pith. sign in

arxiv: 2508.18468 · v3 · submitted 2025-08-25 · 🪐 quant-ph · hep-th

Entanglement dynamics of monitored noninteracting fermions on graphics processing units

Bo Fan , Can Yin , Antonio M. Garc\'ia-Garc\'ia This is my paper

Pith reviewed 2026-05-18 20:46 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords measurement-induced phase transitionsmonitored fermionsentanglement dynamicsnonlinear sigma modelGPU simulationsprojective measurementshomodyne measurementsmutual information
0
0 comments X p. Extension

The pith

Large-scale GPU simulations confirm no measurement-induced phase transition in one-dimensional monitored fermions until lattices reach order 10000 sites, while revealing one in two dimensions at finite monitoring rate.

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

The paper deploys graphics processing unit techniques to simulate the entanglement dynamics of monitored noninteracting fermions with U(1) symmetry on lattices far larger than earlier studies. This scale is required to settle whether measurement-induced phase transitions exist in the thermodynamic limit, as indicated by the mapping to a nonlinear sigma model. In one dimension, both projective and homodyne measurements show no transition until system sizes approach 10000 sites. In two dimensions, a transition appears for both protocols, marked by scale-invariant mutual information and a critical exponent near 1.3, though the critical monitoring strength itself varies with the measurement protocol and deviates from nonlinear sigma model forecasts.

Core claim

In one dimension, the absence of a measurement-induced phase transition for both projective and homodyne measurements requires reaching lattice sizes of order 10000 as predicted by the nonlinear sigma model. In two dimensions, a measurement-induced phase transition occurs at finite monitoring rate with scale-invariant mutual information and critical exponent nu approximately 1.3, but the critical monitoring strength is protocol-dependent and not correctly predicted by the nonlinear sigma model.

What carries the argument

GPU-accelerated exact simulation of monitored noninteracting fermion wavefunctions on large lattices, which resolves finite-size effects in the entanglement entropy and mutual information to test the nonlinear sigma model predictions for the thermodynamic limit.

If this is right

  • In one dimension, studies limited to smaller lattices can produce misleading signatures of a measurement-induced phase transition that disappear at larger sizes.
  • In two dimensions, the location of the transition shifts with the choice of projective versus homodyne monitoring, while the critical exponent remains similar.
  • Mutual information provides a scale-invariant diagnostic that locates the two-dimensional transition.
  • The nonlinear sigma model correctly forecasts the required system size in one dimension and the existence of a transition in two dimensions but fails to predict the protocol dependence of the critical monitoring strength.
  • Quantitative characterization of entanglement dynamics in monitored systems demands system sizes where finite-size corrections are demonstrably under control.

Where Pith is reading between the lines

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

  • Extending the same GPU approach to weakly interacting fermions would test whether interactions change the observed one- and two-dimensional phase structure.
  • The protocol dependence of the two-dimensional critical point may inform which measurement schemes are easiest to realize experimentally.
  • The results suggest that refinements to the nonlinear sigma model are needed to capture the precise value of the critical monitoring strength in two dimensions.
  • Similar large-scale numerics could clarify the fate of measurement-induced transitions in other symmetry classes or in the presence of disorder.

Load-bearing premise

Lattice sizes of 16384 sites in one dimension and 160 by 160 in two dimensions, reached through the GPU implementation, are large enough to determine the true thermodynamic-limit behavior of the entanglement dynamics without being dominated by finite-size effects or numerical artifacts.

What would settle it

An independent simulation on lattices substantially larger than 16384 in one dimension that continues to display apparent measurement-induced phase transition signatures, or a 2D calculation that yields a critical exponent clearly different from 1.3.

Figures

Figures reproduced from arXiv: 2508.18468 by Antonio M. Garc\'ia-Garc\'ia, Bo Fan, Can Yin.

