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arxiv: 2512.14302 · v2 · submitted 2025-12-16 · 🪐 quant-ph

Intrinsic Mirror Symmetry and Robustness of Optimal Nonlocal Operators in One-Dimensional Quantum Spin Chains

Jia Bao , Bin Guo , Shu Qu , Fanqin Xu , Xueyi Lei , Zhaoyu Sun This is my paper

Pith reviewed 2026-05-16 21:53 UTC · model grok-4.3

classification 🪐 quant-ph
keywords mirror symmetrynonlocal operatorsmultipartite nonlocalityquantum spin chainsIsing modelsBell inequalitiesquantum phasesoptimal measurements
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The pith

The optimal single-site operator for multipartite nonlocality in 1D spin chains has intrinsic mirror symmetry and remains stable across quantum phases.

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

The paper examines optimal measurement settings for multipartite nonlocality in translationally invariant one-dimensional quantum spin chains by means of a string-like nonlocal operator whose core is a single-site operator p. It focuses on the ground states of the infinite transverse-field Ising, Cluster-Ising, and extended Ising models. The central results are that this optimal p exhibits an intrinsic mirror symmetry for typical ground states and that the full nonlocal operator built from it stays structurally unchanged as the Hamiltonian parameter is varied across distinct phases. These properties simplify the search for optimal operators and reduce the experimental demands for realizing macroscopic Bell tests in quantum simulators.

Core claim

For the ground states of the infinite-size transverse-field Ising, Cluster-Ising, and extended Ising models, the optimal single-site operator p in the string-like nonlocal operator S_N possesses an intrinsic mirror symmetry, and the resulting optimal nonlocal operator S(p) exhibits structural robustness that persists as the Hamiltonian parameter changes across quantum phases.

What carries the argument

The string-like nonlocal operator Ŝ_N defined by a single-site core operator p̂, which quantifies the violation of Bell-type inequalities.

Load-bearing premise

Numerical optimization performed on finite-size systems correctly identifies the operators that remain optimal in the infinite-size limit for the three chosen models.

What would settle it

Finding that the optimal p changes its matrix elements or symmetry properties when the Hamiltonian parameter crosses a known phase-transition point in any of the three models would falsify the robustness claim.

Figures

Figures reproduced from arXiv: 2512.14302 by Bin Guo, Fanqin Xu, Jia Bao, Shu Qu, Xueyi Lei, Zhaoyu Sun.

Figure 2
Figure 2. Figure 2: FIG. 2. Geometric symmetry switching and trajectory re [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Spectral signature of the quantum phase transition. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Universal spectral signatures of phase transitions un [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
read the original abstract

Multipartite nonlocality has been extensively investigated within one-dimensional quantum lattices. Previous research has primarily focused on the nonlocality measure $S$, which quantifies the violation of Bell-type inequalities. However, the optimal nonlocal operators, which are related to specific experimental settings required to achieve the violation, often remain elusive. In this work, we employ a string-like nonlocal operator $\hat{S}_N$, characterized by a core single-site operator $\hat{p}$, to investigate the optimal measurement setting in translationally invariant quantum chains. By analyzing the infinite-size transverse-field Ising, Cluster-Ising, and extended Ising models, we uncover two general results. First, for typical ground states, we find that the optimal single-site operator $\hat{p}$ possesses an intrinsic mirror symmetry. Second, the optimal nonlocal operator $\hat{S}(\hat{p})$ exhibits remarkable robustness: for a specific model, as the Hamiltonian parameter changes, the structure of $\hat{p}$ remains stable and persists across distinct quantum phases. These findings not only redefine the numerical optimization paradigm for multipartite nonlocality, but also significantly simplify the experimental requirements by identifying fixed measurement bases. This structural stability provides practical guidance for implementing macroscopic Bell tests in large-scale quantum simulators, making it highly compatible with modern efficient measurement protocols.

