Pith. sign in

REVIEW 4 minor 85 references

Measurement-Induced Phase Transition in State Estimation of Chaotic Systems and the Directed Polymer

T0 review · 0 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that the Shannon entropy of a Bayesian observer tracking a chaotic system grows at a rate equal to the negative part of a single control parameter, with a universal square-root-time scaling at the transition.

desk verdict A clean, mostly self-contained mapping of Bayesian state estimation on a tree to directed polymer freezing, with exact scaling; the tree premise for real chaos is the only genuinely soft spot. read the letter →

arxiv 2501.00547 v2 pith:3OD5YFFW submitted 2024-12-31 cond-mat.stat-mech

Federico Gerbino , Guido Giachetti , Pierre Le Doussal , Andrea De Luca This is my paper · ORCID
classification cond-mat.stat-mech MSC 82B2682B4437D45 PACS 05.40.-a05.45.-a
keywords measurement-inducedphasetransitionstateestimationchaoticsystemsBayesianinferencedirectedpolymerCayleytreeShannonentropyLyapunovexponent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a solvable model of a measurement-induced phase transition (MIPT) in a classical chaotic system, with no quantum mechanics and no postselection. In the model, the chaotic trajectory is a directed walk on a Cayley tree with branching ratio $K=e^{\lambda \Delta t}$, and at each time step the observer performs independent noisy measurements of every site, updating a Bayesian posterior distribution. The paper claims that the long-time growth of the Shannon entropy of this posterior is governed by a single control parameter $v = D_{\mathrm{KL}}(P_1\|P_0)/\Delta t - \lambda$: the growth rate is $|v|$ for $v \le 0$ and $0$ for $v > 0$. Exactly at the threshold the entropy grows as $\sqrt{t}$, with a universal scaling function that the authors derive in closed form and verify numerically. Because the observer's measurements do not perturb the classical state, the predicted transition is in principle directly observable, and it pinpoints the same critical location as directed-polymer freezing while having different critical properties.

What carries the argument

The central object is the mapping from Bayesian filtering on the tree to the directed polymer (DP) partition function. Unnormalized weights $z_j^{(\tau)}$ update multiplicatively, $z_j^{(\tau+1)} = B(a_j^{(\tau+1)})\, z_{\lceil j/K\rceil}^{(\tau)}$ with $B(a)=P_1(a)/(K P_0(a))$, so the normalization $Z^{(\tau)}$ is exactly the partition function of a DP on a Cayley tree. The observer's average Shannon entropy is then the difference of two expectations expressible through $\langle Z^{(\tau)} \ln Z^{(\tau)}\rangle_0$, and the self-similarity of the tree gives the closed recursion $G_{\tau+1}(y) = \langle G_\tau(y - \ln B(a))^K\rangle_0$ for the Laplace transform $G_\tau(y) = \langle \exp(-e^{-y} Z^{(\tau)})\rangle_0$. In the continuum limit this recursion becomes a KPP (Kolmogorov–Petrovsky–Piskunov) reaction-diffusion equation for the traveling wave, and the entropy-facing object $u_t(y) = e^y(1-G_\tau(y))$ obeys a drifted-diffusion equation whose drift is $v$; at criticality the nonlinear term acts as a reflecting wall at the origin, and the resulting diffusion-process calculation yields the universal scaling function $S(\eta)$. This machinery is what carries the exact rate (12) and the scaling form (13)–(14).

