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REVIEW 3 major objections 4 minor 58 references

Multiparty Entanglement Microscopy of Quantum Ising models in 1d, 2d and 3d

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Genuine multipartite entanglement between three or four neighbouring Ising spins peaks near the quantum critical point, weakens sharply with dimension, and disappears when the spins are not adjacent in 2d and 3d.

desk verdict Useful dataset and a few new results on multipartite entanglement in the TFIM, but the abstract overstates the vanishing of GME for non-adjacent subregions. read the letter →

arxiv 2412.12533 v2 pith:6YO4LG7M submitted 2024-12-17 cond-mat.str-el

Liuke Lyu , Menghan Song , Ting-Tung Wang , Zi Yang Meng , William Witczak-Krempa This is my paper · ORCID
classification cond-mat.str-el
keywords genuinemultipartiteentanglementtransverse-fieldIsingmodelquantummicroscopyMonteCarlotomographycriticalpointbiseparabilityGMEconcurrence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper maps where genuine multipartite entanglement (GME) lives in the transverse-field Ising model in one, two, and three dimensions by reconstructing the full quantum state of small blocks of three or four neighbouring spins. It finds that these microscopic blocks share GME over broad portions of the phase diagram, that the entanglement peaks in the paramagnetic phase near the quantum critical point, and that its strength drops sharply as the spatial dimension grows. It also finds that once the spins in a block are not all adjacent, GME is no longer detected in 2d and 3d, with many non-adjacent geometries certified to be biseparable, meaning the entanglement is extremely short-ranged. On the formal side, the paper completes a previously incomplete proof that the computable criterion $I_2$ lower-bounds the genuine multipartite concurrence for every product state, and it shows how the lattice symmetries of the state reduce the optimization needed to evaluate $I_2$. If these results hold, GME becomes a practical, microscope-scale diagnostic of quantum criticality and dimensionality in equilibrium spin systems.

What carries the argument

The load-bearing object is the $I_2$ criterion, defined as a maximization over product states of a two-copy quantity built from swap operators on the doubled Hilbert space; a strictly positive value certifies genuine multipartite entanglement, and because it is a maximization any positive value found is already a certificate. The paper supplies the missing step in the proof that $2I_2$ bounds the genuine multipartite concurrence, showing that for any product state the maximum reduces without loss of generality to the |000111\rangle product state, so the bound holds generally. The numerical machinery is the tomography of small reduced density matrices (up to four spins) from quantum Monte Carlo and exact diagonalization, with parity, $Z_2$, and lattice permutation symmetries used to reduce the product-state search space to a few angles. A second central identity is the reduction of the $N$-party GME concurrence for uniform states to $\sqrt{1-\langle \sigma^x\rangle^2-\langle \sigma^z\rangle^2}$, turning the most collective entanglement into a local magnetization measurement.

What would settle it

Run the same tomography and $I_2$ maximization on a 24x24 square-lattice transverse-field Ising model at the critical field $h\approx 3.044$ for a disjoint three-site geometry not covered by Table II, for example sites 1, 5 and 9 forming a diagonal chain two lattice steps long. If $I_2$ evaluates to a strictly positive number at a precision where the reduced-density-matrix errors are below $10^{-5}$, the claim that all non-adjacent subregions in 2d are free of GME is refuted; conversely, certified biseparability for every such geometry would support it.

