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REVIEW 4 major objections 5 minor 101 references

This paper argues that, in the no-click limit, monitoring local or non-local observables of a 1+1D Z2 lattice gauge theory leaves the late-time bipartite entanglement entropy independent of system size, so no measurement-induced phase trans

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2026-08-04 05:36 UTC pith:KTGTUOCO

load-bearing objection A careful, reproducible numerical study of a non-Hermitian filtered Z2 gauge theory, but the no-click measurement framing is under-justified and the nonlocal results contain a direct contradiction. the 4 major comments →

arxiv 2603.28877 v2 pith:KTGTUOCO submitted 2026-03-30 quant-ph cond-mat.str-elhep-lathep-th

Effects of monitoring on entanglement dynamics for 1+1D mathbb Z₂ lattice gauge theory

Nilachal Chakrabarti , Nisa Ara , Neha Nirbhan , Arpan Bhattacharyya , Indrakshi Raychowdhury This is my paper
classification quant-ph cond-mat.str-elhep-lathep-th MSC 81P4081T2581Q12 PACS 03.65.Ud11.15.Ha
keywords Z2 lattice gauge theorystaggered fermionsentanglement entropymeasurement-induced phase transitionno-click limitnon-Hermitian Hamiltoniantensor networkquantum Zeno effect
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies how continuous post-selected measurements affect entanglement growth in the simplest 1+1D Z2 lattice gauge theory coupled to staggered fermions. Starting from a strong-coupling vacuum, the authors evolve with a non-Hermitian effective Hamiltonian and track the bipartite von Neumann entropy for systems up to 256 sites. They find that, for measurements of local electric flux, local particle-antiparticle density, and a non-local mesonic hopping term, the late-time saturation value of the entanglement entropy is the same for L=64, 128, and 256. This is presented as evidence that no measurement-induced phase transition occurs in the no-click limit, within the scanned range of measurement rates and couplings. The result matters because quantum simulations of gauge theories need to distinguish genuine measurement effects from finite-size artifacts.

Core claim

We study the post-selected no-click dynamics of a 1+1D Z2 lattice gauge theory by evolving a normalized density matrix with H_eff = H - iγH_meas, where H_meas is the measured physical operator: electric flux τ^Z, particle-antiparticle density (-1)^j σ^Z, or the gauge-invariant hopping/meson term. For all three measured operators, the late-time saturation value of the bipartite entanglement entropy is independent of system size for L=64, 128, and 256, so we conclude that there is no measurement-induced phase transition in the no-click limit. Local measurements produce a quantum-Zeno-like decrease of the saturation value as the measurement rate increases; non-local mesonic measurements produce

What carries the argument

The central mechanism is the non-Hermitian effective Hamiltonian H_eff = H_0 - iγH_1, where H_1 is the measured observable, combined with the normalized no-click evolution ρ(t) ∝ e^{-iH_eff† t} ρ(0) e^{iH_eff t}. This turns monitoring into deterministic imaginary couplings: (1-iγ)τ^Z for flux, (μ-iγ)(-1)^j σ^Z for density, and (x-iγ) for the mesonic hopping term. The bipartite entanglement entropy is computed from matrix-product-state time evolution, using the Schmidt coefficients across a central bond.

Load-bearing premise

The calculation assumes that adding an imaginary term proportional to the measured operator, rather than to its square or projector, is the correct no-click limit of continuous measurement; if that replacement is not the actual post-selected measurement dynamics, the simulated states do not correspond to real monitored trajectories.

What would settle it

Simulate the same quenches with the full stochastic Schrödinger equation for continuous measurement of τ^Z and the density operator, using jump operators of the form (1-O)/2 or O^2/2, and compare the late-time saturation entropy for L=64, 128, and 256. If a size-dependent saturation value appears in that direct simulation, the paper's conclusion that no measurement-induced phase transition exists would be refuted.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Within the no-click limit, a monitored Z2 gauge theory shows no entanglement phase transition: the late-time saturation entropy remains size-independent up to L=256.
  • Measurements make the entanglement saturate at late times, in contrast to unmonitored evolution from the strong-coupling vacuum, where the entropy keeps oscillating without saturating.
  • Increasing the measurement rate suppresses the late-time saturation entropy for local measurements, a quantum-Zeno-like effect.
  • Non-local mesonic measurements show a qualitatively different early-time dynamics, with a peak appearing for larger measurement rates, before the entropy saturates.
  • The size-independence of the saturation entropy holds in both strong-coupling (x<1) and weak-coupling (x>1) regimes, with the saturation entropy growing linearly with the coupling x.

