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arxiv: 2604.21513 · v2 · submitted 2026-04-23 · 🪐 quant-ph · cond-mat.stat-mech

Quantum jump correlations in long-range dissipative spin systems via cluster and cumulant expansions

Giulia Salatino , Anna Delmonte , Zejian Li , Rosario Fazio , Alberto Biella This is my paper

Pith reviewed 2026-05-14 21:58 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords quantum jumpslong-range interactionsdissipative spin systemsnonequilibrium phasesquantum trajectoriescumulant expansioncluster mean-fieldphase transitions
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The pith

Correlations among quantum jumps distinguish phases in long-range dissipative spins.

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

The paper examines how the statistics of quantum jump trajectories can identify nonequilibrium phases in systems of spins with long-range dissipative interactions. Standard master equation approaches yield only average behaviors, but tracking individual detection events exposes spatial and temporal correlations that differ between phases. By combining a tilted Lindbladian for counting statistics with cluster mean-field for short distances and cumulant expansions for longer ranges, the authors compute jump correlations and waiting-time distributions. These quantities display clear distinctions between the paramagnetic and ferromagnetic regimes and show unique dynamics near the transition point. This approach offers a way to probe collective phenomena through observable jump events rather than solely through expectation values.

Core claim

In a dissipative spin model with long-range interactions undergoing a paramagnetic-to-ferromagnetic transition, the full counting statistics of quantum jumps, computed via tilted Lindbladian combined with cluster mean-field and cumulant expansions, show that jump correlations carry signatures of the phases and distinct features across the transition, as do waiting-time distributions.

What carries the argument

Tilted Lindbladian for full counting statistics of quantum jumps, supplemented by cluster mean-field for short-range correlations and cumulant expansion for long-range structure.

If this is right

  • Quantum jump correlations serve as phase indicators in long-range open quantum systems.
  • Waiting-time distributions of jumps provide additional dynamical information across the transition.
  • Trajectory-resolved observables probe collective behavior beyond mean-field averages.
  • Long-range interactions shape the nonequilibrium dynamics visible through correlation patterns.

Where Pith is reading between the lines

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

  • Experimental platforms with controllable long-range couplings could measure these jump statistics directly to identify phases.
  • The techniques might apply to other open many-body models where steady-state averages alone do not resolve phases.
  • Higher-order cumulants could extend the approach to capture critical fluctuations near the transition.

Load-bearing premise

The cluster mean-field and cumulant expansion techniques accurately capture the short- and long-range structure of jump correlations for the chosen model and parameter regime.

What would settle it

Exact quantum trajectory simulations on small systems that show mismatch between the computed jump correlation functions and the predicted phase-dependent signatures.

Figures

Figures reproduced from arXiv: 2604.21513 by Alberto Biella, Anna Delmonte, Giulia Salatino, Rosario Fazio, Zejian Li.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Sketch of the dissipative phase diagram in the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: In panel (a), representing the ferromagnetic regime at γtf = 20 the light circular shape represents a peak in the probability distribution P(n1, n2) around the values n1, n2 ≃ 15. In this phase, the peak moves to￾wards higher values of n1 and n2 with time and the joint distribution progressively broadens. On the other hand, in panel (b), representing the paramagnetic regime, we can again see the artifact o… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Evolution of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Growth rate of the covariance of the number of jumps [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Steady-state rate of change of the covariance between the numbers of jumps on sites 1 and 1 + [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Inverse of (a) average (b) variance of waiting time [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Inverse of (a) average (b) variance of waiting time [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Cluster mean field results for the steady-state magnetization [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
read the original abstract

We characterize nonequilibrium phases in long-range dissipative spin systems through the statistical properties of quantum jump trajectories. While the average dynamics governed by the Lindblad master equation provides access to steady-state expectation values of order parameters, the quantum trajectory framework reveals features encoded in the spatial and temporal correlations of detection events. Focusing on a model exhibiting a paramagnetic-to-ferromagnetic phase transition, we investigate the full counting statistics of quantum jumps using a tilted Lindbladian approach. We combine this with cluster mean-field and cumulant expansion techniques, which allow us to capture, respectively, the short- and long-range structure of jump correlations. In addition, we study the waiting-time distributions of detection events. We show that quantum jump correlations display clear signatures of the underlying phases and reveal distinct dynamical features across the transition. Our results highlight the potential of trajectory-resolved observables as probes of collective behavior in open quantum many-body systems and provide new insights into the role of long-range interactions in shaping nonequilibrium dynamics.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The manuscript characterizes nonequilibrium phases in long-range dissipative spin systems by analyzing statistical properties of quantum jump trajectories. It combines the tilted Lindbladian formalism with cluster mean-field and cumulant expansion techniques to compute full counting statistics and waiting-time distributions for a model exhibiting a paramagnetic-to-ferromagnetic transition, claiming that jump correlations exhibit clear phase signatures and distinct dynamical features across the transition.

