{"id":"cb3641c8-cef3-41e9-b524-944fa1603e80","arxiv_id":"2608.06194","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"First-detection-triggered work extraction satisfies quantum fluctuation relations whose only protocol-dependent correction is the mean first-detection time of the time-reversed dynamics.","lead":"This paper derives two quantum fluctuation relations for a quantum engine that extracts work the first time a repeated measurement detects a target state. The key correction to the standard quantum Jarzynski equality is a logarithmic term involving the mean first-detection time of the time-reversed process.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (20) replaces the post-quench Hamiltonian by ΔH, so the averaged exponential is for W_Δ=ΔH−H0(0), not for the extracted work; as written it does not establish the claimed extracted-work fluctuation relation.","rationale":"The reader located the weakest assumption in the open-system generalization. I partially disagree: the Section V claim for Eq. (18) is plausibly covered by the same algebraic proof, provided the total Hilbert space is finite-dimensional and the projected partition function is finite. The more load-bearing issue is in the second fluctuation relation. The paper's strongest advertised result includes 'extracted work alone,' and Eq. (20) is the only support for the bound (22) and for the abstract's statement that two work fluctuation relations characterize extraction. But the appendix constructs Eq. (20) by the substitution H1→ΔH in the TPM formula, so the exponential average is for the operator ΔH−H0(0), not for the energy lost by the device in the quench. This is not a commutator technicality: in a common eigenbasis the two exponentials differ, and in the noncommuting two-spin model the detection state is not an eigenstate of ΔH. Thus the claimed extracted-work fluctuation relation is not established. The Jensen bound may still be salvageable if Eq. (20) is reinterpreted as a relation for −w'_ex, but the central claim needs revision. This keeps the paper at conditional rather than accept; I do not see a fatal flaw in Eq. (18) or in the first bound.","tokens_in":10148,"tokens_out":38688,"duration_ms":330407,"concrete_test":"For the two-spin model of Fig. 2 (e.g., h=1, J=0.5, β=1, τ=0.5), compute numerically the left side of Eq. (20) as the detection-conditioned average of e^{−β(ΔH−H0(0))} and compare it with the detection-conditioned two-point-measurement average sum_n sum_{k0,k1} e^{−β(ε_{0,k0}−ε_{1,k1}+⟨H0⟩_0)} |⟨ε_{1,k1}|Π G^{n−1} U_τ|ε_{0,k0}⟩|^2 e^{−β ε_{0,k0}}/Z_0, where the final measurement is a projective measurement of H1 after the quench. If the two values disagree, Eq. (20) is not a fluctuation relation for the physical extracted work of the quench; if they agree, the concern is refuted.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central theorem for the isolated case rests on two displayed identities. Eq. (18) is a faithful first-detection analogue of the TPM Jarzynski identity and its algebraic proof in the appendix is internally coherent. The problem is Eq. (20). The appendix says it follows by replacing H1 with ΔH=H1−H0 in Eq. (A6). That replacement changes the final measured observable from the post-quench Hamiltonian H1 to the operator difference H1−H0. The random variable whose exponential is thereby averaged is W_Δ=ε_Δ−ε_0 (eigenvalue of ΔH minus initial H0 energy), not the extracted work of the quench, W_ex=E_0−E_1, nor the per-detection energy decrease −⟨ΔH⟩_det. The detected state is generally not an eigenstate of ΔH; in the paper's two-spin model [ΔH,H0]=[−2hσ_z^b,−Jσ_x^aσ_x^b−hσ_z^a]≠0. In the commuting case, for a common eigenstate with energies ε_0,ε_1, the RHS factor gives e^{−β(ε_1−2ε_0)}, whereas the physical extracted-work exponential e^{−βW_ex} with W_ex=ε_0−ε_1 gives e^{β(ε_1−ε_0)}. These differ, so Eq. (20) is not a fluctuation relation for the extracted work as defined in Eq. (16). The Jensen bound may survive as a bound on w_ex through the relation for −w'_ex, but the second central claim needs either correction or a clear redefinition of the measured fluctuating quantity.