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Extended reading notes

Core claim

The central claim is that genuine multipartite entanglement in the transverse-field Ising model is a local, dimension-sensitive phenomenon that is strongest when three or four spins sit next to each other in one dimension, and is concentrated in the paramagnetic phase just before the quantum critical point. This is established by performing tomography of the reduced density matrices of at most four spins, using the exact solution in 1d, quantum Monte Carlo in 2d and 3d, and exact diagonalization as a check, then evaluating the genuinely multipartite concurrence, its $I_2$ lower bound, and a weaker $W_1$ witness. The paper proves that $2I_2$ is a lower bound on the genuine multipartite concurrence for any product state in the defining maximization, completing a gap left in the original proof; it also proves that $I_2$ is strictly stronger than $W_1$ because $W_1$ corresponds to restricting the optimization to local-unitary transforms of a fixed product state. For adjacent triples and quadruples, $I_2$ is positive across wide ranges of the transverse field in all dimensions and reaches a maximum near the critical point, while for non-adjacent subregions in 2d and 3d no GME is found and a finite list of geometries is certified biseparable. Finally, for all $N$ spins together, the GME concurrence of a uniform state reduces to a single-site observable, $\sqrt{1-\langle \sigma^x\rangle^2-\langle \sigma^z\rangle^2}$, which in the ferromagnetic phase is large but fragile: it is killed by one local $\sigma^z$ measurement, while the post-measurement GMC peaks near the critical point.

Load-bearing premise

The only load-bearing assumption is that the numerical biseparability certification used for a finite list of separated geometries is reliable and that no untested separated geometry in 2d or 3d has a weak form of the entanglement; if either fails, the claim that genuine multipartite entanglement vanishes for all non-adjacent spins collapses.

Editorial extensions

If this is right

  • For neighbouring three- and four-spin blocks, GME is present over large portions of the phase diagram in 1d, 2d, and 3d, so genuine multipartite entanglement is an ordinary, measurable property of quantum Ising matter near criticality.
  • In 2d and 3d, non-adjacent blocks show no GME, and the listed geometries are certified biseparable; if the pattern is general, entangled correlations at criticality remain effectively short-ranged even when two-point correlations are long-ranged.
  • The global concurrence formula means the collective GME of any uniform pure state can be read off from single-site magnetization, without solving the exponentially large partition optimization.
  • The completed $I_2$ lower bound makes GME certification robust: any optimization attempt that lands on a positive $I_2$ value is already a proof, not a heuristic, for 3- and 4-spin blocks.
  • The observed singular scaling of $dI_2/dh$ near the critical point extends the known critical scaling of bipartite entanglement measures to a genuine multipartite measure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the short-range pattern of GME could serve as a sharper diagnostic of classical versus quantum correlations at criticality, since long-range order-parameter correlations would not be accompanied by long-range multipartite entanglement.
  • Inference: the completed $I_2$ proof applies beyond Ising spins, so any quantum Monte Carlo sampled reduced density matrix with enough symmetry to tame the optimization can be certified for GME at the same cost, opening fermionic or frustrated models to the same microscopy.
  • Inference: the post-measurement global concurrence is the more physical measure of collective entanglement in the ordered phase, since a single local projective measurement already destroys the cat-state GME; experiments should therefore compare pre- and post-selected entanglement when probing ordered phases.
  • Inference: a testable extension is to compute $I_2$ for the untested disjoint geometries in 3d with improved reduced-density-matrix precision below $10^{-5}$; a positive value in any such geometry would replace the 'all non-adjacent' statement with a list of short-ranged exceptions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper applies entanglement microscopy to the transverse-field Ising model in d=1,2,3, obtaining reduced density matrices of up to four sites via quantum Monte Carlo, exact diagonalization, and the exact 1d solution. It evaluates bipartite negativity and several genuine multipartite entanglement criteria (CGME, I2, W1), reporting that 3- and 4-spin GME for adjacent sites is present over wide field ranges, peaks near the quantum critical point, and decreases rapidly with dimension. It also reports that GME vanishes for non-adjacent subregions in 2d and 3d, identifies logarithmic singular scaling of dI2/dh in 1d, derives a formula for the N-party GMC in terms of the single-site transverse magnetization, and claims completion of a proof that I2 lower-bounds the GMC.