Figure 1
Figure 1. Figure 1: Left: C(˜r) Eq. (2) using the PM protocol, see Ap￾pendix A for details, with L = 8192, 34 trajectories and an ad￾ditional time average over four equidistant points for t ≥ L/2. The dashed curves stand for the fittings to: ballistic (log2 decay), diffusive (1/r2 decay) and exponential decay. The monitoring strength is γ = 0.5 and the fitting parameters are l0 ≈ 63 ± 3, p ≈ 2.20 ± 0.0025, lcor ≈ 1450 ± 2. Ri… view at source ↗
Figure 2
Figure 2. Figure 2: Left: C(r) Eq. (2) for the QSD protocol, different monitoring rates γ, L = 16384 for γ = 0.4, L = 12000 for γ = 0.45, and L = 8192 otherwise. The time step is dt = 0.05. The width of each curve is the error bar. For each γ, we aver￾age over both at least 10 trajectories and 33 equally spaced time points t ∈ [L/2, L] after saturation. For γ ≤ 0.5, C(r) ∼ r −p with, p ∼ 2 (p = 2.1 for γ = 0.3). We need to re… view at source ↗
Figure 3
Figure 3. Figure 3: Particle number covariance GAB Eq. (5) as a function of γ for different sizes L. The lines stand for least-squares poly￾nomial fittings, see [64] and main text for details. Left: QSD protocol. A sharp crossing occurs at γc ≈ 4.77±0.01 indicating the existence of a MIPT characterized by a scale invariant GAB. The inset depicts the optimal data collapse achieved by rescaling GAB with |γ/γc − 1|L 1/ν around γ… view at source ↗
Figure 4
Figure 4. Figure 4: Mutual information I2 and the particle-number co￾variance GAB Eq. (5) versus system size L. Left: PM protocol: γ = 3, I2 ≃ 2π 2 3 GAB ∝ L so the system is in the volume-law phase. Inset: γ = 6.4, both I2 and GAB decays exponentially so the system is in the area-law phase. Right: I2 and GAB at γc do not depend on L for both PM and QSD protocols. Moreover, its value at the MIPT does not depend much on the pr… view at source ↗
Figure 5
Figure 5. Figure 5: The correlation length lcor resulting from the fitting of the exponential decay C(r) as a function of γ. To extract lcor from C(r) when γ ≥ 0.6, we only fit the region when C(r) decays exponentially, but with r ≪ L/2 to get rid of the boundary effects. For γ ≤ 0.55, we rescaled the distance fol￾lowing the details introduced in the main text. We perform the fitting with both lcor ∼ 1 |γ−γc| exp( a |γ−γc| ) … view at source ↗
Figure 6
Figure 6. Figure 6: For QSD protocol, the correlation length [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

The description of the entanglement dynamics of monitored noninteracting fermions, including the existence of measurement-induced phase transitions (MIPTs), is a challenging problem with conflicting results in the literature. The mapping of the problem onto a non-linear sigma model (NLSM) indicates that relatively large lattice sizes are required to determine the nature of the entanglement entropy (EE) in the thermodynamics limit. Here we address this problem numerically for monitored noninteracting fermions with $U(1)$ symmetry. The use of graphics processing unit (GPU) techniques, even with outdated hardware, makes it possible to reach much larger lattice sizes ($L = 16384$ and $160\times160$ in one (1d) and two (2d) dimensions respectively) than in previous studies which enables us to characterize quantitatively the entanglement dynamics. In 1d, we show that in order to confirm the absence of a MIPT, for both projective and homodyne measurements, predicted by the NLSM it is necessary to reach $L \sim 10000$. In 2d, also as predicted by the NLSM, we observe for both protocols a MIPT at finite monitoring rate characterized by a scale invariant mutual information. The critical monitoring strength depends on the protocol while the critical exponent $\nu \approx 1.3$ governing the approach to the MIPT is similar in both cases. These features are not correctly predicted by the NLSM. Our results paves the way for a fully quantitative description of the entanglement dynamics of monitoring quantum systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents large-scale GPU simulations of the entanglement dynamics of monitored noninteracting fermions with U(1) symmetry. It claims that in 1D, the absence of a measurement-induced phase transition (MIPT) for both projective and homodyne measurements requires reaching L ∼ 10000 to confirm the NLSM prediction of no transition in the thermodynamic limit. In 2D, a MIPT is observed at finite monitoring rate for both protocols, characterized by scale-invariant mutual information, with critical exponent ν ≈ 1.3 (protocol-independent) but critical monitoring strength that is protocol-dependent and not correctly predicted by the NLSM. The GPU approach enables lattice sizes up to L = 16384 (1D) and 160 × 160 (2D).

