{"id":"3105b0f8-e9bb-421a-a17d-6124fb9221d6","arxiv_id":"2411.13761","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"Shannon permutation entropy and two statistical complexity measures reveal three regimes, quasi-quantum, mesoscopic, and classical, in a semiclassical oscillator's approach to the classical limit, sharpening earlier Poincaré-section results.","lead":"This paper maps how a small quantum system coupled to a classical environment becomes classical, using order-pattern statistics of its time series to split the journey into three stages. A generalist reader might care because the method localizes a mesoscopic crossover region that standard phase-space plots blur, in both conservative and dissipative settings.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The three-zone boundaries are computed from one branch of the constant-E, constant-I manifold; because I and E do not fix the initial state, the reported classical-limit values and Er intervals may be artifacts of that branch choice.","rationale":"Reader's weakest_assumption is exactly this initial-condition representativeness, and I agree it is the load-bearing point. The paper's Er axis is constructed by varying I along a single curve through the constant-E phase space; the curve is not derived from any ergodic or typicality argument. Because the term ecl_q A²x² couples the reservoir amplitude to the x-variance, different ways of satisfying the same invariants produce different trajectories and hence different permutation distributions, so the zone boundaries are not obviously functions of Er alone. The proposed check is decisive: if the complementary branch reproduces the same boundaries and classical limits, the concern is resolved; if not, the central claim must be restricted to the specific initial-condition family. I do not see a more fundamental flaw: the Bandt-Pompe implementation appears standard, d=5/τ=1 is reasonable for N=20000, and the qualitative three-zone picture is consistent with the Poincaré sections. Hence the verdict stays conditional, unchanged from the reader.","tokens_in":11775,"tokens_out":9565,"duration_ms":89630,"concrete_test":"Re-run Figs. 1 and 2 for the complementary branch x²(0)=E/ωq+0.98√((E/ωq)^2−I), p²(0)=I/x²(0), re-deriving PA(0) from energy conservation, with A(0)=L(0)=0, d=5, τ=1, N=20000, η=0 and η=0.05. Also run one L(0)≠0 state per I with the same E and I. If the three-zone boundaries (2.8/104.8 and 3.6/123.0) and the classical-limit values (H_cl_c=0.17676, H_cl_d=0.17207, etc.) move by more than the small-band variation visible in the insets, the boundaries are artifacts of the branch choice rather than properties of the I→0 transition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that H, C_JS, and C_LMC in Er reveal universal zones with boundaries 2.8<Er<104.8 (conservative) and 3.6<Er<123.0 (dissipative). Section 5 fixes E=0.6, A(0)=L(0)=0 and chooses ⟨x²⟩(0)=E/ωq−0.98√((E/ωq)^2−I), with ⟨p²⟩(0)=I/⟨x²⟩(0), and PA(0) determined by energy conservation. For fixed I (i.e., fixed Er) the initial conditions are not unique: the same E and I allow the complementary branch ⟨x²⟩(0)=E/ωq+0.98√((E/ωq)^2−I), and a continuum of L(0)≠0 states. In the I→0 limit the degeneracy is even starker: I=0 only requires x²p²−L²/4=0, so E can be partitioned between x² and the A-reservoir in infinitely many ways (e.g., x²→0.012, p²→0, PA²→1.188 vs. x²→1.188, p²→0, PA²→0.012 for E=0.6, ωq=1). The classical targets H_cl and C_cl are computed from one particular partition, and the nonlinear interaction is ecl_q A²x², so the dynamics and ordinal statistics depend on how the variance is apportioned. Nothing in Eqs. (2)-(6) shows that the three-zone structure or its boundary values is invariant under this choice. The reported precision (e.g., 104.8, 123.0) is therefore not established for the classical transition as an Er-parametrized phenomenon.