Significance. If the main claims hold, this is a valuable systematic dataset on multipartite entanglement across archetypal quantum critical points in one, two, and three dimensions. The strengths are the convergence checks between QMC and ED, the use of the exact 1d solution, the public release of the sampled RDMs on GitHub, and the demonstration that I2 detects GME in 2d and 3d where the W1 criterion fails. The proposed global GMC formula and the symmetry-adapted I2 parametrizations are conceptually attractive and should be useful for future work. The universal vanishing claim and the completed-proof claim, however, currently outrun the evidence presented.

major comments (3)
  1. [§IVB and Abstract] The abstract and Section IVB state that GME disappears for all non-adjacent subregions in 2d and 3d, but the evidence consists of the finite set of geometries in Table II plus, for some subregions, biseparability certificates from the iterative algorithm of Ref. [48]. For subregion 126 no biseparability certificate is given; the text only says that the estimated CGME is 'consistent with a biseparable state', and the geometric-distance upper bounds are not entanglement witnesses. Since I2≤0 does not prove biseparability, and since the 1d case already exhibits weak disjoint GME below the I2 threshold (Ref. [25], acknowledged in this section), the universal statement goes beyond the evidence. Please either prove a general statement or rephrase the claim to 'no GME is detected in the tested non-adjacent geometries'.
  2. [Appendix VIIC] The claimed completion of the proof that 2I2≤CGME depends on the assertion 'fijk≤0 for (i,j,k)≠(1,1,1)', which is stated without proof. In addition, the displayed expression for I2[|ψ⟩⟨ψ|,|000xyz⟩] replaces |a000 Σ aijk xi yj zk| by a sum of absolute values, which is an upper bound rather than an equality; this step needs justification. The formula also appears to contain a typo, since 'a0j0a0j0' should presumably be 'a0j0 ai0k'. Until these steps are supplied, the claim of a complete proof is not established. This gap does not invalidate the positive I2 detections, because I2>0 is itself an established sufficient condition for GME.
  3. [§V, Eq. (12)] The global GMC formula assumes that the minimum over all bipartitions is attained by a single-site partition. The argument that adding sites to Ai always increases the concurrence is heuristic, and the numerical verification is reported only for lattices with N≤20, while Fig. 7 uses 25, 25, and 27 spins for 1d, 2d, and 3d respectively. Please provide a proof for site-transitive pure states or verify the assumption for the actual lattice sizes used; otherwise Eq. (12) should be presented as a conjectured lower/upper bound rather than an exact expression.
minor comments (4)
  1. [Appendix VIIC] The definition of fijk contains a repeated 'a0j0a0j0' term; the intended term is likely 'a0j0 ai0k', and the surrounding derivation should be rechecked for further notational typos.
  2. [Section IVB] For the (2,2) configuration in 2d, the ED and QMC sudden-death distances differ (ED gives r=2 with E(2)=0 on 6×5, while QMC gives a small value consistent with zero); the text should state explicitly whether this is a finite-size or convergence effect.
  3. [Appendix VIID] The caption of Fig. 11 reports a scaling exponent 0.89 for ⟨σx⟩, but the main text uses this to infer dSvN/dh∼|h−hc|−0.11; the sign convention should be stated clearly so that readers do not confuse the order-parameter exponent with the derivative exponent.
  4. [General] The GitHub repository link in Ref. [41] should be accompanied by a version identifier or retrieval date so that the released RDM data can be cited reproducibly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central GME detections and global-GMC formula are computed from independently sampled reduced density matrices, not derived from the claims themselves.