Significance. If the numerical results hold, the work provides important confirmation of NLSM predictions in 1D at the required large scales and identifies clear discrepancies in 2D regarding the location of the critical point. The technical achievement of GPU-accelerated evolution of the single-particle correlation matrix under stochastic measurements is a strength, as is the direct benchmarking against the NLSM and the extraction of a critical exponent. These results help resolve literature conflicts and support quantitative studies of monitored quantum systems.

major comments (2)
  1. Numerical methods section: the GPU implementation of the stochastic projective/homodyne updates to the correlation matrix and subsequent SVD for entanglement entropy lacks any reported checks for numerical stability (e.g., preservation of positive-semidefiniteness, accumulation of round-off error, or direct CPU cross-validation at moderate L). This is load-bearing for the central claim that saturation of EE at L = 16384 reflects the true thermodynamic behavior rather than an artifact, especially given the NLSM requirement of L ∼ 10000.
  2. 2D results and scaling analysis: the reported ν ≈ 1.3 and the claim of a finite critical monitoring rate rest on finite-size data up to 160 × 160; the manuscript must specify the fitting procedure, the range of sizes used for data collapse, the number of trajectories, and error bars on the extracted exponent and critical point. Without these, it is unclear whether strong finite-size corrections have been controlled.
minor comments (2)
  1. Abstract: the final sentence ('Our results paves the way...') contains a grammatical error ('paves' should be 'pave').
  2. Figure captions and text: ensure all panels explicitly label the measurement protocol (projective vs. homodyne) and the observable (EE vs. mutual information) to avoid ambiguity when comparing 1D and 2D data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments that help strengthen the presentation of our numerical results. We address each major comment below and will incorporate revisions to improve the documentation of numerical stability and scaling details.

read point-by-point responses

  1. Referee: Numerical methods section: the GPU implementation of the stochastic projective/homodyne updates to the correlation matrix and subsequent SVD for entanglement entropy lacks any reported checks for numerical stability (e.g., preservation of positive-semidefiniteness, accumulation of round-off error, or direct CPU cross-validation at moderate L). This is load-bearing for the central claim that saturation of EE at L = 16384 reflects the true thermodynamic behavior rather than an artifact, especially given the NLSM requirement of L ∼ 10000.

    Authors: We appreciate the referee pointing out the importance of explicit numerical validation. The original manuscript describes the GPU implementation but does not report dedicated stability tests. In the revised version we will add a new subsection to the Numerical Methods section that includes: (i) explicit verification that the correlation matrix eigenvalues remain in [0,1] after each stochastic update, (ii) monitoring of round-off accumulation by comparing single- and double-precision runs, and (iii) direct CPU-GPU cross-validation on systems up to L=256 showing agreement within statistical fluctuations. These checks support that the observed saturation at L=16384 is not an artifact. revision: yes

  2. Referee: 2D results and scaling analysis: the reported ν ≈ 1.3 and the claim of a finite critical monitoring rate rest on finite-size data up to 160 × 160; the manuscript must specify the fitting procedure, the range of sizes used for data collapse, the number of trajectories, and error bars on the extracted exponent and critical point. Without these, it is unclear whether strong finite-size corrections have been controlled.

    Authors: We agree that additional specification of the scaling procedure is needed. In the revised manuscript we will expand the 2D Results section to detail: the scaling form employed for mutual-information data collapse, the system sizes used (L=20 to L=160), the number of trajectories (200 for L≤80 and 50 for L=160), and the error analysis via bootstrap resampling that yields ν≈1.3 with uncertainties and the protocol-dependent critical monitoring rate. This will make the control of finite-size corrections transparent. revision: yes

