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REVIEW 2 major objections 4 minor 48 references

Local memory of a single-site observer tracks how fast the global particle number can be learned, diverging in the fuzzy phase and becoming finite once charge is sharp.

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2026-07-10 23:21 UTC pith:X27NX5YP

load-bearing objection Solid, clean demonstration that local Markov order tracks charge learnability in the SEP; the L≤24 collapses are the soft spot but do not sink the claim. the 2 major comments →

arxiv 2607.06689 v1 pith:X27NX5YP submitted 2026-07-07 cond-mat.stat-mech nlin.CGquant-ph

Local Markov Order and Global Inference in Many-Body Dynamics

Thomas Iadecola This is my paper
classification cond-mat.stat-mech nlin.CGquant-ph
keywords symmetric exclusion processconditional mutual informationMarkov ordercharge sharpeninglearnability transitionmonitored hydrodynamicsconserved chargesmemory timescale
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.

This paper asks how long a local observer's measurement record remembers the past when the underlying many-body system conserves total particle number. In the classical symmetric exclusion process, an observer who watches only one site finds that the time needed for that record to become Markovian is set by how well they know the global charge. When the charge is fixed and known, memory decays on the ordinary hydrodynamic timescale. When the charge is unknown, learning it from the local record becomes the slowest channel, stretching the Markovianization time. Adding bulk measurements drives a charge-learnability transition: below the transition the memory time grows with system size; above it the charge is learned at a finite rate and the local record Markovianizes on an order-one timescale. The result ties local memory directly to global inference and shows that charge-conserving hydrodynamics keeps local records non-Markovian in the thermodynamic limit unless the charge can be learned quickly.

Core claim

In the monitored symmetric exclusion process the conditional mutual information of a single-site observer's record decays exponentially with a memory timescale that tracks the timescale on which the observer can learn the global particle number. That timescale diverges polynomially with system size in the charge-fuzzy phase and remains finite in the charge-sharp phase.

What carries the argument

Temporal conditional mutual information I(A:C|B) of the probe-site record, whose exponential decay defines the Markovianization timescale; this quantity is shown to be controlled by the posterior entropy of the global charge conditioned on the same record.

Load-bearing premise

The claim that finite-size scaling collapses of the conditional mutual information on lattices up to size 24 correctly identify the thermodynamic-limit memory timescale and its link to the known charge-sharpening transition.

What would settle it

Compute or measure the conditional mutual information of a single-site record in the monitored exclusion process for substantially larger systems (or in the continuum hydrodynamic limit) and check whether the extracted dynamical exponent still jumps from size-dependent growth below the sharpening point to an O(1) timescale above it.

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

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

2 major / 4 minor

Summary. The paper studies how a conserved global charge shapes the Markov order of a local observer's measurement record in the classical one-dimensional symmetric exclusion process (SEP). An observer continuously monitors a single probe site; the conditional mutual information (CMI) I(A:C|B) between the first and last outcomes, conditioned on the intermediate record of length m, is used to extract a Markovianization timescale τ. In a fixed charge sector the CMI decays exponentially with m/L^{2} (z=2), consistent with diffusion. When the total particle number is unknown a priori, an additional slow channel appears whose timescale tracks the decay of the posterior charge entropy H(N|B) and scales as ~L^{3}. Adding bulk measurements at rate p drives the known charge-sharpening transition: below p♯ the effective dynamical exponent of both CMI and H(N|B) flows from 2 toward 1, while above p♯ both quantities decay on an L-independent timescale. The central claim is that local memory tracks global charge learnability.

Significance. If the finite-size extrapolations hold, the work cleanly links two previously separate ideas—hydrodynamic memory and measurement-induced charge sharpening—within a single classical stochastic process. The CMI is an independent, well-defined information-theoretic observable; its exact evaluation via the SEP transfer matrix (Eq. 2) and the clean decomposition of the hidden-charge CMI (End Matter, Eq. 7) are technical strengths. The numerical protocol is transparent and the scaling collapses are reported with uncertainties. The result is of interest both to classical nonequilibrium statistical mechanics and to the quantum community studying monitored U(1) circuits, where the SEP is the large-onsite-dimension limit. The connection between local Markov order and global inference is a useful conceptual contribution even if some of the thermodynamic-limit exponents remain provisional.