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a feedback protocol in which a quantum system evolves unitarily, is projectively measured at intervals τ, and the first detection of a prescribed subspace triggers an instantaneous quench H0→H1. The author defines a detection-conditioned ensemble and derives two integral fluctuation relations. The first, Eq. (18), states that the detection-conditioned average of e^{-βW_tot} equals (Z1/Z0)Σ_n S_n[ρ~1] = e^{-β(ΔF−T log⟨ñ⟩_{ρ~1})}, with S_n the survival probability in the time-reversed first-detection process. The second, Eq. (20), is claimed to be a fluctuation relation for the extracted work, with ΔH=H1−H0 replacing H1. Jensen's inequality yields bounds for w_tot and w_ex. The paper also introduces detection-averaged ergotropy and illustrates the bounds on a two-spin model. An appendix proves Eq. (18) by a trace identity connecting forward work cumulants to backward survival probabilities.","tokens_in":10568,"tokens_out":23194,"duration_ms":198332,"significance":"If Eq. (18) is correct, the paper gives a genuine quantum analogue of the Jarzynski equality for first-detection feedback, with the attractive feature that the information-theoretic correction is the mean first-detection time of the time-reversed process, an experimentally accessible quantity. The appendix trace algebra is compact and internally coherent, and the connection between forward work statistics and backward survival probabilities is elegant. The extracted-work relation, however, has a substantive identification problem that must be fixed before the paper's central claims can be accepted as stated.","major_comments":[{"comment":"Eq. (20) is not a fluctuation relation for the extracted work defined in Eq. (16). The appendix obtains Eq. (20) by replacing H1 with ΔH in Eq. (A6); this changes the final measured observable from H1 to ΔH. The random variable whose exponential is averaged is therefore W_Δ = ε_Δ − ε_0, where ε_Δ is an eigenvalue of ΔH and ε_0 an eigenvalue of H0, not the per-detection extracted work W_ex = ε_0 − ε_1. Even in the commuting case the two differ: for a common eigenstate with energies ε_0 and ε_1, Eq. (20) contains e^{-β(ε_1−2ε_0)}, whereas the extracted-work exponential would be e^{-β(ε_0−ε_1)}. The Jensen bound (22) on w_ex can still be derived from Eq. (20) because the average of W_Δ equals −w_ex − ⟨H0⟩_0, but the manuscript should state this explicitly and should not present Eq. (20) as an extracted-work fluctuation relation. In addition, the text defines the average of the 'shifted extracted work' as w'_ex = w_ex + ⟨H0⟩_0, while the variable actually exponentiated in Eq. (20) has the opposite mean, −w'_ex; this sign inconsistency should be corrected.","section":"§IV, Eq. (20), and Appendix V"},{"comment":"The statement that Eq. (18) 'holds without modification' when the device is coupled to a bath is asserted without proof. The appendix derivation is written for a closed system with a projector on the full Hilbert space; when the measurement projects only on S and the bath is traced out, it is not immediate that the same trace identity survives. The manuscript should either supply the joint-system derivation or explicitly mark this as a conjecture rather than a derived result.","section":"§V (open-system generalization)"},{"comment":"The claim that, when S∞[ρin]>0, the results 'remain unchanged' after normalizing P_n or projecting onto the bright subspace is not demonstrated and appears incompatible with the derivation of Eq. (18), which relies on the cancellation e^{βH0}ρ0 = 1/Z0; after projection this cancellation is lost. Since the main theorems are stated under the assumption S∞[ρin]=0, the caveat should be clarified, proved, or removed.","section":"§II, dark-state caveat"}],"minor_comments":[{"comment":"The displayed equality f_n = (Z1/Z0) S_n[ρ~1] should read S_{n-1}[ρ~1]; otherwise the subsequent sum does not match the index in Eq. (18).","section":"Appendix, Eq. (A11)"},{"comment":"The free energy ΔF in Eq. (18) is defined with the projected partition function Z1 = Tr[Π e^{-βH1}Π], not the full partition function Tr[e^{-βH1}]; this should be emphasized to avoid confusion with the standard free-energy difference.","section":"§IV, after Eq. (17)"},{"comment":"The claim that the ergotropy bound