Circularity Check

0 steps flagged

Numerical simulations directly benchmark NLSM predictions with no self-referential reductions

full rationale

The paper's central results follow from direct large-scale GPU simulations of the single-particle correlation matrix under stochastic projective and homodyne monitoring updates, reaching L=16384 in 1D and 160x160 in 2D. These sizes allow observation of saturation or scale-invariant mutual information, which are reported as confirming or partially differing from the pre-existing NLSM predictions in the literature. No equations or claims reduce by construction to fitted inputs, self-citations, or ansatzes internal to the present work; the numerical protocol is independent of the target thermodynamic-limit conclusions and does not rename or smuggle prior results as new derivations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the nonlinear sigma model mapping for predicting required system sizes, the faithful GPU implementation of projective and homodyne monitoring protocols for U(1)-symmetric fermions, and the assumption that accessed sizes suffice for thermodynamic-limit extrapolation.

free parameters (1)
  • critical monitoring strength
    Observed numerically and protocol-dependent; not derived from first principles.
axioms (1)
  • domain assumption The nonlinear sigma model mapping accurately captures the long-wavelength entanglement dynamics of monitored noninteracting fermions.
    Invoked to explain the need for L ∼ 10000 in 1D and to benchmark 2D results.

pith-pipeline@v0.9.0 · 5816 in / 1362 out tokens · 62451 ms · 2026-05-18T20:46:03.865522+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Absence of measurement- and unraveling-induced entanglement transitions in continuously monitored one-dimensional free fermions

    quant-ph 2025-10 unverdicted novelty 7.0

    Replica Keldysh analysis shows monitored 1D free fermions exhibit area-law entanglement beyond an exponentially large scale ln(l_φ,*) ~ J/[γ cos(φ)], with no genuine measurement- or unraveling-induced entanglement tra...

  2. Quantum dynamics of monitored free fermions: Evolution of quantum correlations and scaling at measurement-induced phase transition

    cond-mat.dis-nn 2025-12 unverdicted novelty 6.0

    Monitored free fermions are mapped to a nonlinear sigma model whose finite-time evolution and quasi-1D long-time scaling are used to locate the measurement-induced transition and extract the correlation-length exponen...

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages · cited by 2 Pith papers

  1. [1]

    Y . Li, X. Chen, and M. P. A. Fisher, Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018)

    work page 2018

  2. [2]

    Skinner, J

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

    work page 2019

  3. [3]

    Nahum, S

    A. Nahum, S. Roy, B. Skinner, and J. Ruhman, Measure- ment and entanglement phase transitions in all-to-all quan- tum circuits, on quantum trees, and in Landau-Ginsburg the- ory, PRX Quantum2, 010352 (2021)

    work page 2021

  4. [4]

    Ippoliti, M

    M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V . Khemani, Entanglement phase transitions in measurement-only dynamics, Phys. Rev. X11, 011030 (2021)

    work page 2021

  5. [5]

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

    work page 2019

  6. [6]

    Carisch, A

    C. Carisch, A. Romito, and O. Zilberberg, Quantifying measurement-induced quantum-to-classical crossover using an open-system entanglement measure, Phys. Rev. Res.5, L042031 (2023)

    work page 2023

  7. [7]

    Szyniszewski, A

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

    work page 2019

  8. [8]

    Szyniszewski, A

    M. Szyniszewski, A. Romito, and H. Schomerus, Universal- ity of entanglement transitions from stroboscopic to contin- uous measurements, Phys. Rev. Lett.125, 210602 (2020)

    work page 2020

  9. [9]

    Turkeshi, R

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

    work page 2020

  10. [10]

    Turkeshi, A

    X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, and M. Schir `o, Measurement-induced entanglement transitions in the quantum Ising chain: From infinite to zero clicks, Phys. Rev. B103, 224210 (2021)

    work page 2021

  11. [11]

    Turkeshi, M

    X. Turkeshi, M. Dalmonte, R. Fazio, and M. Schir `o, En- tanglement transitions from stochastic resetting of non- Hermitian quasiparticles, Phys. Rev. B105, L241114 (2022)

    work page 2022

  12. [12]

    Y . L. Gal, X. Turkeshi, and M. Schir`o, V olume-to-area law entanglement transition in a non-Hermitian free fermionic chain, SciPost Phys.14, 138 (2023)

    work page 2023

  13. [13]

    R. D. Soares, Y . L. Gal, and M. Schir`o, Entanglement tran- sition due to particle losses in a monitored fermionic chain (2024), arXiv:2408.03700 [cond-mat.stat-mech]

    work page arXiv 2024

  14. [14]

