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When trajectory-based bounds fail: information thermodynamics under noisy feedback

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read In a noisy feedback engine, the simple instantaneous mutual-information bound becomes the tightest constraint on extracted work once measurement noise grows.

desk verdict A serious experiment-plus-theory paper whose headline crossover is credible but rests on finite-memory entropy estimates that need independent validation. read the letter →

arxiv 2607.27299 v1 pith:JUECWDA5 submitted 2026-07-29 cond-mat.stat-mech cond-mat.mes-hall

Natalia Ruiz-Pino , Ludovic Bellon , Antonio Prados This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.mes-hall PACS 05.70.Ln05.40.-a89.70.Cf
keywords informationthermodynamicsfeedbackcontrolmeasurementnoiseunavailabletransferentropyMarkovianmutualunderdampedoscillatornon-Markovian
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

This paper asks which information-theoretic refinement of the second law remains most useful when feedback control is realistic: measurements are noisy and control decisions are temporally correlated. In an underdamped micro-cantilever engine, the authors compare three bounds on extracted work—transfer entropy, unavailable information, and Markovian mutual information—and show that none is universally optimal. The central finding is that measurement noise degrades trajectory-based bounds (transfer entropy and unavailable information) much faster than the instantaneous Markovian mutual-information bound, so the Markovian bound becomes the tightest over a broad noise range. The paper also shows that the sign change of the Markovian bound tracks the noise level at which work extraction ceases, and develops finite-memory estimators that make non-Markovian information rates experimentally accessible. If correct, it means the thermodynamic value of a given information measure is not intrinsic but depends on how noise degrades the trajectory statistics it relies on.

What carries the argument

The argument is carried by three information-theoretic bounds on extracted work per cycle: the transfer-entropy rate I_c, the unavailable-information correction I_u − I, and the Markovian mutual-information term −Δ_m I. The first two depend on the full control sequence (trajectory-based); the third depends only on instantaneous pre- and post-measurement correlations. The load-bearing theoretical result is the inequality −Δ_m I ≥ −I_c for both perfect and imperfect measurements (Eq. 10), plus the numerically and experimentally established crossover between I_u − I and −Δ_m I. Estimating the trajectory-based rates requires finite-memory estimators, which plateau once the memory length exceeds

What would settle it

Compute or measure the three bounds under the same noise model but with a measurement variable that is not the control decision (e.g., a threshold on a combination of position and velocity, or a multi-level output). If the unavailable-information bound remains tighter than the Markovian mutual-information bound at high noise, the predicted crossover is specific to y=c rather than a general property of trajectory-based bounds.

Watch

Extended reading notes

Core claim

On the paper's own terms: in an underdamped feedback engine with Markovian measurements and non-Markovian control, the three standard second-law refinements—transfer entropy, unavailable information, and Markovian mutual information—are all valid inequalities but have no universal ordering. For perfect measurements the unavailable-information bound is the tightest, but as measurement noise grows the trajectory-dependent unavailable-information bound degrades fastest, and the instantaneous Markovian mutual-information bound becomes the tightest over a broad range of noise. The transfer-entropy bound always remains looser than the Markovian one, while the unavailable-information bound can even

Load-bearing premise

The paper's headline generalization rests on the minimal model in which the measurement outcome is identified with the control variable (y=c); the authors note that another coarse-graining could quantitatively change the information values, so the crossover could in principle be an artifact of that choice.

Editorial extensions

If this is right

  • In noisy regimes, the simpler instantaneous mutual-information bound is tighter than the unavailable-information bound, so information engines need not be analyzed with full trajectory statistics.
  • The unavailable-information bound, exactly tight for idealized perfect measurements in overdamped systems, does not survive under measurement noise as the tightest constraint.
  • The sign of the Markovian mutual-information bound correctly locates the noise threshold at which work extraction stops, giving an experimentally useful diagnostic.
  • Trajectory-dependent bounds (transfer entropy and unavailable information) require increasingly detailed statistics that measurement noise selectively degrades; no single information measure is universally optimal.
  • Finite-memory estimators allow direct experimental estimation of entropy and unavailable-information rates in non-Markovian control sequences, enabling comparisons that were previously out of reach.