Where Pith is reading between the lines

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

  • If the linear-imaginary-coupling model is a faithful no-click limit, the absence of a measurement-induced phase transition may persist in full stochastic trajectories; but including quantum jumps could still produce a transition, which the authors themselves list as a future direction.
  • The early-time peak seen under mesonic measurement may be a signature of measurement-induced string-breaking dynamics; a natural test is to vary the meson string length and see whether the peak height and position track the confining length scale.
  • A stricter check would be to compare this linear model with the standard no-click Hamiltonian built from the square of the jump operator; such a comparison would show whether the size-independence is specific to the linear replacement or a more general property of monitored gauge theories.
  • The size-independent saturation value could serve as a practical benchmark for quantum simulators of Z2 gauge theory, giving a known late-time entanglement target to compare against experimental or hardware noise.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies the post-selected no-click dynamics of a 1+1D Z2 lattice gauge theory coupled to staggered fermions, using non-Hermitian effective Hamiltonians in which the measured observable appears as a linear imaginary coupling. Three measurements are considered: local electric flux, local particle-antiparticle density, and a nonlocal mesonic hopping term. Using MPS/TDVP simulations for systems up to L=256, the authors report that the late-time bipartite entanglement entropy saturates under measurement and that its saturation value is independent of system size for both local and nonlocal measurements, which they interpret as the absence of a measurement-induced phase transition (MIPT) in the no-click limit. They also document a quantum-Zeno-like decrease of the saturation entropy with measurement rate and an early-time peak for nonlocal monitoring.

Significance. If the effective Hamiltonian is accepted as a faithful no-click measurement model, the paper extends the study of MIPTs from spin chains and random circuits to a concrete lattice gauge theory, with technically demanding numerical results. The authors benchmark their MPS calculations against exact diagonalization, check convergence in bond dimension and cutoff, explicitly verify Gauss's law during time evolution, and make the code and data publicly available. The negative result for a size-dependent entanglement transition in this gauge-theoretic setting is a potentially useful data point for the monitored-LGT literature. The distinct behavior observed for nonlocal measurements is also interesting. However, the significance is contingent on the measurement model being properly derived and on the nonlocal and small-γ evidence being presented completely.

major comments (4)
  1. [Sec. 2.2; Eqs. (4.1)-(4.3)] The effective Hamiltonian is asserted as H_eff = H0 - iγH1 with the measured operator appearing linearly, and this is called the no-click limit with citations [79-88]. No jump operators or unraveling are specified. In the standard no-click limit of continuous monitoring, H_eff = H - (i/2)Σ_m L_m†L_m; for a Hermitian Pauli observable O with L_m ∝ O, this is proportional to O^2 = I (or to sums of projectors), not to iγO. Obtaining Eqs. (4.1)-(4.3) requires a specific choice of jump operators and a post-selected null-outcome sector that is not stated. As written, the simulated states are not demonstrably the post-selected states of a measurement of the stated observables, so the size-independent saturation cannot be attributed to MIPT without this derivation.
  2. [Sec. 4.2 and Fig. 9] The nonlocal measurement results in Fig. 9 are mostly obtained by restricting the mesonic-hopping measurement to subsystem A only, explicitly excluding the link across the bipartition. The text states that 'the conclusions remain true even when we measure the hopping term throughout the system,' but no such data are shown. Monitoring only one side of the partition is a different protocol and may not probe the entanglement transition across the cut. Please provide the full-system data or explicitly restrict the claim to the subsystem-local protocol.
  3. [Sec. 4.2, Fig. 10; Sec. 4.1] For small γ, the paper reports that the entanglement entropy does not saturate within the simulated time window (T=100). The 'absence of MIPT' conclusion is based only on runs that do saturate. A volume-law phase with slow dynamics would also show non-saturation on the accessible timescale, and the size-independence of saturated values cannot distinguish it. Please present a finite-time scaling analysis of S(t,L) at fixed small γ, or restrict the conclusion to the regime in which saturation is actually observed.
  4. [Sec. 4.1, bullet 3] The statement 'Hence, there is no MIPT' is stronger than what the presented evidence supports. The data cover three system sizes (L=64,128,256) at a single observation time T=100, with no scaling collapse or statistical uncertainty quantification. Given the non-saturation issue for small γ, the abstract's more cautious wording ('providing no evidence of a measurement-induced phase transition-like phenomenon') is appropriate and should be used in the main text.
minor comments (5)
  1. [Sec. 2.2] Typo: 'non-click limit' should be 'no-click limit'.
  2. [Eq. (4.2)] The last sum uses the index pair i,i+1 while the rest of the equation uses j; please make the index convention consistent.
  3. [Sec. 4.1, text near Figs. 7-8] The sentence 'From Figs (7) and (10)' appears to reference the wrong figure for the local-operator coupling dependence; it should likely be Fig. 8.
  4. [Sec. 5, first paragraph] The text says 'We first computed EE for the model described by (4.1)' when discussing the no-measurement case; the no-measurement Hamiltonian is (2.5), while (4.1) contains the measurement term.
  5. [Figs. 5, 6, 14, 15] The fitted functional forms and coefficients are presented only for L=64 and a fixed t_sat; please clarify in the text/captions that these are heuristic fits and not claimed to be universal or used for scaling inference.