is an upper bound for the other two quantities is stated as expected but not derived; a brief justification would help the reader.","section":"Fig. 2 and surrounding text"}],"recommendation":"major_revision","confidential_remarks":"The algebraic derivation of Eq. (18) is sound and worth publishing, but the extracted-work claim in Eq. (20) is misidentified as a fluctuation relation for the physical extracted work. This is a load-bearing issue for the paper's second central claim, not a cosmetic one, and should be resolved before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the central total-work relation, Eq. (18), is real. The trace algebra in the appendix is coherent, the backward state is supported on the detection subspace, and the correction term -T log <n_tilde> is a clean, experimentally accessible quantity. That alone is a useful addition to quantum thermodynamics. Second, the companion relation for the extracted work, Eq. (20), does not hold as stated. The stress-test note is right: replacing H1 by ΔH changes the measured operator from the post-quench Hamiltonian to H1-H0. The random variable whose exponential is averaged is W_Δ = ε_Δ - ε_0, not the extracted work W_ex = ε_0 - ε_1. In the commuting case the factors differ by e^{-β(ε1-2ε0)} versus e^{β(ε1-ε0)}. The 'shifted extracted work' defined in the text has average w'_ex = wex + <H0>0, but the relation actually concerns the negative of (wex + ε0) on the same trajectory, so both the sign and the shift are off. This is not a minor notation issue; it invalidates the second central claim and the bound in Eq. (22) as written.\n\nWhat is good: the first-detection ensemble and the survival-probability sum are handled carefully; the comparison with Ref. [15] is fair; the dark-state discussion is honest; and the claim that only the mean first-detection time of the reversed process is needed, rather than full trajectory statistics, is genuinely new. The citation pattern is clean and the example illustrates the bound, though the parameters J,h for Fig. 2 are not specified, which makes the numerics hard to reproduce.\n\nSoft spots, in proportion: (i) Eq. (20) is load-bearing; either correct it to a relation for the actual extracted work or explicitly state that it concerns a different quantity. (ii) The open-system generalization in Section V is asserted without proof; Eq. (18) may survive, but Eq. (20) is deferred and should be qualified. (iii) The abstract and introduction overstate the result while Eq. (20) is in its present form. (iv) There are small typos in the appendix, notably the missing Uτ in Eqs. (A1)-(A5), that need cleaning but do not affect the theorem.\n\nVerdict: Eq. (18) deserves publication and citation; Eq. (20) needs a major correction. This paper is for readers working on measurement-feedback thermodynamics and first-detection protocols. A serious editor should send it to peer review, with a clear instruction that the extracted-work relation must be fixed or removed. That is a conditional acceptance, not a reject.","headline":"Eq. (18) is a solid new first-detection Jarzynski relation, but Eq. (20) as written is not a fluctuation relation for the extracted work and needs a major correction.","tokens_in":11007,"tokens_out":16995,"would_cite":true,"duration_ms":131870,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"First-detection time rewrites the quantum Jarzynski equality","keywords":["quantum fluctuation relations","first-detection time","Jarzynski equality","quantum information engine","repeated projective measurements","work extraction","ergotropy","feedback control"],"falsifier":"Run the protocol on a single qubit with $H_0 = -h\\sigma_z$, detection projector $\\Pi = |\\downarrow\\rangle\\langle\\downarrow|$, and final Hamiltonian $H_1 = +h\\sigma_z$, at several sampling intervals $\\tau$ and inverse temperature $\\beta$; measure $\\langle e^{-\\beta W_{\\rm tot}}\\rangle_{\\rm det}$ from the detection-conditioned work statistics and compare it with $(Z_1/Z_0)\\sum_{n=0}^\\infty S_n[\\tilde{\\varrho}_1]$ computed from time-reversed dynamics. A reproducible disagreement beyond statistical error would falsify eq. (18).","tokens_in":9986,"feed_emoji":"⏱️","tokens_out":8979,"duration_ms":72007,"temperature":0.7,"pith_summary":"This