major comments (2)
  1. The thermodynamic-limit claim that τ tracks the learnability timescale across the sharpening transition rests entirely on χ^{2} collapses of CMI and H(N|B) for L≤24 (Figs. 1–4, scaling forms (6) and (10)). Residual finite-size corrections, crossover effects near p♯, or incomplete sampling of rare records could still produce an apparent tracking that does not survive L o∞. The paper itself notes that pure hydrodynamics already supplies a z=2 channel; the claim that the slower learning channel dominates and becomes O(1) for p>p♯ is therefore only as secure as the L o∞ extrapolation. Additional data at larger L (or a controlled analytic argument for the sharp-phase gap) would substantially strengthen the central result.
  2. In the fuzzy phase the extracted z_eff flows continuously from 2 to 1 as p o p♯ (Fig. 3(d)). It is not clear whether this is a true asymptotic exponent or a prolonged crossover. A more systematic finite-size scaling analysis that includes the known location of p♯ (or an independent estimate of p♯ from the same data) would clarify whether the memory timescale truly inherits the ballistic scaling expected throughout the fuzzy phase.
minor comments (4)
  1. The End Matter comparison of MI and CMI (Figs. 5–6) is useful but the claimed small-m power laws (~1/m for MI, ~1/m^{2} for CMI) are only guides; a short statement that the accessible window is too small for a definitive exponent would avoid over-interpretation.
  2. Notation for the posterior p(N|b) (Eq. 8) and the mixed-charge initial state could be stated more compactly; the present wording is slightly repetitive.
  3. Fig. 1(b) schematic would benefit from an explicit label of the axes (p vs. τ) and a brief note that p♯ is taken from prior literature.
  4. A short remark on the computational cost of exact transfer-matrix evaluation for L=24 would help readers assess the feasibility of modest extensions.

Circularity Check

0 steps flagged

No circularity: CMI and posterior entropy are independently computed observables whose timescales are observed to track each other and the known charge-sharpening transition.

full rationale

The paper's central claim is an observed numerical tracking between two independently defined and computed quantities: the conditional mutual information I(A:C|B) of the local probe record (Eqs. 3–5) and the posterior charge entropy H(N|B) (Eq. 9). Both are obtained by exact transfer-matrix evaluation of the SEP probability (Eq. 2) on sampled trajectories for L≤24; neither is defined in terms of the other, nor is either fitted and then re-labeled a prediction. The identity (Eq. 7) relating fixed-charge and hidden-charge CMI is a standard chain-rule decomposition used only for interpretation, not as a definitional identity that forces the result. Finite-size scaling forms (6) and (10) are conventional ansätze whose exponents are extracted by χ^{2} collapse; the extracted z_eff and au(L) are reported as data, not as first-principles predictions. The charge-sharpening point p♯≈0.4 is taken from the external literature (Refs. 23–26, none by the present sole author) solely as a reference location against which the new memory timescale is compared; it is not used to derive or force the CMI decay. There are no self-citations, uniqueness theorems, or smuggled ansätze that close a logical loop. The derivation is therefore self-contained numerical observation, free of the circularity patterns listed.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The central claim rests on the standard definition of the SEP, the information-theoretic definition of conditional mutual information, and the previously established existence of a charge-sharpening transition under bulk monitoring. No new particles, forces, or conserved quantities are postulated. The only free parameters are the fitted dynamical exponents and the scaling-function parameters that appear in the finite-size collapses; they are not used to define the claim but only to quantify it.