Circularity Check

0 steps flagged

No significant circularity: the no-MIPT claim is a direct MPS simulation observation; the fitted γ-dependence is post-hoc, and the only self-citation is background.

full rationale

The paper's central claim is a direct numerical observation, not a quantity derived from a fit or from a self-citation. In Sec. 4.1, the conclusion 'the late-time saturation value of EE is independent of system size (here, we consider three lattice sizes: L=64,128,256)' is based on the simulated entropy values themselves; no scaling exponent or functional form was fitted and then reused as the predicted output. The fitted f(γ) curves in Figs. 5b, 6b, 14b and 15b are descriptive fits to the γ-dependence at fixed L and do not enter the size-scaling analysis that supports the no-MIPT claim. The only author self-citation, [92] ('Dynamics of monitored SSH model in Krylov space...'), appears in the background sentence 'MIPT has also been observed in this special limit [89–92]' and is not load-bearing; no uniqueness theorem or derivation is imported from it. The no-click effective Hamiltonians (4.1)-(4.3) are stipulated monitored models: the linear -iγO insertion is not derived from the standard Lindblad no-click evolution, which is a modeling/correctness caveat rather than a circularity. The paper itself flags this as a restricted setup in Sec. 5 ('our studies are done in a special no-click limit'). The numerical size-independence result is not equivalent by construction to the input Hamiltonians, so there is no circular step.

Axiom & Free-Parameter Ledger

7 free parameters · 6 axioms · 0 invented entities

The central numerical claim rests on a conventional spin mapping, a gauge-invariant sector restriction, and a specific no-click modeling assumption (linear imaginary coupling) that is not derived. The descriptive fits add fitted coefficients but do not drive the no-MIPT conclusion; the MPS truncation and saturation-time choices are numerical assumptions checked only partially.

free parameters (7)
  • Functional-fit coefficients for electric-flux saturation vs γ (x=0.5, L=64) = a=-0.00329217, b=1.64527, c=-0.00391089, d=0.0301029
    Post-hoc fit f(γ)=a e^{-bγ}+cγ+d to late-time saturation values in Fig. 5; descriptive, not used to establish the no-MIPT claim.
  • Functional-fit coefficients for density saturation vs γ (x=0.5, L=64) = a=0.0437349, b=0.188269, c=-0.0149996
    Post-hoc fit f(γ)=a e^{-bγ}+c in Fig. 6; descriptive.
  • Functional-fit coefficients for electric-flux saturation vs γ (x=1.5, L=64) = a=-0.000869448, b=-0.0010084, c=0.0767559
    Quadratic fit f(γ)=aγ²+bγ+c in Fig. 14; descriptive.
  • Functional-fit coefficients for density saturation vs γ (x=1.5, L=64) = a=-0.000806145, b=-0.00821252, c=0.0787908
    Quadratic fit f(γ)=aγ²+bγ+c in Fig. 15; descriptive.
  • MPS bond dimension and truncation cutoff = D=1000; χ=10^-8
    Numerical hyperparameters chosen by hand; convergence checked only for L=64 (Appendix A), while the central size-independence claim uses L=64,128,256.
  • Trotter time step = δ=0.1
    Chosen for TDVP evolution; convergence checked via ED at L=8 and D/cutoff at L=64.
  • Nominal saturation time t_sat = T=100
    Used to read off 'late-time saturation values'; for nonlocal monitoring at small γ the authors state EE does not saturate by this time (Sec. 4.2), so the 'across the range' no-MIPT statement is not fully supported.
axioms (6)
  • standard math Jordan-Wigner transformation maps the fermionic Z2 gauge theory to the spin Hamiltonian (2.5).
    Used throughout to represent the theory as an MPS; standard and non-controversial.
  • domain assumption Physical states are restricted to the gauge-invariant subspace satisfying G_i|ψ⟩=|ψ⟩ (Eq. 2.4), and the dynamics remain in this sector.
    The paper says Gauss law is checked numerically, but this is an assumption about the simulation sector.
  • ad hoc to paper No-click monitoring is equivalent to H_eff=H_0-iγH_1 with the measured operator appearing linearly (Eqs. 4.1-4.3).
    Load-bearing modeling choice; not derived from a specific jump/unraveling and not the generic -iγO² no-click form.
  • domain assumption The late-time saturation value can be read off at t=100 for all configurations.
    Acknowledged to fail for nonlocal small γ; affects the central null claim.
  • domain assumption MPS truncation at D=1000 and cutoff 10^-8 is sufficient for L=128,256 at all γ.
    Convergence tests in Appendix A are for L=64 only.
  • domain assumption Bipartite von Neumann entropy of the normalized conditional state is the right diagnostic for MIPT detection.
    Standard in the cited MIPT literature; reasonable.