What would settle it

Simulate the same Bayesian estimation protocol on a deterministic chaotic system that is not a tree, such as a one- or two-dimensional coupled map lattice, and measure the Shannon entropy of the filter's posterior as a function of time for measurement strengths above the threshold $v>0$. The paper's claim predicts a vanishing entropy growth rate and, at $v=0$, the universal $\sqrt{t}$ growth; observing linear entropy growth for all measurement strengths, or a critical scaling that depends on the details of $P_0$ and $P_1$, would show the transition is an artefact of the tree geometry.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is the exact entropy growth rate of Eq. (12): $s = \lim_{t\to\infty}\langle S_t\rangle / t = |v|$ for $v \le 0$ and $s=0$ for $v>0$, with $v = D_{\mathrm{KL}}(P_1\|P_0)/\Delta t - \lambda$ measuring the balance between the information content of each measurement and the chaotic spreading of trajectories. In the chaotic or weak-measurement phase the observer's uncertainty still grows exponentially but with a reduced effective Lyapunov exponent $|v|$; in the strong-measurement phase the estimated distribution localizes onto an $\mathcal{O}(1)$ number of sites and the entropy saturates. In the critical window the paper provides the exact universal scaling form $\langle S_t\rangle \simeq (\sigma^2/v)\, S(v\sqrt{t/(2\sigma^2)})$ with $S(\eta) = (\tfrac12+\eta^2)\operatorname{erf}\eta - \eta^2 + (\eta/\sqrt{\pi}) e^{-\eta^2}$, which at $v=0$ reduces to $\langle S_t\rangle \simeq \sigma\sqrt{2t/\pi}$. The transition point coincides with the freezing transition of the directed polymer on the Cayley tree, but the MIPT is dominated by rare polymer configurations and its critical properties differ, as the numerical solutions of the exact recursion confirm.

Load-bearing premise

The load-bearing premise is that a chaotic system's uncertainty spread is exactly a tree with branches multiplying like $e^{\lambda \Delta t}$, and that the observer independently measures every branch tip at every step; the paper itself notes the tree is only a qualitative model for real chaotic systems at sufficiently high spatial dimension.

Editorial extensions

If this is right

  • A Bayesian observer of a tree-like chaotic system gains a sharp threshold: for measurement strengths below $v=0$ (i.e., $D_{\mathrm{KL}}/\Delta t < \lambda$) the estimated distribution continues to spread exponentially, while above the threshold it localizes and the entropy rate is exactly zero.
  • In the weak-measurement phase the entropy rate is exactly $\lambda - D_{\mathrm{KL}}/\Delta t$, so each unit of Kullback–Leibler information per unit time reduces the effective Lyapunov exponent linearly until it vanishes.
  • At the critical point the entropy grows like $\sigma \sqrt{2t/\pi}$, and the full crossover from linear growth to saturation in time is described by the single universal function $S(\eta)$, with the same curve in discrete- and continuous-time versions of the model.
  • The transition location coincides with the directed-polymer freezing transition on the tree, showing that a classical state-estimation transition and a known disordered-system transition can share a critical point while belonging to different universality classes.
  • Because the measurements are classical, no postselection is required, making this MIPT prediction testable without the exponential overhead that complicates quantum MIPT experiments.

Reading between the lines

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

  • An extension the paper leaves implicit: if the same Bayes-reweighting logic holds on other geometries, the coincidence with directed-polymer freezing suggests that in high-dimensional coupled chaotic systems the filtering transition should sit at the same condition $D_{\mathrm{KL}}/\Delta t = \lambda$, with the tree appearing as the effective mean-field geometry.
  • The paper's scaling function gives a ready-made finite-time crossover prediction: for $v$ near zero, the entropy should deviate from its asymptotic linear or constant behavior on the time scale $t_v \sim |v|^{-2}$, which is directly measurable in particle-filter or Kalman-type estimators before their exponential cost becomes prohibitive.
  • A practical design implication left implicit by the authors is that only $D_{\mathrm{KL}}(P_1\|P_0)$ per measurement matters for the phase boundary; redesigning observables to increase this divergence at fixed measurement cost is equivalent to tuning the Lyapunov exponent downward, so smart observable selection and stronger chaos are on the same footing.
  • In the quantum analogue, Born's rule replaces Bayes' rule and the measurement back-action cannot be ignored; this framework suggests that a similar control parameter combining Lyapunov growth with measurement information would set the entanglement phase boundary, a connection the paper does not pursue.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper introduces a solvable model of Bayesian state estimation for a chaotic system, represented as a directed random walk on a Cayley tree with branching ratio K=e^{λΔt}. At each time step, the observer measures every site independently with outcome distributions P1 (occupied) and P0 (empty). The authors map the average over the true trajectory and measurement outcomes to a P0-average reweighted by the directed polymer (DP) partition function, and then study the Shannon entropy of the observer's posterior distribution. They derive an exact growth-rate transition: s = lim_{t→∞} ⟨S_t⟩/t equals |v| for v≤0 and 0 for v>0, where v=D_KL(P1||P0)/Δt−λ. Near the transition they obtain a universal scaling function S(η) for the subleading growth, Eq. (14). Numerical solution of the exact recursion Eq. (8) and Monte Carlo simulations of the particle and posterior support the predictions. The paper presents the results as a toy model and discusses its connection to the freezing transition of the directed polymer.