full rationale

The paper's central detections of genuine multipartite entanglement are obtained by evaluating the sufficient criterion I2>0 on reduced density matrices sampled by QMC, ED, or exact solution. Since I2>0 certifies GME by the externally established criterion of Refs. [35,36], these positive detections are self-contained and do not presuppose the conclusions. The places where the paper goes further—the extrapolation from I2≤0 to 'absence of GME' for disjoint subregions (Sec. IVB, Table II) and the statement that a numerical GMC estimate for subregion 126 is 'consistent with a biseparable state'—are evidential overstatements and possible correctness gaps, but not circular reductions: they do not define the target quantity in terms of the measured quantity. The global GMC formula CGME=sqrt(1-<σx>^2-<σz>^2) (Sec. V, Eq. 12) is derived from uniformity and a single-site reduced density matrix, with the single-site-minimum assumption explicitly flagged as numerically verified for N≤20; this is an extrapolation, not an equivalence-by-construction between input and output. The scaling statements (Sec. IVA and Appendix D) invoke the authors' earlier general scaling result [1], but the fitted coefficients are post hoc fits to the 1d I2 data and do not force the main detections; the 2d/3d scaling expectations are explicitly not extracted from data. The Appendix VIIC 'completion' of the I2 bound relies on an unproved inequality fijk≤0, but that is an omitted proof step, not a circular one. Overall, no step reduces a derived result to its own input or to a self-citation chain.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claims rest on two unproven structural assumptions: the single-site bipartition optimality for the global GMC and the completeness of the biseparability certification for disjoint subregions. The fitted scaling coefficients are post hoc but do not determine the main conclusions. The QMC-to-groundstate approximation and the unproved inequality in the I2 proof are additional load-bearing premises.

free parameters (4)
  • alpha_I2_3spin = 0.11
    Coefficient in the logarithmic divergence dI2/dh = -alpha log|h-hc| + beta for the 1d three-spin subregion, fitted in Fig. 9 over log2|h-hc| in [-8,-7].
  • alpha_I2_4spin = 0.07
    Same fit for the four-spin subregion in 1d.
  • alpha_EN_1plus1 = 0.23
    Linear fit coefficient for dE/dh for the 1+1 spin negativity in 1d (Fig. 10).
  • alpha_EN_2plus2 = 0.44
    Linear fit coefficient for dE/dh for the 2+2 spin negativity in 1d (Fig. 10).
assumptions (5)
  • domain assumption The scaling of entanglement measure derivatives near the QCP follows dM/dh = alpha |h-hc|^(Delta_epsilon_nu - 1) from Ref. [1].
    Used in Section IVA to interpret the divergence of dI2/dh and dE/dh; it is imported from the authors' earlier work and its applicability to I2 is assumed.
  • domain assumption The single-site bipartition gives the minimal concurrence for uniform pure states, so the global GMC reduces to sqrt(2(1 - Tr rho_i^2)).
    Section V, Eq. (12); the authors verify numerically for N≤20 but do not prove it for general N.
  • domain assumption The iterative biseparability certification algorithm of Ref. [48] provides a valid certificate of biseparability for the tested subregions.
    Section IVB and Table II: the conclusion that GME vanishes for disjoint subregions relies on this algorithm applied to a finite set of geometries.
  • domain assumption The QMC sampling with beta = L (1d/2d) or beta = 2L (3d) yields groundstate RDMs to the stated precision.
    Section IIC; the validity of the RDM tomography for subregions depends on this thermal-to-groundstate approximation.
  • ad hoc to paper In the I2 proof, the inequality fijk <= 0 for (i,j,k) != (1,1,1) holds.
    Appendix VIIC: this unproved inequality is required to reduce the maximization over |xyz> to |111>; no derivation is given and the printed expression contains apparent typos.

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Pith. "Pith review of Multiparty Entanglement Microscopy of Quantum Ising models in 1d, 2d and 3d." pith.science (2026). https://pith.science/paper/6YO4LG7M

@misc{pith2026241212533,
  author       = {Pith},
  title        = {Pith review of: Multiparty Entanglement Microscopy of Quantum Ising models in 1d, 2d and 3d},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6YO4LG7M}},
  note         = {Machine review of arXiv:2412.12533}
}
abstract