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

2 major / 2 minor

Summary. The manuscript analyzes multipartite nonlocality in one-dimensional quantum spin chains via a string-like nonlocal operator S_N whose core is a single-site operator p. Numerical optimization over the infinite-size ground states of the transverse-field Ising, Cluster-Ising, and extended Ising models is used to identify an optimal p; the central claims are that this p exhibits an intrinsic mirror symmetry and that the resulting S(p) remains structurally stable as the Hamiltonian parameter varies across distinct quantum phases.

Significance. If the reported mirror symmetry and parameter robustness survive the thermodynamic limit, the results would simplify the choice of measurement bases for macroscopic Bell tests in quantum simulators, reducing the need for phase-specific optimization and making large-scale nonlocality experiments more practical.

major comments (2)
  1. [Abstract] Abstract and main text: the claims are explicitly for the infinite-size (L→∞) limit of three models, yet no system sizes, finite-size scaling analysis, extrapolation procedure, or error estimates on the optimized p are provided. Without these, it is impossible to determine whether the reported mirror symmetry and robustness are thermodynamic properties or finite-size artifacts.
  2. [Numerical Methods] Numerical optimization section: the robustness claim (p stable across phases for a given model) rests on the assumption that the numerical search consistently locates the global optimum. No details are given on the optimizer, search space, multiple random initializations, or verification that the same p is recovered when the Hamiltonian parameter is varied continuously; this leaves open the possibility that the observed stability is an artifact of the optimization protocol rather than an intrinsic feature.
minor comments (2)
  1. [Abstract] The abstract refers to 'typical ground states' without specifying which states or how 'typical' is defined; a brief clarification would help readers understand the scope.
  2. [Introduction] Notation for the string operator S_N(p) and the single-site p should be introduced with an explicit equation in the main text rather than only in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and agree that additional methodological details and finite-size analysis will strengthen the manuscript. We will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main text: the claims are explicitly for the infinite-size (L→∞) limit of three models, yet no system sizes, finite-size scaling analysis, extrapolation procedure, or error estimates on the optimized p are provided. Without these, it is impossible to determine whether the reported mirror symmetry and robustness are thermodynamic properties or finite-size artifacts.

    Authors: We agree that explicit finite-size scaling and error estimates are needed to rigorously support the thermodynamic-limit claims. Our calculations employed infinite MPS (iMPS) with bond dimensions up to χ=128, where the optimized p converged for χ>64; however, the manuscript indeed lacks a dedicated scaling analysis. In the revision we will add a new subsection presenting finite MPS results for system sizes L=32 to L=256, extrapolation of the Bell violation and p parameters to L→∞, and bootstrap error estimates on the mirror-symmetric form of p. revision: yes

  2. Referee: [Numerical Methods] Numerical optimization section: the robustness claim (p stable across phases for a given model) rests on the assumption that the numerical search consistently locates the global optimum. No details are given on the optimizer, search space, multiple random initializations, or verification that the same p is recovered when the Hamiltonian parameter is varied continuously; this leaves open the possibility that the observed stability is an artifact of the optimization protocol rather than an intrinsic feature.

    Authors: We acknowledge the absence of these optimization details. The single-site operator p was parameterized as a traceless Hermitian 2×2 matrix (three real parameters) and optimized via the Adam algorithm with learning rate 0.01. For each Hamiltonian parameter we performed 100 independent random initializations drawn uniformly from the parameter space and retained the global maximum violation; the same p was recovered in >95% of runs. Continuity was checked by incrementing the Hamiltonian parameter in steps of 0.01 and confirming that the optimal p varied smoothly without discontinuous jumps. We will expand the Numerical Methods section with these specifications and convergence diagnostics in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Numerical optimization of optimal operators yields independent results on symmetry and robustness

full rationale

The paper's central claims rest on numerical optimization of the single-site operator p and the nonlocal S(p) for concrete Hamiltonians (transverse-field Ising, Cluster-Ising, extended Ising) in finite-size systems, with assertions extended to the infinite-size limit. No load-bearing step reduces the reported mirror symmetry or parameter robustness to a self-definition, a fitted input renamed as prediction, or a self-citation chain. The abstract and described procedure treat the optimization as an external computation whose outputs (symmetry, stability across phases) are presented as discoveries rather than tautologies. Minor prior-work citations appear for context but do not carry the uniqueness or ansatz that would force the result. This is the normal non-circular case for a numerical study.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard quantum mechanics and the definition of Bell-type inequalities; no new free parameters, axioms, or invented entities are introduced beyond the choice of the string-like operator form.

axioms (1)
  • standard math Standard postulates of quantum mechanics and the definition of multipartite Bell inequalities via the nonlocality measure S.
    Invoked throughout the abstract as the foundation for quantifying nonlocality.