Reading between the lines

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

  • If the mechanism is generic, the same crossover should appear for other measurement variables and noise models; a direct test would be to compute the bounds with a different binary measurement (e.g., based on velocity or a threshold other than the origin) and check whether the unavailable-information bound remains the tightest at high noise.
  • The result suggests that feedback protocols designed to exploit temporal correlations should be evaluated with noise-aware information measures; optimizing for zero-noise tightness may be misleading.
  • The finite-memory plateau method could be transferred to other experimental systems with non-Markovian feedback, including biological sensors, to quantify how much trajectory information survives noise.
  • A possible extension: for non-uniform noise distributions or multi-state measurements, the crossover noise level may shift; mapping the crossover as a function of noise model would test the claimed generality.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies an underdamped feedback-controlled micro-cantilever with binary position measurements, where the measured outcome directly sets the control variable (c = ±1). It compares three second-law refinements — the transfer-entropy bound, the unavailable-information bound, and the Markovian mutual-information bound — as functions of measurement noise Δx and sampling interval Δt_m, using both experimental data and Kramers-equation-based theory. The central claim is that none of the three bounds is universally optimal: at low noise the unavailable-information bound is tightest, but as measurement noise increases, trajectory-dependent bounds (especially unavailable information) degrade faster than the instantaneous Markovian bound, producing a crossover in which the Markovian bound becomes tightest over a broad noise range. The paper interprets this as evidence for a general limitation of trajectory-statistics-based information-thermodynamic descriptions in realistic noisy feedback.

Significance. If the central claim is correct, the paper makes a valuable contribution: it shows that the tightness of thermodynamic information bounds is not an intrinsic property of the information measure but depends on how measurement noise interacts with temporal correlations. It also provides a methodological advance by estimating non-Markovian entropy and unavailable-information rates directly from experimental data, and it gives analytic results in the long-sampling-time limit. A notable strength is that the theoretical curves in Fig. 1 are computed from Kramers-equation solutions with parameters fixed by the experimental setup, with no free parameter fitted to the measured work. The conclusions are of broad interest to stochastic thermodynamics and experimental information engines. However, the headline crossover depends on the accuracy of finite-memory entropy-rate estimators, and the generality of the claim rests on a single coarse-graining choice; both points need additional support before the central claim is fully established.

