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REVIEW 3 major objections 4 minor 37 references

OTM work values are exactly conditional averages of TTM work, so coherence hidden in standard two-time measurement data can be recovered by classical post-processing without changing the experiment.

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2026-08-01 19:28 UTC pith:WGAECY22

load-bearing objection The TTM-to-OTM identity is correct and the transition analysis is a genuine fold-catastrophe result, but the unbiased-meter assumption is a calibration requirement and the 'coherence from TTM' claim is oversold. the 3 major comments →

arxiv 2607.16951 v1 pith:WGAECY22 submitted 2026-07-18 quant-ph

Operational Relation Between One-Time and Two-Time Work Protocols and Measurement-Resolution-Induced Transitions in Quantum Work Statistics

Daniel Alonso , Antonia Ruiz-Garc\'ia This is my paper
classification quant-ph
keywords quantum work statisticstwo-time measurement protocolone-time measurement protocolfinite measurement resolutioncoherence extractionstatistical transitionJarzynski relationtwo-level system
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.

The paper claims that the two standard protocols for defining work in quantum thermodynamics — the one-time measurement (OTM) and two-time measurement (TTM) — are not independent: every OTM work value is exactly the conditional average of TTM work over the second measurement outcome. Because TTM is the routine experimental protocol, coherence information that was thought to require OTM or ancilla schemes is already latent in ordinary TTM data and can be exposed by classical post-processing. For a driven two-level system with a finite-resolution Gaussian meter, the paper identifies a resolution-driven statistical transition: when the dynamical factor A1(t) is negative and the meter resolution σ falls below a critical value, the OTM work distribution develops integrable square-root divergent peaks whose separation vanishes near the critical point with universal exponent 3/2. The same dynamical quantities determine the relative entropy of coherence, so coherence can be quantified using only energy measurements. These predictions are shown to be accessible in nitrogen-vacancy center experiments with current parameters.

Core claim

The central claim is Eq. (12): W^(1)(f1,t) = ∫ df2 W^(2)(f2,f1) P_F(f2|f1). In words, the one-time measurement work associated with a first outcome f1 is the average of the two-time measurement work over the distribution of second outcomes f2. This follows because the conditional mean of the second meter outcome is the true energy expectation at time t, provided the meter is unbiased. Consequently, standard TTM runs — which already record pairs (f1,f2) — contain enough statistics to reconstruct OTM work values and, for two-level systems, the relative entropy of coherence of the driven state. For a Gaussian meter and a two-level system with A1(t) < 0, the conditional work function W(t,f) beco

What carries the argument

The load-bearing identity is Eq. (12), which converts TTM data into OTM work values by a conditional average over the second outcome; it relies on the meter being unbiased. The statistical transition is carried by the conditional work function W(t,f), whose extrema coalesce at σc; near such a cubic inflection point, extrema separate as ε^{1/2} and work values as ε^{3/2}, giving the universal scaling. Coherence enters through the populations of the final state, which for a two-level system are fixed by the products A1(t)Δp and μ²(t), both extractable from the work distribution.

Load-bearing premise

The load-bearing premise is that the energy meter is unbiased — for every eigenstate, the mean readout equals its energy eigenvalue; the quantitative exponents additionally assume the meter's response is Gaussian.

What would settle it

Use a meter with a known, controllable offset in its readout (mean outcome μ+δ for an eigenstate of energy μ) and compare direct OTM work values with those reconstructed from TTM via Eq. (12); the two would disagree, exposing the bias assumption. Alternatively, for A1(t) > 0 the work distribution should stay smooth — observing square-root peaks there would refute the transition criterion.

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

If this is right

  • Standard TTM experimental runs already contain OTM work statistics and coherence information; no modification to the measurement apparatus is required.
  • OTM work distributions can be reconstructed offline from TTM histograms by binning f2 and averaging per f1.
  • At intermediate meter resolution, the OTM work distribution of a driven two-level system can develop non-analytic square-root peaks, not just smooth Gaussians.
  • Peak separation near the critical resolution obeys a universal 3/2 power law, giving a parameter-free experimental signature.
  • The relative entropy of coherence in the final energy basis can be extracted from energy measurements alone for two-level systems.

Where Pith is reading between the lines

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

  • Inference: The mechanism — non-monotonic conditional work causing Jacobian singularities — is generic, so multilevel systems should show analogous non-analytic features whenever W(t,f) develops extrema; the exponents should remain 3/2 and 1/2, though the locus of critical points will be richer.
  • Inference: The same post-processing could make TTM data useful for coherence metrology in settings where ancilla-based schemes are impractical; a natural test is to compare coherence inferred from TTM post-processing against direct interferometric measurements.
  • Inference: If a meter has known bias, the discrepancy in Eq. (12) could be used as an in-situ calibration of the meter's response.
  • Inference: The coherence-dependent Jarzynski correction implies that the post-processed data also yields estimates of the coherence contribution to fluctuation relations, which might be used to witness non-equilibrium coherence.