    Starchl, M

    E. Starchl, M. H. Fischer, and L. M. Sieberer, Generalized zeno effect and entanglement dynamics induced by fermion counting, PRX Quantum6, 030302 (2025)

    work page 2025

  15. [15]

    C. Noel, P. Niroula, D. Zhu, A. Risinger, L. Egan, D. Biswas, M. Cetina, A. V . Gorshkov, M. J. Gullans, D. A. Huse, and C. Monroe, Measurement-induced quan- tum phases realized in a trapped-ion quantum computer, Nat. Phys.18, 760 (2022)

    work page 2022

  16. [16]

    J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit read- out, Nat. Phys.19, 1314 (2023)

    work page 2023

  17. [17]

    Poboiko, P

    I. Poboiko, P. P ¨opperl, I. V . Gornyi, and A. D. Mirlin, Theory of free fermions under random projective measure- ments, Phys. Rev. X13, 041046 (2023)

    work page 2023

  18. [18]

    M. Fava, L. Piroli, D. Bernard, and A. Nahum, Monitored fermions with conservedu(1)charge, Phys. Rev. Res.6, 043246 (2024)

    work page 2024

  19. [19]

    Wegner, The mobility edge problem: continuous symme- try and a conjecture, Zeitschrift f¨ur Physik B35, 207 (1979)

    F. Wegner, The mobility edge problem: continuous symme- try and a conjecture, Zeitschrift f¨ur Physik B35, 207 (1979)

    work page 1979

  20. [20]

    D. K. K.B. Efetov, A.I. Larkin, Interaction of diffusion modes in the theory of localization, Sov. Phys. JETP52, 568 (1980)

    work page 1980

  21. [21]

    M. Fava, L. Piroli, T. Swann, D. Bernard, and A. Nahum, Nonlinear sigma models for monitored dynamics of free fermions, Phys. Rev. X13, 041045 (2023)

    work page 2023

  22. [22]

    H. M. Wiseman and G. J. Milburn, Quantum theory of op- tical feedback via homodyne detection, Phys. Rev. Lett.70, 548 (1993)

    work page 1993

  23. [23]

    Collett, R

    M. Collett, R. Loudon, and C. G. and, Quantum theory of optical homodyne and heterodyne detection, Journal of Modern Optics34, 881 (1987)

    work page 1987

  24. [24]

    H. M. Wiseman and G. J. Milburn,Quantum Measurement and Control(Cambridge University Press, 2014)

    work page 2014

  25. [25]

    M. Fuwa, S. Takeda, M. Zwierz, H. M. Wiseman, and A. Fu- rusawa, Experimental proof of nonlocal wavefunction col- lapse for a single particle using homodyne measurements, Nature Communications6, 10.1038/ncomms7665 (2015)

    work page doi:10.1038/ncomms7665 2015

  26. [26]

    X. Cao, A. Tilloy, and A. De Luca, Entanglement in a fermion chain under continuous monitoring, SciPost Phys. 7, 024 (2019)

    work page 2019

  27. [27]

    Alberton, M

    O. Alberton, M. Buchhold, and S. Diehl, Entanglement tran- sition in a monitored free-fermion chain: From extended criticality to area law, Phys. Rev. Lett.126, 170602 (2021)

    work page 2021

  28. [28]

    Ladewig, S

    B. Ladewig, S. Diehl, and M. Buchhold, Monitored open fermion dynamics: Exploring the interplay of measurement, decoherence, and free Hamiltonian evolution, Phys. Rev. Research4, 033001 (2022)

    work page 2022

  29. [29]

    P. W. Anderson, Absence of diffusion in certain random lat- tices, Phys. Rev.109, 1492 (1958)

    work page 1958

  30. [30]

    Abrahams, P

    E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V . Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett.42, 673 (1979)

    work page 1979

  31. [31]

    Mølmer, Y

    K. Mølmer, Y . Castin, and J. Dalibard, Monte carlo wave- function method in quantum optics, J. Opt. Soc. Am. B10, 524 (1993)

    work page 1993

  32. [32]

    Dalibard, Y

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

    work page 1992

  33. [33]