major comments (3)
  1. [Section I, "minimal model" with y = c; Section V Discussion] The crossover in Fig. 1 relies on the plateau values of H_M and J_M being accurate estimators of the true entropy rates H and J. For a fixed M, plug-in conditional-entropy estimators have downward finite-sample bias, while the true conditional entropy decreases as M increases; these two opposing biases can cancel and produce a plateau at a value that is not the true H. If H is overestimated, then I_u - I = J - H is underestimated (too negative), making the unavailable-information bound appear looser exactly in the noise regime where the crossover is claimed. The theoretical solid lines in Fig. 1 are computed with the same finite-memory estimators, so they do not independently validate the plateau. For the shortest sampling interval (Δt_m/τ_rel = 0.22), the usable memory is particularly limited and truncation bias is most dangerous. The manuscript should provide an independent validation
  2. [Section II, Methods, "Finite-memory estimators"; Fig. 6] The abstract and Discussion claim a 'general limitation' of trajectory-based information-theoretic descriptions under measurement noise. However, all quantitative results are computed within the minimal model in which the measurement outcome is identified with the control variable, y = c. The Discussion itself concedes that other coarse-grainings 'could quantitatively affect the values of the information measures.' The generality is therefore asserted from a mechanism argument, not demonstrated. To support the headline claim, the paper should either test at least one alternative measurement variable or coarse-graining (e.g., a multi-threshold or continuous measurement variable) and show that the same qualitative crossover appears, or provide an analytic argument independent of the specific choice of y. Absent that, the conclusion should be restricted to the minimal model or explicitly fr
  3. [Methods, "Finite-memory estimators", Eq. (32); Section IV A] The finite-memory approximation of the backward process, Eq. (31), conditions on the time-reversed forward sequence only through a memory window of length M+1. The unavailable-information estimator J_M is therefore sensitive not only to the forward-memory truncation but also to the initial-condition prescription for the backward process, P(Γ_1^R | c†_k) = P(Γ_k | c_k). The text states that backward trajectories were generated for each forward trajectory, but it is not explained how finite-length forward trajectories affect the convergence of J_M to its asymptotic rate. If boundary or initial-condition effects decay more slowly than the memory window, the plateau in J_M could be systematically biased. The authors should provide a boundary-effect analysis or demonstrate that J_M is insensitive to the backward initial condition.
minor comments (5)
  1. [Fig. 6 caption] Please specify the number of feedback cycles used for each experimental point, the meaning of the error bars, and the normalization of Δx (in units of σ). The statement that error bars are invisible is not sufficient; a representative error bar or an inset would help the reader assess statistical accuracy.
  2. [Methods, "Finite-memory estimators", Eq. (32)] Please define M_max explicitly and state the criterion used to mark points as 'crossed out' (insufficient statistics). Also report the sample size N in the caption or in the Methods, since the plateau identification is central to the claims.
  3. [Section II, Eqs. (4) and (7)] The overline notation is used both for per-cycle averages and for time-reversed sequences (⃗ y†). This is occasionally confusing, for example in Eqs. (4) and (7). A different symbol for time reversal would improve readability.
  4. [References/Footnotes] The footnote defining singular measurements appears after the reference list; it should be placed as a proper footnote in the main text or integrated into the Methods.
  5. [Data availability] The statement 'Data supporting this study will be available in an open public repository upon acceptance' is acceptable for a journal submission, but given the central role of the experimental estimators, a statement about code availability for the estimators (e.g., for H_M and J_M) would strengthen reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the three information bounds are evaluated from measured control/position sequences and from Kramers-equation solutions with parameters fixed by the setup, without fitting to the extracted work; the central crossover is a computed comparison, not an input.

full rationale

The paper's central claim is the relative behavior of three second-law refinements under measurement noise. The bounds in Eq. (7) are not fitted to the measured work: the Markovian mutual information is computed from stationary Kramers solutions and measured pre-/post-measurement distributions, while transfer-entropy and unavailable-information rates are estimated from recorded control sequences and backward experiments. The crossover in Fig. 1 therefore emerges from direct evaluation of the quantities appearing in the bounds, and no parameter is tuned to force agreement with βW. The minimal model y=c is an explicit modeling choice, not a result derived from itself; the Discussion's claim of generality is an extrapolation, but the quantitative crossover is demonstrated within the stated model. Finite-memory estimation could introduce statistical bias, but that is a correctness/estimation concern, not an equivalence-by-construction between input and output. Self-citations to Refs. [30,33] are used for auxiliary ordering relations and motivation, not for the unavailable-information-vs-Markovian crossover, which the paper explicitly states is not covered by any known ordering. No self-definitional step, fitted-input-as-prediction, or author-imported uniqueness theorem carries the central derivation.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

Physical parameters (gamma = 1/Q_f ~ 1/7, L = 1, Delta_x, Delta_t_m) are controlled by the apparatus, not fitted to the measured work, so the theoretical curves carry no explicit fit freedom for the crossover. The genuine free choices are estimation hyperparameters (memory M, kNN k, regularisation constants): they can bias the reported H, J, Delta_m I but cannot manufacture the crossover, which also appears in the long-sampling Markovian limit with analytic expressions (Eqs. 37-39). No invented entities are introduced. The main axioms are the standard second-law refinements from the cited literature and the flat-noise measurement model; the underdamped extensions of the theorems are asserted in the main text and deferred to the SI.