pith-pipeline@v1.3.0-alltime-deepseek · 17091 in / 18164 out tokens · 192562 ms · 2026-08-04T05:36:38.480195+00:00 · methodology

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Cite this review

Pith. "Pith review of Effects of monitoring on entanglement dynamics for $1+1$D $\mathbb Z_2$ lattice gauge theory." pith.science (2026). https://pith.science/paper/KTGTUOCO

@misc{pith2026260328877,
  author       = {Pith},
  title        = {Pith review of: Effects of monitoring on entanglement dynamics for $1+1$D $\mathbb Z_2$ lattice gauge theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KTGTUOCO}},
  note         = {Machine review of arXiv:2603.28877}
}
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read the original abstract

The $(1+1)$-dimensional $\mathbb Z_2$ gauge theory is the simplest model that allows for quantum simulation to probe the fundamental aspects of a gauge theory coupled with dynamical fermions. To reliably benchmark such a system, it is crucial to understand the non-unitary quantum dynamics arising from effective non-Hermitian evolution and post-selected monitoring protocols. This work focuses on the post-selected non-Hermitian filtering dynamics of a $\mathbb Z_2$ gauge theory, where the non-Hermitian terms are associated with local and non-local gauge-invariant operators naturally present in the theory. We interpret the resulting dynamics as post-selected filtering, where different operator sectors are coupled to loss channels with different rates. This gives a unified framework for both the local electric flux and particle-number terms and the non-local mesonic hopping term. Tensor network calculations are performed to probe the effect of the filtering for larger lattice sizes (up to 256-site systems). Using Matrix Product State calculations, the dynamics of entanglement entropy are studied as a function of the filtering rate and the coupling constant. We find that, under both local and non-local filtering, the late-time saturation value of the bipartite entanglement entropy remains independent of system size, providing no evidence of a measurement-induced phase transition-like phenomenon in the post-selected dynamics across the range of filtering strengths, evolution times, and system sizes considered here.

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Works this paper leans on

101 extracted references · 26 linked inside Pith

  1. [1]

    Kogut,An introduction to lattice gauge theory and spin systems,Rev

    J.B. Kogut,An introduction to lattice gauge theory and spin systems,Rev. Mod. Phys.51 (1979) 659

    1979

  2. [2]

    Banerjee, M

    D. Banerjee, M. Dalmonte, M. Müller, E. Rico, P. Stebler, U.-J. Wiese et al.,Atomic quantum simulation of dynamical gauge fields coupled to fermionic matter: From string breaking to evolution after a quench,Phys. Rev. Lett.109(2012) 175302

    2012

  3. [3]

    Zohar, J.I

    E. Zohar, J.I. Cirac and B. Reznik,Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices,Reports on Progress in Physics79(2015) 014401

    2015

  4. [4]

    Martinez, C.A

    E.A. Martinez, C.A. Muschik, P. Schindler, D. Nigg, A. Erhard, M. Heyl et al.,Real-time dynamics of lattice gauge theories with a few-qubit quantum computer,Nature534(2016) 516

    2016

  5. [5]

    B. Yang, H. Sun, R. Ott, H.-Y. Wang, T.V. Zache, J.C. Halimeh et al.,Observation of gauge invariance in a 71-site bose–hubbard quantum simulator,Nature587(2020) 392

    2020

  6. [6]

    Zohar,Quantum simulation of lattice gauge theories in more than one space dimension—requirements, challenges and methods,Phil

    E. Zohar,Quantum simulation of lattice gauge theories in more than one space dimension—requirements, challenges and methods,Phil. Trans. A. Math. Phys. Eng. Sci. 380(2021) 20210069 [2106.04609]. – 20 –

    Pith/arXiv arXiv 2021

  7. [7]

    Zhou, G.-X

    Z.-Y. Zhou, G.-X. Su, J.C. Halimeh, R. Ott, H. Sun, P. Hauke et al.,Thermalization dynamics of a gauge theory on a quantum simulator,Science377(2022) 311

    2022

  8. [8]

