{"id":"1b3930e2-1d40-4cfc-8bd3-8ef07d91516a","arxiv_id":"2512.11078","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Memory-based quantum-jump feedback is mapped to a Markovian Lindblad equation on an enlarged space, enabling full counting statistics of any counting observable.","lead":"This paper shows that a quantum system steered by feedback based on the last detected quantum jump can be recast as a larger Markovian system, and that standard full-counting-statistics tools then apply. That lets one compute average currents, noise, and power spectra for feedback-controlled quantum thermal machines, demonstrated on a three-level maser.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Classical reference model asserted to match quantum feedback for arbitrary drive strength without proof; invalidates application figures if false, though Result (2) appears sound.","rationale":"I examined the central claim (Result 2) directly. The derivation of the extended Lindblad equation (27) from the feedback master equation (9) is algebraically correct, and the equality of the extended stochastic charge with the system's stochastic charge is a direct consequence of the one-to-one correspondence between system jumps L_k(q) and extended jumps L_{k,q}=L_k(q)⊗|k⟩⟨q|. The instrument-form assumption (M_0(k), M_q(k)) is a clearly stated scope restriction of the protocol; it is not an internal inconsistency and does not constitute a serious objection to the central claim within that scope. The genuinely weak point is the classical reference model introduced in Sec. IV B and again in Sec. IV C. The paper asserts, without proof, that the diagonal classical model with a single incoherent rate γ_c exactly reproduces the populations of the coherently driven quantum maser for all drive strengths. This is a known approximation that is generally valid only in the secular/weak-drive limit. The paper extends it to strong drive (λ≫γ) and uses the agreement with the quantum curves to support the feedback work-extraction results. If the classical model fails in this regime, the quantitative comparisons in Figs. 3–5 are misleading, though the quantum computations themselves remain correct. The reader's CONDITIONAL verdict appropriately flags this missing derivation/validity bound; my analysis agrees that the central framework is sound but the application needs a stated validity range or a proof. Therefore no change to the reader's verdict is warranted.","tokens_in":22093,"tokens_out":16348,"duration_ms":149785,"concrete_test":"Compute the steady-state populations of the full quantum feedback master equation (27) with Hamiltonian (50)–(52) and jump operators (43)–(44) for a strong-drive case (e.g., λ/γ=10, n̄_l=0.3, n̄_r=8, Δ=0) and compare with the steady-state populations of the classical feedback model (Eq. (46) with γ_c transitions conditioned on memory E_r). If the population vectors differ, the classical model does not reproduce the quantum power in that regime; if they match, the unsupported claim is validated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central framework (Result 2) is internally consistent: the extended Lindblad equation (27) is correctly derived from Eq. (9), and the equality Ñ(t)=N(t) follows by construction from the jump operators L_{k,q}=L_k(q)⊗|k⟩⟨q|. The instrument-form assumption (Appendix A) is a stated scope limitation, not a hidden flaw. The genuinely unsupported step is in the application: Sec. IV B 1 and IV C 1 assert that the classical reference model of Eq. (46), with the incoherent rate γ_c = 2λ²Γ/(Δ²+Γ²) of Eq. (47), reproduces the populations of the coherently driven quantum maser for arbitrary drive strengths. This equivalence is a secular/weak-drive approximation in the literature (Ref. [44]); no derivation or validity condition is given here. The paper then claims 'both classical and quantum masers yield the same steady-state power' (Fig. 3a) and uses this to support the feedback work-extraction demonstration. For strong drive (λ≫γ), the quantum populations are coupled to coherences that the diagonal classical rate equation cannot represent, so the claimed agreement is not assured. This does not undermine Result (2) but does call into question the quantitative curves in Figs. 3–5 and the associated conclusions about feedback-enhanced power.