Entanglement microscopy reveals the true quantum correlations among the microscopic building blocks of many-body systems [Nat. Commun. 16, 96 (2025)]. Using this approach, we study the multipartite entanglement of the quantum Ising model in 1d, 2d, and 3d. We first obtain the full reduced density matrix (tomography) of subregions that have at most 4 sites via quantum Monte Carlo, exact diagonalization, and the exact solution in 1d. We then analyze both bipartite and genuine multipartite entanglement (GME) among the sites in the subregion. To do so, we use a variety of measures including the negativity, as well as a true measure of GME: the genuinely multipartite concurrence (or GME concurrence), and its computationally cheaper lower bound, $I_2$. We provide a complete proof that $I_2$ bounds the GME concurrence, and show how the symmetries of the state simplify its evaluation. For adjacent sites, we find 3- and 4-spin GME present across large portions of the phase diagram, reaching maximum near the quantum critical point. In 1d, we identify the singular scaling of the derivative $dI_2/dh$ approaching the critical point. We observe a sharp decrease of GME with increasing dimensionality, coherent with the monogamous nature of entanglement. Furthermore, we find that GME disappears for subregions consisting of non-adjacent sites in both 2d and 3d, offering a stark illustration of the short-ranged nature of entanglement in equilibrium quantum matter arXiv:2402.06677. Finally, we analyze the most collective form of entanglement by evaluating the GME concurrence among all spins in the lattice, which can be obtained from a simple observable: the single-site transverse magnetization. This global concurrence is larger in 1d compared to 2d/3d, but it is relatively less robust against perturbations such as local measurements.

Figures

Figures reproduced from arXiv: 2412.12533 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figures from the paper (10 more)
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison of the GME measures [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The derivative of the Log Negativity, [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. (Left) [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. (Left) [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Log-log plot of [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

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Works this paper leans on

58 extracted references · 31 canonical work pages

  1. [25]

    Hofmann, A

    M. Hofmann, A. Osterloh, and O. Gühne, Scaling of gen- uinemultiparticleentanglementclosetoaquantumphase transition, Phys. Rev. B89, 134101 (2014)

    2014

  2. [48]

    Kampermann, O

    H. Kampermann, O. Gühne, C. Wilmott, and D. Bruß, Algorithm for characterizing stochastic local operations and classical communication classes of multiparticle en- tanglement, Phys. Rev. A86, 032307 (2012)

    2012

  3. [1]

    T.-T. Wang, M. Song, L. Lyu, W. Witczak-Krempa, and Z. Y. Meng, Entanglement microscopy and tomography in many-body systems, Nature Communications16, 96 (2025)

    2025

  4. [2]

    Parez and W

    G. Parez and W. Witczak-Krempa, The fate of entangle- ment, arXiv e-prints , arXiv:2402.06677 (2024)

    arXiv 2024

  5. [3]

    Entanglement R\'{e}nyi Negativity of Interacting Fermions from Quantum Monte Carlo Simulations

    F.-H. Wang and X. Y. Xu, Entanglement Rényi Negativ- ity of Interacting Fermions from Quantum Monte Carlo Simulations, arXiv e-prints , arXiv:2312.14155 (2023)

    work page Pith review arXiv 2023

  6. [4]

    Mao, Y.-M

    B.-B. Mao, Y.-M. Ding, and Z. Yan, Sampling reduced densitymatrixtoextractfinelevelsofentanglementspec- trum, arXiv e-prints , arXiv:2310.16709 (2023)

    arXiv 2023

  7. [5]

    Braiorr-Orrs, M

    B. Braiorr-Orrs, M. Weyrauch, and M. V. Rakov, Numer- icalStudiesofEntanglementPropertiesinOne-andTwo- Dimensional Quantum Ising and XXZ Models, Ukrainian Journal of Physics61, 613 (2019)

    work page 2019

  8. [6]

    M. Song, J. Zhao, Z. Y. Meng, C. Xu, and M. Cheng, Extracting subleading corrections in entanglement en- tropy at quantum phase transitions, SciPost Phys.17, 010 (2024)

    2024

Show all 58 references
  1. [7]

    Osterloh, L

    A. Osterloh, L. Amico, G. Falci, and R. Fazio, Scaling of entanglement close to a quantum phase transition, Na- ture416, 608 (2002)

    2002

  2. [8]

    Parez and W

    G. Parez and W. Witczak-Krempa, Entanglement nega- tivity between separated regions in quantum critical sys- tems, Phys. Rev. Res.6, 023125 (2024)