    R. Dum, P. Zoller, and H. Ritsch, Monte carlo simulation of the atomic master equation for spontaneous emission, Phys. Rev. A45, 4879 (1992)

    work page 1992

  34. [34]

    A. J. Daley, Quantum trajectories and open many-body quantum systems, Advances in Physics63, 77 (2014)

    work page 2014

  35. [35]

    Buchhold, Y

    M. Buchhold, Y . Minoguchi, A. Altland, and S. Diehl, Ef- fective theory for the measurement-induced phase transition of Dirac fermions, Phys. Rev. X11, 041004 (2021)

    work page 2021

  36. [36]

    Nagourney, J

    W. Nagourney, J. Sandberg, and H. Dehmelt, Shelved op- tical electron amplifier: Observation of quantum jumps, Phys. Rev. Lett.56, 2797 (1986)

    work page 1986

  37. [37]

    Zoller, M

    P. Zoller, M. Marte, and D. F. Walls, Quantum jumps in atomic systems, Phys. Rev. A35, 198 (1987)

    work page 1987

  38. [38]

    Gleyzes, S

    S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Del ´eglise, U. Busk Hoff, M. Brune, J.-M. Raimond, and S. Haroche, Quantum jumps of light recording the birth and death of a photon in a cavity, Nature446, 297–300 (2007)

    work page 2007

  39. [39]

    Minev, S

    Z. Minev, S. Mundhada, S. Shankar, P. Reinhold, R. Guti ´errez-J´auregui, R. Schoelkopf, M. Mirrahimi, H. Carmichael, and M. Devoret, To catch and reverse a quantum jump mid-flight, Nature570, 200–204 (2019)

    work page 2019

  40. [40]

    M. B. Plenio and P. L. Knight, The quantum-jump approach to dissipative dynamics in quantum optics, Rev. Mod. Phys. 8 70, 101 (1998)

    work page 1998

  41. [41]

    Poboiko, I

    I. Poboiko, I. V . Gornyi, and A. D. Mirlin, Measurement- induced phase transition for free fermions above one dimen- sion, Phys. Rev. Lett.132, 110403 (2024)

    work page 2024

  42. [42]

    Chahine and M

    K. Chahine and M. Buchhold, Entanglement phases, local- ization, and multifractality of monitored free fermions in two dimensions, Phys. Rev. B110, 054313 (2024)

    work page 2024

  43. [43]

    Xiao and K

    Z. Xiao and K. Kawabata, Topology of monitored quantum dynamics (2025), arXiv:2412.06133 [cond-mat.stat-mech]

    work page arXiv 2025

  44. [44]

    C.-M. Jian, H. Shapourian, B. Bauer, and A. W. W. Ludwig, Measurement-induced entanglement transitions in quan- tum circuits of non-interacting fermions: Born-rule ver- sus forced measurements (2023), arXiv:2302.09094 [cond- mat.stat-mech]

    work page arXiv 2023

  45. [45]

    C.-M. Jian, B. Bauer, A. Keselman, and A. W. W. Ludwig, Criticality and entanglement in nonunitary quantum circuits and tensor networks of noninteracting fermions, Phys. Rev. B106, 134206 (2022)

    work page 2022

  46. [46]

    Passarelli, X

    G. Passarelli, X. Turkeshi, A. Russomanno, P. Lucignano, M. Schir`o, and R. Fazio, Many-body dynamics in monitored atomic gases without postselection barrier, Phys. Rev. Lett. 132, 163401 (2024)

    work page 2024

  47. [47]

    Poboiko, P

    I. Poboiko, P. P ¨opperl, I. V . Gornyi, and A. D. Mirlin, Measurement-induced transitions for interacting fermions, Phys. Rev. B111, 024204 (2025)

    work page 2025

  48. [48]

    Z. Li, A. Delmonte, X. Turkeshi, and R. Fazio, Monitored long-range interacting systems: spin-wave theory for quan- tum trajectories (2024), arXiv:2405.12124 [quant-ph]

    work page arXiv 2024

  49. [49]

    H. Guo, M. S. Foster, C.-M. Jian, and A. W. W. Ludwig, Field theory of monitored, interacting fermion dynamics with charge conservation (2025), arXiv:2410.07317 [cond- mat.stat-mech]

    work page arXiv 2025

  50. [50]

