pith. sign in

arxiv: 2606.31165 · v1 · pith:OO4HKXIOnew · submitted 2026-06-30 · 🪐 quant-ph

Entanglement Structure Across mathbb{Z}_n Phase Transitions in 1D Rydberg Atom Arrays

Hyeonjun Yeo , Kabgyun Jeong , Hyunchul Nha This is my paper

Pith reviewed 2026-07-01 05:36 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum entanglementRydberg atomsZ_n phase transitionsconcurrenceFourier analysissymmetry breakingone-dimensional chainsmany-body systems
0
0 comments X

The pith

Fourier analysis of pairwise concurrence across a Rydberg chain detects Z_n phase transitions by revealing periodic entanglement patterns.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors investigate how entanglement between atom pairs changes as a one-dimensional Rydberg array passes through phases with Z_n symmetry. They replace or supplement the usual local-density order parameter with an entanglement-structure factor obtained by taking the Fourier transform of the total concurrence summed at each site. This factor is shown to carry signatures of the ordered phases and the transitions between them. The same concurrence values can be extracted in current analog Rydberg experiments via site-selective erasure combined with shaped laser pulses. The work therefore positions entanglement structure as a practical diagnostic for symmetry breaking in driven many-body systems.

Core claim

The entanglement-structure factor, constructed from the Fourier transform of total site concurrence, exhibits distinct features that mark the emergence and transitions of Z_n-ordered phases in the Rydberg chain, providing an alternative characterization to the conventional local-density (magnetization) order parameter.

What carries the argument

entanglement-structure factor: the Fourier transform of the total concurrence (summed over all other sites) evaluated at each site, whose peaks or periodicity directly reflect the Z_n symmetry breaking.

If this is right

  • The same concurrence-based factor can be measured directly in analog Rydberg arrays using site-selective erasure and parametrized pulses.
  • Entanglement patterns supply an independent diagnostic for symmetry-breaking transitions that does not rely on single-site density.
  • Multipartite entanglement can be probed through its pairwise reductions in a manner that tracks phase structure across the chain.
  • The approach extends the utility of two-qubit concurrence beyond bipartite settings to global phase identification.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may generalize to other lattice models where conventional order parameters are hard to access experimentally.
  • Comparing the factor's scaling with system size could quantify how entanglement length scales with the interaction range.
  • If the factor remains robust under weak disorder, it could serve as a diagnostic in imperfect experimental arrays.
  • Extension to two-dimensional Rydberg lattices would test whether the Fourier approach captures higher-dimensional symmetry breaking.

Load-bearing premise

The Fourier transform of total site concurrence produces clear, interpretable signatures of Z_n order that are at least as informative as the local-density order parameter, without needing extra tuning of analysis windows or interaction ranges.

What would settle it

Numerical or experimental data in which the entanglement-structure factor remains featureless across a known Z_n transition point while the local-density order parameter shows a clear jump or peak.

Figures

Figures reproduced from arXiv: 2606.31165 by Hyeonjun Yeo, Hyunchul Nha, Kabgyun Jeong.

Figure 1
Figure 1. Figure 1: FIG. 1: Magnetization and concurrence profiles of representative phases on 1D Rydberg atom chain. In each part, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Site-resolved concurrence decay at ∆ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Contour plots for Fourier transform (a) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Finite-size scaling for [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison of two cuts into the period-4 ordered phase. In (a), [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Infidelity 1 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows why we adopt AC /TC as the entangle￾ment structure factor in the main text. In [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Representative individual concurrences between two single atoms as a function of ∆ [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Detailed [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Wave-vector comparison for the two [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Representative negativity values (a) and (d): [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: Finally, Fig. 14 shows that the feature visible [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Plot of the next nearest neighbor (NNN) [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Color maps of (a,b) local pairwise concurrence and (c) [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: (a) Negativity of bipartitions formed from 2 [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
read the original abstract

Multipartite quantum entanglement plays a crucial role in the emergence of different quantum phases and their transitions in quantum many-body systems. It is of general interest to know what sort of analysis on quantum entanglement can bring us a profound insight to understand the rich dynamics of quantum many-body systems. In this work we study the characteristics of quantum entanglement in relation to $\mathbb{Z}_n$-ordered phases emerging under a varied strength of 1-dim Rydberg interaction. We propose an approach based on the structure of pair-wise entanglement across the Rydberg chain using two-qubit concurrence as an entanglement measure. We define an entanglement-structure factor via Fourier analysis of total concurrence at each site and address $\mathbb{Z}_n$ phase transitions in comparison with the conventional order-parameter based on local density, i.e. magnetization. We also discuss how the required two-qubit concurrence can be measured in analog Rydberg atom arrays using site-selective erasure and parametrized laser pulses. Our investigation suggests that an entanglement-structure-based approach can provide a powerful tool in analyzing symmetry-breaking in quantum phase transitions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies multipartite entanglement in 1D Rydberg atom arrays across Z_n-ordered phases that arise as the Rydberg interaction strength is varied. It introduces an entanglement-structure factor constructed from the Fourier transform of the total site concurrence (with two-qubit concurrence as the entanglement measure) and compares its ability to detect the phase transitions against the conventional local-density (magnetization) order parameter. The work also outlines an experimental protocol for extracting the required concurrences via site-selective erasure and parametrized laser pulses in analog Rydberg simulators. The central claim is that an entanglement-structure-based approach can serve as a powerful tool for analyzing symmetry-breaking quantum phase transitions.