free parameters (2)
  • effective dynamical exponent z_eff (and related eta, φ)
    Extracted by χ^{2} collapse of CMI and posterior-entropy data for each p; values such as z = 2.00(1), 2.9(1), φ = 2.7(1) are reported. They quantify the claim but do not define it.
  • bulk measurement probability p and estimated critical point p♯ ≈ 0.4
    p is the control parameter; p♯ is taken from prior literature with a protocol-dependent adjustment noted in footnote 36. The precise location of p♯ is not re-derived from first principles in this work.
axioms (4)
  • domain assumption The one-dimensional symmetric exclusion process with the stated even/odd transfer-matrix updates is a faithful model of classical charge-conserving stochastic dynamics.
    Standard; invoked from the Model and setup section onward.
  • domain assumption Conditional mutual information I(A:C|B) vanishing exponentially with record length m diagnoses finite Markov order of the local measurement process.
    Standard information-theoretic definition; used to define the memory timescale au via Eq. (5).
  • domain assumption A charge-sharpening (learnability) transition exists in the monitored SEP at a finite bulk measurement rate p♯.
    Taken from prior literature (Refs. 23–26); used as the reference point for the memory-time transition.
  • ad hoc to paper Finite-size scaling collapses on L ≤ 24 correctly capture the thermodynamic-limit dynamical exponents that control Markovianization.
    Implicit in the interpretation of all scaling plots (Figs. 1–4); not independently verified at larger sizes.

pith-pipeline@v1.1.0-grok45 · 16293 in / 2966 out tokens · 30395 ms · 2026-07-10T23:21:12.824754+00:00 · methodology

0 comments
read the original abstract

We consider how the presence of conserved charges affects memory in a classical stochastic process, the symmetric exclusion process, with an observer constantly measuring a single site. We find that the observer's measurement record becomes Markovian (i.e., loses memory) on a timescale that depends on their knowledge of the global charge, namely the total particle number. In particular, when the global charge is unknown a priori, the observer's time series Markovianizes on a timescale constrained by their ability to learn it from their measurement record. Augmenting the observer's record with bulk measurements drives a charge-learnability transition between charge-fuzzy and -sharp phases. We show that the memory timescale tracks the learnability timescale, diverging in the fuzzy phase and remaining finite in the sharp phase.

Figures

Figures reproduced from arXiv: 2607.06689 by Thomas Iadecola.

Figure 2
Figure 2. Figure 2: FIG. 2. Decay of the CMI in the unmonitored SEP for an [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Markovianization and charge learning in the monitored SEP. (a) Decay of the CMI at half filling for bulk measurement [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Decay of the hidden-charge CMI (a) and the pos [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Decay of the unconditioned MI in the unmonitored [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Collapsed data on logarithmic axes for (a) the MI and [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 6 internal anchors

  1. [1]

    Mori, Transport, collective motion, and brownian mo- tion, Progress of Theoretical Physics33, 423–455 (1965)

    H. Mori, Transport, collective motion, and brownian mo- tion, Progress of Theoretical Physics33, 423–455 (1965)

    work page 1965

  2. [2]

    Zwanzig,Nonequilibrium statistical mechanics(Ox- ford university press, 2001)

    R. Zwanzig,Nonequilibrium statistical mechanics(Ox- ford university press, 2001)

    work page 2001

  3. [3]

    Grabert,Projection operator techniques in nonequilib- rium statistical mechanics(Springer, 2006)

    H. Grabert,Projection operator techniques in nonequilib- rium statistical mechanics(Springer, 2006)

    work page 2006

  4. [4]

    B. J. Alder and T. E. Wainwright, Decay of the velocity autocorrelation function, Phys. Rev. A1, 18 (1970)

    work page 1970

  5. [5]

    Michaels and I

    I. Michaels and I. Oppenheim, Long-time tails and brow- nian motion, Physica A: Statistical Mechanics and its Applications81, 221–240 (1975)

    work page 1975

  6. [6]

    Franosch, M

    T. Franosch, M. Grimm, M. Belushkin, F. M. Mor, G. Foffi, L. Forr´ o, and S. Jeney, Resonances arising from hydrodynamic memory in brownian motion, Nature478, 85–88 (2011)

    work page 2011

  7. [7]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The theory of open quantum systems(OUP Oxford, 2002)

    work page 2002

  8. [8]

    Foligno, T

    A. Foligno, T. Zhou, and B. Bertini, Temporal entan- glement in chaotic quantum circuits, Phys. Rev. X13, 041008 (2023)

    work page 2023

  9. [9]

    Vilkoviskiy, M

    I. Vilkoviskiy, M. Sonner, Q. C. Huang, W. W. Ho, A. Lerose, and D. A. Abanin, Temporal entanglement transition in chaotic quantum many-body dynamics (2025), arXiv:2511.03846

    work page arXiv 2025

  10. [10]

