{"id":"0cf75cb4-84ab-4c43-aee8-c526d8ba6d88","arxiv_id":"2608.09705","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In the quantum kicked top, two Wasserstein-based diagnostics on geometric quantum states, distinguishability and state-space coverage, grow with interaction strength and show parity-dependent finite-size behavior.","lead":"This paper introduces two new measures of complexity for small interacting quantum systems: how quickly nearby states become distinguishable, and how much of the possible state space they explore over time. Tested on a standard model of interacting spins, both measures grow with interaction strength and show a clear difference between integer and half-integer spin systems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central parity claim rests on computational-basis GQS; if parity ordering of Γ and S1 flips under environment-basis rotation, the abstract's integer-vs-half-integer claim is an artifact.","rationale":"The reader's weakest assumption—computational-basis conditioning may not reveal intrinsic complexity—is exactly the most load-bearing concern. The GQS framework intentionally encodes more than ρ_S, but the extra information is basis-dependent, and the diagnostics Γ and S1 inherit that dependence. The paper is transparent about this limitation, which is a credit, but the abstract's unqualified parity claim goes beyond what is currently supported. The proposed test is cheap and decisive: rotating the environment basis leaves the global unitary dynamics and ρ_S unchanged while changing the GQS, so any change in the parity ordering directly indicts the computational-basis choice as the source of the reported integer/half-integer difference. Other concerns (no error bars, finite-time averaging, lack of quantitative baseline against established chaos diagnostics) are real but secondary; they affect interpretation and significance rather than the validity of the central parity assertion. Because the concern is addressable by a computational check and the paper already flags basis dependence as open, CONDITIONAL remains the appropriate verdict: the claims are plausible and well-documented but not yet robust to the representation choice on which they rest.","tokens_in":24892,"tokens_out":6350,"duration_ms":59747,"concrete_test":"Recompute Γ and S1 for the same global kicked-top trajectories used in Figs. 10–11, e.g., L=4 and L=5, κ=2.5, initial spin-coherent state (θ,φ)=(π/2+0.5, π/2), T=200 kicks. For each time step, re-expand the global state in (i) the computational basis, (ii) the Hadamard basis on each environment qubit, and (iii) 10 randomly drawn local unitary bases on the environment. Compute Γ and S1 in each basis. Determine whether the parity ordering Γ_4 > Γ_5 and S1_4 > S1_5 persists across all bases. If it flips in any basis, the central parity claim is basis-dependent and the abstract must be qualified; if it persists, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that environment-size dependence is \"structured by parity symmetry, with integer-spin systems often exhibiting greater sensitivity and state-space coverage\"—is computed entirely in the GQS representation defined by conditioning on the computational environment basis (Sec. II.E, Eqs. 15–19). The paper explicitly acknowledges this dependence in Sec. IV.B: \"All values reported below are computed using conditioning in the computational environment basis and are therefore basis dependent,\" and lists basis dependence as an open direction in the Conclusion. Because Γ and S1 act on the GQS measure rather than on the reduced density matrix ρ_S, they are not intrinsic functions of the subsystem state: the same global state and the same ρ_S yield different GQS ensembles, and hence different Γ and S1, under different environment bases (Fig. 2 illustrates this for different environment sizes already). The abstract's unqualified parity statement would fail if a different natural basis—Hadamard, random local unitaries, or the Schmidt basis—reverses the integer-spin vs half-integer-spin ordering. A secondary confound strengthens this worry: the L=3 vs L=4 comparison mixes parity with a change in interaction graph connectivity (all-to-all vs ring, noted in Sec. V.B.3), though the L=5/6 and L=7/8 pairs are cleaner. The load-bearing question is whether the parity effect is a property of the kicked-top dynamics or a representational artifact of the computational-basis conditioning.