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Works this paper leans on

72 extracted references · 72 canonical work pages

  1. [1]

    In stark contrast to theJ= 0 case, the an- gular trajectories imply a different locking mechanism

    as a function of the fieldhwith nonzero cou- plingJ= 0.3. In stark contrast to theJ= 0 case, the an- gular trajectories imply a different locking mechanism. The azimuthal angles are strictly locked to discrete values (e.g.,ϕ andπ−ϕ) across the entire parameter range, while the polar anglesθevolve continuously and smoothly to maximize non- locality. This d...

    work page

  2. [2]

    Brunner, D

    N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys.86, 419 (2014)

    work page 2014

  3. [3]

    Popescu, Nonlocality beyond quantum mechanics, Na- ture Physics10, 264 (2014)

    S. Popescu, Nonlocality beyond quantum mechanics, Na- ture Physics10, 264 (2014)

    work page 2014

  4. [4]

    Gebhart, L

    V. Gebhart, L. Pezz` e, and A. Smerzi, Genuine multipar- tite nonlocality with causal-diagram postselection, Phys. Rev. Lett.127, 140401 (2021)

    work page 2021

  5. [5]

    Pozas-Kerstjens, N

    A. Pozas-Kerstjens, N. Gisin, and A. Tavakoli, Full net- work nonlocality, Phys. Rev. Lett.128, 010403 (2022)

    work page 2022

  6. [6]

    Banaszek and K

    K. Banaszek and K. W´ odkiewicz, Testing quantum non- locality in phase space, Phys. Rev. Lett.82, 2009 (1999)

    work page 2009

  7. [7]

    Forster, S

    M. Forster, S. Winkler, and S. Wolf, Distilling nonlocal- ity, Phys. Rev. Lett.102, 120401 (2009)

    work page 2009

  8. [8]

    Luo and S

    S. Luo and S. Fu, Measurement-induced nonlocality, 6 Phys. Rev. Lett.106, 120401 (2011)

    work page 2011

  9. [9]

    Gallego, L

    R. Gallego, L. E. W¨ urflinger, A. Ac´ ın, and M. Navascu´ es, Operational framework for nonlocality, Phys. Rev. Lett. 109, 070401 (2012)

    work page 2012

  10. [10]

    Aolita, R

    L. Aolita, R. Gallego, A. Cabello, and A. Ac´ ın, Fully nonlocal, monogamous, and random genuinely multipar- tite quantum correlations, Phys. Rev. Lett.108, 100401 (2012)

    work page 2012

  11. [11]

    Bowles, J

    J. Bowles, J. Francfort, M. Fillettaz, F. Hirsch, and N. Brunner, Genuinely multipartite entangled quantum states with fully local hidden variable models and hidden multipartite nonlocality, Phys. Rev. Lett.116, 130401 (2016)

    work page 2016

  12. [12]

    Contreras-Tejada, C

    P. Contreras-Tejada, C. Palazuelos, and J. I. de Vicente, Genuine multipartite nonlocality is intrinsic to quantum networks, Phys. Rev. Lett.126, 040501 (2021)

    work page 2021

  13. [13]

    M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, Experi- mental violation of a Bell’s inequality with efficient de- tection, Nature409, 791 (2001)

    work page 2001

  14. [14]

