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arxiv: 2405.00416 · v2 · submitted 2024-05-01 · 🪐 quant-ph · cond-mat.stat-mech· cond-mat.str-el

Entanglement and fidelity across quantum phase transitions in locally perturbed topological codes with open boundaries

Harikrishnan K J , Amit Kumar Pal This is my paper

Pith reviewed 2026-05-24 01:55 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcond-mat.str-el
keywords quantum phase transitionfidelity susceptibilityentanglement witnessKitaev codetopological codeIsing modelopen boundariescolor code
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The pith

Perturbed Kitaev codes on open cylinders undergo quantum phase transitions marked by power-law diverging fidelity susceptibility and logarithmically diverging entanglement witnesses.

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

The paper examines topological-to-non-topological quantum phase transitions in the Kitaev code embedded on a wide cylinder with open boundaries under local magnetic fields and Ising-type spin interactions. Fidelity susceptibility is shown to diverge as a power law at these transitions, with quantum critical points located through finite-size scaling analysis. The results are verified by mapping the perturbed code exactly to the two-dimensional Ising model with nearest- and next-nearest-neighbor interactions and computing single-site magnetization as an order parameter via quantum Monte Carlo, while also revealing an odd-even dichotomy in the occurrence of transitions for odd versus even circumferences under pure Ising perturbation. A local entanglement witness operator is constructed to bound localizable entanglement on the vertical non-trivial loop, and its first derivative is shown to diverge logarithmically at the transition, with finite-size scaling performed; the same pattern appears in the locally perturbed color code.

Core claim

In locally perturbed topological codes with open boundaries, the fidelity susceptibility displays a power-law divergence across the quantum phase transition, enabling location of the quantum critical points via finite-size scaling. This is verified by an exact mapping to the two-dimensional Ising model with nearest- and next-nearest-neighbor interactions, where single-site magnetization computed by quantum Monte Carlo serves as an independent order parameter locating the same points. The first derivative of the expectation value of a local entanglement witness operator designed to capture a lower bound to localizable entanglement on the vertical non-trivial loop exhibits a logarithmic diver

What carries the argument

The exact mapping of the perturbed Kitaev code on the cylinder to the 2D Ising model with nearest- and next-nearest-neighbor interactions, which supplies an independent order parameter (single-site magnetization) computed by quantum Monte Carlo to confirm the critical points found from fidelity susceptibility.

If this is right

  • Quantum critical points are located by finite-size scaling of the power-law divergence in fidelity susceptibility.
  • An odd-even dichotomy appears in the Kitaev ladder: the quantum phase transition occurs only for odd or even values of the circumference under pure Ising perturbation.
  • The topological phase exhibits higher robustness against local perturbations when one boundary direction is left open.
  • The first derivative of the local entanglement witness expectation value diverges logarithmically at the transition, and the same divergence pattern holds for the perturbed color code.

Where Pith is reading between the lines

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

  • The local witness construction may be adapted to bound entanglement across transitions in other stabilizer codes or lattice gauge theories.
  • The reported increase in robustness with open boundaries suggests that boundary engineering could raise effective noise thresholds in topological quantum memories.
  • Finite-size scaling of both fidelity susceptibility and witness derivatives could be applied to locate transitions in higher-dimensional or non-stabilizer topological models where exact mappings are unavailable.

Load-bearing premise

The locally perturbed Kitaev code on the cylinder can be exactly mapped to the 2D Ising model with nearest- and next-nearest-neighbor interactions so that magnetization locates the same critical points.

What would settle it

A direct computation in which the quantum critical points obtained from finite-size scaling of fidelity susceptibility fail to coincide with those obtained from single-site magnetization in the mapped Ising model.

Figures

Figures reproduced from arXiv: 2405.00416 by Amit Kumar Pal, Harikrishnan K J.