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a geometric, state-based framework for diagnosing dynamical complexity in open quantum systems. A mixed reduced state of a subsystem is represented not only as a density matrix but as an environment-conditioned ensemble of pure states on complex projective space, called a geometric quantum state (GQS). Two diagnostics are introduced: a distinguishability measure Γ, defined as the time-averaged logarithmic growth of the Wasserstein distance (with Fubini–Study cost) between nearby GQSs, and a State-Space Coverage Index S1, comparing the time-aggregated GQS to the uniform measure on CP^{d-1}. Both are applied to the quantum kicked top with a single-qubit subsystem and L−1 environment qubits for L=3,...,8. The paper reports that both diagnostics generally grow with interaction strength κ, that their dependence on environment size is structured by parity symmetry (integer total spin / even L often showing larger sensitivity and coverage than half-integer / odd L), and that recurrent regimes can either preserve distinguishability (κ=2πj) or show distinguishability growth despite recurrence (κ=πj). The presentation includes phase-space maps of both diagnostics over initial spin-coherent states and qualitative comparisons with classical Lyapunov maps.","tokens_in":25123,"tokens_out":15541,"duration_ms":139158,"significance":"If the central claims hold, the framework adds a genuinely geometric, state-resolved diagnostic axis complementary to OTOCs, Loschmidt echo, and Krylov complexity, with the appealing feature of working directly on CP^{d-1} with a classical optimal-transport metric. The manuscript is unusually transparent: numerical parameters are tabulated (Table I), code is provided, claims are hedged with 'generally' and 'often', the basis dependence of the GQS is explicitly acknowledged, and Appendix F's analysis of recurrent regimes shows that Γ is not a trivial monotone proxy of interaction strength. The cleanest nontrivial finding is the parity structuring of Γ and S1 across L=3,...,8, together with the systematic initial-condition scans of Appendix G. The main risk is that the headline parity effect is established only in a single environment basis and without control for the growing number of GQS support points, so its status as a property of the kicked-top dynamics rather than of the chosen GQS conditioning is not yet settled.","major_comments":[{"comment":"The headline parity claim — that integer-spin systems 'often exhibit greater sensitivity and state-space coverage' than half-integer-spin systems — is computed entirely from GQSs defined by conditioning the global wavefunction on the computational environment basis (Eqs. (15)-(19)). The paper states in Sec. IV.B that all reported values are 'therefore basis dependent', yet no test is made of whether the parity ordering of Γ and S1 survives a change of environment basis. Because Γ and S1 act on the measure QS rather than on the reduced density matrix ρS, a unitary rotation of the environment basis changes both diagnostics while leaving the reduced state and all subsystem observables invariant. I recommend a concrete robustness test: repeat the L=3,...,8 computations with the Hadamard basis, with random single-qubit local unitaries applied to the environment, and with the Schmidt basis, and report whether the even-odd ordering of Γ and S1 persists. If the ordering flips, the abstract's parity statement should be rephrased as a property of the computational-basis conditioning rather than of the kicked-top dynamics.","section":"Abstract; Sec. IV.B; Eqs. (15)-(19)"},{"comment":"The coverage index S1 = 1 − W1(νS, σ)/W1(δ_{Z0}, σ) is computed from a time-aggregated measure νS that is a sum of at most d_E × T_S point masses (Eqs. (19) and (34)). Since σ is the continuous uniform measure on CP^1, the equality S1 = 1 is unattainable in this construction, and the value of S1 reflects not only how broadly the dynamics spread probability but also how many support points the GQS possesses. Because d_E = 2^{L−1} doubles between each consecutive pair of system sizes, the parity comparison in Fig. 10 (e.g., S1 = 0.95 for L=4 versus 0.56 for L=3, and 0.91 for L=8 versus 0.56 for L=7) is confounded with the exponential growth of the number of conditioning branches; the same caveat applies to Γ, whose transport plans involve d_E atoms per state. The non-monotonic gaps (ΔS1 ≈ 0.39, 0.05, 0.35 for the 4/3, 6/5, and 8/7 pairs) show that resolution alone does not determine the pattern, but without controls the parity claim is not quantitatively separated from this mechanical effect. I recommend a matched-support control: compute S1 and Γ for L=4, 6, 8 while conditioning on a randomly chosen subset of d_E' = 2^{L−2} environment basis states, matching the support size of the neighboring odd-L systems, and verify that the parity ordering persists.","section":"Sec. IV.B; Eqs. (34)-(35); Fig. 10; Table I"},{"comment":"For L=3 the all-to-all J_z^2 interaction coincides with a three-site nearest-neighbor ring, whereas for every L>3 the same collective term couples non-nearest-neighbor qubits. The most dramatic parity contrast in the data is the L=3 versus L=4 pair (S1 = 0.56 versus 0.95 at κ=2.5), which simultaneously changes both parity and interaction-graph connectivity; the text itself attributes a 'pronounced increase in complexity' to the connectivity change. The parity claim therefore rests most heavily on the 5/6 and 7/8 pairs, and for the 5/6 pair the S1 gap is only 0.05. Please present the parity analysis with the 3/4 pair explicitly flagged as mixing two effects, and report the clean pairs (5/6 and 