    2024

  3. [9]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, Journal of Statistical Mechanics: Theory and Experiment2004, P06002 (2004)

    2004

  4. [10]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entan- glement in many-body systems, Rev. Mod. Phys.80, 517 (2008)

    2008

  5. [11]

    M. B. Hastings, I. González, A. B. Kallin, and R. G. Melko, Measuring Renyi Entanglement Entropy in Quan- tum Monte Carlo Simulations, Phys. Rev. Lett.104, 157201 (2010)

    2010

  6. [12]

    Humeniuk and T

    S. Humeniuk and T. Roscilde, Quantum Monte Carlo calculation of entanglement Rényi entropies for generic quantum systems, Phys. Rev. B86, 235116 (2012)

    2012

  7. [13]

    Grover, Entanglement of Interacting Fermions in Quantum Monte Carlo Calculations, Phys

    T. Grover, Entanglement of Interacting Fermions in Quantum Monte Carlo Calculations, Phys. Rev. Lett. 111, 130402 (2013)

    2013

  8. [14]

    Alba, Entanglement negativity and conformal field theory: a Monte Carlo study, Journal of Statistical Me- chanics: Theory and Experiment2013, P05013 (2013)

    V. Alba, Entanglement negativity and conformal field theory: a Monte Carlo study, Journal of Statistical Me- chanics: Theory and Experiment2013, P05013 (2013)

    2013

  9. [15]

    Laflorencie, Quantum entanglement in condensed matter systems, Physics Reports646, 1 (2016), quantum entanglement in condensed matter systems

    N. Laflorencie, Quantum entanglement in condensed matter systems, Physics Reports646, 1 (2016), quantum entanglement in condensed matter systems

    2016

  10. [16]

    Alba, Out-of-equilibrium protocol for Rényi entropies via the Jarzynski equality, Phys

    V. Alba, Out-of-equilibrium protocol for Rényi entropies via the Jarzynski equality, Phys. Rev. E95, 062132 (2017)

    2017

  11. [17]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement- Induced Phase Transitions in the Dynamics of Entangle- ment, Phys. Rev. X9, 031009 (2019)

    2019

  12. [18]

    D’Emidio, Entanglement Entropy from Nonequilib- rium Work, Phys

    J. D’Emidio, Entanglement Entropy from Nonequilib- rium Work, Phys. Rev. Lett.124, 110602 (2020)

    2020

  13. [19]

    Zhao, Y.-C

    J. Zhao, Y.-C. Wang, Z. Yan, M. Cheng, and Z. Y. Meng, Scaling of Entanglement Entropy at Deconfined Quan- tum Criticality, Phys. Rev. Lett.128, 010601 (2022)

    2022

  14. [20]

    M. Song, J. Zhao, M. Cheng, C. Xu, M. M. Scherer, L. Janssen, and Z. Y. Meng, Deconfined quantum criti- cality lost, arXiv e-prints , arXiv:2307.02547 (2023)

    2023 arXiv

  15. [21]

    Z. Deng, L. Liu, W. Guo, and H.-Q. Lin, Diagnosing Quantum Phase Transition Order and Deconfined Criti- cality via Entanglement Entropy, Phys. Rev. Lett.133, 100402 (2024)

    2024

  16. [22]

    Zhang, G

    X. Zhang, G. Pan, B.-B. Chen, K. Sun, and Z. Y. Meng, Integralalgorithmofexponentialobservablesforinteract- ing fermions in quantum Monte Carlo simulations, Phys. Rev. B109, 205147 (2024)

    2024

  17. [23]

    X. Zhou, Z. Y. Meng, Y. Qi, and Y. Da Liao, Incremen- tal SWAP operator for entanglement entropy: Applica- tion for exponential observables in quantum Monte Carlo simulation, Phys. Rev. B109, 165106 (2024)

    2024

  18. [24]

    S. M. Giampaolo and B. C. Hiesmayr, Genuine multi- partite entanglement in theXYmodel, Phys. Rev. A 88, 052305 (2013)

    2013

  19. [26]