Figure 1
Figure 1. Figure 1: FIG. 1. Kitaev code defined on the (a) square and the (b) triangular lattices with open (periodic) boundary condition along the vertical [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Color code defined on the (a) honeycomb and the (b) square-octagonal lattices with open (periodic) boundary condition along the [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Kitaev’s toric code on an [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Variations of the (a) energy gap, [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Construction of witness operators corresponding to (a) Kitaev code on square lattice with [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
read the original abstract

We investigate the topological-to-non-topological quantum phase transitions (QPTs) occurring in the Kitaev code under local perturbations in the form of local magnetic field and spin-spin interactions of the Ising-type using fidelity susceptibility (FS) and entanglement as the probes. We assume the code to be embedded on the surface of a wide cylinder of height $M$ and circumference $D$ with $M\ll D$. We demonstrate a power-law divergence of FS across the QPT, and determine the quantum critical points (QCPs) via a finite-size scaling analysis. We verify these results by mapping the perturbed Kitaev code to the 2D Ising model with nearest- and next-nearest-neighbor interactions, and computing the single-site magnetization as order parameter using quantum Monte-Carlo technique. We also demonstrate a finite size odd-even dichotomy in the occurrence of the QPT in the Kitaev ladder with respect to the odd and even values of $D$, when the system is perturbed with only Ising interaction. Our results also indicate a higher robustness of the topological phase of the Kitaev code against local perturbations if the boundary is made open along one direction. We further consider a local entanglement witness operator designed specifically to capture a lower bound to the localizable entanglement on the vertical non-trivial loop of the code. We show that the first derivative of the expectation value of the witness operator exhibits a logarithmic divergence across the QPT, and perform the finite-size scaling analysis. We demonstrate similar behaviour of the expectation value of the appropriately constructed witness operator also in the case of locally perturbed color code with open boundaries.

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

1 major / 1 minor

Summary. The paper examines topological-to-non-topological quantum phase transitions in the Kitaev code (and color code) on wide cylinders (M ≪ D) with open boundaries under local magnetic-field and Ising perturbations. It reports power-law divergence of fidelity susceptibility, locates QCPs via finite-size scaling, verifies the QCPs by mapping to the 2D Ising model with NNN interactions and computing single-site magnetization via QMC, notes an odd-even dichotomy for Ising perturbations on ladders, and shows logarithmic divergence in the first derivative of a local entanglement-witness operator (with analogous FSS).

Significance. If the mapping and numerical procedures hold, the results would strengthen understanding of the stability of topological codes to local perturbations, especially the enhanced robustness under open boundaries, and would support fidelity susceptibility and local entanglement witnesses as practical probes. The cross-check with an independent order parameter (QMC magnetization) and explicit finite-size scaling constitute a methodological strength.

major comments (1)
  1. [Verification via mapping to 2D Ising model] Verification via mapping (abstract and the section describing the mapping to the 2D Ising model): the claim that the locally perturbed Kitaev code on the open-boundary cylinder maps exactly onto the bulk 2D Ising model with nearest- and next-nearest-neighbor interactions requires explicit demonstration that no residual boundary operators are generated by the perturbations. Any such terms would shift the location or finite-size scaling of the magnetization jump relative to the FS-derived QCPs, undermining the asserted independent confirmation.
minor comments (1)
  1. [Entanglement witness section] The definition and construction of the local entanglement witness operator (and its relation to localizable entanglement on the vertical loop) would benefit from an explicit operator expression or diagram in the main text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying a point that requires clarification to strengthen the verification of our mapping. We address the major comment below.

read point-by-point responses

  1. Referee: Verification via mapping (abstract and the section describing the mapping to the 2D Ising model): the claim that the locally perturbed Kitaev code on the open-boundary cylinder maps exactly onto the bulk 2D Ising model with nearest- and next-nearest-neighbor interactions requires explicit demonstration that no residual boundary operators are generated by the perturbations. Any such terms would shift the location or finite-size scaling of the magnetization jump relative to the FS-derived QCPs, undermining the asserted independent confirmation.