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper investigates the classical transition in a nonlinear semiclassical system described by a Hamiltonian with quantum operators (x,p) and classical variables (A,PA), with a coupling term ecl_q A^2 x^2. The authors compute the normalized permutation entropy H, the LMC statistical complexity, and the Jensen-Shannon statistical complexity from time series of ⟨x^2⟩(t) using the Bandt-Pompe method, for both conservative (η=0) and dissipative (η=0.05) regimes. They define a relative energy Er = |E|/(I^{1/2}ωq) based on the motion invariant I, which measures the deviation from classicality, and study the limit Er→∞ (I→0). They report three zones in Er: a quasi-quantum zone (Er≈1–2.8 conservative), a transitional/mesoscopic zone (2.8<Er<104.8 conservative; 3.6<Er<123.0 dissipative), and a classical zone where the quantifiers converge to values computed from the classical analog system. They conclude that these information quantifiers provide more precise characterization of the classical transition than the Poincaré sections of Ref. [30].","tokens_in":12161,"tokens_out":6644,"duration_ms":58516,"significance":"If the reported three-zone structure is robust, the paper offers a concrete demonstration that permutation-based information quantifiers can serve as sensitive probes of the semiclassical-to-classical crossover, complementing dynamical methods. Strengths include the explicit specification of the model and initial conditions, the use of the established ordpy package, and the parallel treatment of conservative and dissipative regimes. The convergence to the classical analog values is visually coherent and the qualitative distinction between zones is consistent across the three quantifiers. However, because the central quantitative claims (zone boundaries and classical-limit values) are tied to a single, non-unique family of initial states, the significance is provisional pending sensitivity analysis.","major_comments":[{"comment":"The paper selects ⟨x^2⟩(0)=E/ωq−0.98√((E/ωq)^2−I) with ⟨L⟩(0)=A(0)=0. For fixed E and I, this does not uniquely specify the initial state: one may choose the complementary branch with the plus sign, or states with L(0)≠0, and for I=0 there is a continuum of partitions of E between x^2 and the A-reservoir satisfying I=0 and energy conservation. The dynamics depend on this partition through the coupling term ecl_q A^2⟨x^2⟩. Consequently, the reported boundaries (2.8<Er<104.8, 3.6<Er<123.0) and the classical-limit values H_cl, C_cl may be artifacts of the particular branch chosen. The paper does not provide any invariance argument or numerical evidence that the three-zone structure is independent of this choice. Please test the sensitivity of the curves and boundary values to: (i) the plus branch, (ii) the coefficient 0.98 varied over a range, (iii) nonzero L(0), and (iv) the sign of PA(0). If the zones shift or disappear, the central claim needs to be reformulated.","section":"Section 5, Eqs. (2)–(6)"},{"comment":"The boundaries are identified visually from single curves with no quantitative criterion, no error bars, and no bootstrap over initial conditions or time-window choices. Given that the curves are nearly flat over large Er intervals, small numerical noise or finite-length effects could shift the apparent \"jump\" points. The reported precision (e.g., 104.8, 123.0) is not supported without a defined threshold or a statistical measure of the transition. Please provide a reproducible criterion (e.g., crossing of a tolerance band around the classical value, or a derivative-based change point) and report the resulting uncertainty.","section":"Figs. 1 and 2; Section 5"}],"minor_comments":[{"comment":"There is a typographical comma in \"(ωq + ecl_q , A^2) x^2\"; it should read \"(ωq + ecl_q A^2) x^2\".","section":"Eq. (4), third line"},{"comment":"\"Er0\" should be \"Er\" for consistency with the rest of the text.","section":"Fig. 3 caption"},{"comment":"\"Smax = log(nπ) y 0 ≤ H ≤ 1\" contains a Spanish \"y\"; it should be \"and\".","section":"Section 4.1"},{"comment":"The paper uses both \"mesoscopic\" and \"mezoscopic\"; please standardize the spelling.","section":"Throughout"},{"comment":"The choice of the factor 0.98 in the initial condition is not justified; if no derivation is available, state explicitly that it is an arbitrary but fixed choice and discuss its role in the results.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is largely a follow-up to the authors' own earlier work (Refs. [24–30]), and the novelty rests on the application of permutation quantifiers to this specific model. The main technical risk is the initial-condition degeneracy described in major comment 1; if the sensitivity analysis shows strong dependence, the paper's quantitative conclusions would not be supported. The paper is likely within scope for a physics journal but would benefit from a more critical treatment of its numerical claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a competent follow-up to the authors' Ref. [30]. The genuinely new part is the application of Bandt-Pompe permutation entropy and LMC/JS statistical complexity to this particular semiclassical Hamiltonian, in both conservative and dissipative regimes. That is worth doing: the quantifiers do separate the quasi-quantum, transitional, and classical zones more cleanly than the Poincaré sections did, and the figures support that qualitative claim. The setup is transparent—equations, parameters, initial conditions, and the ordpy settings are all specified—and the d=6 check is a reasonable robustness gesture.