    Wang, W.-Y

    H.-Y. Wang, W.-Y. Zhang, Z. Yao, Y. Liu, Z.-H. Zhu, Y.-G. Zheng et al.,Interrelated thermalization and quantum criticality in a lattice gauge simulator,Phys. Rev. Lett.131 (2023) 050401

    2023

  9. [9]

    Desaules, G.-X

    J.-Y. Desaules, G.-X. Su, I.P. McCulloch, B. Yang, Z. Papić and J.C. Halimeh,Ergodicity Breaking Under Confinement in Cold-Atom Quantum Simulators,Quantum8(2024) 1274

    2024

  10. [10]

    Mueller, T

    N. Mueller, T. Wang, O. Katz, Z. Davoudi and M. Cetina,Quantum computing universal thermalization dynamics in a (2+1)D Lattice Gauge Theory,Nature Commun.16(2025) 5492 [2408.00069]

    arXiv 2025

  11. [11]

    Zhang, Y

    W.-Y. Zhang, Y. Liu, Y. Cheng, M.-G. He, H.-Y. Wang, T.-Y. Wang et al.,Observation of microscopic confinement dynamics by a tunable topologicalθ-angle,Nature Physics21 (2025) 155

    2025

  12. [12]

    Osborne, I.P

    J.J. Osborne, I.P. McCulloch, B. Yang, P. Hauke and J.C. Halimeh,Large-scale 2+1d u(1) gauge theory with dynamical matter in a cold-atom quantum simulator,Communications Physics8(2025) 273

    2025

  13. [13]

    M. Kang, S. Kim, Y. Qian, P.M. Neves, L. Ye, J. Jung et al.,Measurements of the quantum geometric tensor in solids,Nature Physics21(2025) 110 [2412.17809]

    Pith/arXiv arXiv 2025

  14. [14]

    Halimeh, N

    J.C. Halimeh, N. Mueller, J. Knolle, Z. Papić and Z. Davoudi,Quantum simulation of out-of-equilibrium dynamics in gauge theories,2509.03586

    Pith/arXiv arXiv

  15. [15]

    Wegner,Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters,Journal of Mathematical Physics12(1971) 2259

    F.J. Wegner,Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters,Journal of Mathematical Physics12(1971) 2259

    1971

  16. [16]

    Fradkin and L

    E.H. Fradkin and L. Susskind,Order and Disorder in Gauge Systems and Magnets,Phys. Rev. D17(1978) 2637

    1978

  17. [17]

    Fradkin and S.H

    E. Fradkin and S.H. Shenker,Phase diagrams of lattice gauge theories with higgs fields, Phys. Rev. D19(1979) 3682

    1979

  18. [18]

    Senthil and M.P.A

    T. Senthil and M.P.A. Fisher,Z2 gauge theory of electron fractionalization in strongly correlated systems,Phys. Rev. B62(2000) 7850

    2000

  19. [19]

    Lai and O.I

    H.-H. Lai and O.I. Motrunich,Majorana spin liquids on a two-leg ladder,Phys. Rev. B84 (2011) 235148

    2011

  20. [20]

    Kormos, M

    M. Kormos, M. Collura, G. Takács and P. Calabrese,Real-time confinement following a quantum quench to a non-integrable model,Nature Physics13(2017) 246 [1604.03571]

    Pith/arXiv arXiv 2017

  21. [21]

    Smith, J

    A. Smith, J. Knolle, D.L. Kovrizhin and R. Moessner,Disorder-free localization,Phys. Rev. Lett.118(2017) 266601

    2017

  22. [22]

    Smith, J

    A. Smith, J. Knolle, R. Moessner and D.L. Kovrizhin,Dynamical localization inZ2 lattice gauge theories,Phys. Rev. B97(2018) 245137

    2018

  23. [23]

    Frank, E

    J. Frank, E. Huffman and S. Chandrasekharan,Emergence of gauss’ law in aZ2 lattice gauge theory in 1+1 dimensions,Physics Letters B806(2020) 135484

    2020

  24. [24]

    Borla, R

    U. Borla, R. Verresen, F. Grusdt and S. Moroz,Confined phases of one-dimensional spinless fermions coupled toZ2 gauge theory,Phys. Rev. Lett.124(2020) 120503. – 21 –

    2020

  25. [25]

    Zohar, A

    E. Zohar, A. Farace, B. Reznik and J.I. Cirac,Digital quantum simulation ofZ2 lattice gauge theories with dynamical fermionic matter,Phys. Rev. Lett.118(2017) 070501 [1607.03656]

    Pith/arXiv arXiv 2017

  26. [26]

    Grusdt and L

    F. Grusdt and L. Pollet,Z2 parton phases in the mixed-dimensionalt−J z model,Phys. Rev. Lett.125(2020) 256401

    2020

  27. [27]