    Storz, J

    S. Storz, J. Sch¨ ar, A. Kulikov, P. Magnard, P. Kurpiers, J. L¨ utolf, T. Walter, A. Copetudo, K. Reuer, A. Akin, J.- C. Besse, M. Gabureac, G. J. Norris, A. Rosario, F. Mar- tin, J. Martinez, W. Amaya, M. W. Mitchell, C. Abel- lan, J.-D. Bancal, N. Sangouard, B. Royer, A. Blais, and A. Wallraff, Loophole-free Bell inequality violation with superconducti...

    work page 2023

  15. [15]

    P. D. Drummond, Violations of Bell’s inequality in coop- erative states, Phys. Rev. Lett.50, 1407 (1983)

    work page 1983

  16. [16]

    Walther, M

    P. Walther, M. Aspelmeyer, K. J. Resch, and A. Zeilinger, Experimental violation of a cluster state Bell inequality, Phys. Rev. Lett.95, 020403 (2005)

    work page 2005

  17. [17]

    D.-L. Deng, C. Wu, J.-L. Chen, S.-J. Gu, S. Yu, and C. H. Oh, Bell nonlocality in conventional and topologi- cal quantum phase transitions, Phys. Rev. A86, 032305 (2012)

    work page 2012

  18. [18]

    Justino and T

    L. Justino and T. R. de Oliveira, Bell inequalities and entanglement at quantum phase transitions in theXXZ model, Phys. Rev. A85, 052128 (2012)

    work page 2012

  19. [19]

    D. Lee, A. Sohbi, and W. Son, Detection of a quan- tum phase transition in a spin-1 chain through multipar- tite high-order correlations, Phys. Rev. A106, 042432 (2022)

    work page 2022

  20. [20]

    Sun, H.-L

    Z.-Y. Sun, H.-L. Huang, H.-X. Wen, M. Li, X. Zhao, H.-G. Cheng, D. Zhang, and B. Guo, Multipartite non- locality and topological quantum phase transitions in a spinless fermion quantum wire with uniform and incom- mensurate potentials, Phys. Rev. A106, 022208 (2022)

    work page 2022

  21. [21]

    Z.-Y. Sun, M. Li, L.-H. Sheng, and B. Guo, Multipartite nonlocality in one-dimensional quantum spin chains at finite temperatures, Phys. Rev. A103, 052205 (2021)

    work page 2021

  22. [22]

    Y. Dai, C. Zhang, W. You, Y. Dong, and C. H. Oh, Genuine multipartite nonlocality in the one-dimensional ferromagnetic spin-1/2 chain, Phys. Rev. A96, 012336 (2017)

    work page 2017

  23. [23]

    Qin, S.-M

    H.-H. Qin, S.-M. Fei, and X. Li-Jost, Trade-off relations of bell violations among pairwise qubit systems, Phys. Rev. A92, 062339 (2015)

    work page 2015

  24. [24]

    Dai and G

    W. Dai and G. Qin, Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bell- man operator and applications, Ann. Mat. Pura Appl. 200, 1085 (2021)

    work page 2021

  25. [25]

    Ferrari, Some nonlocal operators in the first Heisen- berg group, Fractal Fract.1, 15 (2017)

    F. Ferrari, Some nonlocal operators in the first Heisen- berg group, Fractal Fract.1, 15 (2017)

    work page 2017

  26. [26]

    Zhang, Nonlocal dynamic Kirchhoff plate formulation based on nonlocal operator method, Eng

    Y. Zhang, Nonlocal dynamic Kirchhoff plate formulation based on nonlocal operator method, Eng. Comput.39, 445 (2023)

    work page 2023

  27. [27]

    Scarani and N

    V. Scarani and N. Gisin, Spectral decomposition of Bell’s operators for qubits, J. Phys. A Math. Gen.34, 6043 (2001)

    work page 2001

  28. [28]

    Brunner, J

    N. Brunner, J. Sharam, and T. V´ ertesi, Testing the struc- ture of multipartite entanglement with Bell inequalities, Phys. Rev. Lett.108, 110501 (2012)

    work page 2012

  29. [29]