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers a general feedback protocol for open quantum systems in which the last detected quantum-jump channel is stored in a memory and subsequently used to condition the system Hamiltonian and/or jump operators. The first main result is that the memory-resolved master equation (Eq. (9)) can be rewritten as a Lindblad master equation (Eq. (27)) for a hybrid classical-quantum state that includes the memory as a classical degree of freedom. The second main result is that, with a suitable choice of counting weights, the extended stochastic charge of the hybrid system coincides with the system's stochastic charge, so that standard full-counting-statistics tools can be applied to jump-based feedback protocols. The framework is applied to a three-level maser coupled to two thermal baths, where a feedback protocol that turns on the drive only after a |2>->|1> emission is shown to suppress refrigeration cycles and yield positive work output, and the authors compute the corresponding power, noise, and power spectrum.","tokens_in":22378,"tokens_out":33587,"duration_ms":343976,"significance":"If correct, this is a useful formal development: it gives a Markovian embedding of a class of non-Markovian feedback protocols and extends full counting statistics to that class. The appendices contain a self-contained derivation of Eq. (9) and of Result (1), and Result (2) follows by construction. The application to a three-level maser is physically interesting and demonstrates the utility of the formalism. The central formal results appear sound; the principal uncertainty lies in the application section's use of a classical reference model.","major_comments":[{"comment":"The classical reference model is asserted to reproduce the populations of the coherently driven quantum maser for arbitrary drive strength. The replacement of a coherent drive by the incoherent rate gamma_c = 2 lambda^2 Gamma/(Delta^2+Gamma^2) is a secular/adiabatic-elimination procedure; the time-dependent statement that both rho_t and sigma_t have the same populations is not generally true, because coherent transients (Rabi oscillations) are not captured by the diagonal rate equation. If the intended claim is only the stationary populations, this should be stated explicitly and justified, either by a derivation or by a precise citation to the relevant proof in Ref. [44]. The application's quantitative comparison of classical and quantum feedback curves relies on this equivalence, so this missing support is load-bearing for the application, although it does not affect the central formal","section":"Sec. IV B 1, Eqs. (46)-(47); Sec. IV C 1; Figs. 3-5"},{"comment":"The two-time correlation function, the noise, and the power spectrum are quoted as standard FCS results without derivation. This is acceptable for specialist readers, but the relation between the extended superoperators tilde J and tilde H and the textbook formulas should be made explicit, in particular the treatment of the delta(tau) term in Eq. (39) and the ordering of the superoperators in the time-ordered correlation. A short derivation or a more precise reference would remove ambiguity.","section":"Sec. III C, Eqs. (39)-(41)"}],"minor_comments":[{"comment":"\"It proofs our second main result\" should read \"It proves our second main result.\"","section":"Eq. (31)"},{"comment":"The phrase \"this classical system is such that both states rho_t and sigma_t have the same populations\" should be qualified as \"in the steady state\" or replaced by a statement about the stationary populations, to avoid the false time-dependent reading.","section":"Sec. IV B 1"},{"comment":"Gamma is called the \"net decoherence rate.\" It would be helpful to state explicitly that this is the dephasing rate of the 0-1 coherence and to explain why emission channels from |2> do not contribute to it; this connects directly to the validity of the classical reference model.","section":"Sec. IV B 1, Eq. (47)"},{"comment":"The inset in Fig. 3(a) is mentioned in the text but is not labeled in the figure; please add a label for clarity.","section":"Fig. 3(a)"},{"comment":"The notation \\bar n is used both for the Bose-Einstein distributions \\bar n_l, \\bar n_r and for the sum \\bar n = \\bar n_l + \\bar n_r in Appendix C. Please introduce a distinct symbol for the sum.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The formal content of the paper is solid: Results (1) and (2) are derived carefully and the Markovian-embedding idea is publishable. The main issue is that the application section's classical-vs-quantum equivalence is both overstated (time-domain claim) and under-supported (no derivation or precise validity condition). Since this equivalence underpins several figures and the stated agreement between classical and quantum steady-state power, I would ask the authors to add a proof or a clearly qualified steady-state statement before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is Result 2: attaching counting weights to the extended Lindblad equation gives you the full counting statistics of any jump-counting observable under memory-based feedback. That is useful and, as far as I can tell, correct. The earlier parts—the memory-resolved master equation (Eq. 9) and the hybrid representation (Result 1)—are reworkings of the group's prior work, but they are re-derived cleanly in Appendices A and B. The derivation of Result 2 is by construction once you accept the instrument form; no circularity, no fitted parameters. The application to a three-level maser demonstrates that the machinery produces closed-form expressions for power, noise, and spectra, and the equations in Appendix C are consistent with the framework.