free parameters (3)
  • Memory length M (estimator truncation) = probed M=1..9; plateau regions shown in Fig. 6, not quantified numerically
    H and J are read off the plateaus of H_M and J_M; the truncation and the exclusion of M >= M_max points (crossed out in Fig. 6) set the central estimate of the unavailable-information bound.
  • k (kNN estimator parameter) = 3
    Standard compromise between local resolution and statistical stability for the mixed discrete-continuous mutual information estimator (Methods).
  • Bootstrap and regularisation parameters = not specified in the main text
    Regularised plug-in estimators [39,40] and bootstrap resampling are used for H and J; the actual parameter values are deferred to the SI.
assumptions (7)
  • domain assumption Cantilever = single underdamped harmonic mode obeying Kramers dynamics (Eqs. 11-14) with harmonic potential V = 0.5(x-cL)^2
    Underlies all theoretical predictions (solid lines in Fig. 1); standard for this platform and supported by prior experiments [16,31].
  • domain assumption Flat measurement noise: c_n = sgn(x_n + eta_n), eta uniform on [-Delta_x, Delta_x], giving Theta of Eq. (18)
    Defines the noise model used for all information measures; matches the implemented injection. A different noise statistic would change the quantitative crossover.
  • domain assumption The three second-law refinements (Eqs. 3, 4, 5) remain valid for underdamped dynamics with non-Markovian control sequences
    Taken from Refs [22,26,27,30]; the underdamped extensions are asserted, with proofs deferred to the absent SI (Section II).
  • domain assumption Ordering beta W >= -Delta_m I >= -I_c (Eq. 10) holds underdamped
    Load-bearing for the claim that the Markovian bound is always tighter than transfer entropy; overdamped proof in authors' Ref [30], underdamped proof only in SI.
  • domain assumption Control-sequence correlations are finite-ranged, memory proportional to tau_rel/Delta_t_m, so H_M and J_M plateau for M <= M_max
    Needed to compute H and J from finite samples (Fig. 6); if correlations outlived the accessible memory, the unavailable-information bound would be biased.
  • domain assumption Long-time periodic state P(Gamma, c, t+Delta_t_m) = P(Gamma, c, t) with stationary per-measurement distributions
    Justifies per-cycle rates; supported by Fig. 4 and standard for periodically repeated feedback.
  • domain assumption Backward-process initial condition P^R(Gamma^R_1 | c-dagger) = P(Gamma_k | c_k) correctly implements the Potts unavailable information
    A wrong backward ensemble would directly bias I_u (Eq. 4); the main text states the prescription but gives no cross-validation beyond the long-sampling analytic limit.

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Pith. "Pith review of When trajectory-based bounds fail: information thermodynamics under noisy feedback." pith.science (2026). https://pith.science/paper/JUECWDA5

@misc{pith2026260727299,
  author       = {Pith},
  title        = {Pith review of: When trajectory-based bounds fail: information thermodynamics under noisy feedback},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JUECWDA5}},
  note         = {Machine review of arXiv:2607.27299}
}
read the original abstract

Information engines exploit feedback to extract work from thermal fluctuations, extending the second law of thermodynamics through information-theoretic bounds. While several such bounds have been proposed, their relative performance under realistic conditions---where measurements are noisy and feedback is temporally correlated---remains largely unclear. Here, we experimentally and theoretically investigate this problem in an underdamped feedback-controlled system with Markovian measurements but non-Markovian control sequences. We compare three representative bounds derived from transfer entropy, unavailable information, and Markovian mutual information, and find that none is universally optimal. Instead, measurement noise preferentially affects information measures that rely on detailed trajectory statistics, while leaving quantities based on instantaneous correlations comparatively robust. As a consequence, trajectory-dependent bounds deteriorate rapidly, giving rise to a crossover in which the Markovian mutual-information bound becomes tighter than the unavailable-information bound over a broad range of measurement noise. Our results reveal a general limitation of information-theoretic descriptions that rely on detailed trajectory statistics in realistic settings and provide a unified perspective on information thermodynamics beyond idealised feedback protocols.

Figures

Figures reproduced from arXiv: 2607.27299 by the authors.

Figure 1
Figure 1. FIG. 1. Comparison between the extracted work and the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Schematic representation of perfect and imperfect [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Forward and backward processes for a feedback-controlled information engine, illustrated for an overdamped particle [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Position probability distributions immediately be [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic representation of the forward and back [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Convergence of the finite-memory estimators for [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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