{"id":"7e0a10af-4ff0-46ab-906f-dd355a9e735d","arxiv_id":"2607.09855","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Wigner-Fisher information of Gaussian states splits into passive (Wigner-entropy-rate) and ergotropic (displacement/squeezing) contributions that constrain phase-space trajectories and explain the ergotropic Mpemba effect.","lead":"The paper decomposes the Wigner-Fisher information of Gaussian quantum states into passive and ergotropic parts, linking extractable work to statistical geometry in phase space. This explains the ergotropic Mpemba effect as anomalous passive-state evolution and ties thermodynamics to information geometry for continuous-variable systems.","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the Gaussian restriction as the weakest assumption and correctly judges that it does not undermine the claim inside the domain the paper addresses. The mathematics of the decomposition is parameter-free and fully analytic; the Mpemba application is a consistent illustration rather than a necessary pillar. Because no load-bearing flaw appears, the ACCEPT verdict stands.","tokens_in":18013,"tokens_out":409,"duration_ms":5439,"concrete_test":"Independently recompute I_W from the definition (integral of W(\theta)[\theta ln W]^{2}) for a concrete numerical trajectory of a squeezed-displaced thermal state (e.g., the parameters of Fig. 2) and verify that it equals the sum of the closed-form expressions (9)–(12) to machine precision; any discrepancy larger than 10^{-10} would indicate an algebraic error in Appendix B.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the exact analytic split I_W(t)=I_π(t)+I_E(t) for single-mode Gaussian states under the stated dynamics, with I_π=2Ṡ_W^{2} fixed by the passive-state Wigner entropy rate (Eqs. 9–12, App. B). The derivation follows directly from the closed-form WFI (Eq. 2 / App. A), the 2\times2 matrix identity Tr{A^{2}}= (Tr A)^{2}-2 det A, and the fact that det Θ=det Θ_π for any Gaussian state. All algebraic steps check; the Gaussian restriction is stated up front and is precisely the regime in which the Wigner function remains positive and the covariance fully determines the state. No internal inconsistency or hidden assumption that would invalidate the split inside that regime was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript derives a closed-form expression for the Wigner-Fisher information of single-mode Gaussian states (Eq. 2) and shows that it decomposes exactly as I_W(t)=I_π(t)+I_E(t). The passive piece I_π=2 Ṗ_W^{2} is fixed solely by the Wigner-entropy rate of the associated passive state (Eq. 9), while the ergotropic remainder I_E=I_d+I_s+I_ds quantifies the geometric contributions of displacement, squeezing and their cross term (Eqs. 10–12). The derivation rests on the positivity of the Wigner function, the 2\times2 matrix identity Tr{A^{2}}=(Tr A)^{2}-2 det A, and the Lyapunov equation for the covariance under weak-coupling GKLS dynamics. The same framework is used to re-interpret the ergotropic Mpemba effect as an anomalous trajectory of the passive state on the thermal manifold, with supporting analytic expressions and numerical illustrations.","tokens_in":18223,"tokens_out":729,"duration_ms":6020,"significance":"If correct, the work supplies a concrete, parameter-free bridge between ergotropy and information geometry for continuous-variable systems. The analytic split is fully elementary (Appendices A–C) and immediately yields geometric interpretations of statistical length and velocity in terms of extractable work and entropy production. The Mpemba application demonstrates that the decomposition is not merely formal: it isolates the non-monotonic passive-state dynamics that underlie the anomalous discharge. Within the Gaussian regime the results are therefore both theoretically clean and practically useful for quantum-battery and metrology analyses.","major_comments":[],"minor_comments":[{"comment":"In the abstract and introduction the phrase “Wigner entropy rate of the associated passive state” is used; a one-sentence reminder that S_W coincides with the Rényi-2 entropy for Gaussian states would help non-specialist readers.","section":null},{"comment":"Figure 3 panels (c,d) show non-monotonic η_∞,π and V_π for the squeezed passive state; adding a short remark that these quantities are computed from the instantaneous passive occupation (Eq. 18) rather than from a physical trajectory of the master equation would avoid possible misreading.","section":null},{"comment":"Eq. (16) for the pure-displacement length L_d(t) is exact only when the passive state is the equilibrium state; a parenthetical note clarifying this restriction would improve readability.","section":null},{"comment":"A few typographical inconsistencies appear (e.g., “ergotropic” vs. “ergotropy”, missing spaces around “×” in matrix dimensions); a light copy-edit pass would suffice.","section":null}],"recommendation":"accept","confidential_remarks":"The Gaussian restriction is stated clearly and is essential for the closed-form results; the paper does not over-claim generality. The Mpemba application re-uses a previously published effect but supplies a genuinely new geometric explanation. Fit for a Letter-style venue is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The real news here is a clean, fully analytic decomposition: for single-mode Gaussians the Wigner-Fisher information splits exactly as I_W = I_π + I_E, with the passive piece fixed by the Wigner-entropy rate of the associated passive state (I_π = 2Ṡ_W^{2}) and the ergotropic piece collecting displacement, squeezing, and their cross term. That link between extractable work and statistical length/velocity on the phase-space manifold is new, and the appendices actually deliver the algebra.\n\nWhat they do well is keep the math elementary and checkable. Appendix A recovers the known closed-form WFI from the KL expansion of positive Wigner functions; Appendix B uses the 2×2 identity Tr{A^{2}}=(Tr A)^{2}-2 det A plus det Θ=det Θ_π to isolate the passive term; Appendix C ties the ergotropy loss rate to the difference of Wigner entropy productions. The Mpemba application then shows that the anomalous passive-state trajectory (non-monotonic L_π, negative Π_π) is what drives the faster discharge of squeezed charge. Figures are consistent with the formulas, and the Gaussian restriction is stated up front rather than hidden.