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arxiv: 2604.15268 · v1 · submitted 2026-04-16 · 🪐 quant-ph · cond-mat.stat-mech

Recognition: unknown

Assembling Extensive Quantum Fisher Information in Stabilizer Systems

Arnau Lira-Solanilla , Sreemayee Aditya , Xhek Turkeshi , Silvia Pappalardi
Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:26 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords quantum Fisher informationstabilizer codesstring order parametersmonitored quantum systemsquantum metrologytoric codecluster codesnonlocal observables
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The pith

A mapping from stabilizer generators to dual Ising spins turns hidden nonlocal order into observables with extensive quantum Fisher information density.

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

The paper introduces a systematic framework to construct nonlocal observables possessing extensive quantum Fisher information density inside stabilizer codes. It achieves this by mapping the generators onto dual Ising spins so that the spins' correlators directly equal string order parameters. This step converts previously inaccessible hidden nonlocal order into a quantity usable for quantum metrology. The method is applied to monitored cluster codes and the toric code, where the QFI scaling changes from extensive, supported by long-range string order, to intensive when single-site measurements compete. A reader would care because the construction supplies a concrete route to harness metrological resources in many-body systems protected by stabilizer constraints.

Core claim

The construction maps stabilizer generators to dual Ising spins whose correlators equal string order parameters, converting hidden nonlocal order into a metrologically accessible observable. Applying this to monitored cluster codes and the toric code, the work identifies transitions in the QFI scaling from an extensive regime, where long-range string order prevails, to an intensive one driven by competing single-site measurements.

What carries the argument

The mapping of stabilizer generators onto dual Ising spins whose correlators are identified with string order parameters.

If this is right

  • Long-range string order in monitored cluster codes directly supplies extensive QFI density.
  • The toric code exhibits a transition to intensive QFI scaling when single-site measurements dominate.
  • String order parameters become concrete, metrologically usable observables inside stabilizer systems.
  • Nonlocal observables constructed this way remain accessible for sensing tasks in the extensive regime.

Where Pith is reading between the lines

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

  • The same mapping technique could be tested on other stabilizer codes such as the color code to check whether extensive QFI appears under analogous monitoring.
  • If the link between string order and QFI holds, it may guide the design of measurement protocols that preserve metrological advantage in topological codes.
  • One could examine whether the framework extends to time-dependent monitoring rates to locate the precise boundary between extensive and intensive QFI regimes.

Load-bearing premise

That the mapping from stabilizer generators to dual Ising spins produces correlators that directly yield extensive QFI density without further assumptions on measurement dynamics or code details.

What would settle it

An explicit computation or measurement in the toric code under competing single-site monitoring that shows the QFI density remains extensive even after the associated string order parameter has decayed.

Figures

Figures reproduced from arXiv: 2604.15268 by Arnau Lira-Solanilla, Silvia Pappalardi, Sreemayee Aditya, Xhek Turkeshi.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic of the framework to find extensive QFI [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: QFI density of the monitored one-dimensional cluster [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Examples of the dual spin operators used in a square [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: QFI density of the monitored two-dimensional cluster [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Examples of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: QFI density of the monitored toric code with [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Additional data of the QFI density in the 1D cluster code. (a) Scaling exponent extracted from the QFI density [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: (a) [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

We introduce a systematic framework to construct nonlocal observables with extensive quantum Fisher information (QFI) density in stabilizer codes. The construction maps stabilizer generators to dual Ising spins whose correlators equal string order parameters, converting hidden nonlocal order into a metrologically accessible observable. Applying this to monitored cluster codes and the toric code, we identify transitions in the QFI scaling from an extensive regime, where long-range string order prevails, to an intensive one driven by competing single-site measurements.

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 / 3 minor

Summary. The manuscript introduces a systematic framework to construct nonlocal observables with extensive quantum Fisher information (QFI) density in stabilizer codes. The central construction maps stabilizer generators to dual Ising spins whose correlators are identified with string order parameters, thereby converting hidden nonlocal order into a metrologically usable observable. The framework is applied to monitored cluster codes and the toric code, where transitions in QFI scaling are identified: from an extensive regime (long-range string order) to an intensive regime under competing single-site measurements.