    Schweizer, F

    C. Schweizer, F. Grusdt, M. Berngruber, L. Barbiero, E. Demler, N. Goldman et al., Floquet approach toZ2 lattice gauge theories with ultracold atoms in optical lattices,Nature Physics15(2019) 1168 [1901.07103]

    Pith/arXiv arXiv 2019

  28. [28]

    Surace, P.P

    F.M. Surace, P.P. Mazza, G. Giudici, A. Lerose, A. Gambassi and M. Dalmonte,Lattice gauge theories and string dynamics in rydberg atom quantum simulators,Phys. Rev. X10 (2020) 021041

    2020

  29. [29]

    Bañuls et al.,Simulating Lattice Gauge Theories within Quantum Technologies,Eur

    M.C. Bañuls et al.,Simulating Lattice Gauge Theories within Quantum Technologies,Eur. Phys. J. D74(2020) 165 [1911.00003]

    Pith/arXiv arXiv 2020

  30. [30]

    Borla, R

    U. Borla, R. Verresen, J. Shah and S. Moroz,Gauging the Kitaev chain,SciPost Phys.10 (2021) 148

    2021

  31. [31]

    Halimeh, L

    J.C. Halimeh, L. Homeier, C. Schweizer, M. Aidelsburger, P. Hauke and F. Grusdt, Stabilizing lattice gauge theories through simplified local pseudogenerators,Phys. Rev. Res.4 (2022) 033120 [2108.02203]

    Pith/arXiv arXiv 2022

  32. [32]

    Mildenberger, W

    J. Mildenberger, W. Mruczkiewicz, J.C. Halimeh, Z. Jiang and P. Hauke,Confinement in a Z2 lattice gauge theory on a quantum computer,Nature Phys.21(2025) 312 [2203.08905]

    arXiv 2025

  33. [33]

    Davoudi, N

    Z. Davoudi, N. Mueller and C. Powers,Towards quantum computing phase diagrams of gauge theories with thermal pure quantum states,Phys. Rev. Lett.131(2023) 081901

    2023

  34. [34]

    Kebrič, U

    M. Kebrič, U. Schollwöck and F. Grusdt,Mean-field theory of 1+1DZ2 lattice gauge theory with matter,SciPost Phys.20(2026) 017 [2404.02890]

    arXiv 2026

  35. [35]

    Chen, J.C

    I.-C. Chen, J.C. Getelina, K. Pollock, A. Khindanov, S. Sen, Y.-X. Yao et al.,Classical and quantum simulations of 1+1-dimensionalZ2 gauge theory at finite temperature and density, Commun. Phys.8(2025) 375 [2407.11949]

    Pith/arXiv arXiv 2025

  36. [36]

    Alexandrou, A

    C. Alexandrou, A. Athenodorou, K. Blekos, G. Polykratis and S. Kühn,Realizing string breaking dynamics in aZ2 lattice gauge theory on quantum hardware,Phys. Rev. D112 (2025) 114506

    2025

  37. [37]

    Pichler, E

    T. Pichler, E. Rico, M. Dalmonte, S. Montangero and P. Zoller,Real-time dynamics in lattice gauge theories with tensor networks,Physical Review X6(2016) 011023

    2016

  38. [38]

    Knaute, M

    J. Knaute, M. Feuerstein and E. Zohar,Entanglement and confinement in lattice gauge theory tensor networks,JHEP02(2024) 174 [2401.01930]

    Pith/arXiv arXiv 2024

  39. [39]

    Fromm, O

    M. Fromm, O. Philipsen, M. Spannowsky and C. Winterowd,SimulatingZ2 lattice gauge theory with the variational quantum thermalizer,EPJ Quant. Technol.11(2024) 20 [2306.06057]

    Pith/arXiv arXiv 2024

  40. [40]

    Casini, M

    H. Casini, M. Huerta and J.A. Rosabal,Remarks on entanglement entropy for gauge fields, Phys. Rev. D89(2014) 085012

    2014

  41. [41]

    Bañuls, K

    M.C. Bañuls, K. Cichy, J.I. Cirac, K. Jansen and S. Kühn,Tensor Networks and their use for Lattice Gauge Theories,PoSLA TTICE2018(2018) 022 [1810.12838]. – 22 –

    Pith/arXiv arXiv 2018

  42. [42]

    Magnifico, M

    G. Magnifico, M. Dalmonte, P. Facchi, S. Pascazio, F.V. Pepe and E. Ercolessi,Real Time Dynamics and Confinement in theZn Schwinger-Weyl lattice model for 1+1 QED, Quantum4(2020) 281 [1909.04821]

    Pith/arXiv arXiv 2020

  43. [43]