Significance. If the proposed factor indeed yields clear Z_n signatures that are at least as informative as the local-density order parameter, the work would supply a new diagnostic that directly probes entanglement structure rather than local observables. The experimental measurement proposal adds immediate relevance for current Rydberg-array platforms. The modest phrasing of the conclusion is appropriate and avoids overstating the results.

minor comments (2)
  1. [Abstract] The abstract refers to 'total concurrence at each site' without specifying whether this is a sum over all pairs involving the site or a different aggregation; a one-sentence clarification in the main text would remove ambiguity.
  2. Figure captions (or the methods section) should explicitly state the system sizes, interaction ranges, and Fourier-window choices used in the illustrative comparisons so that readers can assess whether post-hoc tuning was required.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript and for the positive assessment of its significance. The recommendation of minor revision is noted. No major comments appear in the report, so we have no specific points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript defines an entanglement-structure factor as the Fourier transform of site-resolved total concurrence and compares its signatures of Z_n order to the conventional local-density (magnetization) order parameter. No equations, fitted parameters, or self-citations are shown that would make any claimed prediction or result equivalent to its inputs by construction. The central claim is framed as a proposal for an analysis tool rather than a derivation that reduces to prior fitted quantities or author-specific uniqueness theorems. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The proposal rests on standard quantum information definitions (concurrence) and Fourier analysis; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Two-qubit concurrence is a valid entanglement monotone for the Rydberg system.
    Invoked when defining the entanglement measure from pair-wise reduced density matrices.
  • domain assumption Fourier analysis of site-resolved concurrence will produce peaks at wave-vectors corresponding to Z_n order.
    Central to the definition of the entanglement-structure factor.
invented entities (1)
  • entanglement-structure factor no independent evidence
    purpose: To quantify periodic entanglement patterns across Z_n transitions
    Newly defined observable constructed from concurrence; no independent evidence provided in abstract.

pith-pipeline@v0.9.1-grok · 5725 in / 1350 out tokens · 21176 ms · 2026-07-01T05:36:07.657837+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

70 extracted references · 4 canonical work pages · 2 internal anchors

  1. [1]

    The basic components of Ry- dberg Hamiltonian are two internal energy levels, ground state|g⟩and Rydberg state|r⟩

    and scalability [17, 38]. The basic components of Ry- dberg Hamiltonian are two internal energy levels, ground state|g⟩and Rydberg state|r⟩. The Rydberg state is a highly excited state of an atom which strongly interacts with other atoms in Rydberg states onµm length-scale. A Rydberg system withNatoms is expressed as H= NX i=1 Ω 2 (|gi⟩ ⟨ri|+|ri⟩ ⟨gi|)−∆ˆ...

    2025

  2. [2]

    S. L. Sondhi, S. Girvin, J. Carini, and D. Shahar, Reviews of modern physics69, 315 (1997)

    1997

  3. [3]

    Sachdev, Physics world12, 33 (1999)

    S. Sachdev, Physics world12, 33 (1999)

    1999

  4. [4]

    Sachdev,Quantum Phase Transitions(Cambridge University Press, 2011)

    S. Sachdev,Quantum Phase Transitions(Cambridge University Press, 2011)

    2011

  5. [5]

    Kitaev and J

    A. Kitaev and J. Preskill, Physical review letters96, 110404 (2006)

    2006

  6. [6]

    Wen, Reviews of Modern Physics89, 041004 (2017)

    X.-G. Wen, Reviews of Modern Physics89, 041004 (2017)

    2017

  7. [7]

    T. J. Osborne and M. A. Nielsen, Quantum Information Processing1, 45 (2002)

    2002

  8. [8]

    T. J. Osborne and M. A. Nielsen, Physical Review A66, 032110 (2002)

    2002

  9. [9]

    Osterloh, L

    A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature 416, 608 (2002)

    2002

  10. [10]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Reviews of mod- ern physics82, 277 (2010)

    2010

  11. [11]

    Bianchi, L

    E. Bianchi, L. Hackl, M. Kieburg, M. Rigol, and L. Vid- mar, PRX Quantum3, 030201 (2022)

    2022

  12. [12]