\n\nSoft spots are real but proportionate. Everything lives inside Gaussian states under Gaussianity-preserving GKLS dynamics; once the Wigner function can go negative or the covariance no longer determines the state, both the closed form and the clean split evaporate. The underlying WFI itself is essentially the SLD QFI for Gaussians, so the novelty is the thermodynamic reading, not a new metric. The Mpemba discussion re-uses their earlier PRL result and is interpretive rather than predictive. None of that breaks the central claim inside the stated regime.\n\nThis is for people already working on continuous-variable quantum batteries, geometric thermodynamics, or open Gaussian systems. They will get a usable dictionary between ergotropy and statistical geometry. It is not required reading outside that circle, but it is serious enough that a referee should see it. I would accept it for peer review and expect light-to-moderate revision mainly on framing and scope.","headline":"Clean analytic split of Wigner-Fisher into passive and ergotropic pieces for Gaussians; solid subfield bridge, not a paradigm shift.","tokens_in":18781,"tokens_out":539,"would_cite":true,"duration_ms":5716,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"The extractable work stored in a Gaussian state splits the phase-space Fisher information into a passive piece fixed by entropy rate and an ergotropic piece fixed by displacement and squeezing.","keywords":["Wigner-Fisher information","ergotropy","Gaussian states","information geometry","passive states","Mpemba effect","continuous-variable quantum thermodynamics"],"falsifier":"Prepare a single-mode Gaussian state that is both displaced and squeezed, measure its Wigner function at successive times under weak thermal damping, reconstruct the empirical Fisher information from the trajectory, and check whether it equals the sum of the analytically predicted passive and ergotropic pieces; any statistically significant mismatch falsifies the decomposition.","tokens_in":18949,"feed_emoji":"⚡","tokens_out":679,"duration_ms":6032,"temperature":0.7,"pith_summary":"This paper shows that the Wigner-Fisher information of a single-mode Gaussian state can be written exactly as the sum of a passive contribution and an ergotropic contribution. The passive term is completely fixed by the rate of change of the Wigner entropy of the associated passive (thermal) state; the ergotropic term records how displacement and squeezing change the statistical velocity and length of the trajectory on the manifold of Wigner functions. Because ergotropy is the maximum work extractable by unitary operations, the decomposition directly links the thermodynamic resource of extractable work to the geometry of the state’s evolution. As a concrete illustration the authors re-examine the ergotropic Mpemba effect and show that the anomalous discharge of squeezed thermal batteries is the geometric signature of a non-monotonic passive-state trajectory that temporarily behaves as if it were non-Markovian. The result therefore supplies a single geometric language in which energy extraction, entropy production and statistical speed can be compared for continuous-variable systems.","feed_headline":"Extractable work splits the Fisher information of Gaussian states","feed_subtitle":"Passive entropy rate plus displacement and squeezing fix the statistical path of a quantum battery","key_machinery":"The ergotropic decomposition of the Wigner-Fisher information: I_W = I_π + I_d + I_s + I_ds, derived from the mean-vector and covariance-matrix formula for the Fisher information and the Lyapunov equation of the open-system dynamics.","core_discovery":"For single-mode Gaussian states the Wigner-Fisher information admits the exact decomposition I_W(t)=I_π(t)+I_E(t), where the passive term I_π(t)=2Ṡ_W^{2}(t) is fixed solely by the Wigner-entropy rate of the associated passive state and the ergotropic remainder I_E=I_d+I_s+I_ds quantifies the geometric effect of displacement and squeezing resources.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Ergotropy and passive entropy split Wigner-Fisher info of Gaussian states","Wigner-Fisher information decomposes into passive rate and ergotropic geometry","Extractable work constrains phase-space statistical paths of Gaussian states","Passive entropy rate plus displacement-squeezing fix Gaussian Fisher length","Ergotropic Mpemba effect arises from anomalous passive-state geometry"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The whole analysis assumes that every state remains Gaussian and that the open dynamics preserves that Gaussianity, so that a positive Wigner function and a two-by-two covariance matrix completely determine both the Fisher information and the passive–ergotropic split.","fun_headline_variants_meta":{"raw":{"variants":["Ergotropy and passive entropy split Wigner-Fisher info of Gaussian states","Wigner-Fisher information decomposes into passive rate and ergotropic geometry","Extractable work constrains phase-space statistical paths of Gaussian states","Passive entropy rate plus displacement-squeezing fix Gaussian Fisher length","Ergotropic Mpemba effect arises from anomalous passive-state geometry"]},"model":"grok-4.5","effort":"low","cost_usd":0.006208,"raw_usage":{"total_tokens":1596,"prompt_tokens":746,"num_sources_used":0,"completion_tokens":97,"cost_in_usd_ticks":62080000,"prompt_tokens_details":{"text_tokens":746,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":753,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":746,"tokens_out":97,"duration_ms":6927,"temperature":1.0,"reasoning_tokens":753,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T15:00:48.351450+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Prepare a single-mode Gaussian state that is both displaced and squeezed, measure its Wigner function at successive times under weak thermal damping, reconstruct the empirical Fisher information from the trajectory, and check whether it equals the sum of the analytically predicted passive and ergotropic pieces; any statistically significant mismatch falsifies the decomposition.","supporting_citations":[],"review_version":1}