Significance. If the mapping and variance scaling are rigorously derived, the work provides a concrete method to engineer metrologically useful observables from stabilizer structure and string order, with direct relevance to quantum sensing in many-body systems and measurement-induced phases. The explicit applications to two codes and the reported scaling transition constitute falsifiable predictions that strengthen the contribution.

minor comments (3)
  1. [Abstract] Abstract: the central claim of extensive QFI density relies on the variance of the constructed observable scaling linearly with system size; a one-sentence reminder of the pure-state relation QFI = 4 Var(O) would help readers connect the string-order construction to the metrological figure of merit.
  2. The mapping from stabilizer generators to dual Ising spins is described at a high level; an explicit example (e.g., for the smallest cluster or toric-code plaquette) with the resulting Hamiltonian or correlator would improve reproducibility.
  3. Notation for the string order parameters and dual spins should be introduced once and used consistently; occasional redefinition of symbols across sections risks confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment. We are pleased that the central construction—mapping stabilizer generators to dual Ising spins to convert hidden nonlocal order into observables with extensive QFI density—is recognized as providing a concrete method with relevance to quantum sensing and measurement-induced phases. The applications to monitored cluster codes and the toric code, along with the identified scaling transitions, are highlighted as strengthening the contribution. As no specific major comments were raised in the report, we will implement minor revisions to improve clarity, presentation, and any minor technical details as recommended.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a mapping from stabilizer generators to dual Ising spins with correlators identified as string order parameters, then applies it to monitored cluster and toric codes to observe QFI scaling transitions. This is a constructive framework rather than a derivation that reduces by construction to fitted inputs, self-citations, or tautological definitions. No equations or steps in the provided abstract and description show a prediction that is statistically forced by the inputs themselves or that relies on load-bearing self-citations for its validity. The QFI relation for pure states (QFI = 4 Var(O)) and the properties of long-range string order are standard and independent of the specific mapping chosen here. The derivation chain is self-contained against external benchmarks in stabilizer code theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that stabilizer generators can be faithfully represented as dual Ising spins whose two-point functions reproduce string order parameters; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Stabilizer generators map to dual Ising spins such that their correlators equal string order parameters
    This mapping is the load-bearing step that converts hidden order into a metrologically usable observable.

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Reference graph

Works this paper leans on

67 extracted references · 2 canonical work pages

  1. [1]

    Gottesman,Stabilizer codes and quantum error cor- rection, Ph.D

    D. Gottesman,Stabilizer codes and quantum error cor- rection, Ph.D. thesis, California Institute of Technology (1997)

    1997

  2. [2]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information, 10th ed. (Cambridge Univer- sity Press, 2010)

    2010

  3. [3]

    Preskill, Quantum2, 79 (2018)

    J. Preskill, Quantum2, 79 (2018)

    2018

  4. [4]

    Raussendorf and H

    R. Raussendorf and H. J. Briegel, Phys. Rev. Lett.86, 5188 (2001)

    2001

  5. [5]

    A. Y. Kitaev, Ann. Phys.303, 2 (2003)

    2003

  6. [6]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Phys. Rev. X9, 031009 (2019)

    2019

  7. [7]

    Y. Li, X. Chen, and M. P. Fisher, Phys. Rev. B98, 205136 (2018)

    2018

  8. [8]

    Y. Li, X. Chen, and M. P. Fisher, Phys. Rev. B100, 134306 (2019)

    2019

  9. [9]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Annu. Rev. Condens. Matter Phys.14, 335 (2023)

    2023

  10. [10]

    A. C. Potter and R. Vasseur, Entanglement dynamics in 6 hybrid quantum circuits, inEntanglement in Spin Chains (Springer International Publishing, 2022) p. 211

    2022

  11. [11]

    Turkeshi and P

    X. Turkeshi and P. Sierant, Phys. Rev. Lett.132, 140401 (2024)

    2024

  12. [12]

    Y. Bao, S. Choi, and E. Altman, Phys. Rev. B101, 104301 (2020)

    2020

  13. [13]

    Jian, Y.-Z

    C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, Phys. Rev. B101, 104302 (2020)

    2020

  14. [14]

    M. J. Gullans and D. A. Huse, Phys. Rev. X10, 041020 (2020)

    2020

  15. [15]

    Sierant, M

    P. Sierant, M. Schir` o, M. Lewenstein, and X. Turkeshi, Phys. Rev. B106, 214316 (2022)

    2022

  16. [16]

    C. Noel, P. Niroula, D. Zhu, A. Risinger, L. Egan, D. Biswas, M. Cetina, A. V. Gorshkov, M. J. Gullans, D. A. Huse, and C. Monroe, Nat. Phys.18, 760 (2022)