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

3 major / 4 minor

Summary. The paper derives an operational identity between the one-time measurement (OTM) and two-time measurement (TTM) work protocols, Eq. (12), showing that OTM work values can be reconstructed from TTM data by classical post-processing under an unbiased-meter condition. For a driven two-level system with a Gaussian energy meter, the paper then identifies a resolution-driven statistical transition when A_1(t)<0: at σ_c = sqrt(−A_1 μ) the conditional work function W(t,f) becomes multivalued, leading to square-root divergent peaks in P_W(w), with ΔW ~ ε^{3/2} and P_W(w) ~ (σ_c−σ)^{−1/4}|w−w_i|^{−1/2}. The paper further claims that these same work statistics determine the relative entropy of coherence, and it proposes an NV-center experiment where both sides of the transition are observable.

Significance. If correct, the central identity and the transition analysis are significant: they unify two standard work-probability definitions, show that TTM data contain coherence information that is latent but extractable, and provide concrete, measurable signatures with a claimed universal exponent. The derivation of Eq. (12) is transparent, the exact closed-form expression Eq. (36) is consistent with the subsequent expansion, and the numerical verification in Fig. 5 supports the ε^{3/2} scaling. The NV-center parameter estimates make the predictions experimentally relevant. The main weaknesses are that the operational claim relies on an unbiased-meter condition that is not flagged as a calibration requirement, and the coherence-extraction procedure is not fully operationalized.

major comments (3)
  1. [§II, Eqs. (11)–(12)] The central identity W^{(1)}(f_1,t)=∫df_2 W^{(2)}(f_2,f_1)P_F(f_2|f_1) is derived by using ∫df f G_σ(f|μ)=μ for each energy eigenvalue. This is a strong, resolution-dependent calibration condition. If a real meter has a bias b(μ,σ), then ∫df f G_σ(f|μ)=μ+b(μ,σ), and the reconstructed quantity becomes Tr[(Ĥ(t)+B_σ(t))ρ(t|f_1)] − f_1, which is not the OTM work. All downstream results — σ_c, the peak positions, the ε^{3/2} scaling, and the coherence extraction — inherit this error. The manuscript does not state this unbiasedness as a testable requirement nor discuss how to calibrate the meter. The abstract's claim 'without any modification of the experimental setup' is therefore stronger than the derivation supports. Please add an explicit statement of the condition, a calibration/bias-correction discussion, or restrict the scope of the claim.
  2. [§IV.C–D, Eqs. (33) and (39)] The claim that the exponent 3/2 is 'universal' is established only for a Gaussian meter and a thermal two-level initial state. The heuristic argument that the exponent follows from a cubic inflection point is plausible for a generic meter, but it is not a proof. If the universality claim is meant to apply beyond the Gaussian model, a robustness argument or an explicit derivation for a wider class of POVMs is needed; otherwise the statement should be restricted to the Gaussian-meter case analyzed in the paper.
  3. [§V, Eqs. (46)–(48)] The coherence-extraction claim requires determining A_1(t) and Δp from P_W(w). The text says these are 'determined from the work distribution' but does not provide an explicit inversion or identification procedure, nor an analysis of which regimes make this determination possible. In the weak-measurement limit, for example, the first two moments give ⟨w⟩=(A_1+μ)Δp and Var(w)=σ², so A_1 and Δp are not separately identifiable from the Gaussian envelope alone. Please specify the reconstruction protocol and the regimes in which C_H(t) is identifiable.
minor comments (4)
  1. [§V, Eq. (48)] There is a typographical comma: '1/2 (1 + ∆p, A1(t)/µ2(t))' should be '1/2 (1 + ∆p A1(t)/µ2(t))'. Also, the notation µ2(t) for the largest eigenvalue is confusing because µ_2 already denotes an eigenvalue; please use a clearer symbol such as μ_+(t).
  2. [§IV.D] The text contains a typo: 'non-analitic' should be 'non-analytic'.
  3. [§IV.C] The visibility condition for the non-analytic peaks is discussed qualitatively ('the visibility of the peaks depends on the interplay...'), but a quantitative condition or an explicit formula for |T_c| is not given. Since this is used to determine when the peaks are observable, please provide the precise condition or state where it is derived.
  4. [§III, Eq. (15)] The definitions of ΔC and ΔD_KL refer to objects such as ρ̂(t,f) and ρ̂_D(t,f) that are not explicitly defined in the text. Please define these states or point to the exact equation in Ref. [30] where they are introduced.