Significance. If the approximations are shown to be accurate, the results would demonstrate that trajectory-resolved observables can serve as sensitive probes of collective behavior and long-range interactions in open quantum many-body systems, extending beyond steady-state order parameters from the Lindblad equation.

major comments (1)
  1. [Abstract] The central claim that quantum jump correlations display clear signatures of the phases rests on the cluster mean-field and cumulant expansion accurately capturing short- and long-range structure. However, no convergence checks with cluster size, error estimates, or direct comparisons to trajectory Monte Carlo or exact diagonalization are provided to validate the cumulant truncation near the critical point where collective fluctuations diverge.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for highlighting the need to strengthen the validation of our approximation methods. We address the major comment below and will revise the manuscript to incorporate additional checks and benchmarks.

read point-by-point responses

  1. Referee: [Abstract] The central claim that quantum jump correlations display clear signatures of the phases rests on the cluster mean-field and cumulant expansion accurately capturing short- and long-range structure. However, no convergence checks with cluster size, error estimates, or direct comparisons to trajectory Monte Carlo or exact diagonalization are provided to validate the cumulant truncation near the critical point where collective fluctuations diverge.

    Authors: We agree that explicit convergence checks, error estimates, and benchmarks are important to substantiate the reliability of the cluster mean-field and cumulant expansion results, particularly near the critical point. In the revised manuscript we will add (i) results for the jump correlation functions and waiting-time distributions obtained with increasing cluster sizes to demonstrate convergence of the short-range structure, (ii) a comparison of cumulant expansions truncated at successive orders together with an estimate of the truncation error, and (iii) direct comparisons with exact diagonalization and quantum-trajectory Monte Carlo simulations for small system sizes where these methods remain feasible. These additions will be placed in a new subsection of the methods/results and will be referenced from the abstract and main text. We note that exact or Monte Carlo benchmarks become computationally prohibitive for the system sizes required to resolve long-range correlations near criticality; however, the consistency between the two approximate methods already provides supporting evidence, which the new checks will further quantify. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard methods to concrete model

full rationale

The paper's central claims rest on applying the standard Lindblad master equation, tilted Lindbladian for full counting statistics, and established cluster mean-field plus cumulant expansion techniques to a specific long-range dissipative spin model. No load-bearing step reduces a prediction or phase signature to a fitted parameter, self-defined quantity, or unverified self-citation chain. The reported signatures of paramagnetic-to-ferromagnetic transitions in jump correlations are computed outputs, not inputs by construction. The approach is self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The phase transition itself is treated as a given property of the model.

axioms (1)
  • domain assumption The chosen spin model exhibits a paramagnetic-to-ferromagnetic phase transition under long-range dissipation.
    Invoked as the central setting for studying jump correlations across the transition.

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Forward citations

Cited by 1 Pith paper

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

  1. Dissipation Mechanisms and Dissipative Phase Transitions of two coupled Fully Connected Quantum Ising models

    cond-mat.stat-mech 2026-04 unverdicted novelty 5.0

    Different classes of dissipators in coupled quantum Ising models produce either equilibrium-like relaxation with protocol-dependent dynamics or nonequilibrium steady states featuring reentrant symmetry breaking.

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · cited by 1 Pith paper

  1. [1]

    Tilted Linbladian The tilted LindbladianL χ is connected to the study of the dynamics of the quantum state ˆ˜ρn(t) corresponding to evolution throughnjumps. In practice, the state ˆ˜ρn is obtained by averaging the quantum state over the en- semble of trajectories Γ(t) which containnjumps up to 4 timet: ˆ˜ρn(t) = 1 Ntraj|n P Γ|n | ˜ψΓ|n(t)⟩⟨ ˜ψΓ|n(t)|, whe...

    work page

  2. [2]

    tilted expectation

    Waiting time distribution The last quantity we will consider is thewaiting-time distributionP(t). We focus our study of the waiting-time distribution on the case in which we monitor a single site jof the spin chain, analyzing the statistics over trajec- tories of the time separating two consecutive jumps. Formally, since we do not keep track of the quantu...

    work page

  3. [3]