    2022

  17. [17]

    J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Nat. Phys.19, 1314 (2023)

    2023

  18. [18]

    Google AI Quantum and collaborators, Nature622, 481 (2023)

    2023

  19. [19]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys.80, 517 (2008)

    2008

  20. [20]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys.81, 865 (2009)

    2009

  21. [21]

    Laflorencie, Phys

    N. Laflorencie, Phys. Rep.646, 1 (2016)

    2016

  22. [22]

    S. L. Braunstein and C. M. Caves, Phys. Rev. Lett.72, 3439 (1994)

    1994

  23. [23]

    M. G. Paris, Int. J. Quantum Inf.7, 125 (2009)

    2009

  24. [24]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photonics 5, 222 (2011)

    2011

  25. [25]

    Pezz` e, A

    L. Pezz` e, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Rev. Mod. Phys.90, 035005 (2018)

    2018

  26. [26]

    Pezz´ e and A

    L. Pezz´ e and A. Smerzi, Phys. Rev. Lett.102, 100401 (2009)

    2009

  27. [27]

    T´ oth and I

    G. T´ oth and I. Apellaniz, J. Phys. A: Math. Theor.47, 424006 (2014)

    2014

  28. [28]

    Pezze’ and A

    L. Pezze’ and A. Smerzi, (2014), arXiv:1411.5164 [quant- ph]

    work page arXiv 2014

  29. [29]

    Hyllus, W

    P. Hyllus, W. Laskowski, R. Krischek, C. Schwemmer, W. Wieczorek, H. Weinfurter, L. Pezz´ e, and A. Smerzi, Phys. Rev. A85, 022321 (2012)

    2012

  30. [30]

    T´ oth, Phys

    G. T´ oth, Phys. Rev. A85, 022322 (2012)

    2012

  31. [31]

    Lira-Solanilla, X

    A. Lira-Solanilla, X. Turkeshi, and S. Pappalardi, Phys. Rev. Lett.135, 080401 (2025)

    2025

  32. [32]

    Pezz` e, M

    L. Pezz` e, M. Gabbrielli, L. Lepori, and A. Smerzi, Phys. Rev. Lett.119, 250401 (2017)

    2017

  33. [33]

    Zhang, Y

    Y.-R. Zhang, Y. Zeng, H. Fan, J. Q. You, and F. Nori, Phys. Rev. Lett.120, 250501 (2018)

    2018

  34. [34]

    Zhang, Y

    Y.-R. Zhang, Y. Zeng, T. Liu, H. Fan, J. Q. You, and F. Nori, Phys. Rev. Res.4, 023144 (2022)

    2022

  35. [35]

    Ferro and M

    F. Ferro and M. Fagotti, (2025), arXiv:2503.21905 [quant-ph]

    work page arXiv 2025

  36. [36]

    Zeng and X.-G

    B. Zeng and X.-G. Wen, Phys. Rev. B91, 125121 (2015)

    2015

  37. [37]

    Schuch, D

    N. Schuch, D. P´ erez-Garc´ ıa, and I. Cirac, Phys. Rev. B 84, 165139 (2011)

    2011

  38. [38]

    Affleck and F

    I. Affleck and F. D. M. Haldane, Phys. Rev. B36, 5291 (1987)

    1987

  39. [39]

    Haldane, Physics Letters A93, 464 (1983)

    F. Haldane, Physics Letters A93, 464 (1983)

    1983

  40. [40]

    F. D. M. Haldane, Phys. Rev. Lett.50, 1153 (1983)

    1983

  41. [41]

    Pollmann, E

    F. Pollmann, E. Berg, A. M. Turner, and M. Oshikawa, Phys. Rev. B85, 075125 (2012)

    2012

  42. [42]

    This statement corresponds toM j|ψ⟩=|ψ⟩for eachj= 1, . . . , N

  43. [43]

    Paviglianiti and A

    A. Paviglianiti and A. Silva, Phys. Rev. B108, 184302 (2023)

    2023

  44. [44]

    Lang and H

    N. Lang and H. P. B¨ uchler, Phys. Rev. B102, 094204 (2020)

    2020

  45. [45]

    Lavasani, Y

    A. Lavasani, Y. Alavirad, and M. Barkeshli, Nat. Phys. 17, 342 (2021)

    2021

  46. [46]