    Process Tensor Approaches to Non-Markovian Quantum Dynamics

    J. Keeling, E. M. Stoudenmire, M.-C. Ba˜ nuls, and D. R. Reichman, Process tensor approaches to non-markovian quantum dynamics (2025), arXiv:2509.07661

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  11. [11]

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

    work page 2018

  12. [12]

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

    work page 2019

  13. [13]

    Vasseur, A

    R. Vasseur, A. C. Potter, Y.-Z. You, and A. W. W. Lud- wig, Entanglement transitions from holographic random tensor networks, Physical Review B100, 134203 (2019)

    work page 2019

  14. [14]

    Skinner, J

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

    work page 2019

  15. [15]

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

    work page 2020

  16. [16]

    Zabalo, M

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

    work page 2020

  17. [17]

    Y. Li, Y. Zou, P. Glorioso, E. Altman, and M. P. A. Fisher, Cross entropy benchmark for measurement- induced phase transitions, Phys. Rev. Lett.130, 220404 (2023)

    work page 2023

  18. [18]

    Ippoliti and V

    M. Ippoliti and V. Khemani, Learnability transitions in monitored quantum dynamics via eavesdropper’s classi- cal shadows, PRX Quantum5, 020304 (2024)

    work page 2024

  19. [19]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics43, 4452–4505 (2002)

    work page 2002

  20. [20]

    J. Y. Lee, C.-M. Jian, and C. Xu, Quantum criticality under decoherence or weak measurement, PRX Quantum 4, 030317 (2023)

    work page 2023

  21. [21]

    R. Fan, Y. Bao, E. Altman, and A. Vishwanath, Diag- nostics of mixed-state topological order and breakdown of quantum memory, PRX Quantum5, 020343 (2024)

    work page 2024

  22. [22]

    J. Y. Lee, Exact calculations of coherent information for toric codes under decoherence: Identifying the fun- damental error threshold, Phys. Rev. Lett.134, 250601 (2025)

    work page 2025

  23. [23]

    Agrawal, A

    U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. H. Pixley, S. Gopalakrishnan, and R. Vasseur, Entanglement and charge-sharpening transitions in u(1) 6 symmetric monitored quantum circuits, Phys. Rev. X12, 041002 (2022)

    work page 2022

  24. [24]

    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, 120604 (2022)

    work page 2022

  25. [25]

    Agrawal, J

    U. Agrawal, J. Lopez-Piqueres, R. Vasseur, S. Gopalakr- ishnan, and A. C. Potter, Observing quantum measure- ment collapse as a learnability phase transition, Phys. Rev. X14, 041012 (2024)

    work page 2024

  26. [26]

    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, 200602 (2022)

    work page 2022

  27. [27]

    Gopalakrishnan, E

    S. Gopalakrishnan, E. McCulloch, and R. Vasseur, Mon- itored fluctuating hydrodynamics, Phys. Rev. X16, 011024 (2026)

    work page 2026

  28. [28]

    S. W. P. Kim, C. von Keyserlingk, and A. Lamacraft, Measurement-induced phase transitions in quantum in- ference problems and quantum hidden markov models (2025), arXiv:2504.08888

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  29. [29]

    P¨ utz, S

    M. P¨ utz, S. J. Garratt, H. Nishimori, S. Trebst, and G.-Y. Zhu, Learning transitions in classical ising models and de- formed toric codes, Phys. Rev. Lett.136, 190402 (2026)

    work page 2026

  30. [30]

    Bayesian critical points in classical lattice models

    A. Nahum and J. L. Jacobsen, Bayesian critical points in classical lattice models (2025), arXiv:2504.01264

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  31. [31]

    Spitzer, Interaction of markov processes, Advances in Mathematics5, 246–290 (1970)

    F. Spitzer, Interaction of markov processes, Advances in Mathematics5, 246–290 (1970)

    work page 1970

  32. [32]

    Papapetrou and D

    M. Papapetrou and D. Kugiumtzis, Markov chain order estimation with conditional mutual information, Phys- ica A: Statistical Mechanics and its Applications392, 1593–1601 (2013)

    work page 2013

  33. [33]