Significance. If accepted, this is a valuable exactly solvable example of a classical measurement-induced phase transition, with a clean mapping to the directed polymer on a Cayley tree in the rare-event regime. The derivation is careful and largely self-contained: Eq. (4) is an exact reweighting identity, Eq. (9) is proved in Appendix B, and the scaling function Eqs. (13)–(14) is derived without adjustable parameters and checked against numerics. The paper also establishes that the transition location coincides with the DP freezing point while the critical properties differ, which is a physically interesting distinction. The main limitation, acknowledged by the authors in the Conclusions, is that the Cayley-tree representation with independent per-site measurements is a modeling assumption rather than a derived property of generic chaotic systems; the authors describe it as qualitative for sufficiently high dimension. I regard this as a scope limitation, not an internal inconsistency: the mathematical claims are claims about the tree model, and the broader 'chaotic systems' title should be read through that lens.

minor comments (4)
  1. [Abstract and Conclusions] The first sentence of the abstract and the title state the result for 'chaotic systems', but all proofs concern the Cayley-tree model with independent per-site measurements. Since the Conclusions already call the model a toy model and note that the tree is only a qualitative description for d>2, I recommend making this scope explicit in the abstract as well, e.g. by saying 'for a tree model of a chaotic system'.
  2. [Appendix I] The citation 'following [65]' at the start of Appendix I appears to be incorrect: if the numbering follows the main text, [65] is a reference on classical many-body information dynamics, not the Derrida–Spohn traveling-wave analysis. The intended reference is likely [68] or the footnote [72] of the main text.
  3. [Figure 2 caption] The caption writes 'D(P1||P0)=Var(P1||P0)/2' without defining D; the main text uses D_KL(P1||P0). Please use a consistent notation in the caption.
  4. [Supplemental Fig. S1 and text after Eq. (12)] For v<0, the truncation protocol in the Monte Carlo simulations causes the entropy to saturate at a value proportional to the cutoff τ*, as shown in Fig. S1. The main text's comparison of ⟨S_t⟩ to the linear growth |v|t implicitly relies on subtracting this truncation artifact; it would be helpful to state this explicitly in the main text where the numerical agreement for v<0 is discussed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the entropy-rate transition and scaling function are derived from the model equations and checked against parameter-free numerics.

full rationale

The paper's central claims are derived, not assumed. The Cayley-tree representation is introduced explicitly as a modeling premise ('we assume the state of the system to be one of the nodes of a Cayley tree...'), not imported from a citation. The mapping to the directed polymer follows from the definitions of the measurement likelihoods and the Bayes update (Eqs. (1)-(4)), and the entropy identity Eq. (5) is exact. The first term is computed explicitly in the End Matter as τ(D_KL - ln K) = v t, so the '-v t' contribution to ⟨S_t⟩ is a direct calculation. The second term, ⟨Z ln Z⟩, is analyzed through the KPP front equation (10)-(11) and the integral identity (9); the rate in Eq. (12) then follows from the front motion: for v>0 the linearized front contributes a +v t term that cancels -v t, while for v<0 it saturates, leaving s=|v|. The critical scaling function (14) is obtained from an explicit solution of the reflected-Wiener problem (Eq. (SF.2)), not from a fit. Numerics solve Eq. (8) or simulate the protocol and compare to the analytical S(η) with no adjustable parameters, as shown in Figs. 2-3 and Fig. S1. Self-citations appear only as contextual references (e.g., [54], [56], [69]) and are not load-bearing; the central polymer input [68] is the external Derrida-Spohn result. The unvalidated step is the tree representation of real chaotic dynamics, which the authors themselves call a minimal/qualitative high-dimensional description; this is a scope limitation, not a circular derivation.