paper derives two exact fluctuation relations for a quantum feedback engine whose controller performs repeated projective measurements and stops at the first detection of a specified outcome, then quenches the Hamiltonian to extract work. The first relation, eq. (18), states that the detection-conditioned average of $e^{-\\beta W_{\\rm tot}}$ equals $(Z_1/Z_0)$ times the mean first-detection step of the time-reversed process, which is the standard Jarzynski factor $e^{-\\beta \\Delta F}$ corrected by the entropic term $e^{+\\beta T \\log \\langle \\tilde{n}\\rangle}$. The second relation, eq. (20), is the analogous statement for the extracted work alone. The author shows that Jensen's inequality converts these equalities into second-law-style bounds on the total work and on the extracted work, and illustrates the bounds on a two-spin model where the power output is optimized by tuning the sampling interval. A reader should care because the correction term is experimentally accessible: it requires only the mean detection time of the time-reversed dynamics, not the full statistics of every trajectory.","feed_headline":"First-detection time rewrites the quantum Jarzynski equality","feed_subtitle":"The correction is the mean detection time of the time-reversed process, a single measurable quantity.","key_machinery":"The load-bearing object is the operator $G_\\tau = U_\\tau \\Pi_\\perp$, which evolves the state between null measurements, together with the survival probabilities $S_n$ and the first-detection ensemble. The proof's key step is the identity $f_n = (Z_1/Z_0)S_n[\\tilde{\\varrho}_1]$, which maps forward work contributions to backward survival probabilities through the time-reversal relations $\\theta U_\\tau \\theta^{-1} = \\tilde{U}_\\tau^\\dagger$; summing $S_n$ over $n$ gives the mean detection step $\\langle \\tilde{n}\\rangle$ and converts the series into the compact exponential form of eq. (18).","core_discovery":"The central claim is the exact identity (18): for a system initialised in the thermal state of $H_0$, probed every $\\tau$ by the projector $\\Pi$, and quenched $H_0\\to H_1$ at the first detection event, $\\langle e^{-\\beta W_{\\rm tot}}\\rangle_{\\rm det} = (Z_1/Z_0)\\sum_{n=0}^\\infty S_n[\\tilde{\\varrho}_1] = e^{-\\beta(\\Delta F - T\\log\\langle \\tilde{n}\\rangle_{\\tilde{\\varrho}_1})}$. Here $S_n$ is the survival probability of the time-reversed first-detection process starting from $\\tilde{\\varrho}_1 = \\Pi e^{-\\beta H_1}\\Pi/Z_1$, and $\\langle \\tilde{n}\\rangle$ is its mean first-detection step; the analogous relation (20) holds for the extracted work with $\\Delta H = H_1 - H_0$ in place of $H_1$. The proof rewrites the detection-conditioned work distribution (A2), identifies each term $f_n$ with $(Z_1/Z_0)S_n[\\tilde{\\varrho}_1]$, and uses the telescoping sum $\\sum S_n = \\langle \\tilde{n}\\rangle$. If correct, the work statistics of the forward feedback protocol are fully encoded in the first-detection-time statistics of the backward protocol.","pith_inferences":["One could test eq. (18) on a single qubit by measuring forward detection-click statistics and backward survival probabilities, since the relation predicts the exponential work average without any two-point energy measurement at each step.","The presence of dark states, for which $\\langle \\tilde{n}\\rangle$ diverges, implies parameter regions where the entropic correction makes the work bound arbitrarily loose; a natural check is whether the power output actually degrades near those $\\tau$ values or is rescued by imperfect measurements.","Because $S_n$ is defined through powers of $G_\\tau$, the fluctuation relation ties thermodynamic performance to the spectral properties of a non-normal operator; extending this to continuous monitoring would connect the result to first-passage-time statistics for quantum trajectories."],"forward_implications":["With a trivial detector, $\\Pi = I$, the mean detection step is one and eq. (18) reduces exactly to the standard quantum Jarzynski equality, so the new relation generalises the QJE to first-detection feedback.","Jensen's inequality applied to eq. (18) yields the lower