\n\nThe weak point is the application's classical reference model. In Secs. IV B 1 and IV C 1, the paper asserts that the incoherent rate γ_c = 2λ²Γ/(Δ²+Γ²) reproduces the quantum populations for arbitrary drive strength, but no derivation or validity condition is given. This equivalence is a secular/weak-drive approximation in the cited literature (Ref. [44]); for strong drive, the quantum coherences could break the diagonal classical model. Since the quantum/classical curves in Figs. 3–5 all rely on this assertion, the quantitative claims about feedback-enhanced power are on shakier ground than the rest of the paper. I don't see evidence that this undermines Result 2—the extended-space construction and the FCS mapping hold regardless—but the application should be patched: either prove the equivalence under stated conditions or restrict the parameter range.\n\nThe instrument form (Eqs. A4–A5) limits the protocol to jump detection without generalized measurements; the paper acknowledges this by construction, so it is a scope limitation, not a hidden flaw. The FCS formulas (39)–(41) are standard and quoted without derivation, which is acceptable for a methods paper.\n\nBottom line: this is a paper for people working on feedback control in quantum thermodynamics and quantum jump theory. The central methodological contribution is worth a serious referee; send it to review with a request to fix the classical-model equivalence.","headline":"Result 2 gives a clean FCS handle on jump-based feedback; the maser application is undermined by an unproven classical-model equivalence.","tokens_in":22879,"tokens_out":8724,"would_cite":true,"duration_ms":83752,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.65.Yz","05.40.-a","42.50.Lc"],"model":"deepseek-v4-flash","headline":"Jump-based feedback with memory can be rewritten exactly as a Lindblad master equation in an enlarged classical–quantum space, making its full counting statistics computable.","keywords":["quantum jumps","feedback control","full counting statistics","hybrid classical-quantum dynamics","Lindblad master equation","three-level maser","stochastic work","non-Markovian dynamics"],"falsifier":"Construct a jump-monitored protocol in which the no-jump instrument M_0(k) is not of the Lindblad no-jump form—for example, it contains a measurement-induced backaction or depends on the time since the last jump—and compare the exact two-time counting statistics from a trajectory simulation with the predictions of Eq. (27); any disagreement beyond O(δt²) falsifies the claimed equivalence. A simpler target: a protocol where the jump operators depend on the total number of previous jumps of the same channel, not just the last channel, and show the extended Lindblad equation fails to reproduce th","tokens_in":21976,"feed_emoji":"⚛️","tokens_out":6251,"duration_ms":59789,"temperature":0.7,"pith_summary":"The paper sets out to show that feedback protocols which act on the last detected quantum jump—storing that jump in a classical memory and letting it control the Hamiltonian or jump operators—do not need bespoke non-Markovian tools. Its first result is that the joint system–memory state obeys an ordinary Markovian Lindblad equation on a hybrid classical-quantum Hilbert space. Its second result is that, with the natural choice of extended weights, the extended stochastic charge equals the original counting observable, so standard full-counting-statistics formulas give the average current, noise, correlations, and power spectrum under feedback. The authors demonstrate the framework on a three-level maser, where jump-based feedback selects engine cycles and produces positive work even when the unmonitored machine would refrigerate. A sympathetic reader should take the paper as providing a bridge between feedback control and full counting statistics, with analytical handles on thermodynamic quantities that previously required trajectory-by-trajectory treatment.","feed_headline":"Feedback that stores the last jump maps to a Lindblad master equation","feed_subtitle":"Counting currents, noise, correlations, and spectra under jump-based feedback follow from standard formulas.","key_machinery":"The load-bearing object is the hybrid classical-quantum state ρ_sm(t)=Σ_k ϱ_t(k)⊗|k⟩⟨k|, with the classical register |k⟩⟨k| encoding the last detected jump and ϱ_t(k) the memory-resolved, unnormalized system state. The extended jump operators L_{k,q}=L_k(q)⊗|k⟩⟨q| simultaneously implement the quantum jump in channel k and the memory update q→k; this is what converts a non-Markovian feedback dynamics into a Markovian Lindblad equation and makes the counting statistics of the physical jumps identical to those of the extended Markov process.","core_discovery":"The central claim is Result (1) and