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

The model has no free parameters fitted to data; v and σ² are computed directly from the assumed measurement distributions (P0, P1) and the Lyapunov exponent λ (in the Gaussian example, v=µ²/2 - ln 2). The central derivation rests on five axioms listed above: the tree representation of chaos, the uniform random trajectory, the continuum scaling, the standard DP/KPP results, and the wall approximation at criticality. The wall approximation is the most delicate, but its predictions are tested against numerical solutions of the exact recursion Eq. (8).

assumptions (5)
  • domain assumption The chaotic uncertainty growth is represented by a Cayley tree with branching ratio K=e^{λΔt}; the maximum Lyapunov exponent λ is identified with the branching rate.
    Introduced in the model section where K=e^{λΔt} and λ is identified with the maximal Lyapunov exponent. This is the core modeling assumption linking chaos to the tree geometry.
  • domain assumption The true trajectory is uniformly random among all paths, and measurement outcomes are independent across sites given the occupancy.
    Used in Eqs. (1) and (3) to define the average over trajectories and outcomes; it produces the uniform average over the K^τ leaves and the factorization that leads to the DP mapping.
  • ad hoc to paper The continuum limit Δt→0 with P1-P0=O(√Δt) is universal, with only D_KL and its variance (σ²) surviving.
    Appendix D derives the scaling D_KL = σ² Δt/2 and Var = σ² Δt; this is a specific scaling assumption that justifies the continuous-time KPP equation (10) and the universal form of the scaling function.
  • standard math The KPP traveling-wave and front-selection results of Derrida-Spohn apply, including the freezing transition at v=0.
    Relies on Ref. [68] for the DP on a tree and the KPP front velocity formulas; used to identify the freezing transition and the n=0 free energy.
  • ad hoc to paper At criticality, the nonlinear term in Eq. (11) acts as a reflecting wall at Y=0, and the initial condition becomes a step function θ(Y) in the scaled variables.
    The paper states: 'U_T(Y) satisfies drifted diffusion for Y>0 but with a wall imposing U_T(Y≤0)=0.' This wall/step approximation is the basis of the exact scaling function S(η); it is argued to be exact in the v→0 limit but is a non-trivial simplification.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Measurement-Induced Phase Transition in State Estimation of Chaotic Systems and the Directed Polymer." pith.science (2026). https://pith.science/paper/3OD5YFFW

@misc{pith2026250100547,
  author       = {Pith},
  title        = {Pith review of: Measurement-Induced Phase Transition in State Estimation of Chaotic Systems and the Directed Polymer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3OD5YFFW}},
  note         = {Machine review of arXiv:2501.00547}
}
read the original abstract

We introduce a solvable model of a measurement-induced phase transition (MIPT) in a deterministic but chaotic dynamical system with a positive Lyapunov exponent. In this setup, an observer only has a probabilistic description of the system but mitigates chaos-induced uncertainty through repeated measurements. Using a minimal representation via a branching tree, we map this problem to the directed polymer (DP) model on the Cayley tree, although in a regime dominated by rare events. By studying the Shannon entropy of the probability distribution estimated by the observer, we demonstrate a phase transition distinguishing a chaotic phase with reduced Lyapunov exponent from a strong-measurement phase where uncertainty remains bounded. Remarkably, the location of the MIPT transition coincides with the freezing transition of the DP, although the critical properties differ. We provide an exact, universal scaling function describing the entropy growth in the critical regime. Numerical simulations confirm our theoretical predictions, highlighting a simple yet powerful framework to explore measurement-induced transitions in classical chaotic systems.

Figures

Figures reproduced from arXiv: 2501.00547 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Scaling limit for the discrete model (see Caption of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

85 extracted references · 37 canonical work pages

  1. [1]

    Indeed, a comparison with the numerics performed on the discrete model shows perfect agreement (see Fig. 3). Conclusions —In this Letter, we considered the com- petition between exponential spreading of uncertainty and Bayesian updating of information by repeated mea- sures. We introduced a toy model in terms of the directed FIG. 3. Scaling limit for the ...