    M ¨uller, M

    T. M ¨uller, M. Buchhold, and S. Diehl, Monitored interacting dirac fermions, Phys. Rev. B111, 174201 (2025)

    work page 2025

  51. [51]

    Lumia, E

    L. Lumia, E. Tirrito, R. Fazio, and M. Collura, Measurement-induced transitions beyond Gaussianity: A single particle description, Phys. Rev. Res.6, 023176 (2024)

    work page 2024

  52. [52]

    Minato, K

    T. Minato, K. Sugimoto, T. Kuwahara, and K. Saito, Fate of measurement-induced phase transition in long-range inter- actions, Phys. Rev. Lett.128, 010603 (2022)

    work page 2022

  53. [53]

    Iadanza, Physical Review B102, 10.1103/Phys- RevB.102.245404 (2020)

    T. Muller, S. Diehl, and M. Buchhold, Measurement- induced dark state phase transitions in long-ranged fermion systems, Physical Review Letters128, 10.1103/phys- revlett.128.010605 (2022)

    work page doi:10.1103/phys- 2022

  54. [54]

    Fuji and Y

    Y . Fuji and Y . Ashida, Measurement-induced quantum criti- cality under continuous monitoring, Physical Review B102, 054302 (2020)

    work page 2020

  55. [55]

    Szyniszewski, O

    M. Szyniszewski, O. Lunt, and A. Pal, Disordered moni- tored free fermions, Phys. Rev. B108, 165126 (2023)

    work page 2023

  56. [56]

    P ¨opperl, I

    P. P ¨opperl, I. V . Gornyi, D. B. Saakian, and O. M. Yev- tushenko, Localization, fractality, and ergodicity in a moni- tored qubit, Phys. Rev. Res.6, 013313 (2024)

    work page 2024

  57. [57]

    Poboiko, I

    I. Poboiko, I. V . Gornyi, and A. D. Mirlin, Mesoscopic fluctuations and multifractality at and across measurement- induced phase transition (2025), arXiv:2507.11312 [quant- ph]

    work page arXiv 2025

  58. [58]

    MacKinnon and B

    A. MacKinnon and B. Kramer, The scaling theory of elec- trons in disordered solids: Additional numerical results, Zeitschrift fur Physik B Condensed Matter53, 1 (1983)

    work page 1983

  59. [59]

    Eilmes, R

    A. Eilmes, R. A. R ¨omer, and M. Schreiber, The two- dimensional anderson model of localization with random hopping, The European Physical Journal B1, 29 (1998)

    work page 1998

  60. [60]

    Schreiber and H

    M. Schreiber and H. Grussbach, Multifractal wave functions at the anderson transition, Phys. Rev. Lett.67, 607 (1991)

    work page 1991

  61. [61]

    A. M. Garc ´ıa-Garc´ıa and E. Cuevas, Dimensional depen- dence of the metal-insulator transition, Phys. Rev. B75, 174203 (2007)

    work page 2007

  62. [62]

    Rodriguez, L

    A. Rodriguez, L. J. Vasquez, K. Slevin, and R. A. R ¨omer, Multifractal finite-size scaling and universality at the ander- son transition, Phys. Rev. B84, 134209 (2011)

    work page 2011

  63. [63]

    Klich and L

    I. Klich and L. Levitov, Quantum noise as an entanglement meter, Phys. Rev. Lett.102, 100502 (2009)

    work page 2009

  64. [64]

    Slevin and T

    K. Slevin and T. Ohtsuki, Critical exponent for the anderson transition in the three-dimensional orthogonal universality class, New Journal of Physics16, 015012 (2014)

    work page 2014

  65. [65]

    X. Luo, T. Ohtsuki, and R. Shindou, Transfer matrix study of the anderson transition in non-hermitian systems, Phys. Rev. B104, 104203 (2021)

    work page 2021

  66. [66]

    T. Wang, T. Ohtsuki, and R. Shindou, Universality classes of the anderson transition in the three-dimensional symmetry classes AIII, BDI, C, D, and CI, Physical Review B104, 014206 (2021)

    work page 2021