    Irmejs, M.C

    R. Irmejs, M.C. Banuls and J.I. Cirac,Quantum simulation ofZ2 lattice gauge theory with minimal resources,Phys. Rev. D108(2023) 074503 [2206.08909]

    Pith/arXiv arXiv 2023

  44. [44]

    Mathew, N

    E. Mathew, N. Gupta, S.V. Kadam, A. Bapat, J. Stryker, Z. Davoudi et al., Tensor-network toolbox for probing dynamics of non-Abelian gauge theories,PoS LA TTICE2024(2025) 472 [2501.18301]

    Pith/arXiv arXiv 2025

  45. [45]

    Skinner, J

    B. Skinner, J. Ruhman and A. Nahum,Measurement-induced phase transitions in the dynamics of entanglement,Phys. Rev. X9(2019) 031009

    2019

  46. [46]

    Y. Li, X. Chen and M.P.A. Fisher,Quantum zeno effect and the many-body entanglement transition,Phys. Rev. B98(2018) 205136

    2018

  47. [47]

    Y. Li, X. Chen and M.P.A. Fisher,Measurement-driven entanglement transition in hybrid quantum circuits,Phys. Rev. B100(2019) 134306

    2019

  48. [48]

    Chan, R.M

    A. Chan, R.M. Nandkishore, M. Pretko and G. Smith,Unitary-projective entanglement dynamics,Phys. Rev. B99(2019) 224307

    2019

  49. [49]

    Boorman, M

    T. Boorman, M. Szyniszewski, H. Schomerus and A. Romito,Diagnostics of entanglement dynamics in noisy and disordered spin chains via the measurement-induced steady-state entanglement transition,Phys. Rev. B105(2022) 144202

    2022

  50. [50]

    Biella and M

    A. Biella and M. Schiró,Many-Body Quantum Zeno Effect and Measurement-Induced Subradiance Transition,Quantum5(2021) 528

    2021

  51. [51]

    Szyniszewski, A

    M. Szyniszewski, A. Romito and H. Schomerus,Universality of entanglement transitions from stroboscopic to continuous measurements,Phys. Rev. Lett.125(2020) 210602

    2020

  52. [52]

    Barratt, U

    F. Barratt, U. Agrawal, A.C. Potter, S. Gopalakrishnan and R. Vasseur,Transitions in the Learnability of Global Charges from Local Measurements,Phys. Rev. Lett.129(2022) 200602 [2206.12429]

    Pith/arXiv arXiv 2022

  53. [53]

    Zabalo, J.H

    A. Zabalo, J.H. Wilson, M.J. Gullans, R. Vasseur, S. Gopalakrishnan, D.A. Huse et al., Infinite-randomness criticality in monitored quantum dynamics with static disorder,Phys. Rev. B107(2023) L220204 [2205.14002]

    Pith/arXiv arXiv 2023

  54. [54]

    Barratt, U

    F. Barratt, U. Agrawal, S. Gopalakrishnan, D.A. Huse, R. Vasseur and A.C. Potter,Field theory of charge sharpening in symmetric monitored quantum circuits,Phys. Rev. Lett.129 (2022) 120604

    2022

  55. [55]

    Lunt and A

    O. Lunt and A. Pal,Measurement-induced entanglement transitions in many-body localized systems,Phys. Rev. Research2(2020) 043072

    2020

  56. [56]

    Zabalo, M.J

    A. Zabalo, M.J. Gullans, J.H. Wilson, R. Vasseur, A.W.W. Ludwig, S. Gopalakrishnan et al.,Operator scaling dimensions and multifractality at measurement-induced transitions, Phys. Rev. Lett.128(2022) 050602

    2022

  57. [57]

    Iaconis and X

    J. Iaconis and X. Chen,Multifractality in nonunitary random dynamics,Phys. Rev. B104 (2021) 214307

    2021

  58. [58]

    Sierant, G

    P. Sierant, G. Chiriacò, F.M. Surace, S. Sharma, X. Turkeshi, M. Dalmonte et al., Dissipative Floquet Dynamics: from Steady State to Measurement Induced Criticality in Trapped-ion Chains,Quantum6(2022) 638. – 23 –

    2022

  59. [59]

    Y. Bao, S. Choi and E. Altman,Theory of the phase transition in random unitary circuits with measurements,Phys. Rev. B101(2020) 104301

    2020

  60. [60]

    S. Choi, Y. Bao, X.-L. Qi and E. Altman,Quantum error correction in scrambling dynamics and measurement-induced phase transition,Phys. Rev. Lett.125(2020) 030505

    2020

  61. [61]

    Szyniszewski, A

    M. Szyniszewski, A. Romito and H. Schomerus,Entanglement transition from variable-strength weak measurements,Phys. Rev. B100(2019) 064204