    Authors: We agree that an explicit demonstration is required to rigorously establish that the mapping to the bulk 2D Ising model (with NNN interactions) is free of residual boundary operators that could affect the QCP location or scaling. The current manuscript presents the mapping but does not include this explicit check. In the revised manuscript we will add an appendix containing the perturbative expansion of the local perturbations on the open cylinder, showing that no relevant boundary operators are generated beyond those already accounted for in the bulk mapping. This will confirm consistency between the fidelity-susceptibility QCPs and the QMC magnetization results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; verification relies on external mapping and QMC

full rationale

The derivation uses FS divergence and finite-size scaling to locate QCPs, then verifies via an explicit mapping of the perturbed Kitaev code on the cylinder to the 2D Ising model (with NNN interactions) whose magnetization is obtained from independent QMC simulations. No step reduces a claimed prediction to a fitted parameter or self-citation by construction; the mapping and QMC are presented as external, established techniques. The entanglement-witness derivative is likewise computed directly from the model without internal redefinition. This is the normal case of a self-contained calculation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of the Kitaev-to-Ising mapping and on standard finite-size scaling assumptions; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The locally perturbed Kitaev code on the cylinder geometry maps exactly onto the 2D Ising model with nearest- and next-nearest-neighbor interactions.
    Invoked to justify using single-site magnetization from QMC as an independent order parameter that locates the same QCPs.
  • standard math Finite-size scaling of fidelity susceptibility and witness derivatives can be used to extract quantum critical points in the thermodynamic limit.
    Standard assumption in quantum phase transition studies; invoked for the scaling analysis.

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Reference graph

Works this paper leans on

64 extracted references · 64 canonical work pages · 3 internal anchors

  1. [1]

    (10) and (11))

    First, a non linear transformation of both abscissa and ordinate is performed as per the scaling formula (see Eqs. (10) and (11)). For example,χ(g)→(χ(g m)− χ(g))/χ(g)andg→D νg(g−g m)in the case of fi- delity susceptibility under field perturbation

    work page

  2. [2]

    Next, assuming a polynomialχ g of order eight as the finite scaling function [41, 42], we numerically per- form the least-square fitting of the re-scaled data to the finite scaling functionχ(g)

    work page

  3. [3]

    We achieve this by numerical optimization tools

    In the last step, we minimize the error in the least- square fitting by fine-tuning the critical exponents and critical perturbation strengths involved in the scaling function. We achieve this by numerical optimization tools. For the last step, we exploit the connection between the perturbed toric code and the transverse-field Ising model to make a judici...

    work page

  4. [4]

    The subset{S α}containsnindependent and commut- ing stabilizers

    work page

  5. [5]

    Thenreduced Pauli operatorsdefined as{S Ω α }= {T rΩ(Sα)}, where Ωis the set of qubits outsideΩ, are independent, and mutually commute. 14 0.40 0.45 0.50 0.55 0.6 0.8 1.0 1.2 1.4 D = 3 D = 4 D = 5 D = 6 D = 7 D = 8 D = 9 λ 0 c 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 D = 3 D = 4 D = 5 D = 6 D = 7 D = 8 D = 9 λ 0 c (a) 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.2...

    work page

  6. [6]

    All the reduced single qubit Pauli operators,S Ω,i α , on every qubit outsideΩmutually commute

    work page

  7. [7]

    dimensionalpseudoincidence matrix(M) of the subsystemΩ, defined as Mi,j = ( 1, if{S Ω,i α ,S Ω,i α′ }= 0, 0, if[S Ω,i α ,S Ω,i α′ ] = 0, (D1) is a rankn−1matrix. Here,S Ω,i α represents the single qubit reduced Pauli operator of the stabilizerS Ω α cor- responding to qubiti∈Ωandj∈[1, ( n 2)]is an index corresponding to a pair{α, α ′}that runs through all ...

    work page

  8. [8]

    9(a)), S1 =X 1X2,(D3) S2 =Z 1.(D4)

    For Kitaev code on square lattice withn= 2 (Fig. 9(a)), S1 =X 1X2,(D3) S2 =Z 1.(D4)

    work page

  9. [9]

    9(b)), S1 =X 1X2,(D5) S2 =Z 1,(D6) S3 =Z 1Z2 (D7)

    For Kitaev code on triangular lattice withn= 3 (Fig. 9(b)), S1 =X 1X2,(D5) S2 =Z 1,(D6) S3 =Z 1Z2 (D7)

    work page

  10. [10]