Result (2). For a feedback protocol in which the last detected jump channel k is stored and controls both the Hamiltonian H(k) and the jump operators L_q(k), the memory-resolved states ϱ_t(k) evolve via Eq. (9); the authors prove that the bipartite density matrix ρ_sm(t)=Σ_k ϱ_t(k)⊗|k⟩⟨k| satisfies the Markovian Lindblad equation (27) with H=Σ_k H(k)⊗|k⟩⟨k| and L_{k,q}=L_k(q)⊗|k⟩⟨q|. Because there is a one-to-one map between jumps in the original system and jumps in the extended space, the extended stochastic charge with weights ν̃_kq=ν_k is exactly the physical stochastic charge N(t)=Σ_k ν_k N_k(t). Hence all full-counting-statistics machinery—current, no","pith_inferences":["The causal-memory update construction in the appendix is not restricted to the last jump; the same hybrid embedding should extend, for example, to protocols that also depend on the time elapsed since the last jump, at the price of a larger classical register—a direction the paper leaves open.","Because the extended process is a genuine Markovian Lindblad dynamics, the framework is a natural setting for deriving feedback-modified fluctuation theorems or thermodynamic uncertainty relations in the extended space; the paper itself does not state these.","A possible testable extension is to apply the same generator-based formulas to feedback that modifies jump operators rather than only Hamiltonians, such as voltage-gated energy-gap control in quantum dots, and compare predicted current noise with photon-counting experiments.","The exact coincidence of extended and physical counting statistics suggests that any protocol whose instruments are of the no-jump/jump Lindblad form can be assigned a Markovian price equal to the dimension of the memory space, which could guide experimental design."],"forward_implications":["Any counting observable of a jump-based feedback protocol—average current, noise, two-point correlations, power spectrum—can be obtained from the extended Lindblad generator using standard full-counting-statistics formulas.","Feedback protocols with last-jump memory are Markovianizable: the memory acts as a finite classical register, so no trajectory ensemble simulation is needed for steady-state statistics.","In the three-level maser, the protocol yields always-positive steady-state power, meaning jump information is converted into work even when the unmonitored machine would refrigerate; the same formulas give the fluctuations of that power.","The framework supplies closed analytical expressions for feedback steady states, populations, and currents, enabling quantitative design of feedback thermal machines.","In the examined regime, feedback reduces the noise of the stochastic work because suppressing refrigeration cycles reduces the number of possible jump trajectories."],"fun_headline_variants":["Last-jump memory feedback becomes a Lindblad master equation","Counting statistics for jump-based feedback via Lindblad dynamics","Stored jump feedback unlocks full counting statistics tools","Feedback with memory: exact Lindblad equation for jump protocols","From last-jump memory to Lindblad: full counting made exact"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The framework assumes that every feedback action can be written in the instrument form M_0(k)=1+δt L_0(k) and M_q(k)=δt J_q(k), with a memory-dependent Lindblad generator; if a protocol uses generalized measurements, delayed responses, or non-Markovian bath memory, the central equivalence embodied in Eq. (9) and Result (1) breaks down.","fun_headline_variants_meta":{"raw":{"variants":["Last-jump memory feedback becomes a Lindblad master equation","Counting statistics for jump-based feedback via Lindblad dynamics","Stored jump feedback unlocks full counting statistics tools","Feedback with memory: exact Lindblad equation for jump protocols","From last-jump memory to Lindblad: full counting made exact"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000448,"raw_usage":{"total_tokens":2109,"prompt_tokens":765,"completion_tokens":1344,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":1264}},"tokens_in":509,"tokens_out":1344,"duration_ms":9259,"temperature":1.0,"reasoning_tokens":1264,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T16:57:00.936296+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a jump-monitored protocol in which the no-jump instrument M_0(k) is not of the Lindblad no-jump form—for example, it contains a measurement-induced backaction or depends on the time since the last jump—and compare the exact two-time counting statistics from a trajectory simulation with the predictions of Eq. (27); any disagreement beyond O(δt²) falsifies the claimed equivalence. A simpler target: a protocol where the jump operators depend on the total number of previous jumps of the same channel, not just the last channel, and show the extended Lindblad equation fails to reproduce th","supporting_citations":[],"review_version":1}