    +γ 4 16J4m∗x 4γ2 . Since both the average and the variance have the denom- inator multiplied bym ∗ x, we notice that how the distribu- tion approaches the uniform one yields an infinite aver- age and variance. Using these facts, we can characterize the ferromagnetic phase as the region with finite average waiting time and variance, and the paramagnetic ph...

    work page 2020

  4. [4]

    Fazio, J

    R. Fazio, J. Keeling, L. Mazza, and M. Schir` o, Many- body open quantum systems, SciPost Phys. Lect. Notes , 99 (2025)

    work page 2025

  5. [5]

    Minganti, Fabrizio and Biella, Alberto, Open quantum systems - a brief introduction, Cahiers de l´IPa2, 2 (2026)

    work page 2026

  6. [6]

    T. E. Lee, H. H¨ affner, and M. C. Cross, Antiferromag- netic phase transition in a nonequilibrium lattice of ryd- berg atoms, Phys. Rev. A84, 031402 (2011)

    work page 2011

  7. [7]

    C.-K. Chan, T. E. Lee, and S. Gopalakrishnan, Limit- cycle phase in driven-dissipative spin systems, Phys. Rev. A91, 051601 (2015)

    work page 2015

  8. [8]

    Landa, M

    H. Landa, M. Schir´ o, and G. Misguich, Multistability of driven-dissipative quantum spins, Phys. Rev. Lett.124, 043601 (2020)

    work page 2020

  9. [9]

    Biella, J

    A. Biella, J. Jin, O. Viyuela, C. Ciuti, R. Fazio, and D. Rossini, Linked cluster expansions for open quantum systems on a lattice, Phys. Rev. B97, 035103 (2018)

    work page 2018

  10. [10]

    J. Jin, A. Biella, O. Viyuela, C. Ciuti, R. Fazio, and D. Rossini, Phase diagram of the dissipative quantum ising model on a square lattice, Phys. Rev. B98, 241108 (2018)

    work page 2018

  11. [11]

    D. A. Paz, B. E. Maves, N. A. Kamar, A. Safavi- Naini, and M. Maghrebi, Entanglement, information and non-equilibrium phase transitions in long-range open quantum ising chains, arXiv preprint arXiv:2410.05370 (2024)

    work page arXiv 2024

  12. [12]

    Defenu, A

    N. Defenu, A. Lerose, and S. Pappalardi, Out-of- equilibrium dynamics of quantum many-body systems with long-range interactions, Physics Reports Out-of- Equilibrium Dynamics of Quantum Many-Body Systems with Long-Range Interactions,1074, 1 (2024)

    work page 2024

  13. [13]

    Minganti, A

    F. Minganti, A. Biella, N. Bartolo, and C. Ciuti, Spec- tral theory of liouvillians for dissipative phase transitions, Phys. Rev. A98, 042118 (2018)

    work page 2018

  14. [14]

    E. M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D. Lukin, and J. I. Cirac, Dissipative phase transition in a central spin system, Phys. Rev. A86, 012116 (2012)

    work page 2012

  15. [15]

    Mølmer, Y

    K. Mølmer, Y. Castin, and J. Dalibard, Monte carlo wave-function method in quantum optics, Journal of the Optical Society of America B10, 524 (1993)

    work page 1993

  16. [16]

    A. J. Daley, Quantum trajectories and open many-body quantum systems, Advances in Physics63, 77 (2014), https://doi.org/10.1080/00018732.2014.933502

    work page doi:10.1080/00018732.2014.933502 2014

  17. [17]

    G. T. Landi, M. J. Kewming, M. T. Mitchison, and P. P. Potts, Current fluctuations in open quantum sys- tems: Bridging the gap between quantum continuous measurements and full counting statistics, PRX Quan- tum5, 020201 (2024)

    work page 2024

  18. [18]

    Fitzner, F

    E. Fitzner, F. Carnazza, F. Carollo, and I. Lesanovsky, Detecting nonequilibrium phase transitions via con- tinuous monitoring of space-time trajectories and autoencoder-based clustering (2026), arXiv:2602.17341 [quant-ph]

    work page arXiv 2026

  19. [19]

    Yamamoto and R

    K. Yamamoto and R. Hamazaki, Measurement-induced crossover of quantum jump statistics in postselection-free many-body dynamics (2026), arXiv:2503.02418 [cond- mat.stat-mech]

    work page arXiv 2026

  20. [20]

    Yamamoto and R

    K. Yamamoto and R. Hamazaki, Anomalous waiting- time distributions in postselection-free quantum many- body dynamics under continuous monitoring (2026), arXiv:2604.00358 [cond-mat.stat-mech]

    work page arXiv 2026

  21. [21]