    In the cases of color code on honeycomb and square- octagonal lattices (Figs. 9(c)-(d)) both havingn= 4, S1 =X 1X2,(D8) S2 =Z 1,(D9) S3 =Z 1Z2,(D10) S4 =Z 1Z2Z3.(D11) While constructing the pseudoincidence matrixM, choosing a particular ordering for the pairing such that the firstn−1indices correspond to the pairings involving theS Ω 1 component, i.e.,{{α...

    work page

  11. [11]

    The third condition is satisfied due to the freedom in choosing different plaquette recombinations such that all the reduced single qubit Pauli operators,S Ω,i α , on every qubit outsideΩmutually commute

    work page

  12. [12]

    Thus, by the theorem in [35], all four necessary and sufficient conditions are satisfied, and{S α}constructs a local witness operator for the subsystemΩ

    Finally, the fourth condition is satisfied, as we directly see rank(Mn) =n−1. Thus, by the theorem in [35], all four necessary and sufficient conditions are satisfied, and{S α}constructs a local witness operator for the subsystemΩ. Appendix E: Lower bound to localizable genuine multipartite entanglement with geometric measure The set ofreduced Pauli opera...

    work page

  13. [13]

    Sachdev,Quantum phase transitions(Cambridge Univer- sity Press, Cambridge, 2011)

    S. Sachdev,Quantum phase transitions(Cambridge Univer- sity Press, Cambridge, 2011)

    work page 2011

  14. [14]

    The renormalization group and theepsilonexpansion,

    K. G. Wilson and J. Kogut, “The renormalization group and theepsilonexpansion,” Physics Reports12, 75–199 (1974); L. D. Landau, E. M. Lifshitz, and E. M. Pitaevskii,Statistical Physics(Butterworth-Heinemann, New York, 1999)

    work page 1974

  15. [15]

    Fidelity, dynamic struc- ture factor, and susceptibility in critical phenomena,

    W.-L. You, Y .-W. Li, and S.-J. Gu, “Fidelity, dynamic struc- ture factor, and susceptibility in critical phenomena,” Phys. Rev. E76, 022101 (2007); N. T. Jacobson, S. Garnerone, S. Haas, and P. Zanardi, “Scaling of the fidelity susceptibility 16 in a disordered quantum spin chain,” Phys. Rev. B79, 184427 (2009); S.-J. GU, “Fidelity approach to quantum ph...

    work page 2007

  16. [16]

    Entangle- ment in many-body systems,

    L. Amico, R. Fazio, A. Osterloh, and V . Vedral, “Entangle- ment in many-body systems,” Rev. Mod. Phys.80, 517–576 (2008); J. I. Latorre and A. Riera, “A short review on entangle- ment in quantum spin systems,” Journal of Physics A: Math- ematical and Theoretical42, 504002 (2009); G. D. Chiara and A. Sanpera, “Genuine quantum correlations in quantum many-...

    work page 2008

  17. [17]

    Entanglement in a simple quantum phase transition,

    T. J. Osborne and M. A. Nielsen, “Entanglement in a simple quantum phase transition,” Phys. Rev. A66, 032110 (2002); A. Osterloh, L. Amico, G. Falci, and R. Fazio, “Scaling of en- tanglement close to a quantum phase transition,” Nature416, 608–610 (2002)

    work page 2002

  18. [18]

    Geometric entanglement and quan- tum phase transition in generalized cluster-xy models,

    A. Deger and T.-C. Wei, “Geometric entanglement and quan- tum phase transition in generalized cluster-xy models,” Quan- tum Information Processing18, 326 (2019)

    work page 2019

  19. [19]

    Ground-state degeneracy of the fractional quantum hall states in the presence of a ran- dom potential and on high-genus riemann surfaces,

    X. G. Wen and Q. Niu, “Ground-state degeneracy of the fractional quantum hall states in the presence of a ran- dom potential and on high-genus riemann surfaces,” Phys. Rev. B41, 9377–9396 (1990); X.-G. Wen, “Topolog- ical orders and edge excitations in fractional quantum hall states,” Advances in Physics44, 405–473 (1995), https://doi.org/10.1080/00018739...