bound $w_{\\rm tot} \\ge \\Delta F - T\\log\\langle \\tilde{n}\\rangle$ on the total work, and applied to eq. (20) yields an upper bound on the extracted work; both bounds contain the same first-detection entropic correction.","Evaluating either bound requires only the mean first-detection step of the time-reversed process, not reconstruction of the full forward and backward trajectory statistics.","For the two-spin example, the extracted-work bound implies a bound on the average power $\\dot{w}_{\\rm ex} = w_{\\rm ex}/(\\tau\\langle n\\rangle)$, and the numerical maxima of the power occur near the minima of the mean detection time.","The paper asserts that eq. (18) survives coupling to a bath unchanged when the measurement and quench act only on the system of interest, whereas eq. (20) is deferred to future work for open systems."],"supporting_citations":[{"why":"Supplies the quantum first-detection formalism: the operator $G_\\tau = U_\\tau \\Pi_\\perp$, first-detection probabilities $P_n$, survival probabilities $S_n$, and the notion of dark states.","marker":"[20]"},{"why":"One of the founding derivations of the quantum Jarzynski equality that eq. (18) generalizes and reduces to when $\\Pi = I$.","marker":"[23]"},{"why":"Provides the two-projective-measurement scheme and quantum fluctuation theorem that define the work PDF used in the proof.","marker":"[24]"},{"why":"Establishes the TPM work distribution and the statement that work is not an observable, justifying the two-time correlation approach.","marker":"[25]"},{"why":"Gives the compact two-time correlation form $\\langle e^{-\\beta H_1}e^{\\beta H_0}\\rangle$ that the paper adapts to the first-detection ensemble.","marker":"[27]"},{"why":"Supplies the first-detection statistics of quantum walks, backing the survival-probability identities and the divergent-detection-time behavior used in the example.","marker":"[28]"},{"why":"Provides the time-reversal operator properties used in the proof to map forward work statistics to backward survival probabilities.","marker":"[29]"}],"fun_headline_variants":["First-detection time modifies quantum Jarzynski equality","Exact quantum work relations from first-detection feedback","Detection-time statistics encode quantum work extraction","Correction to Jarzynski: mean first-detection time","First-detection processes yield new quantum fluctuation relations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes perfectly unitary evolution between the projective measurements; the exact equalities (18) and (20) are proved only for a closed system, and the paper's claim that eq. (18) holds unchanged for an open system is stated without proof.","fun_headline_variants_meta":{"raw":{"variants":["First-detection time modifies quantum Jarzynski equality","Exact quantum work relations from first-detection feedback","Detection-time statistics encode quantum work extraction","Correction to Jarzynski: mean first-detection time","First-detection processes yield new quantum fluctuation relations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1575,"prompt_tokens":922,"completion_tokens":653,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":538,"completion_tokens_details":{"reasoning_tokens":579}},"tokens_in":538,"tokens_out":653,"duration_ms":5941,"temperature":1.0,"reasoning_tokens":579,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:35:13.711087+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the protocol on a single qubit with $H_0 = -h\\sigma_z$, detection projector $\\Pi = |\\downarrow\\rangle\\langle\\downarrow|$, and final Hamiltonian $H_1 = +h\\sigma_z$, at several sampling intervals $\\tau$ and inverse temperature $\\beta$; measure $\\langle e^{-\\beta W_{\\rm tot}}\\rangle_{\\rm det}$ from the detection-conditioned work statistics and compare it with $(Z_1/Z_0)\\sum_{n=0}^\\infty S_n[\\tilde{\\varrho}_1]$ computed from time-reversed dynamics. A reproducible disagreement beyond statistical error would falsify eq. (18).","supporting_citations":[{"cited_title":"Talkner, E","cited_arxiv_id":null,"evidence_quote":"Establishes the TPM work distribution and the statement that work is not an observable, justifying the two-time correlation approach."}],"review_version":2}