\n\nThe soft spots are about the precision of the headline numbers. The quoted boundaries (2.8, 104.8, 123.0) are read from single trajectories with no error bars, no bootstrap, and no sensitivity analysis in N, d, τ, or integration time. The classical benchmark values are computed from one reference trajectory; that is not circular, but the boundaries are inferred from the same curves that define them, so the precision is overstated.\n\nThere is also a more substantive gap, and it is the one the stress test flags. Section 5 fixes E=0.6 and chooses one initial-condition family: L(0)=A(0)=0 and ⟨x²⟩(0)=E/ωq − 0.98√((E/ωq)²−I). For fixed I and E, the state is not unique. The complementary plus-sign branch exists and gives different x² for the same I, and L(0)=0 is a choice rather than a consequence. As I→0, E can be split between x² and the A-reservoir in infinitely many ways, and since the interaction is ecl_q A²x², the ordinal statistics should depend on that split. The paper offers no argument that the three-zone structure or the specific Er boundaries are invariant under that choice. That is not a fatal objection—the zones may well be robust—but it means the central quantitative claim is not yet backed.\n\nWho is this for? People using permutation quantifiers in semiclassical or regular dynamical systems, and the authors' own line of work. It is a solid application study, not a methodological advance. I would send it to peer review and ask for revisions: add uncertainty quantification on the boundaries, and test at least one other initial-condition branch to show the zones are not an artifact of the chosen path. If that check fails, the paper becomes a much weaker claim.","headline":"A clean follow-up application of permutation quantifiers to a specific semiclassical model; the three-zone structure is plausible, but the quoted boundaries need uncertainty quantification and an initial-condition invariance check before they can be taken literally.","tokens_in":12724,"tokens_out":3123,"would_cite":false,"duration_ms":29904,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Permutation entropy and statistical complexity divide the semiclassical transition into three sharply delimited energy zones, locating boundaries more precisely than Poincaré sections.","keywords":["semiclassical systems","classical limit","Bandt-Pompe permutation entropy","statistical complexity","LMC complexity","Jensen-Shannon complexity","dissipative dynamics","relative energy"],"falsifier":"Take the same model and the same three quantifiers, but scan to the classical limit along a different one-parameter family of initial conditions that still satisfies $I\\to0$—for example, starting with nonzero $\\langle\\hat{L}\\rangle(0)$ or redistributing the initial energy between $\\langle\\hat{x}^2\\rangle$ and $\\langle\\hat{p}^2\\rangle$—and check whether the zone boundaries in $E_r$ remain at 2.8 and 104.8 (conservative) and 3.6 and 123.0 (dissipative).","tokens_in":11553,"feed_emoji":"⚛️","tokens_out":11775,"duration_ms":79845,"temperature":0.7,"pith_summary":"This paper studies a semiclassical model in which a quantum oscillator interacts nonlinearly with two classical variables, and asks how its dynamics becomes classical as a motion invariant linked to the uncertainty principle shrinks to zero. The authors compute three information-theoretic measures—permutation entropy and two statistical complexities—from the time series $\\langle \\hat{x}^2\\rangle(t)$ using the Bandt–Pompe ordinal-pattern method, in both conservative and dissipative regimes. All three quantifiers partition the classicality parameter $E_r$ into the same three regions: a quasi-quantum zone, a transitional (mesoscopic) zone, and a convergence zone where the semiclassical values reach the classical analog values. The paper's claim is that these information quantifiers locate the zone boundaries more precisely than the Poincaré sections used in earlier work.","feed_headline":"Three zones mark the quantum-to-classical transition","feed_subtitle":"Permutation entropy and statistical complexity place the same boundaries at relative energies 2.8–104.8 and 3.6–123.0.","key_machinery":"The load-bearing object is the Bandt–Pompe permutation method: each window of $d$ consecutive time-series values (here $d=5$, $\\tau=1$) is replaced by the permutation of indices that sorts the values, yielding a probability distribution over the $d!