    O. F. Sylju˚ asen, Physical Review A68, 060301 (2003)

    2003

  13. [13]

    Osterloh, G

    A. Osterloh, G. Palacios, and S. Montangero, Physical review letters97, 257201 (2006)

    2006

  14. [14]

    T. R. de Oliveira, G. Rigolin, M. C. de Oliveira, and E. Miranda, Physical Review A—Atomic, Molecular, and Optical Physics77, 032325 (2008)

    2008

  15. [15]

    Saffman, T

    M. Saffman, T. G. Walker, and K. Mølmer, Reviews of modern physics82, 2313 (2010)

    2010

  16. [16]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, et al., Nature551, 579 (2017)

    2017

  17. [17]

    Semeghini, H

    G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kali- nowski, R. Samajdar, et al., Science374, 1242 (2021)

    2021

  18. [18]

    Ebadi, T

    S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Se- meghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pich- ler, W. W. Ho, et al., Nature595, 227 (2021)

    2021

  19. [19]

    Scholl, M

    P. Scholl, M. Schuler, H. J. Williams, A. A. Eberharter, D. Barredo, K.-N. Schymik, V. Lienhard, L.-P. Henry, T. C. Lang, T. Lahaye, et al., Nature595, 233 (2021)

    2021

  20. [20]

    Zhang, S

    J. Zhang, S. H. Cant´ u, F. Liu, A. Bylinskii, B. Braver- man, F. Huber, J. Amato-Grill, A. Lukin, N. Gemelke, A. Keesling, et al., Nature Communications16, 712 (2025)

    2025

  21. [21]

    Bluvstein, H

    D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, et al., Nature604, 451 (2022)

    2022

  22. [22]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, et al., Nature626, 58 (2024)

    2024

  23. [23]

    B. W. Reichardt, A. Paetznick, D. Aasen, I. Basov, J. M. Bello-Rivas, P. Bonderson, R. Chao, W. van Dam, M. B. Hastings, A. Paz, et al., arXiv preprint arXiv:2411.11822 (2024)

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  24. [24]

    Fendley, K

    P. Fendley, K. Sengupta, and S. Sachdev, Physical Re- view B69, 075106 (2004)

    2004

  25. [25]

    Samajdar, W

    R. Samajdar, W. W. Ho, H. Pichler, M. D. Lukin, and S. Sachdev, Physical Review Letters124, 103601 (2020)

    2020

  26. [26]

    Chepiga and F

    N. Chepiga and F. Mila, Nature Communications12, 414 (2021)

    2021

  27. [27]

    M. J. O’Rourke and G. K.-L. Chan, Nature Communica- tions14, 5397 (2023)

    2023

  28. [28]

    Reini´ c, D

    N. Reini´ c, D. Jaschke, D. Wanisch, P. Silvi, and S. Mon- tangero, Physical Review Research6, 033322 (2024)

    2024

  29. [29]

    Samajdar, S

    R. Samajdar, S. Choi, H. Pichler, M. D. Lukin, and S. Sachdev, Physical Review A98, 023614 (2018)

    2018

  30. [30]

    Chepiga and F

    N. Chepiga and F. Mila, Physical review letters122, 017205 (2019)

    2019

  31. [31]

    Rader and A

    M. Rader and A. M. L¨ auchli, arXiv preprint arXiv:1908.02068 (2019)

    work page arXiv 1908

  32. [32]

    X.-J. Yu, S. Yang, J.-B. Xu, and L. Xu, Physical Review B106, 165124 (2022)

    2022

  33. [33]

    S.-A. Liao, J. Zhang, and L.-P. Yang, Physical Review B 111, 165154 (2025)

    2025

  34. [34]

    Soto-Garcia and N

    J. Soto-Garcia and N. Chepiga, Physical Review Re- search7, 013215 (2025)

    2025

  35. [35]

    Verresen, M

    R. Verresen, M. D. Lukin, and A. Vishwanath, Physical Review X11, 031005 (2021)

    2021

  36. [36]

    Samajdar, D

    R. Samajdar, D. G. Joshi, Y. Teng, and S. Sachdev, Physical Review Letters130, 043601 (2023)

    2023

  37. [37]

    Yan, Y.-C

    Z. Yan, Y.-C. Wang, R. Samajdar, S. Sachdev, and Z. Y. Meng, Physical Review Letters130, 206501 (2023)

    2023

  38. [38]

    Giudici, M

    G. Giudici, M. D. Lukin, and H. Pichler, Physical Review Letters129, 090401 (2022)

    2022

  39. [39]

    H. J. Manetsch, G. Nomura, E. Bataille, X. Lv, K. H. Leung, and M. Endres, Nature647, 60 (2025)

    2025

  40. [40]