    work page doi:10.1080/00018739500101566 1990

  20. [20]

    Topological entanglement entropy of a bose–hubbard spin liquid,

    S. V . Isakov, M. B. Hastings, and R. G. Melko, “Topological entanglement entropy of a bose–hubbard spin liquid,” Nature Physics7, 772–775 (2011); S. Yan, D. A. Huse, and S. R. White, “Spin-liquid ground state of the ¡i¿s¡/i¿ = 1/2 kagome heisenberg antiferromagnet,” Science332, 1173–1176 (2011), https://www.science.org/doi/pdf/10.1126/science.1201080

    work page doi:10.1126/science.1201080 2011

  21. [21]

    Topologi- cal quantum memory,

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topologi- cal quantum memory,” J. Math. Phys.43, 4452–4505 (2002); A. Kitaev, “Fault-tolerant quantum computation by anyons,” Ann. Phys.303, 2 – 30 (2003); “Anyons in an exactly solved model and beyond,” Ann. Phys.321, 2 – 111 (2006)

    work page 2002

  22. [22]

    Topological quantum distillation,

    H. Bombin and M. A. Martin-Delgado, “Topological quantum distillation,” Phys. Rev. Lett.97, 180501 (2006); “Topological computation without braiding,” Phys. Rev. Lett.98, 160502 (2007)

    work page 2006

  23. [23]

    Ex- ploring topological phases with quantum walks,

    T. Kitagawa, M. S. Rudner, E. Berg, and E. Demler, “Ex- ploring topological phases with quantum walks,” Phys. Rev. A 82, 033429 (2010); A. Grudka, M. Karczewski, P. Kurzy´nski, J. W´ojcik, and A. W ´ojcik, “Topological invariants in quantum walks,” Phys. Rev. A107, 032201 (2023)

    work page 2010

  24. [24]

    Fidelity in topological quantum phases of matter,

    S. Garnerone, D. Abasto, S. Haas, and P. Zanardi, “Fidelity in topological quantum phases of matter,” Phys. Rev. A79, 032302 (2009); S. Panahiyan, W. Chen, and S. Fritzsche, “Fidelity susceptibility near topological phase transitions in quantum walks,” Phys. Rev. B102, 134111 (2020)

    work page 2009

  25. [25]

    Topological entanglement entropy,

    A. Kitaev and J. Preskill, “Topological entanglement entropy,” Phys. Rev. Lett.96, 110404 (2006); M. Levin and X.-G. Wen, “Detecting topological order in a ground state wave function,” Phys. Rev. Lett.96, 110405 (2006)

    work page 2006

  26. [26]

    Non-abelian anyons and topological quantum computation,

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-abelian anyons and topological quantum computation,” Rev. Mod. Phys.80, 1083–1159 (2008)

    work page 2008

  27. [27]

    Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension,

    S. Trebst, P. Werner, M. Troyer, K. Shtengel, and C. Nayak, “Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension,” Phys. Rev. Lett.98, 070602 (2007); J. Vidal, S. Dusuel, and K. P. Schmidt, “Low-energy effective theory of the toric code model in a parallel magnetic field,” Phys. Rev. B79, 033109 (2009); S. Dusuel, M...

    work page 2007

  28. [28]

    Kitaev- ising model and the transition between topological and ferro- magnetic order,

    V . Karimipour, L. Memarzadeh, and P. Zarkeshian, “Kitaev- ising model and the transition between topological and ferro- magnetic order,” Phys. Rev. A87, 032322 (2013)

    work page 2013

  29. [29]

    Robustness of topological quantum codes: Ising perturbation,

    M. H. Zarei, “Robustness of topological quantum codes: Ising perturbation,” Phys. Rev. A91, 022319 (2015)

    work page 2015

  30. [30]

    Robustness of a topological phase: Topological color code in a parallel magnetic field,