    Sang and T

    S. Sang and T. H. Hsieh, Phys. Rev. Res.3, 023200 (2021)

    2021

  47. [47]

    Klocke and M

    K. Klocke and M. Buchhold, Phys. Rev. B106, 104307 (2022)

    2022

  48. [48]

    Hauke, M

    P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Nat. Phys.12, 778 (2016)

    2016

  49. [49]

    Zabalo, M

    A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrish- nan, D. A. Huse, and J. H. Pixley, Phys. Rev. B101, 060301 (2020)

    2020

  50. [50]

    Zabalo, M

    A. Zabalo, M. J. Gullans, J. H. Wilson, R. Vasseur, A. W. W. Ludwig, S. Gopalakrishnan, D. A. Huse, and J. H. Pixley, Phys. Rev. Lett.128, 050602 (2022)

    2022

  51. [51]

    Lavasani, Y

    A. Lavasani, Y. Alavirad, and M. Barkeshli, Phys. Rev. Lett.127, 235701 (2021)

    2021

  52. [52]

    An equivalent picture can be done using only plaquette operatorsB p, which shows the potential of this protocol despite this freedom of choice

    Such a subset consists of all star operatorsA s with no support on boundary spins. An equivalent picture can be done using only plaquette operatorsB p, which shows the potential of this protocol despite this freedom of choice

  53. [53]

    P. M. Poggi and M. H. Mu˜ noz-Arias, Quantum8, 1229 (2024)

    2024

  54. [54]

    Sharma and E

    V. Sharma and E. J. Mueller, Phys. Rev. A112, 012411 (2025)

    2025

  55. [55]

    Fr´ erot and T

    I. Fr´ erot and T. Roscilde, Phys. Rev. Lett.121, 020402 (2018)

    2018

  56. [56]

    Costa de Almeida and P

    R. Costa de Almeida and P. Hauke, Phys. Rev. Res.3, L032051 (2021)

    2021

  57. [57]

    Vitale, A

    V. Vitale, A. Rath, P. Jurcevic, A. Elben, C. Branciard, and B. Vermersch, PRX Quantum5, 030338 (2024)

    2024

  58. [58]

    Laurell, A

    P. Laurell, A. Scheie, C. J. Mukherjee, M. M. Koza, M. Enderle, Z. Tylczynski, S. Okamoto, R. Coldea, D. A. Tennant, and G. Alvarez, Phys. Rev. Lett.127, 037201 (2021)

    2021

  59. [59]

    Scheie, P

    A. Scheie, P. Laurell, A. M. Samarakoon, B. Lake, S. E. Nagler, G. E. Granroth, S. Okamoto, G. Alvarez, and D. A. Tennant, Phys. Rev. B103, 224434 (2021)

    2021

  60. [60]

    Klocke and M

    K. Klocke and M. Buchhold, Phys. Rev. X13, 041028 (2023)

    2023

  61. [61]

    Passarelli, X

    G. Passarelli, X. Turkeshi, A. Russomanno, P. Lucig- nano, M. Schir` o, and R. Fazio, Phys. Rev. Lett.132, 163401 (2024)

    2024

  62. [62]

    Z. Li, A. Delmonte, X. Turkeshi, and R. Fazio, Nature Communications16, 4329 (2025)

    2025

  63. [63]

    M. Yu, Y. Liu, P. Yang, M. Gong, Q. Cao, S. Zhang, H. Liu, M. Heyl, T. Ozawa, N. Goldman, and J. Cai, npj Quantum Info.8, 56 (2022)

    2022

  64. [64]

    M. M. Rams, P. Sierant, O. Dutta, P. Horodecki, and J. Zakrzewski, Phys. Rev. X8, 021022 (2018)

    2018

  65. [65]

    Y. Chu, S. Zhang, B. Yu, and J. Cai, Phys. Rev. Lett. 126, 010502 (2021)

    2021

  66. [66]

    Lira-Solanilla, S

    A. Lira-Solanilla, S. Aditya, X. Turkeshi, and S. Pap- palardi, Code and data will be available at publications (2025)

    2025

  67. [67]

    J. W. Essam, Reports on Progress in Physics43, 833 (1980). 7 End matter Scaling analysis of QFI densityTo quantify the transitions in QFI scaling reported in the main text, we extract scaling exponents by fittingf Q as a function of system size at each measurement rate. The results for the 1D cluster model are summarized in Fig. 7a. In the 1D cluster code...

    1980