    Racca, F

    E. Racca, F. Laio, D. Poggi, and L. Ridolfi, Test to de- termine the markov order of a time series, Phys. Rev. E 75, 011126 (2007)

    work page 2007

  34. [34]

    Taranto, F

    P. Taranto, F. A. Pollock, S. Milz, M. Tomamichel, and K. Modi, Quantum markov order, Phys. Rev. Lett.122, 140401 (2019)

    work page 2019

  35. [35]

    Taranto, S

    P. Taranto, S. Milz, F. A. Pollock, and K. Modi, Struc- ture of quantum stochastic processes with finite markov order, Phys. Rev. A99, 042108 (2019)

    work page 2019

  36. [36]

    Note that Ref. [24] quotesp ♯ ≈0.2, but their monitoring protocol allows bulk measurements between each layer of gates, and therefore has approximately twice as many measurements perpas our protocol

    work page

  37. [37]

    Chiribella, G

    G. Chiribella, G. M. D’Ariano, and P. Perinotti, Quan- tum circuit architecture, Phys. Rev. Lett.101, 060401 (2008)

    work page 2008

  38. [38]

    Chiribella, G

    G. Chiribella, G. M. D’Ariano, and P. Perinotti, Theo- retical framework for quantum networks, Phys. Rev. A 80, 022339 (2009)

    work page 2009

  39. [39]

    Oreshkov, F

    O. Oreshkov, F. Costa, and v. Brukner, Quantum corre- lations with no causal order, Nature Communications3, 10.1038/ncomms2076 (2012)

    work page doi:10.1038/ncomms2076 2012

  40. [40]

    F. A. Pollock, C. Rodr´ ıguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Non-markovian quantum processes: Complete framework and efficient characteri- zation, Phys. Rev. A97, 012127 (2018)

    work page 2018

  41. [41]

    F. A. Pollock, C. Rodr´ ıguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Operational markov condi- tion for quantum processes, Phys. Rev. Lett.120, 040405 (2018)

    work page 2018

  42. [42]

    Lerose, M

    A. Lerose, M. Sonner, and D. A. Abanin, Influence matrix approach to many-body floquet dynamics, Phys. Rev. X 11, 021040 (2021)

    work page 2021

  43. [43]

    Mesoscopic Regimes of Temporal Entanglement in Ergodic Quantum Systems

    S. Cerezo-Roquebr´ un, J. T. Schneider, S. Carignano, A. Bou-Comas, M. C. Ba˜ nuls, E. L´ opez, and L. Tagli- acozzo, Mesoscopic regimes of temporal entanglement in ergodic quantum systems (2026), arXiv:2605.08356

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  44. [44]

    L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Strong-to-weak spontaneous symmetry break- ing in mixed quantum states, PRX Quantum6, 010344 (2025)

    work page 2025

  45. [45]

    Huang, M

    X. Huang, M. Qi, J.-H. Zhang, and A. Lucas, Hy- drodynamics as the effective field theory of strong-to- weak spontaneous symmetry breaking, Phys. Rev. B111, 125147 (2025)

    work page 2025

  46. [46]

    Hauser, K

    J. Hauser, K. Su, H. Ha, J. Lloyd, T. G. Kiely, R. Vasseur, S. Gopalakrishnan, C. Xu, and M. P. A. Fisher, Strong-to-weak symmetry breaking in open quan- tum systems: From discrete particles to continuum hy- drodynamics (2026), arXiv:2602.16045

    work page arXiv 2026

  47. [47]

    J. Y. Lee, Charge scrambling in strong-to-weak sponta- neous symmetry breaking (2026), arXiv:2605.05288

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  48. [48]

    Y. Tang, P. Kattel, and J. H. Pixley, A mean-field description of strong-to-weak symmetry breaking in the monitored three-dimensional bose-hubbard model (2026), arXiv:2606.02713. End Matter Comparing MI and CMI atp= 0.—To contextualize our finite-size data for the CMI in the unmonitored SEP [Fig. 1(c)], we study here the unconditioned MI between AandC,I(A...

    work page internal anchor Pith review Pith/arXiv arXiv 2026