$ ordinal patterns. From this distribution the paper builds the normalized permutation entropy $H(P)=S(P)/\\log(d!)$, the LMC complexity $C_{\\mathrm{LMC}}=H\\cdot D_2(P,P_e)$ with the Euclidean disequilibrium, and the Jensen–Shannon complexity $C_{\\mathrm{JS}}=H\\cdot D_{\\mathrm{JS}}(P,P_e)/D_{\\mathrm{max}}$. The classicality parameter is the relative energy $E_r=E/(I^{1/2}\\omega_q)$ built from the motion invariant $I$, which encodes the uncertainty-principle deviation from classicality; $E_r=1$ is the fully quantum case and $E_r\\to\\infty$ is the classical limit. The mechanism that carries the argument is that ordinal-pattern statistics of the single observable $\\langle\\hat{x}^2\\rangle(t)$ respond sharply to the qualitative change in dynamics as $I\\to0$, even though the underlying dynamics is regular.","core_discovery":"The central claim is that for the Hamiltonian $\\hat H = \\tfrac{1}{2}[\\omega_q(\\hat{x}^2+\\hat{p}^2) + \\omega_{\\mathrm{cl}}(A^2+P_A^2)\\hat I + e_{\\mathrm{cl}}^q A^2 \\hat{x}^2]$, the approach to the classical limit is not featureless when viewed through information quantifiers. With the relative energy $E_r = E/(I^{1/2}\\omega_q)$, where $I=\\langle\\hat{x}^2\\rangle\\langle\\hat{p}^2\\rangle - \\langle\\hat{L}\\rangle^2/4$ is a motion invariant obeying $I\\ge\\hbar^2/4$, the classical limit is $I\\to0$, i.e. $E_r\\to\\infty$. The paper computes the normalized permutation entropy $H$ and the LMC and Jensen–Shannon complexities $C_{\\mathrm{LMC}}$ and $C_{\\mathrm{JS}}$ from Bandt–Pompe ordinal distributions of $\\langle\\hat{x}^2\\rangle(t)$, scanning $E_r$ through a one-parameter family of initial conditions. Each quantifier shows three zones in $E_r$: a quasi-quantum regime near $E_r\\approx1$, a transitional (mesoscopic) regime ($2.8<E_r<104.8$ conservative; $3.6<E_r<123.0$ dissipative), and a classical-convergence regime where the quantifiers saturate at the values of the purely classical analog, such as $H_{\\mathrm{cl}}^c=0.17676$ and $H_{\\mathrm{cl}}^d=0.17207$. The same boundaries appear for all three quantifiers, and the near-constancy of the disequilibrium terms explains why $H$ and $C_{\\mathrm{JS}}$ share their shape. The paper concludes that these information quantifiers confirm the three zones earlier found with Poincaré sections, but with greater precision.","pith_inferences":["If the boundaries are robust to the scan path, the same Bandt–Pompe protocol could be used to locate decoherence crossovers in other semiclassical models with a conserved uncertainty-bound invariant, even where Poincaré sections are too dense to read by eye.","Because the disequilibrium is nearly constant in $E_r$, the complexity measures mostly rescale the entropy for this system; an implication is that the entropy alone may carry the zone information, and the choice of complexity measure is secondary.","A testable extension is to vary the dissipation strength $\\eta$ (the paper fixes $\\eta=0.05$) and check whether the 3.6 and 123.0 boundaries shift monotonically, which would tie the delay of classicality to the dissipation rate."],"forward_implications":["All three information quantifiers give the same transition boundaries in $E_r$, so the three-zone structure is not specific to one complexity measure.","The semiclassical quantifiers already match the classical analog values at finite $E_r$ (about 104.8 conservative, 123.0 dissipative), so the classical limit is effectively reached before $E_r\\to\\infty$.","The near-constancy of the disequilibrium factors means $H$ and $C_{\\mathrm{JS}}$ differ mainly by scale, while $C_{\\mathrm{LMC}}$ is noisier but shows the same global pattern.","Convergence starts at a smaller $E_r$ in the conservative case than in the dissipative case, so dissipation delays the onset of the classical regime in this model.","The low values of all quantifiers are consistent with the regularity of the underlying dynamics seen in the Poincaré sections."],"supporting_citations":[{"why":"Prior Poincaré-section analysis of the same model that identified the three zones; the present paper refines its boundaries.","marker":"[30]"},{"why":"Introduces the permutation-pattern symbolization used to build every