  2. [2]

    Preskill, Quantum Computing in the NISQ era and beyond, Quantum2, 79 (2018)

    J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum2, 79 (2018)

    2018

  3. [3]

    J. C. Hoke, M. Ippoliti, E. Rosenberg, D. Abanin, R. Acharya, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, J. C. Bardin, A. Bengts- son, G. Bortoli, A. Bourassa,et al., Measurement- induced entanglement and teleportation on a noisy quan- tum processor, Nature622, 481 (2023)

    2023

  4. [4]

    Nahum, S

    A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X8, 021014 (2018)

    2018

  5. [5]

    Christopoulos, P

    A. Christopoulos, P. Le Doussal, D. Bernard, and A. De Luca, Universal out-of-equilibrium dynamics of 1d critical quantum systems perturbed by noise coupled to energy, Phys. Rev. X13, 011043 (2023)

    2023

  6. [6]

    Hruza and D

    L. Hruza and D. Bernard, Coherent fluctuations in noisy mesoscopic systems, the open quantum ssep, and free probability, Phys. Rev. X13, 011045 (2023)

    2023

  7. [7]

    Bernard, Can the macroscopic fluctuation theory be quantized?, Journal of Physics A: Mathematical and The- oretical54, 433001 (2021)

    D. Bernard, Can the macroscopic fluctuation theory be quantized?, Journal of Physics A: Mathematical and The- oretical54, 433001 (2021)

    work page 2021

  8. [8]

    S. Roy, J. T. Chalker, I. V. Gornyi, and Y. Gefen, Measurement-induced steering of quantum systems, Phys. Rev. Res.2, 033347 (2020)

    2020

Show all 85 references
  1. [9]

    O. Lunt, J. Richter, and A. Pal, Quantum simulation using noisy unitary circuits and measurements, inEn- tanglement in Spin Chains: From Theory to Quantum Technology Applications, edited by A. Bayat, S. Bose, and H. Johannesson (Springer, Cham, 2022) pp. 251–284

    2022

  2. [10]

    Botzung, S

    T. Botzung, S. Diehl, and M. M¨ uller, Engineered dissi- pation induced entanglement transition in quantum spin chains: From logarithmic growth to area law, Phys. Rev. B104, 184422 (2021)

    2021

  3. [11]

    C. M. Caves and G. J. Milburn, Quantum-mechanical model for continuous position measurements, Phys. Rev. A36, 5543 (1987)

    1987

  4. [12]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and ther- modynamics, Advances in Physics65, 239 (2016), https://doi.org/10.1080/00018732.2016.1198134

    2016

  5. [13]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col- loquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys.91, 021001 (2019)

    2019

  6. [14]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X7, 031016 (2017)

    2017

  7. [15]

    A. Chan, A. De Luca, and J. T. Chalker, Solution of a minimal model for many-body quantum chaos, Phys. Rev. X8, 041019 (2018)

    2018

  8. [16]

    Bertini, P

    B. Bertini, P. Kos, and T. c. v. Prosen, Entanglement spreading in a minimal model of maximal many-body quantum chaos, Phys. Rev. X9, 021033 (2019)

    2019

  9. [17]

    Skinner, J

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

    2019

  10. [18]

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

    2018

  11. [19]

    Jian, Y.-Z

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

    2020

  12. [20]

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

    2020

  13. [21]

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

    2019

  14. [22]

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

    2020

  15. [23]

    R. Fan, S. Vijay, A. Vishwanath, and Y.-Z. You, Self- organized error correction in random unitary circuits with measurement, Phys. Rev. B103, 174309 (2021)

    2021

  16. [24]

    Li and M

    Y. Li and M. P. A. Fisher, Statistical mechanics of quan- tum error correcting codes, Phys. Rev. B103, 104306 (2021)

    2021

  17. [25]

    Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys

    G. Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys. Rev. Lett.91, 147902 (2003)

    2003

  18. [26]

    Cecile, H

    G. Cecile, H. L´ oio, and J. De Nardis, Measurement- induced phase transitions by matrix product states scal- ing, Phys. Rev. Res.6, 033220 (2024)

    2024

  19. [27]

    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 readout, Nature Physics19, 1314 (2023)

    2023

  20. [28]

    Czischek, G

    S. Czischek, G. Torlai, S. Ray, R. Islam, and R. G. Melko, Simulating a measurement-induced phase transition for trapped-ion circuits, Phys. Rev. A104, 062405 (2021)

    2021

  21. [29]

    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)

    2022

  22. [30]

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

    2020

  23. [31]