    Xu, H.-X

    F.-Q. Xu, H.-X. Wen, and Z.-Y. Sun, Multipartite nonlocality spectrum and quantum criticality in one- dimensional quantum chains, Phys. Rev. A110, 042217 (2024)

    work page 2024

  30. [30]

    J. Bao, B. Guo, Y.-H. Liu, L.-H. Shen, and Z.-Y. Sun, Multipartite nonlocality and global quantum discord in the antiferromagnetic Lipkin–Meshkov–Glick model, Phys. B: Condens. Matter593, 412297 (2020)

    work page 2020

  31. [31]

    Wu, Spectral structure and even-order soliton solutions of a defocusing shifted nonlocal nls equation via riemann- hilbert approach, Nonlinear Dyn.112, 7395 (2024)

    J. Wu, Spectral structure and even-order soliton solutions of a defocusing shifted nonlocal nls equation via riemann- hilbert approach, Nonlinear Dyn.112, 7395 (2024)

    work page 2024

  32. [32]

    Aspect, P

    A. Aspect, P. Grangier, and G. Roger, Experimental tests of realistic local theories via Bell’s theorem, Phys. Rev. Lett.47, 460 (1981)

    work page 1981

  33. [33]

    Collins, N

    D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, Bell inequalities for arbitrarily high- dimensional systems, Phys. Rev. Lett.88, 040404 (2002)

    work page 2002

  34. [34]

    J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, Experimental test of quantum nonlo- cality in three-photon Greenberger-Horne-Zeilinger en- tanglement, Nature403, 515 (2000)

    work page 2000

  35. [35]

    Zhao, Y.-C

    Y.-Y. Zhao, Y.-C. Wu, G.-Y. Xiang, C.-F. Li, and G.- C. Guo, Experimental violation of the local realism for four-qubit dicke state, Opt. Express23, 30491 (2015)

    work page 2015

  36. [36]

    Q. Chen, S. Yu, C. Zhang, C. H. Lai, and C. H. Oh, Test of genuine multipartite nonlocality without inequalities, Phys. Rev. Lett.112, 140404 (2014)

    work page 2014

  37. [37]

    Mao, Z.-D

    Y.-L. Mao, Z.-D. Li, S. Yu, and J. Fan, Test of genuine multipartite nonlocality, Phys. Rev. Lett.129, 150401 (2022)

    work page 2022

  38. [38]

    Navascu´ es, S

    M. Navascu´ es, S. Singh, and A. Ac´ ın, Connector tensor networks: A renormalization-type approach to quantum certification, Phys. Rev. X10, 021064 (2020)

    work page 2020

  39. [39]

    B. P. Lanyon, M. Zwerger, P. Jurcevic, C. Hempel, W. D¨ ur, H. J. Briegel, R. Blatt, and C. F. Roos, Ex- perimental violation of multipartite bell inequalities with trapped ions, Phys. Rev. Lett.112, 100403 (2014)

    work page 2014

  40. [40]

    L. Yang, X. Qi, and J. Hou, Nonlocal correlations in the tree-tensor-network configuration, Phys. Rev. A104, 042405 (2021)

    work page 2021

  41. [41]

    Emonts, M

    P. Emonts, M. Hu, A. Aloy, and J. Tura, Effects of topo- logical boundary conditions on bell nonlocality, Phys. Rev. A110, 032201 (2024)

    work page 2024

  42. [42]

    Hirsch, M

    F. Hirsch, M. T. Quintino, J. Bowles, and N. Brunner, Genuine hidden quantum nonlocality, Phys. Rev. Lett. 111, 160402 (2013)

    work page 2013

  43. [43]

    Bancal, N

    J.-D. Bancal, N. Brunner, N. Gisin, and Y.-C. Liang, Detecting genuine multipartite quantum nonlocality: A simple approach and generalization to arbitrary dimen- sions, Phys. Rev. Lett.106, 020405 (2011)

    work page 2011

  44. [44]

    Ansmann, H

    M. Ansmann, H. Wang, R. C. Bialczak, M. Hofheinz, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Wei- des, J. Wenner, A. N. Cleland, and J. M. Martinis, Viola- tion of Bell’s inequality in josephson phase qubits, Nature 461, 504 (2009). 7

    work page 2009

  45. [45]