Circularity Check

0 steps flagged

No significant circularity: the central TTM–OTM identity is derived from stated meter assumptions, and the transition/coherence results are explicit model calculations.

full rationale

The central relation, Eq. (12), is not an input or a fitted prediction. It is derived in Eq. (11) by directly integrating the TTM work over the second measurement outcome and using the spectral decomposition of H(t) together with the explicitly stated unbiased-meter condition ∫ df f G_σ(f|μ)=μ for each eigenvalue μ. The OTM work in Eq. (10) is defined independently as Tr[H(t)ρ(t|f_1)]−f_1, so the equality is an exact algebraic consequence of the stated assumptions, not a definitional renaming. The unbiased-meter condition is a modeling/calibration assumption, and a real meter with bias would indeed break the reconstruction, but that is a correctness/robustness caveat, not circularity. The subsequent transition analysis begins from the explicit two-level expression W(t,f)=A_1(t) tanh(βμ−μf/σ²)−f; the extrema, critical resolution σ_c=√(−A_1μ), and the exponents 3/2 and −1/2 are obtained by direct calculus and local Taylor expansion around a fold, with no parameter fitted to the predicted quantity. The coherence extraction in Sec. V is an inversion of the work–distribution map onto (Δp, A_1), not an assumed input. Self-citations, notably Ref. [30] for the generalized Jarzynski relation, appear only as background and are not load-bearing for the TTM–OTM identity, the statistical transition, or the coherence reconstruction. Under the stated assumptions the paper is self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No free parameters are fitted to data; the model uses experimental inputs (ω, ω1, δ) from Ref. [11] and a variable measurement resolution σ. The main modeling assumptions are the unbiased Gaussian meter and the diagonal initial state; both are stated explicitly in the text.

axioms (6)
  • domain assumption The POVM meter satisfies ∫ df f G_σ(f|μ) = μ for each eigenvalue μ (unbiased meter)
    Used in the derivation of Eq. (11) to equate the conditional average of TTM work to the energy expectation; if the meter is biased, the OTM-TTM reconstruction fails.
  • domain assumption The meter is modeled by a Gaussian pointer (Eq. 22), G_σ(f|μ) = (2πσ²)^{-1/2} exp[-(f-μ)²/(2σ²)]
    This explicit form is used to obtain Eq. (23) and all subsequent quantitative transition results; a different POVM shape would change the fine details, though the qualitative fold structure is generic.
  • domain assumption Initial state is diagonal in the H(0) eigenbasis and represents a (possibly negative-temperature) thermal state (Eq. 18)
    The transition analysis and coherence extraction are derived under this restriction; Eq. (12) itself holds for general initial states.
  • domain assumption The two-level Hamiltonian is traceless, so μ1=-μ, μ2=μ, and μ2(t) is the largest eigenvalue of H(t)
    Used to derive the closed form of W(t,f) in Eq. (23) and the populations in Eq. (48).
  • standard math Unitary evolution preserves the spectrum of the density operator, so S(ρ(t)) = S(ρ(0))
    Needed for the relative entropy of coherence formula (47), where C = S(ρ_D) - S(ρ(0)) = -Σρ_kk ln ρ_kk + Σ p_i ln p_i.
  • domain assumption The NV-center electronic spin can be modeled as a driven two-level system with the Rabi Hamiltonian (49)-(50), with parameters from Ref. [11]
    Underlies the experimental proposal and the numerical figures.

pith-pipeline@v1.3.0-alltime-deepseek · 11889 in / 28464 out tokens · 252606 ms · 2026-08-01T19:28:47.361000+00:00 · methodology

0 comments
read the original abstract

We establish a direct operational connection between the one-time measurement (OTM) and two-time measurement (TTM) protocols for quantum work statistics, showing that OTM work values, including coherence signatures, can be reconstructed from standard TTM data through classical post-processing without any modification of the experimental setup. This reveals that coherence information commonly attributed to OTM schemes is already latent within TTM data and becomes explicit upon a natural post-processing step. For a driven two-level system subject to finite-resolution energy measurements, this reconstruction unveils a resolution-driven statistical transition in the work distribution: whereas projective and weak measurements yield smooth distributions, intermediate resolutions produce non-analytic structures whose character is determined by the dynamics and a critical meter resolution. We show that the non-analyticity of the OTM work distribution takes the form of square-root divergent peaks at specific work values, and that the separation between these singular points is an experimentally accessible quantity that vanishes at the critical resolution with a universal exponent. The same dynamical quantities that govern the transition also encode the relative entropy of coherence of the driven state, enabling coherence quantification from energy measurements alone. We demonstrate experimental accessibility in a nitrogen-vacancy center platform with parameters drawn from current experiments, where both sides of the transition and the coherence signatures are within reach of existing technology.

Figures

Figures reproduced from arXiv: 2607.16951 by Antonia Ruiz-Garc\'ia, Daniel Alonso.

Figure 1
Figure 1. Figure 1: FIG. 1. Function [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Sign of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Sequence of work probability distributions for decreasing values of [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The same as Fig. 3. In panels (a)-(b): [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Peak separation ∆ [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

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Reference graph

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