    S. S. Jahromi, M. Kargarian, S. F. Masoudi, and K. P. Schmidt, “Robustness of a topological phase: Topological color code in a parallel magnetic field,” Phys. Rev. B87, 094413 (2013); S. S. Jahromi, S. F. Masoudi, M. Kargar- ian, and K. P. Schmidt, “Quantum phase transitions out of a 𭟋2 ×𭟋 2 topological phase,” Phys. Rev. B88, 214411 (2013)

    work page 2013

  31. [31]

    Robust- ness of topological order in the toric code with open bound- aries,

    A. Jamadagni, H. Weimer, and A. Bhattacharyya, “Robust- ness of topological order in the toric code with open bound- aries,” Phys. Rev. B98, 235147 (2018)

    work page 2018

  32. [32]

    Universal shift of the fidelity susceptibility peak away from the critical point of the berezinskii-kosterlitz-thouless quan- tum phase transition,

    L. Cincio, M. M. Rams, J. Dziarmaga, and W. H. Zurek, “Universal shift of the fidelity susceptibility peak away from the critical point of the berezinskii-kosterlitz-thouless quan- tum phase transition,” Phys. Rev. B100, 081108 (2019)

    work page 2019

  33. [33]

    Quantum entanglement,

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys.81, 865–942 (2009)

    work page 2009

  34. [34]

    Entanglement in graph states and its ap- plications,

    M. Hein, W. D ¨ur, J. Eisert, R. Raussendorf, M. Van den Nest, and H. J. Briegel, “Entanglement in graph states and its ap- plications,” inQuantum Computers, Algorithms and Chaos, edited by G. Casati, D. L. Shepelyansky, P. Zoller, and G. Be- nenti (IOS Press, 2005) pp. 115–218

    work page 2005

  35. [35]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, 2010)

    work page 2010

  36. [36]

    Graphi- cal description of the action of local clifford transformations on graph states,

    M. Van den Nest, J. Dehaene, and B. De Moor, “Graphi- cal description of the action of local clifford transformations on graph states,” Phys. Rev. A69, 022316 (2004); N. Lang and H. P. B ¨uchler, “Minimal instances for toric code ground states,” Phys. Rev. A86, 022336 (2012)

    work page 2004

  37. [37]

    Localizing genuine multipartite entan- glement in noisy stabilizer states,

    H. K. J. and A. K. Pal, “Localizing genuine multipartite entan- glement in noisy stabilizer states,” Phys. Rev. A108, 032404 (2023)

    work page 2023

  38. [38]

    Entanglement ver- sus correlations in spin systems,

    F. Verstraete, M. Popp, and J. I. Cirac, “Entanglement ver- sus correlations in spin systems,” Phys. Rev. Lett.92, 027901 (2004); F. Verstraete, M. A. Mart´ın-Delgado, and J. I. Cirac, “Diverging entanglement length in gapped quantum spin sys- tems,” Phys. Rev. Lett.92, 087201 (2004); M. Popp, F. Ver- straete, M. A. Mart ´ın-Delgado, and J. I. Cirac, “Loc...

    work page 2004

  39. [39]

    Estimating localiz- able entanglement from witnesses,

    D. Amaro, M. M ¨uller, and A. K. Pal, “Estimating localiz- able entanglement from witnesses,” New J. Phys.20, 063017 (2018); “Scalable characterization of localizable entangle- ment in noisy topological quantum codes,” New Journal of Physics22, 053038 (2020)

    work page 2018

  40. [40]

    Distinguishing phases via non- markovian dynamics of entanglement in topological quan- tum codes under parallel magnetic field,

    H. K. J. and A. K. Pal, “Distinguishing phases via non- markovian dynamics of entanglement in topological quan- tum codes under parallel magnetic field,” Phys. Rev. A105, 17 052421 (2022)

    work page 2022

  41. [41]

    Quantum discord: A measure of the quantumness of correlations,

    H. Ollivier and W. H. Zurek, “Quantum discord: A measure of the quantumness of correlations,” Phys. Rev. Lett.88, 017901 (2001); L. Henderson and V . Vedral, “Classical, quantum and total correlations,” Journal of Physics A: Mathematical and General34, 6899 (2001)

    work page 2001

  42. [42]