    2019

  62. [62]

    Block, Y

    M. Block, Y. Bao, S. Choi, E. Altman and N.Y. Yao,Measurement-induced transition in long-range interacting quantum circuits,Phys. Rev. Lett.128(2022) 010604

    2022

  63. [63]

    Jian, Y.-Z

    C.-M. Jian, Y.-Z. You, R. Vasseur and A.W.W. Ludwig,Measurement-induced criticality in random quantum circuits,Phys. Rev. B101(2020) 104302

    2020

  64. [64]

    Agrawal, A

    U. Agrawal, A. Zabalo, K. Chen, J.H. Wilson, A.C. Potter, J.H. Pixley et al.,Entanglement and charge-sharpening transitions in u(1) symmetric monitored quantum circuits,Phys. Rev. X12(2022) 041002

    2022

  65. [65]

    Gullans and D.A

    M.J. Gullans and D.A. Huse,Scalable probes of measurement-induced criticality,Phys. Rev. Lett.125(2020) 070606

    2020

  66. [66]

    Sharma, X

    S. Sharma, X. Turkeshi, R. Fazio and M. Dalmonte,Measurement-induced criticality in extended and long-range unitary circuits,SciPost Phys. Core5(2022) 023

    2022

  67. [67]

    Zabalo, M.J

    A. Zabalo, M.J. Gullans, J.H. Wilson, S. Gopalakrishnan, D.A. Huse and J.H. Pixley, Critical properties of the measurement-induced transition in random quantum circuits, Phys. Rev. B101(2020) 060301

    2020

  68. [68]

    Vasseur, A.C

    R. Vasseur, A.C. Potter, Y.-Z. You and A.W.W. Ludwig,Entanglement transitions from holographic random tensor networks,Phys. Rev. B100(2019) 134203

    2019

  69. [69]

    Y. Li, X. Chen, A.W.W. Ludwig and M.P.A. Fisher,Conformal invariance and quantum nonlocality in critical hybrid circuits,Phys. Rev. B104(2021) 104305

    2021

  70. [70]

    Turkeshi, R

    X. Turkeshi, R. Fazio and M. Dalmonte,Measurement-induced criticality in (2 + 1)-dimensional hybrid quantum circuits,Phys. Rev. B102(2020) 014315

    2020

  71. [71]

    Sierant and X

    P. Sierant and X. Turkeshi,Universal behavior beyond multifractality of wave functions at measurement-induced phase transitions,Phys. Rev. Lett.128(2022) 130605

    2022

  72. [72]

    Gullans and D.A

    M.J. Gullans and D.A. Huse,Dynamical purification phase transition induced by quantum measurements,Phys. Rev. X10(2020) 041020

    2020

  73. [73]

    Yamamoto and R

    K. Yamamoto and R. Hamazaki,Localization properties in disordered quantum many-body dynamics under continuous measurement,Phys. Rev. B107(2023) L220201

    2023

  74. [74]

    Han and X

    Y. Han and X. Chen,Measurement-induced criticality inZ2 -symmetric quantum automaton circuits,Phys. Rev. B105(2022) 064306

    2022

  75. [75]

    Wauters, E

    M.M. Wauters, E. Ballini, A. Biella and P. Hauke,Symmetry-protection zeno phase transition in monitored lattice gauge theories,Phys. Rev. B111(2025) 094315

    2025

  76. [76]

    Y. Li, D.S. Bhakuni, Y.-C. Liu and M. Dalmonte,Metastable confinement in Rydberg lattice gauge theories,2602.22890

    arXiv

  77. [77]

    Maeno,Efficient construction ofZ 2 gauge-invariant bases for the Quantum Minimally Entangled Typical Thermal States algorithm,2603.10932

    R. Maeno,Efficient construction ofZ 2 gauge-invariant bases for the Quantum Minimally Entangled Typical Thermal States algorithm,2603.10932. – 24 –

    Pith/arXiv arXiv

  78. [78]

    Rhodes, S

    M.L. Rhodes, S. Pathak and R.W. Chien,Quantum simulation of lattice gauge theories coupled to fermionic matter via anyonic regularization,2603.15820

    Pith/arXiv arXiv

  79. [79]

    Dalibard, Y

    J. Dalibard, Y. Castin and K. Mølmer,Wave-function approach to dissipative processes in quantum optics,Phys. Rev. Lett.68(1992) 580

    1992

  80. [80]

    Carmichael,Quantum trajectory theory for cascaded open systems,Phys

    H.J. Carmichael,Quantum trajectory theory for cascaded open systems,Phys. Rev. Lett.70 (1993) 2273

    1993

Showing first 80 references.