    Levine, A

    H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuleti´ c, H. Pichler, et al., Physical review letters123, 170503 (2019)

    2019

  41. [41]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Reviews of modern physics80, 517 (2008)

    2008

  42. [42]

    M. B. Hastings, Journal of statistical mechanics: theory and experiment2007, P08024 (2007)

    2007

  43. [43]

    Schollw¨ ock, Annals of physics326, 96 (2011)

    U. Schollw¨ ock, Annals of physics326, 96 (2011)

    2011

  44. [44]

    Or´ us, Nature Reviews Physics1, 538 (2019)

    R. Or´ us, Nature Reviews Physics1, 538 (2019)

    2019

  45. [45]

    G¨ uhne and G

    O. G¨ uhne and G. T´ oth, Physics Reports474, 1 (2009), ISSN 0370-1573

    2009

  46. [46]

    W. K. Wootters, Physical Review Letters80, 2245 (1998)

    1998

  47. [47]

    Vidal and R

    G. Vidal and R. F. Werner, Physical Review A65, 032314 (2002)

    2002

  48. [48]

    S. R. White, Physical review letters69, 2863 (1992)

    1992

  49. [49]

    Verstraete, V

    F. Verstraete, V. Murg, and J. I. Cirac, Advances in physics57, 143 (2008)

    2008

  50. [50]

    Fishman, S

    M. Fishman, S. White, and E. M. Stoudenmire, SciPost Physics Codebases p. 004 (2022)

    2022

  51. [51]

    Quantum Optimization for Maximum Independent Set Using Rydberg Atom Arrays

    H. Pichler, S.-T. Wang, L. Zhou, S. Choi, and M. D. Lukin, arXiv preprint arXiv:1808.10816 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  52. [52]

    Ebadi, A

    S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J.-G. Liu, R. Samajdar, et al., Science376, 1209 (2022)

    2022

  53. [53]

    Keesling, A

    A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pich- ler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, et al., Nature568, 207 (2019)

    2019

  54. [54]

    Bak, Reports on Progress in Physics45, 587 (1982)

    P. Bak, Reports on Progress in Physics45, 587 (1982)

    1982

  55. [55]

    Ostlund, Physical Review B24, 398 (1981)

    S. Ostlund, Physical Review B24, 398 (1981)

    1981

  56. [56]

    I. A. Maceira, N. Chepiga, and F. Mila, Physical Review Research4, 043102 (2022)

    2022

  57. [57]

    A. W. Sandvik, inAIP Conference Proceedings(AIP, 2010), vol. 1297, pp. 135–338

    2010

  58. [58]

    M. E. Fisher and M. N. Barber, Physical Review Letters 28, 1516 (1972). 18

    1972

  59. [59]

    Binder, Zeitschrift f¨ ur Physik B Condensed Matter 43, 119 (1981)

    K. Binder, Zeitschrift f¨ ur Physik B Condensed Matter 43, 119 (1981)

    1981

  60. [60]

    Nohadani, S

    O. Nohadani, S. Wessel, and S. Haas, Physical Review B—Condensed Matter and Materials Physics72, 024440 (2005)

    2005

  61. [61]

    Wierschem and P

    K. Wierschem and P. Sengupta, Modern Physics Letters B28, 1430017 (2014)

    2014

  62. [62]

    Bhandari and V

    P. Bhandari and V. Malik, Journal of Physics: Confer- ence Series814, 012005 (2017)

    2017

  63. [63]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of statistical mechan- ics: theory and experiment2004, P06002 (2004)

    2004

  64. [64]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of physics a: mathe- matical and theoretical42, 504005 (2009)

    2009

  65. [65]

    Cesa and H

    F. Cesa and H. Pichler, Physical Review Letters131, 170601 (2023)

    2023

  66. [66]

    Votto, J

    M. Votto, J. Zeiher, and B. Vermersch, Quantum8, 1513 (2024)

    2024

  67. [67]

    Chevallier, J

    C. Chevallier, J. Vovrosh, J. de Hond, M. Dagrada, A. Dauphin, and V. E. Elfving, Physical Review A109, 062604 (2024)

    2024

  68. [68]

    Anand, C

    S. Anand, C. E. Bradley, R. White, V. Ramesh, K. Singh, and H. Bernien, Nature Physics20, 1744 (2024)

    2024

  69. [69]

    A. P. Burgers, S. Ma, S. Saskin, J. Wilson, M. A. Alarc´ on, C. H. Greene, and J. D. Thompson, PRX Quantum3, 020326 (2022)

    2022

  70. [70]

    Wurtz, A

    J. Wurtz, A. Bylinskii, B. Braverman, J. Amato-Grill, S. H. Cantu, F. Huber, A. Lukin, F. Liu, P. Weinberg, J. Long, et al., arXiv preprint arXiv:2306.11727 (2023)

    work page arXiv 2023