    Computing quantum discord is np-complete,

    Y . Huang, “Computing quantum discord is np-complete,” New Journal of Physics16, 033027 (2014)

    work page 2014

  43. [43]

    Ladders in a magnetic field: a strong coupling ap- proach,

    Mila, F., “Ladders in a magnetic field: a strong coupling ap- proach,” Eur. Phys. J. B6, 201–205 (1998); K. Tandon, S. Lal, S. K. Pati, S. Ramasesha, and D. Sen, “Magnetization proper- ties of some quantum spin ladders,” Phys. Rev. B59, 396–410 (1999); E. Ercolessi, “One and quasi-one dimensional spin systems,” Modern Physics Letters A18, 2329–2336 (2003)...

    work page 1998

  44. [44]

    Spin- 1 2 heisenberg ladder: Variation of entanglement and fidelity measures close to quantum criti- cal points,

    A. Tribedi and I. Bose, “Spin- 1 2 heisenberg ladder: Variation of entanglement and fidelity measures close to quantum criti- cal points,” Phys. Rev. A79, 012331 (2009)

    work page 2009

  45. [45]

    Application of a continuous time cluster algorithm to the two-dimensional random quan- tum ising ferromagnet,

    Rieger, H. and Kawashima, N., “Application of a continuous time cluster algorithm to the two-dimensional random quan- tum ising ferromagnet,” Eur. Phys. J. B9, 233–236 (1999)

    work page 1999

  46. [46]

    Mapping the spatial distribution of entanglement in optical lattices,

    E. Alba, G. T´oth, and J. J. Garc´ıa-Ripoll, “Mapping the spatial distribution of entanglement in optical lattices,” Phys. Rev. A 82, 062321 (2010)

    work page 2010

  47. [47]

    Design and experimental perfor- mance of local entanglement witness operators,

    D. Amaro and M. M ¨uller, “Design and experimental perfor- mance of local entanglement witness operators,” Phys. Rev. A 101, 012317 (2020)

    work page 2020

  48. [48]

    Entanglement detection,

    O. G ¨uhne and G. T ´oth, “Entanglement detection,” Phys. Rep. 474, 1–75 (2009)

    work page 2009

  49. [49]

    Quantifying entanglement with wit- ness operators,

    F. G. S. L. Brand ˜ao, “Quantifying entanglement with wit- ness operators,” Phys. Rev. A72, 022310 (2005); F. G. S. L. Brand˜ao and R. O. Viana, “Witnessed entanglement,” Int. J. Quant. Inf.04, 331–340 (2006); J. Eisert, F. G. S. L. Brand ˜ao, and K. M. R. Audenaert, “Quantitative entanglement wit- nesses,” New J. Phys.9, 46 (2007)

    work page 2005

  50. [50]

    Baxter- wu model in a transverse magnetic field,

    S. Capponi, S. S. Jahromi, F. Alet, and K. P. Schmidt, “Baxter- wu model in a transverse magnetic field,” Phys. Rev. E89, 062136 (2014)

    work page 2014

  51. [51]

    Cluster monte carlo simulation of the transverse ising model,

    H. W. J. Bl¨ote and Y . Deng, “Cluster monte carlo simulation of the transverse ising model,” Phys. Rev. E66, 066110 (2002)

    work page 2002

  52. [52]

    Quantum codes on a lattice with boundary

    S. B. Bravyi and A. Y . Kitaev, “Quantum codes on a lattice with boundary,” (1998), arXiv:quant-ph/9811052 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  53. [53]

    Fi- delity susceptibility, scaling, and universality in quantum crit- ical phenomena,

    S.-J. Gu, H.-M. Kwok, W.-Q. Ning, and H.-Q. Lin, “Fi- delity susceptibility, scaling, and universality in quantum crit- ical phenomena,” Phys. Rev. B77, 245109 (2008); W.-C. Yu, H.-M. Kwok, J. Cao, and S.-J. Gu, “Fidelity susceptibility in the two-dimensional transverse-field ising andxxzmodels,” Phys. Rev. E80, 021108 (2009)

    work page 2008

  54. [54]