    Ippoliti and V

    M. Ippoliti and V. Khemani, Postselection-free entangle- ment dynamics via spacetime duality, Phys. Rev. Lett. 126, 060501 (2021)

    2021

  24. [32]

    Vasseur, A

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

    2019

  25. [33]

    M´ ezard, G

    M. M´ ezard, G. Parisi, and M. A. Virasoro,Spin glass theory and beyond, Vol. 9 (World Scientific Publishing Company, 1987)

    1987

  26. [34]

    A. J. Bray and M. A. Moore, Replica theory of quantum spin glasses, Journal of Physics C: Solid State Physics 13, L655 (1980)

    1980

  27. [35]

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

    2019

  28. [36]

    Coppola, E

    M. Coppola, E. Tirrito, D. Karevski, and M. Collura, Growth of entanglement entropy under local projective measurements, Phys. Rev. B105, 094303 (2022)

    2022

  29. [37]

    Alberton, M

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

    2021

  30. [38]

    Buchhold, Y

    M. Buchhold, Y. Minoguchi, A. Altland, and S. Diehl, Ef- fective theory for the measurement-induced phase tran- 7 sition of dirac fermions, Phys. Rev. X11, 041004 (2021)

    2021

  31. [39]

    Lucas, L

    M. Lucas, L. Piroli, J. De Nardis, and A. De Luca, Gen- eralized deep thermalization for free fermions, Phys. Rev. A107, 032215 (2023)

    2023

  32. [40]

    Fidkowski, J

    L. Fidkowski, J. Haah, and M. B. Hastings, How Dynam- ical Quantum Memories Forget, Quantum5, 382 (2021)

    2021

  33. [41]

    L´ oio, A

    H. L´ oio, A. De Luca, J. De Nardis, and X. Turkeshi, Purification timescales in monitored fermions, Phys. Rev. B108, L020306 (2023)

    2023

  34. [42]

    Merritt and L

    J. Merritt and L. Fidkowski, Entanglement transitions with free fermions, Phys. Rev. B107, 064303 (2023)

    2023

  35. [43]

    Evers and A

    F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys.80, 1355 (2008)

    2008

  36. [44]

    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)

    2023

  37. [45]

    Poboiko, P

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

    2023

  38. [46]

    S.-K. Jian, C. Liu, X. Chen, B. Swingle, and P. Zhang, Measurement-induced phase transition in the monitored sachdev-ye-kitaev model, Phys. Rev. Lett.127, 140601 (2021)

    2021

  39. [47]

    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)

    2022

  40. [48]

    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)

    2021

  41. [49]

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

    2020

  42. [50]

    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)

    2020

  43. [51]

    Zhang, S.-K

    P. Zhang, S.-K. Jian, C. Liu, and X. Chen, Emergent Replica Conformal Symmetry in Non-Hermitian SYK 2 Chains, Quantum5, 579 (2021)

    2021

  44. [52]

    Nahum, S

    A. Nahum, S. Roy, B. Skinner, and J. Ruhman, Mea- surement and entanglement phase transitions in all-to- all quantum circuits, on quantum trees, and in landau- ginsburg theory, PRX Quantum2, 010352 (2021)

    2021

  45. [53]

    G. S. Bentsen, S. Sahu, and B. Swingle, Measurement- induced purification in large-n hybrid brownian circuits, Physical Review B104, 094304 (2021)

    2021

  46. [54]

    Lopez-Piqueres, B

    J. Lopez-Piqueres, B. Ware, and R. Vasseur, Mean-field entanglement transitions in random tree tensor networks, Phys. Rev. B102, 064202 (2020)

    2020

  47. [55]

    Giachetti and A

    G. Giachetti and A. De Luca, Elusive phase transi- tion in the replica limit of monitored systems (2023), arXiv:2306.12166 [cond-mat.stat-mech]

    2023 arXiv

  48. [56]

    De Luca, C

    A. De Luca, C. Liu, A. Nahum, and T. Zhou, Universality classes for purification in nonunitary quantum processes (2024), arXiv:2312.17744 [cond-mat.stat-mech]

    2024 arXiv

  49. [57]

