities. For such SDPs, we show that the optimiser matrix is unique and moreover this uniqueness is robust to small deviations from the quantum bound. Further, we consider a generalised scenario in presence of quantum channels and draw analogies in the structure of Bell and sequential inequalities using the pseudo-density matrix formalism. These analogies allow us to show a practical use of maximal violations of sequential inequalities in the form of certification of quantum channels up to isometries.
Current status:
Reports on this Submission
Report
Let me start by confessing that when I had looked at title and abstract of the manuscript, I was expecting to understand more than I really did, and I was a bit surprised that the work is rather at the periphery of my expertise.
If the manuscript got into the wrong channels, the authors should consider revising title and abstract to focus more on their target readership, and they should also consider submitting their work to a more specialized journal.
If, on the other hand, the target is a broader audience (that thus includes myself), I believe that a number of structural changes to the manuscript would be appropriate:
1- After a first very general paragraph of the Introduction, the authors suddenly assume that the reader is familiar with specialist jargon such as "self-testing" and "certifying". I recommend that they add explanations of these terms.
2- Relegating technical proofs to appendices may be a good choice. However, I think that then one should use the opportunity to add more discussion to the main text what the Theorems actually mean and what they are useful for.
3- Conversely, it might be beneficial for the reader if comparatively short appendices were moved into the main text since this would eliminate the overhead for referring back and forth. This concerns in particular appendices C2 and D that each basically just contain a single equation (even if I admit that these equations are long).
4- The purpose of Appendix B is not very clear. The appendix as such is not referenced from anywhere else. Theorems 5 and 6 are needed in Appendix C1, but Theorem 4 is never referenced. Furthermore, no justification for Theorem 4 is provided (neither a proof nor a reference to a source). I feel that some restructuring and/or shortening of this part would be appropriate.
I note that the manuscript seems to have been written with a length limit for the main text in mind. However, SciPost does not have such length limits, thus permitting the extensions and reorganizations suggested above.
Requested changes
Beyond the major points mentioned in the Report, there are a number of technical issues that in my opinion would deserve attention. The combination of the two is summarized in the following list:
1- General: Use the SciPost LaTeX template available at https://scipost.org/SciPostPhys/authoring#manuprep.
2- Add further comments on the interpretation of the Theorems to the main text.
3- Move independent short fragments from the Appendices into the main text.
4- Reorganize and/or shorten Appendix B.
5- The "Introduction" at the beginning of the main text should be a section heading.
6- Add explanation of technical terms such as "self-testing" and "certifying" at the beginning of the second paragraph of the main text.
7- If the "qudit" at the beginning of section I is not a typographic error, this would merit an explanation.
8- The symbol "$\succeq$" should be defined (or explained) before it is first used in Eq. (3) and not only at the beginning of Appendix B.
9- The acronym "PSD" should be defined when it is first used in Remark 1 and not only at the beginning of Appendix A.
10- Typeset scalar products as $\vec{a}\cdot\vec{b}$ rather than $\vec{a}.\vec{b}$ ("\cdot" rather than "." - at least three instances, two a few lines below Eq. (5), one two lines above Eq. (C17)).
11- Fig. 1 uses a symbol ${\cal E}(\rho')$ that does not appear anywhere else (even the caption uses different terminology).
12- Eq. (11) defines $S_3$, but it is not an inequality, contrary what the terminology "Leggett-Garg inequality" suggests.
13- Add DOIs for all references (see again https://scipost.org/SciPostPhys/authoring#manuprep).
14- Eliminate ISSNs from Refs. [6,19,31].
15- Remember updating the references, specifically Refs. [20,34].
16- Remove spurious "m. k." in the name of the second author of Ref. [25].
17- Add article number and/or page numbers in Ref. [31].
18- There is a "$\Pi$" two lines below Eq. (C3) that probably should rather be a "$\pi$".
Report
In the manuscript titled ''Certifying Temporal Correlations'', the authors have considered the scenario involving sequential measurements of binary valued (+1 or -1) observables performed on a single quantum system. Considering no quantum channel acting on the state between sequential measurements in the above scenario, contingent upon using semi-definite programming, it has been shown that the optimal quantum bound of cyclic temporal inequalities can certify the temporal correlation matrix uniquely and this certification is robust to small deviations from the quantum bound. The isometries involved in this certification are also pointed out. Next, considering the case when the state is subjected to some quantum channel between two consecutive measurements, it has been shown that the aforementioned optimal quantum bound can certify the quantum channel up to some isometries. However, the robustness of this channel certification has been left as a future direction.
I believe the mathematical calculations are correct. The problems addressed in the manuscript are quite significant in the sense that these present some important facets of the structure of quantum temporal correlations. Specifically, applying the pseudo density matrix formalism in Sec. III and connecting the Bell scenario with the sequential measurement scenario is really novel. The presentation is good and easy to follow (I must appreciate the excellent literature review in the introduction). Overall, the manuscript is, in principle, publishable. However, before recommending for publication, it would be good if the authors can address the following remarks.
(1) In the usual self testing scheme, quantum state and measurements are certified, which is important for realizing any quantum information and communication task. However, It is not at all clear the physical significance of certifying a temporal correlation matrix. It would be good if the authors can discuss it from an information theoretic perspective and/or from a quantum foundation point of view.
(3) Can one certify the quantum measurements from the certified correlation matrix under some assumptions on the state (like the results of [41], [42])?
(2) It would be beneficial for the authors if some sketches of the proofs of the three theorems can be presented in the main paper.
(3) Appropriate reference should be cited while presenting the N-cycle inequalities, their classical and quantum bounds.