    Computational studies of quantum spin systems,

    A. W. Sandvik, A. Avella, and F. Mancini, “Computational studies of quantum spin systems,” inAIP Conference Proceed- ings(AIP, 2010)

    work page 2010

  55. [55]

    Adiabatic preparation of topo- logical order,

    A. Hamma and D. A. Lidar, “Adiabatic preparation of topo- logical order,” Phys. Rev. Lett.100, 030502 (2008)

    work page 2008

  56. [56]

    Duality in generalized Ising models

    J. B. Kogut, “An introduction to lattice gauge theory and spin systems,” Rev. Mod. Phys.51, 659–713 (1979); F. J. Wegner, “Duality in generalized ising models,” (2014), arXiv:1411.5815 [hep-lat]

    work page internal anchor Pith review Pith/arXiv arXiv 1979

  57. [57]

    Degree of entanglement,

    A. Shimony, “Degree of entanglement,” Annals of the New York Academy of Sciences755, 675–679 (1995); H. Barnum and N. Linden, “Monotones and invariants for multi-particle quantum states,” Journal of Physics A: Mathematical and Gen- eral34, 6787 (2001); T.-C. Wei and P. M. Goldbart, “Geomet- ric measure of entanglement and applications to bipartite and mul...

    work page 1995

  58. [58]

    Estimating en- tanglement measures in experiments,

    O. G ¨uhne, M. Reimpell, and R. F. Werner, “Estimating en- tanglement measures in experiments,” Phys. Rev. Lett.98, 110502 (2007)

    work page 2007

  59. [59]

    Separability criterion for density matrices,

    A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett.77, 1413–1415 (1996); M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A223, 1 – 8 (1996); K. ˙Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, “V olume of the set of separable states,” Phys. Rev. A...

    work page 1996

  60. [60]

    Fault-tolerant quantum computing with color codes

    A. J. Landahl, J. T. Anderson, and P. R. Rice, “Fault- tolerant quantum computing with color codes,” (2011), arXiv:1108.5738 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  61. [61]

    Quan- tum computations on a topologically encoded qubit,

    D. Nigg, M. M ¨uller, E. A. Martinez, P. Schindler, M. Hen- nrich, T. Monz, M. A. Martin-Delgado, and R. Blatt, “Quan- tum computations on a topologically encoded qubit,” Science 345, 302–305 (2014); N. M. Linke, M. Gutierrez, K. A. Landsman, C. Figgatt, S. Debnath, K. R. Brown, and C. Mon- roe, “Fault-tolerant quantum error detection,” Sci. Adv.3, 10 (2017)

    work page 2014

  62. [62]

    Realization of an error-correcting surface code with superconducting qubits,

    Y . Zhao, Y . Ye, H.-L. Huang, Y . Zhang, D. Wu, H. Guan, Q. Zhu, Z. Wei, T. He, S. Cao, F. Chen, T.-H. Chung, H. Deng, D. Fan, M. Gong, C. Guo, S. Guo, L. Han, N. Li, S. Li, Y . Li, F. Liang, J. Lin, H. Qian, H. Rong, H. Su, L. Sun, S. Wang, Y . Wu, Y . Xu, C. Ying, J. Yu, C. Zha, K. Zhang, Y .-H. Huo, C.-Y . Lu, C.-Z. Peng, X. Zhu, and J.-W. Pan, “Reali...

    work page 2022

  63. [63]

    Brower, S

    R. Brower, S. Chandrasekharan, and U.-J. Wiese, “Green’s functions from quantum cluster algorithms11this work is sup- ported in part by funds provided by the us department of energy (doe) under cooperative research agreement de-fc02- 94er40818.” Physica A: Statistical Mechanics and its Appli- cations261, 520–533 (1998)

    work page 1998

  64. [64]

    D. M. Greenberger, M. A. Horne, and A. Zeilinger,Bell’s theorem, quantum theory and conceptions of the universe (Kluwer, Netherlands, 1989)

    work page 1989