    R. Y. Wen, G. Parez, W. Witczak-Krempa, and A. Kempf, Separable ellipsoids around multipartite states, arXiv e-prints , arXiv:2410.05400 (2024)

    2024 arXiv

  20. [27]

    Wu, T.-C

    K.-H. Wu, T.-C. Lu, C.-M. Chung, Y.-J. Kao, and T. Grover, Entanglement Renyi Negativity across a Fi- nite Temperature Transition: A Monte Carlo Study, Phys. Rev. Lett.125, 140603 (2020)

    2020

  21. [28]

    Pfeuty, The one-dimensional Ising model with a trans- verse field, Annals of Physics57, 79 (1970)

    P. Pfeuty, The one-dimensional Ising model with a trans- verse field, Annals of Physics57, 79 (1970)

    1970

  22. [29]

    Hesselmann and S

    S. Hesselmann and S. Wessel, Thermal Ising transi- tions in the vicinity of two-dimensional quantum critical points, Phys. Rev. B93, 155157 (2016)

    2016

  23. [30]

    H. W. J. Blöte and Y. Deng, Cluster Monte Carlo sim- ulation of the transverse Ising model, Phys. Rev. E66, 066110 (2002)

    2002

  24. [31]

    K.Życzkowski, P.Horodecki, A.Sanpera,andM.Lewen- stein, Volume of the set of separable states, Physical Re- view A58, 883 (1998)

    1998

  25. [32]

    Eisert and M

    J. Eisert and M. B. Plenio, A Comparison of entangle- ment measures, J. Mod. Opt.46, 145 (1999)

    1999

  26. [33]

    Vidal and R

    G. Vidal and R. F. Werner, Computable measure of en- tanglement, Phys. Rev. A65, 032314 (2002)

    2002

  27. [34]

    Gühne and G

    O. Gühne and G. Tóth, Entanglement detection, Physics Reports474, 1 (2009)

    2009

  28. [35]

    Ma, Z.-H

    Z.-H. Ma, Z.-H. Chen, J.-L. Chen, C. Spengler, A. Gabriel, and M. Huber, Measure of genuine multipar- tite entanglement with computable lower bounds, Phys. Rev. A83, 062325 (2011)

    2011

  29. [36]

    Huber, F

    M. Huber, F. Mintert, A. Gabriel, and B. C. Hies- mayr, Detection of High-Dimensional Genuine Multipar- tite Entanglement of Mixed States, Phys. Rev. Lett.104, 210501 (2010)

    2010

  30. [37]

    Gühne and M

    O. Gühne and M. Seevinck, Separability criteria for gen- uine multiparticle entanglement, New Journal of Physics 12, 053002 (2010)

    2010

  31. [38]

    S. M. Giampaolo and B. C. Hiesmayr, Genuine multipar- titeentanglementinthecluster-Isingmodel,NewJournal of Physics16, 093033 (2014)

    2014

  32. [39]

    A. W. Sandvik, Stochastic Series Expansion Methods, arXiv:1909.10591

    1909 arXiv

  33. [40]

    J. Zhao, Z. Yan, M. Cheng, and Z. Y. Meng, Higher-form symmetry breaking at Ising transitions, Phys. Rev. Res. 12 3, 033024 (2021)

    2021

  34. [41]

    Song, T.-T

    M. Song, T.-T. Wang, L. Lyu, Z. Y. Meng, and W.Witczak-Krempa,Reduceddensitymatricesforquan- tumIsingmodelatvariousdimensions,https://github. com/songmengh/Ising_RDMs/tree/main

  35. [42]

    Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, andT.Xiang,Coarse-grainingrenormalizationbyhigher- order singular value decomposition, Phys. Rev. B86, 045139 (2012)

    2012

  36. [43]

    Ou and H

    Y.-C. Ou and H. Fan, Monogamy inequality in terms of negativityforthree-qubitstates,Phys.Rev.A75,062308 (2007)

    2007

  37. [44]