    Gerbino, P

    F. Gerbino, P. Le Doussal, G. Giachetti, and A. De Luca, A dyson brownian motion model for weak measurements in chaotic quantum systems, Quantum Reports6, 200 (2024)

    2024

  50. [58]

    Tang and W

    Q. Tang and W. Zhu, Measurement-induced phase transi- tion: A case study in the nonintegrable model by density- matrix renormalization group calculations, Phys. Rev. Res.2, 013022 (2020)

    2020

  51. [59]

    Y. Li, R. Vasseur, M. P. A. Fisher, and A. W. W. Ludwig, Statistical mechanics model for clifford random tensor networks and monitored quantum circuits, Phys. Rev. B 109, 174307 (2024)

    2024

  52. [60]

    Turkeshi, Measurement-induced criticality as a data- structure transition, Phys

    X. Turkeshi, Measurement-induced criticality as a data- structure transition, Phys. Rev. B106, 144313 (2022)

    2022

  53. [61]

    Kumar, K

    A. Kumar, K. Aziz, A. Chakraborty, A. W. W. Ludwig, S. Gopalakrishnan, J. H. Pixley, and R. Vasseur, Bound- ary transfer matrix spectrum of measurement-induced transitions, Phys. Rev. B109, 014303 (2024)

    2024

  54. [62]

    Zabalo, M

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

    2022

  55. [63]

    Nahum and K

    A. Nahum and K. J. Wiese, Renormalization group for measurement and entanglement phase transitions, Phys. Rev. B108, 104203 (2023)

    2023

  56. [64]

    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, 104305 (2021)

    2021

  57. [65]

    Willsher, S.-W

    J. Willsher, S.-W. Liu, R. Moessner, and J. Knolle, Measurement-induced phase transition in a chaotic clas- sical many-body system, Phys. Rev. B106, 024305 (2022)

    2022

  58. [66]

    Pizzi, D

    A. Pizzi, D. Malz, A. Nunnenkamp, and J. Knolle, Bridg- ing the gap between classical and quantum many-body information dynamics, Phys. Rev. B106, 214303 (2022)

    2022

  59. [67]

    Jin and D

    T. Jin and D. G. Martin, Kardar-parisi-zhang physics and phase transition in a classical single random walker under continuous measurement, Phys. Rev. Lett.129, 260603 (2022)

    2022

  60. [68]

    S. W. P. Kim and A. Lamacraft, The planted directed polymer: inferring a random walk from noisy images (2024), arXiv:2404.07263 [cond-mat.stat-mech]

    2024 arXiv

  61. [69]

    Derrida and H

    B. Derrida and H. Spohn, Polymers on disordered trees, spin glasses, and traveling waves, Journal of Statistical Physics51, 817 (1988)

    1988

  62. [70]

    Carpentier and P

    D. Carpentier and P. Le Doussal, Glass transition of a particle in a random potential, front selection in nonlin- ear renormalization group, and entropic phenomena in liouville and sinh-gordon models, Physical review E63, 026110 (2001)

    2001

  63. [71]

    In the language of the DP, the chemical potential and inverse temperature are proportional to the mean and to the r.m.s of lnP 1(a)/(KP 0(a)) respectively

  64. [72]

    See Supplemental material for additional details

  65. [73]

    As obtained by linearizing (10) forh t(y)≪1 [68]. Here vKPP = lim t→+∞ 1 t logZ (τ) is the intensive free energy associated to typical polymer paths (i.e.n= 0).v KPP = 0 corresponds to the high-T phase with ¯h(z)∼e −z,α= 0, andσ > √ 2λto the low-T phase with ¯h(z)∼e − √ 2λz/σ ...

  66. [74]

    Halpin-Healy and Y.-C

    T. Halpin-Healy and Y.-C. Zhang, Kinetic roughening phenomena, stochastic growth, directed polymers and all that. aspects of multidisciplinary statistical mechanics, Physics reports254, 215 (1995)

    1995

  67. [75]

    Cometset al.,Directed polymers in random environ- ments(Springer, 2017)

    F. Cometset al.,Directed polymers in random environ- ments(Springer, 2017)

    2017

  68. [76]

    Gardiner,Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, Proceedings in Life Sciences (Springer-Verlag, 1985)

    C. Gardiner,Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, Proceedings in Life Sciences (Springer-Verlag, 1985). 8

    1985

  69. [77]

    9 End Matter Appendix A: One-point contribution to⟨St⟩ We will now compute the one-point contribution to ⟨St⟩, i.e., the second term in the r.h.s

    It is alsoU T (Y) = Prob(min 0≤τ≤T X(τ)>0) for a Wiener process of drift−ξwithX(0) =Y. 9 End Matter Appendix A: One-point contribution to⟨St⟩ We will now compute the one-point contribution to ⟨St⟩, i.e., the second term in the r.h.s. of Eq. (5). This can be rewritten in terms ...