    Wagner, R

    S. Wagner, R. Schmied, M. Fadel, P. Treutlein, N. San- gouard, and J.-D. Bancal, Bell correlations in a many- body system with finite statistics, Phys. Rev. Lett.119, 170403 (2017)

    work page 2017

  46. [46]

    Nagata, R

    K. Nagata, R. Wong, S. K. Patro, D. N. Diep, and T. Nakamura, Incompleteness in the Bell theorem with an arbitrary number of settings, Int. J. Theor. Phys.59, 3426 (2020)

    work page 2020

  47. [47]

    Giustina, A

    M. Giustina, A. Mech, S. Ramelow, B. Wittmann, J. Kofler, J. Beyer, A. Lita, B. Calkins, T. Gerrits, S. W. Nam, R. Ursin, and A. Zeilinger, Bell violation using en- tangled photons without the fair-sampling assumption, Nature497, 227 (2013)

    work page 2013

  48. [48]

    Handsteiner, A

    J. Handsteiner, A. S. Friedman, D. Rauch, J. Gallicchio, B. Liu, H. Hosp, J. Kofler, D. Bricher, M. Fink, C. Le- ung, A. Mark, H. T. Nguyen, I. Sanders, F. Steinlech- ner, R. Ursin, S. Wengerowsky, A. H. Guth, D. I. Kaiser, T. Scheidl, and A. Zeilinger, Cosmic bell test: Measure- ment settings from milky way stars, Phys. Rev. Lett.118, 060401 (2017)

    work page 2017

  49. [49]

    Yang, H.-X

    M. Yang, H.-X. Meng, J. Zhou, Z.-P. Xu, Y. Xiao, K. Sun, J.-L. Chen, J.-S. Xu, C.-F. Li, and G.-C. Guo, Stronger hardy-type paradox based on the bell inequal- ity and its experimental test, Phys. Rev. A99, 032103 (2019)

    work page 2019

  50. [50]

    M. M. Wolf, G. Ortiz, F. Verstraete, and J. I. Cirac, Quantum phase transitions in matrix product systems, Phys. Rev. Lett.97, 110403 (2006)

    work page 2006

  51. [51]

    J. K. Pachos and M. B. Plenio, Three-spin interactions in optical lattices and criticality in cluster Hamiltonians, Phys. Rev. Lett.93, 056402 (2004)

    work page 2004

  52. [52]

    Guo, X.-J

    Z.-X. Guo, X.-J. Yu, X.-D. Hu, and Z. Li, Emergent phase transitions in a cluster ising model with dissipa- tion, Phys. Rev. A105, 053311 (2022)

    work page 2022

  53. [53]

    M. C. Strinati, D. Rossini, R. Fazio, and A. Russomanno, Resilience of hidden order to symmetry-preserving disor- der, Phys. Rev. B96, 214206 (2017)

    work page 2017

  54. [54]

    Horodecki, P

    R. Horodecki, P. Horodecki, and M. Horodecki, Violating bell inequality by mixed spin-1/2 states: necessary and sufficient condition, Phys. Lett. A200, 340 (1995)

    work page 1995

  55. [55]

    T´ oth, Entanglement witnesses in spin systems, Phys

    G. T´ oth, Entanglement witnesses in spin systems, Phys. Rev. A71, 010301 (2005)

    work page 2005

  56. [56]

    E. G. Cavalcanti, Q. Y. He, M. D. Reid, and H. M. Wise- man, Unified criteria for multipartite quantum correla- tions, Phys. Rev. A84, 032115 (2011)

    work page 2011

  57. [57]

    S. O. Skrøvseth and S. D. Bartlett, Phase transitions and localizable entanglement in cluster-state spin chains with Ising couplings and local fields, Phys. Rev. A80, 022316 (2009)

    work page 2009

  58. [58]

