09/01/2019
Editorial Decision Ms b180617
Multipartite quantum nonlocality in
two-dimensional transverse-field Ising models on $N\times N$ square lattices
by Sun et al.
Dear Dr Sun,
Your manuscript has been carefully reviewed by
our referee(s), who recommended certain revisions to meet EPJB editorial
criteria (please see the enclosed report(s)).
Provided the recommendations are carried out,
your revised version may be returned to the referee(s) or submitted to new
referees. Consequently, it is important to attach a detailed letter of response
to your revised manuscript. The letter should indicate the changes made and an
explanation for the recommendations that were not followed.
Please send the new version of your paper
(preferably a PDF file) via:
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using your author ID: 105889
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Your revised version should be received within 2
months from the date of this letter. Resubmission after that date will be
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Yours sincerely,
Dr Tobias Stauber
Editor for EPJ B
_________________
Report from Referee 9:
Report 1:
The authors discuss an interesting idea, which
will be helpful in the further development of tensor network algorithms for 2D
quantum many-body systems. An approach to these problems is via PEPS
wavefunctions, which, however, pose severe problems when it comes to the
numerical evaluation of relevant quantities, like observables. On the other
hand, MPS are a standard tool for 1D systems. The authors convincingly show how
to transform a PEPS for a 2D sytem to a MPS and in this way extract rather
straight forwardly physically relevant information. I therefore support
publication of the manuscript, after considering the following minor points:
- In the introduction, it would be somewhat
cleaner not to call the entanglement entropy a correlation, as usually
correlations/correlation functions are rather associated to observables, and
the entanglement entropy is not an observable. Also, it would be more precise
to mention that it is known that some systems have log-corrections in the
entropy (e.g., free fermions in 2D), and cite corresponding papers.
- After Eq. (3), it is said "After some
calculations, we may find...". It would be instructive to at least sketch
these calculations, e.g, in an appendix.
- In Fig. 7 it might be helpful to write in the
legend explicitly "PEPS" and "MPS" instead to indicate this
via \phi_0 and \varphi.
- There are some typos throughout the manuscript, which
should be fixed.
Report from Referee 11:
Report 1:
See enclosed file: reviewEPJB.pdf
The paper “Multipartite quantum nonlocality in two-dimensional transverse-field Ising models on NxN square lattices”proposes a technique to transform a PEPS into an MPS structure and then use this transformation to study multipartite quantum correlationsin a 2D system. As a test case, they study the ground state of NxN Ising model which was obtained using PEPS algorithm. They then transform the PEPS ground stateinto an MPS using a variational DMRG-
like algorithm. They do this for different values of N and compute things like Bell correlation
function, entanglement entropy, etc. From the value of the Bell correlation function obtained,they classify the phase transition of the model for different Ns with the different kinds of Bell-
type inequalities they violate. While the paper uses a combination of several known numericaltensor network techniques, it is still novel in the sense that one can use it to classify multipartite entanglement in 2D. For this reason, I would be in favor of its publication in EPJB. However, I have some serious numericalconcerns and a few technical queriesfor the authors before it gets accepted in the journal. I list them below:
[if !supportLists](1) [endif]The definition of Eq. (5) can be confusing. It is better to define both g_l and e_l in terms of A_l’s. Also to be consistent with the literature, the authors can mention that Eq. (7) is not the most general form of expressing the canonical form of an MPS. It should be mentioned here or in Eq. (4) that the ‘lambdamatrices’ are already absorbed in the A’s.
[if !supportLists](2) [endif]Below Eq.(8), the authorswrite the \hat{M}_{\[1\dotsN\]} as an MPO. The authorscan elaborate a little bit on this construction for those who are not familiar with this.
[if !supportLists](3) [endif]Section 3.2: The authors write that calculating two-siteEVs require six effective
environment tensors. This is only true for theCTMRG technique. One can also use other
techniques like boundary MPS, etc.
[if !supportLists](4) [endif]On the same section,the authors discussesthe difficulty in evaluating Bell correlation
function. I do not quite get why the authors can’t simply use a PEPO to represent the M
operator and compute the sandwich. The authors say computing it with PEPS is ‘too difficult’. Do they mean its too expensive? If so, what is the cost?
[if !supportLists](5) [endif]This is the most major concern I have. 2D systems have been studied in the past using MPS, this is the whole idea of 2D DMRG. The problem with this is that it does not capture the true 2D correlations. For example, a nearest neighborcorrelation in PEPS will turn into some long range correlation in the MPS. I have the same concern here. How good is the transformation. The authors show that the distance is of the order of 10^{-15}which is not bad. I would be happy to see if they can provide more figure of merits like a two-point EV to show that the transformation doesn’t change much the structure of the 2D correlations in the model which is very crucial for this paper.
[if !supportLists](6) [endif]Fig. 4 The authorsshould provide more information on the ground state computation. Is it a simple update or full update? What are the trotter steps, etc?
[if !supportLists](7) [endif]I have some major concern with Fig. 11, in particular Fig. 11(c) and (d). How is the entropy computed here? If the partitions are done based on Fig. 10, then the entropy should depend on the number of links you cut and not on the number of nearest neighbor sites.For example in Fig. 10 (a), I and II have the same numberof links cut although the number
of nearest neighborsare different. And how is such a ‘cut’ made when you have an MPS
representation of an PEPS. I feel that the anomaliesmentioned in footnote2, might have to
do with this. I would definitely like to get a clarification on this before the paper gets accepted.
I would definitely like to getsome clarifications on the abovespecially point (5) and(7) before the paper gets acceptedin the journal.
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