  70. [78]

    the first term in the r.h.s

    Finally, by taking the derivative we find Kτ X j=1 D z(τ) j lnz (τ) j E 0 =τ(D KL(P1 ∥P 0)−lnK).(SA.4) Appendix B: Proof of Eq.(9) We want now to estimate the collective contribution to⟨S t⟩, i.e. the first term in the r.h.s. of Eq. (5). First, we express it in terms ofG τ (y)...

  71. [79]

    ( z(τ) j Z (τ) )# Z (τ) = * F

    We now want to evaluate the latter expression in the large-time limit. Settingq→q/ √τ, only the first two orders inqof ω(q) contribute: asω(q) =−iqv∆t−Var(P 1 ∥P 0)q2/2 + O(q3) we see that, for anyt=τ∆t Uτ (y) = e−(y−vt) 2/(2σ2t) √ 2πσ 2t +O(τ −1),(SC.4) and uτ (y) = Z dy′ √ 2...

  72. [80]

    The front velocity is obtained as follows [65]

    V elocity of the front Consider the continuum model (10). The front velocity is obtained as follows [65]. One insertsh t(y)≃ ¯h(z=y−y t), assuming limt→+∞ d dt yt =v KPP in (10) leading to 0 = σ2 2 ¯h′′(z) + (vKPP +λ+ σ2 2 )¯h′(z) +λF( ¯h(z)) (SI.1) which determines ¯h(z) ifv ...

  73. [81]

    The linearized form of the recursion Eq

    Discrete model Consider now the discrete time model. The linearized form of the recursion Eq. (8) withG τ (y) = 1−h τ (y) reads hτ+1 (y) =K⟨h τ (y−B(a))⟩ 0 ,(SI.4) 14 whereB(a) =−lnK+ ln P1(a) P0(a) . Looking for a front solutionh τ (y) = ¯h(y−cτ) with ¯h(z)∼e −µz we find c=c(...

  74. [82]

    With this choice of the probability distributions our control parameter becomesv=D KL(P1 ∥P 0)−lnK=µ 2/2−ln 2, and the critical value ofµcorresponding tov= 0 is thenµ c = √ 2 ln 2

    Montecarlo dynamics of the particle on the tree First, we carry out simulations of the physical single-particle hopping process on a binary Cayley treeK= 2, choosingP 0 ∼ N(0,1) a Gaussian distribution, andP1 ∼ N(µ,1) a shifted Gaussian averaging toµ. With this choice of the p...

  75. [83]

    Accordingly, we can determine the probability distribution for the measurement outcomesa (τ) simply using the conditional probability rule Eq

    We can pick a random trajectoryx τ uniformly distributed among all those on the tree. Accordingly, we can determine the probability distribution for the measurement outcomesa (τ) simply using the conditional probability rule Eq. (1) of the main text. Since all physical traject...

  76. [84]

    G, we can use Eq

    Alternatively, as explained in the main text and in Sec. G, we can use Eq. (SG.10) to generate the measurement outcomesa (τ) from the known probabilitiesp (τ) and Eq. (2) to consequently update the probabilitiesp (τ) → p(τ+1) themselves. As a benchmark, we show the agreement b...

  77. [85]

    (8) We numerically estimate the behavior of the term⟨ZlnZ⟩ 0 in Eq

    Numerical solution of Eq. (8) We numerically estimate the behavior of the term⟨ZlnZ⟩ 0 in Eq. (5) of the main text, solving numerically the recursive equation (8) for its generating function. More explicitly, forK= 2, we evolve the equation uτ+1 (y) = uτ y+ ln 2−ln P1(a) P0(a)...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.