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Long-distance entanglement in Motzkin and Fredkin spin chains

by Luca Dell'Anna

Submission summary

As Contributors: Luca Dell'Anna
Preprint link: scipost_201905_00002v1
Date submitted: 2019-05-12
Submitted by: Dell'Anna, Luca
Submitted to: SciPost Physics
Domain(s): Theoretical
Subject area: Quantum Physics


We derive some entanglement properties of the ground states of two classes of quantum spin chains described by the Fredkin model for half-integer spins and the Motzkin model for integer ones. Since the ground states of the two models are known analytically, we can calculate the entanglement entropy, the negativity and the quantum mutual information exactly. We show, in particular, that these systems exhibit long-distance entanglement, namely two disjoint regions of the chains remain entangled even when the separation is sent to infinity, i.e. these systems are not affected by decoherence. This strongly entangled behavior is consistent with the violation of the cluster decomposition property occurring in the case of colorful versions of the models (with spin larger than 1/2 or 1, respectively), but is also verified for colorless cases (spin 1/2 and 1). Moreover we show that this behavior involves disjoint segments located both at the edges and in the bulk of the chains.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Submission scipost_201905_00002v1 on 12 May 2019

Reports on this Submission

Anonymous Report 1 on 2019-6-11 Invited Report


1. The results are novel and give insights into the entanglement properties of an interesting family of unusual spin chain models.
2. The introduction and summary of results are good and the motivation is well explained.
3. There are very good explanations of the models and derivations, and the manuscript is mostly well self contained.
4. Thorough, pedagogical treatment of the problem, with sensibly divided sections for readability.
5. Good use of pictures to illustrate examples, and the plots are nice and clear.


1. There are very many typos and grammatical errors. In places these are enough to obfuscate the author's meaning.
2. In a few places it isn't clear if the author is deriving something for the first time, or recapitulating a previously known result.
3. Figure 3 is very confusing (at least to me).
4. Some results are not discussed or interpreted, i.e. the peak in the right panel of Fig. 7.


This manuscript reports on some technical results for entanglement properties of ground states in an unusual family of spin chain models.
The Fredkin and Motzkin models are defined by local projective Hamiltonians for lattices of spins with special boundary conditions and three body interaction terms. There has been considerable interest in these models because they are local but have highly unconventional entanglement properties, so results that give new insights into their entanglement are useful.

The author focuses on calculating the negativity and mutual information for disjoint intervals in the ground states of these lattice models. The author shows that even when the intervals are infinitely far apart they remain entangled, and compares the case of intervals at the boundaries of the system, to those deep in the bulk. In the latter case results are found that
contrast with recently proposed continuum limit models.

I think the results will be of general interest in the field of many-body entanglement and worthy of publication, but first several issues should be addressed.

1. In the introduction the author states that for gapped systems the entanglement entropy scales like the area of the boundary between two subsystems. I believe this result is only true for ground states and possibly low lying states in the spectrum. Can the author clarify this?

2. The author refers to 'colorful' and 'colorless' cases in the introduction, without explaining what color means in this context. Just a brief note (as in the abstract) to explain this relates to higher spin variants would avoid confusion for non-specialists.

3. Negativity and mutual information are less familiar entanglement measures (outside of the field of quantum information) than e.g. von Neumann or Renyi entropies, and it would be useful if the author provided a very brief description of their properties and utility, to keep the manuscript self contained.

4. I find Figure 3 really confusing. First, the notation used in the caption for the central region kets doesn't seem to match that used in Eqs. 17 and 18. Presumably this is a simplification to avoid explicit referral to all the $c,\bar{c}$ indices in A and B? Second, the many overlapping lines make it difficult to discern the individual walks. Could the author separate these up, or provide more diagrams to demonstrate e.g. the blue walks. In particular I can't understand how the highest blue point is compatible with walks that touch z=2 once - unless the very first point counts, in which case other blue paths would touch z=2 twice.

5. At the point where they are introduced, or at Eq. 19 where the explicit form is given, it would be useful to sketch a proof and/or give citations for the relation between the coeffs $\mathcal{A}$ and the $\mathcal{D}$ numbers.

6. The author calculates the negativity for both Fredkin and Motzkin models but I don't think there is any interpretation of the results found, nor are there plots. If the negativity is worth calculating, surely the resulting expressions are worthy of some discussion?

7. Comparing Figure 5 and figure 7, there is a local maximum at q=3 for the mutual information between edge spins in the Motzkin model, but the result for the Fredkin case is monotonic. Can the author explain this difference and is there a physical interpretation of what is happening? I understand that the two results will differ at q=1 because the edge spins for the Fredkin chain are necessarily uncorrelated, but I don't see how this relates to the presence or absence of a peak at small q.

8. The results for spins in the bulk of the Fredkin model are shown in Fig. 9. The mutual information as a function of distance from the edge (right panel) shows a series of steps of length two. What causes this and why is it not present for the Motzkin case, Fig 10? In addition, why not include the asymptotic values (or a best fit extrapolation of the curves) on these plots to convince the reader that they really are consistent with Eqs. 134 and 167?

9. The author states in the intro and conclusion that the some of the results are in contradiction to those found in the continuum limit version of the model. Based on this can the author make a concrete statement as to whether the continuum limit model is actually a good representation of the Motzkin and Fredkin spin chains?

10. Finally but importantly, there is a very large number of typos and grammatical errors throughout the entire manuscript. These serve to make the paper confusing and risk leading to misinterpretations of the author's meaning. As an example, at one point the author uses 'ones' when they mean 'once'.

Requested changes

1. Correct spelling and grammar throughout the manuscript.
2. Clarify the statement on entanglement scaling in gapped systems in the intro.
3. Include a very brief description of what 'color' means in the intro.
4. Include a brief description of the properties and meaning of entanglement negativity and mutual information (probably in the introduction).
5. Fix or clarify the panels in Figure 3.
6. Citation (or sketch proof if it is trivial) for Eqs. 19 and 20 (and possibly 66 and 67).
7. When the notation $()^t$ is first used (between Eqs 32 and 33) please indicate that it means to take the 'transpose'.
8. Add some discussion on the meaning of the results found for the negativity (and possibly plot them).
9. Discuss the peak in the right panel of Fig 7.
10. Discuss the staircase effect in Fig. 9 versus Fig. 10 and show the asymptotic values/extrapolations.

  • validity: high
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: below threshold

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