    Coffman, J

    V. Coffman, J. Kundu, and W. K. Wootters, Distributed entanglement, Phys. Rev. A61, 052306 (2000)

    2000

  38. [45]

    Javanmard, D

    Y. Javanmard, D. Trapin, S. Bera, J. H. Bardarson, and M. Heyl, Sharp entanglement thresholds in the logarith- mic negativity of disjoint blocks in the transverse-field Ising chain, New Journal of Physics20, 083032 (2018)

    2018

  39. [46]

    A. O. Pittenger and M. H. Rubin, Convexity and the sep- arability problem of quantum mechanical density matri- ces, Linear Algebra and its Applications346, 47 (2002)

    2002

  40. [47]

    Röthlisberger, J

    B. Röthlisberger, J. Lehmann, and D. Loss, libCreme: An optimization library for evaluating convex-roof entan- glement measures, Computer Physics Communications 183, 155 (2012)

    2012

  41. [49]

    Vedral, M

    V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Quantifying entanglement, Phys. Rev. Lett.78, 2275 (1997)

    1997

  42. [50]

    Pandya, O

    P. Pandya, O. Sakarya, and M. Wieśniak, Hilbert- Schmidt distance and entanglement witnessing, Phys. Rev. A102, 012409 (2020)

    2020

  43. [51]

    E. G. Gilbert, An Iterative Procedure for Computing the Minimum of a Quadratic Form on a Convex Set, SIAM Journal on Control4, 61 (1966)

    1966

  44. [52]

    [50] choses random product states at each iteration to update the separable state

    Ref. [50] choses random product states at each iteration to update the separable state. In contrast, we follow the original deterministic protocol of Gilbert [51], which per- forms an optimization at each iteration to find a product state, leading to faster convergence

  45. [53]

    Jungnitsch, T

    B. Jungnitsch, T. Moroder, and O. Gühne, Taming Mul- tiparticle Entanglement, Phys. Rev. Lett.106, 190502 (2011)

    2011

  46. [54]

    Montakhab and A

    A. Montakhab and A. Asadian, Multipartite entangle- ment and quantum phase transitions in the one-, two-, and three-dimensional transverse-field Ising model, Phys. Rev. A82, 062313 (2010)

    2010

  47. [55]

    T.-T. Wang, M. Song, Z. Y. Meng, and T. Grover, An analog of topological entanglement entropy for mixed states, arXiv e-prints , arXiv:2407.20500 (2024). 13 VII. APPENDIX A. RDMs from Entanglement Microscopy This section presents a selection of RDMs at the QCP in 2d and 3d, ob...

    2024 arXiv

  48. [56]

    2d For the 2d case, we present the RDMs: ED, lattice size6×5,h= 3   0.8323 0 0 0.0413 0 0.0478 0.0979 0 0 0.0539 0.0364 0 0.0399 0 0 0.0074 0 0.0364 0.0459 0 0.0366 0 0 0.0059 0.0413 0 0 0.0041 0 0.0039 0.0060 0 0 0.0399 0.0366 0 0.0453 0 0 0.0061 0.0478 0 0 0.0039 ...

  49. [57]

    3d For 3d, we present the RDMs: ED, lattice size3×3×3,h= 5.2   0.9130 0 0 0.0641 0 0.0279 0.0641 0 0 0.0262 0.0192 0 0.0191 0 0 0.0030 0 0.0192 0.0227 0 0.0192 0 0 0.0026 0.0641 0 0 0.0050 0 0.0024 0.0048 0 0 0.0191 0.0192 0 0.0262 0 0 0.0030 0.0279 0 0 0.0024 0 0.0...

  50. [58]

    xk , where|x k⟩= P i xk i |i⟩

    0x1x2 . . . xk , where|x k⟩= P i xk i |i⟩. However, any |xk⟩can be reduced via a local unitary transformation to the subspace spanned by{|0⟩,|1⟩}. This allows us to restrict |xk⟩to the qubit subspace and extend the lower bound to arbitrary dimensions by following the same step...

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