    Z. Li, A. Delmonte, X. Turkeshi, and R. Fazio, Moni- tored long-range interacting systems: Spin-wave theory for quantum trajectories, Nature Communications16, 4329 (2025)

    work page 2025

  22. [22]

    Delmonte, Z

    A. Delmonte, Z. Li, G. Passarelli, E. Y. Song, D. Bar- berena, A. M. Rey, and R. Fazio, Measurement-induced phase transitions in monitored infinite-range interacting systems, Physical Review Research7, 023082 (2025)

    work page 2025

  23. [23]

    Z. Li, A. Delmonte, and R. Fazio, Emergent determin- istic entanglement dynamics in monitored infinite-range bosonic systems, Physical Review B112, 104315 (2025)

    work page 2025

  24. [24]

    Eisert, M

    J. Eisert, M. van den Worm, S. R. Manmana, and M. Kastner, Breakdown of quasilocality in long-range quantum lattice models, Phys. Rev. Lett.111, 260401 14 (2013)

    work page 2013

  25. [25]

    Hauke and L

    P. Hauke and L. Tagliacozzo, Spread of correlations in long-range interacting quantum systems, Phys. Rev. Lett.111, 207202 (2013)

    work page 2013

  26. [26]

    L. F. Santos, F. Borgonovi, and G. L. Celardo, Coop- erative shielding in many-body systems with long-range interaction, Phys. Rev. Lett.116, 250402 (2016)

    work page 2016

  27. [27]

    Schachenmayer, B

    J. Schachenmayer, B. P. Lanyon, C. F. Roos, and A. J. Daley, Entanglement growth in quench dynamics with variable range interactions, Phys. Rev. X3, 031015 (2013)

    work page 2013

  28. [28]

    Sierant, G

    P. Sierant, G. Chiriac` o, F. M. Surace, S. Sharma, X. Turkeshi, M. Dalmonte, R. Fazio, and G. Pagano, Dis- sipative floquet dynamics: From steady state to measure- ment induced criticality in trapped-ion chains, Quantum 6, 638 (2022)

    work page 2022

  29. [29]

    Russo and T

    F. Russo and T. Pohl, Quantum dissipative continu- ous time crystals, Physical Review Letters135, 110404 (2025)

    work page 2025

  30. [30]

    J. Jin, A. Biella, O. Viyuela, L. Mazza, J. Keeling, R. Fazio, and D. Rossini, Cluster mean-field approach to the steady-state phase diagram of dissipative spin sys- tems, Phys. Rev. X6, 031011 (2016)

    work page 2016

  31. [31]

    Kikuchi, A theory of cooperative phenomena, Phys

    R. Kikuchi, A theory of cooperative phenomena, Phys. Rev.81, 988 (1951)

    work page 1951

  32. [32]

    H. A. Bethe, Statistical theory of superlattices, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences150, 552 (1935), https://royalsocietypublishing.org/rspa/article- pdf/150/871/552/33679/rspa.1935.0122.pdf

    work page arXiv 1935

  33. [33]

    Verstraelen, D

    W. Verstraelen, D. Huybrechts, T. Roscilde, and M. Wouters, Quantum and classical correlations in open quantum spin lattices via truncated-cumulant trajecto- ries, PRX Quantum4, 030304 (2023)

    work page 2023

  34. [34]

    Fricke, Transport equations including many-particle correlations for an arbitrary quantum system: A general formalism, Annals of Physics252, 479 (1996)

    J. Fricke, Transport equations including many-particle correlations for an arbitrary quantum system: A general formalism, Annals of Physics252, 479 (1996)

    work page 1996

  35. [35]

    Kubo, Generalized cumulant expansion method, Jour- nal of the Physical Society of Japan17, 1100 (1962)

    R. Kubo, Generalized cumulant expansion method, Jour- nal of the Physical Society of Japan17, 1100 (1962)

    work page 1962

  36. [36]

    Monroe, W

    C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. W. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano,et al., Programmable quantum sim- ulations of spin systems with trapped ions, Reviews of Modern Physics93, 025001 (2021)

    work page 2021

  37. [37]

    Ritsch, P

    H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Cold atoms in cavity-generated dynamical optical poten- tials, Reviews of Modern Physics85, 553 (2013)

    work page 2013

  38. [38]

    Lahaye, C

    T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The physics of dipolar bosonic quantum gases, Reports on Progress in Physics72, 126401 (2009)

    work page 2009

  39. [39]