probability distribution in the paper.","marker":"[35]"},{"why":"Defines the LMC statistical complexity used as one of the two complexity quantifiers.","marker":"[33]"},{"why":"Defines the Jensen–Shannon statistical complexity used as the other complexity quantifier.","marker":"[34]"},{"why":"Earlier application of the Bandt–Pompe approach to the classical-quantum transition, supplying the methodological precedent.","marker":"[31]"},{"why":"Provides the semiclassical dissipative model and mean-value equations of motion underlying the numerical experiments.","marker":"[24]"},{"why":"Supplies the classical-limit formulation through the motion invariant and the mean-value Hamiltonian dynamics.","marker":"[26]"},{"why":"Provides the numerical implementation of permutation entropy and ordinal-pattern quantifiers used for the time-series analysis.","marker":"[44]"}],"fun_headline_variants":["Permutation entropy finds three zones in quantum-classical transition","Three zones discovered in quantum-classical transition","Quantum-classical transition shows three distinct zones","Entropy and complexity define three zones in quantum-classical transition","Mesoscopic zone emerges between quantum and classical"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reported zone boundaries in $E_r$ (2.8 and 104.8 conservative; 3.6 and 123.0 dissipative) rest on the assumption that the particular one-parameter family of initial conditions used to scan $I\\to0$—with $\\langle\\hat{L}\\rangle(0)=A(0)=0$ and a fixed split of energy between $\\langle\\hat{x}^2\\rangle$ and $\\langle\\hat{p}^2\\rangle$—is representative of the classical transition; if the boundaries shift when that path through phase space is changed, the stated numbers are an artifact of the chosen scan.","fun_headline_variants_meta":{"raw":{"variants":["Permutation entropy finds three zones in quantum-classical transition","Three zones discovered in quantum-classical transition","Quantum-classical transition shows three distinct zones","Entropy and complexity define three zones in quantum-classical transition","Mesoscopic zone emerges between quantum and classical"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000813,"raw_usage":{"total_tokens":3616,"prompt_tokens":1046,"completion_tokens":2570,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":662,"completion_tokens_details":{"reasoning_tokens":2497}},"tokens_in":662,"tokens_out":2570,"duration_ms":912568,"temperature":1.0,"reasoning_tokens":2497,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:54:43.179592+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the same model and the same three quantifiers, but scan to the classical limit along a different one-parameter family of initial conditions that still satisfies $I\\to0$—for example, starting with nonzero $\\langle\\hat{L}\\rangle(0)$ or redistributing the initial energy between $\\langle\\hat{x}^2\\rangle$ and $\\langle\\hat{p}^2\\rangle$—and check whether the zone boundaries in $E_r$ remain at 2.8 and 104.8 (conservative) and 3.6 and 123.0 (dissipative).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Prior Poincaré-section analysis of the same model that identified the three zones; the present paper refines its boundaries."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the permutation-pattern symbolization used to build every probability distribution in the paper."},{"cited_title":"L., & Calbet, X","cited_arxiv_id":null,"evidence_quote":"Defines the LMC statistical complexity used as one of the two complexity quantifiers."},{"cited_title":"W., Martin, M","cited_arxiv_id":null,"evidence_quote":"Defines the Jensen–Shannon statistical complexity used as the other complexity quantifier."},{"cited_title":"M., Mart ´ ın, M","cited_arxiv_id":null,"evidence_quote":"Earlier application of the Bandt–Pompe approach to the classical-quantum transition, supplying the methodological precedent."},{"cited_title":"M., Plastino, A., & Proto, A","cited_arxiv_id":null,"evidence_quote":"Provides the semiclassical dissipative model and mean-value equations of motion underlying the numerical experiments."},{"cited_title":"M., Plastino, A., & Proto, A","cited_arxiv_id":null,"evidence_quote":"Supplies the classical-limit formulation through the motion invariant and the mean-value Hamiltonian dynamics."},{"cited_title":"A., & Ribeiro, H","cited_arxiv_id":null,"evidence_quote":"Provides the numerical implementation of permutation entropy and ordinal-pattern quantifiers used for the time-series analysis."}],"review_version":1}