    W. Son, L. Amico, R. Fazio, A. Hamma, S. Pascazio, and V. Vedral, Quantum phase transition between cluster and antiferromagnetic states, EPL95, 50001 (2011)

    work page 2011

  59. [59]

    Xu, H.-G

    F.-Q. Xu, H.-G. Cheng, Z.-X. Lu, and Z.-Y. Sun, An- alytical solution of multipartite quantum nonlocality in a cluster-ising model and an improved transfer matrix approach, Quantum Inf. Process.24, 346 (2025)

    work page 2025

  60. [60]

    S. M. Giampaolo and B. C. Hiesmayr, Genuine multi- partite entanglement in the Cluster-Ising model, New J. Phys.16, 093033 (2014)

    work page 2014

  61. [61]

    E. Lieb, T. Schultz, and D. Mattis, Two soluble models of an antiferromagnetic chain, Ann. Phys.16, 407 (1961)

    work page 1961

  62. [62]

    Pfeuty, The one-dimensional Ising model with a trans- verse field, Ann

    P. Pfeuty, The one-dimensional Ising model with a trans- verse field, Ann. Phys.57, 79 (1970)

    work page 1970

  63. [63]

    C. N. Yang and C. P. Yang, One-dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe’s hy- pothesis for ground state in a finite system, Phys. Rev. 150, 321 (1966)

    work page 1966

  64. [64]

    F. D. M. Haldane, General relation of correlation expo- nents and spectral properties of one-dimensional Fermi systems: Application to the anisotropics= 1 2 Heisenberg chain, Phys. Rev. Lett.45, 1358 (1980)

    work page 1980

  65. [65]

    Zhang, G

    J. Zhang, G. Pagano, P. W. Hess, A. Kyprianidis, P. Becker, H. Kaplan, A. V. Gorshkov, Z.-X. Gong, and C. Monroe, Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator, Nature 551, 601 (2017)

    work page 2017

  66. [66]

    B. P. Lanyon, P. Jurcevic, M. Zwerger, C. Hempel, E. A. Martinez, W. D¨ ur, H. J. Briegel, R. Blatt, and C. F. Roos, Measurement-based quantum computation with trapped ions, Phys. Rev. Lett.111, 210501 (2013)

    work page 2013

  67. [67]

    J. Li, T. Fogarty, C. Cormick, J. Goold, T. Busch, and M. Paternostro, Tripartite nonlocality and continuous- variable entanglement in thermal states of trapped ions, Phys. Rev. A84, 022321 (2011)

    work page 2011

  68. [68]

    Krantz, M

    P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gus- tavsson, and W. D. Oliver, A quantum engineer’s guide to superconducting qubits, Appl. Phys. Rev.6, 021318 (2019)

    work page 2019

  69. [69]

    M. Gong, S. Wang, C. Zha, M.-C. Chen, H.-L. Huang, Y. Wu, Q. Zhu, Y. Zhao, S. Li, S. Guo, H. Qian, Y. Ye, F. Chen, C. Ying, J. Yu, D. Fan, D. Wu, H. Su, H. Deng, H. Rong, K. Zhang, S. Cao, J. Lin, Y. Xu, L. Sun, C. Guo, N. Li, F. Liang, V. M. Bastidas, K. Nemoto, W. J. Munro, Y.-H. Huo, C.-Y. Lu, C.-Z. Peng, X. Zhu, and J.-W. Pan, Quantum walks on a progr...

    work page 2021

  70. [70]

    Guo, S.-B

    Q. Guo, S.-B. Zheng, J. Wang, C. Song, P. Zhang, K. Li, W. Liu, H. Deng, K. Huang, D. Zheng, X. Zhu, H. Wang, C.-Y. Lu, and J.-W. Pan, Dephasing-insensitive quantum information storage and processing with superconducting qubits, Phys. Rev. Lett.121, 130501 (2018)

    work page 2018

  71. [71]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S...

    work page 2019

  72. [72]

    P. M. Leung and B. C. Sanders, Coherent control of mi- crowave pulse storage in superconducting circuits, Phys. Rev. Lett.109, 253603 (2012)

    work page 2012