    J. L. Bohn, A. M. Rey, and J. Ye, Cold molecules: Progress in quantum engineering of chemistry and quan- tum matter, Science357, 1002 (2017)

    work page 2017

  40. [40]

    Weimer, M

    H. Weimer, M. M¨ uller, I. Lesanovsky, P. Zoller, and H. P. B¨ uchler, A Rydberg quantum simulator, Nature Physics 6, 382 (2010)

    work page 2010

  41. [41]

    Ferioli, A

    G. Ferioli, A. Glicenstein, I. Ferrier-Barbut, and A. Browaeys, A non-equilibrium superradiant phase tran- sition in free space, Nature Physics19, 1345 (2023)

    work page 2023

  42. [42]

    ˇZunkoviˇ c, M

    B. ˇZunkoviˇ c, M. Heyl, M. Knap, and A. Silva, Dy- namical quantum phase transitions in spin chains with long-range interactions: Merging different concepts of nonequilibrium criticality, Physical review letters120, 130601 (2018)

    work page 2018

  43. [43]

    F. J. Dyson, Existence of a phase-transition in a one-dimensional Ising ferromagnet, Communications in Mathematical Physics12, 91 (1969)

    work page 1969

  44. [44]

    H. J. Carmichael,Statistical methods in quantum optics 2: Non-classical fields(Springer, 2008)

    work page 2008

  45. [45]

    H. M. Wiseman and G. J. Milburn,Quantum measure- ment and control(Cambridge university press, 2009)

    work page 2009

  46. [46]

    Jacobs,Quantum measurement theory and its appli- cations(Cambridge University Press, 2014)

    K. Jacobs,Quantum measurement theory and its appli- cations(Cambridge University Press, 2014)

    work page 2014

  47. [47]

    Wilczek, Quantum time crystals, Physical review let- ters109, 160401 (2012)

    F. Wilczek, Quantum time crystals, Physical review let- ters109, 160401 (2012)

    work page 2012

  48. [48]

    Khemani, R

    V. Khemani, R. Moessner, and S. Sondhi, A brief history of time crystals, arXiv preprint arXiv:1910.10745 (2019)

    work page arXiv 1910

  49. [49]

    Sacha,Time crystals, Vol

    K. Sacha,Time crystals, Vol. 114 (Springer, 2020)

    work page 2020

  50. [50]

    Iemini, A

    F. Iemini, A. Russomanno, J. Keeling, M. Schir` o, M. Dal- monte, and R. Fazio, Boundary time crystals, Physical review letters121, 035301 (2018)

    work page 2018

  51. [51]

    Viotti, M

    L. Viotti, M. Huber, R. Fazio, and G. Manzano, Quan- tum time crystal clock and its performance, Physical Re- view Letters136, 110401 (2026)

    work page 2026

  52. [52]

    Tosca, F

    J. Tosca, F. Carnazza, L. Giacomelli, and C. Ciuti, Efficient Variational Dynamics of Open Quantum Bosonic Systems via Automatic Differentiation (2025), arXiv:2507.14076

    work page arXiv 2025

  53. [53]

    Tosca, Z

    J. Tosca, Z. Li, F. Carnazza, and C. Ciuti, Variational Dynamics of Open Quantum Spin Systems in Phase Space (2026), arXiv:2604.01165

    work page arXiv 2026

  54. [54]

    J. A. Maki, A. Berti, I. Carusotto, and A. Biella, Monte Carlo matrix-product-state approach to the false vacuum decay in the monitored quantum Ising chain, SciPost Phys.15, 152 (2023)

    work page 2023

  55. [55]

    Tirrito, A

    E. Tirrito, A. Santini, R. Fazio, and M. Collura, Full counting statistics as probe of measurement-induced transitions in the quantum ising chain, SciPost Physics 15, 096 (2023)

    work page 2023

  56. [56]

    Turkeshi, A

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

    work page 2021

  57. [57]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement- induced phase transitions in the dynamics of entangle- ment, Physical Review X9, 031009 (2019)

    work page 2019

  58. [58]

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

    work page 2019

  59. [59]

    X. Cao, A. Tilloy, and A. De Luca, Entanglement in a fermion chain under continuous monitoring, SciPost Physics7, 024 (2019)

    work page 2019

  60. [60]

    Mattes, I

    R. Mattes, I. Lesanovsky, and F. Carollo, Long-Range In- teracting Systems Are Locally Noninteracting, Physical Review Letters134, 070402 (2025)

    work page 2025