7/8) separately, so that the abstract's parity statement is not implicitly supported by the connectivity jump.","section":"Sec. V.B.3; Figs. 8 and 10"}],"minor_comments":[{"comment":"The distinguishability measure is averaged over M=200 perturbations, but no error bars, standard errors, or convergence tests are reported. Given that the L=5 versus L=6 S1 gap is only 0.05, I recommend reporting the spread over perturbations (and over initial conditions) to allow the strength of the parity ordering to be assessed.","section":"Table I; Figs. 5 and 11"},{"comment":"The text states that T is chosen according to the numerically estimated recurrence timescale, but Table I lists a single value (T=200) for all runs. Please clarify how T was set for each regime and report the sensitivity of Γ and S1 to T, T_S, and the perturbation scale ε=0.2.","section":"Sec. IV.A; Table I"},{"comment":"The sentence 'A positive Γ signals sensitivity and complex dynamics' overstates the interpretation: Appendix F shows positive Γ for recurrent κ=πj dynamics that the paper itself classifies as a low-sensitivity regime. I suggest rewording to 'indicates growth of ensemble distinguishability' and reserving the complexity interpretation for the joint (Γ, S1) classification.","section":"Sec. IV.A; Appendix F"},{"comment":"The claimed parallelism between the quantum Γ phase-space maps and the classical λ_max maps is supported only visually. I suggest adding a quantitative comparison, such as a spatial correlation coefficient between the two maps over the 100×100 grid, to substantiate the relationship.","section":"Fig. 5(c)-(f); Sec. V.A.1"},{"comment":"There are several notational slips: 'Throughout this following we focus on the case p=1' is missing a noun; Sec. IV.B and Fig. 3 use 'Sp' while the numerics report 'S1'; and in Eq. (32) the order of the time average and the perturbation average should be made explicit for clarity.","section":"Sec. III; Sec. IV.B; Eq. (32)"},{"comment":"The non-uniqueness of convex decompositions of a density matrix (Eq. (14)) is stated without reference; citing the standard ensemble-purification results (e.g., Schrödinger or Hughston–Jozsa–Wootters) would help readers place the GQS construction relative to known results.","section":"Sec. II.D"}],"recommendation":"major_revision","confidential_remarks":"To the editor: this is a good-faith, well-hedged contribution with reproducible numerics and an honest discussion of its own limitations. The main gap is that the paper's distinctive new claim — parity structuring of the diagnostics — is not yet robustly separated from representational choices (environment basis, number of GQS support points) and from the L=3 connectivity change. The required tests are numerical and within scope, so I recommend major revision rather than rejection. One further point: the paper builds on the authors' own GQS program (refs. [24-27]); given that, the novelty of Γ and S1 relative to that program could be stated more explicitly for readers outside the group."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear —,\n\nThe short version: this is a useful, honestly written addition to the quantum-complexity toolkit. The two diagnostics—Γ, a Wasserstein distinguishability measure, and S1, a state-space coverage index—are genuinely new, cleanly defined, and the kicked-top application at L=3–8 gives a nice finite-size picture. Code and data are public, the numerical parameters are explicit, and the paper is careful to say when trends are 'generally' rather than universal. Send it to a serious referee; the caveat below is the one I’d want settled.\n\nThe real value is the geometric state-based viewpoint. Conditioning on an environment basis turns the reduced state into a probability measure on CP^1, and Γ and S1 then track sensitivity and spreading of that measure. The qualitative match between the Γ phase-space maps and the classical Lyapunov exponent maps is striking, and the recurrence appendix (κ=πj vs 2πj) is a good control: it shows positive Γ does not automatically mean chaos, which makes the paper more credible. That is the kind of negative-result check I like.\n\nThe soft spot is the basis dependence of the central parity claim. The authors state plainly in Sec. IV.B that all values are computed in the computational environment basis and are therefore basis dependent, and they list basis dependence as an open direction in the Conclusion. That honesty is good, but the abstract's parity sentence (integer spin often showing greater sensitivity and coverage) carries no caveat. The reader's worry is legitimate: if a different natural basis—say, the Schmidt basis or random local unitaries—reverses the parity ordering, the headline result is an artifact. I’d note that the computational basis is not arbitrary here: it is the eigenbasis of J_z^2, the coupling term that drives the interactions, so the conditioning is dynamically motivated. Still, the authors should test at least one other basis (Hadamard or random local unitaries) and either show the ordering survives or qualify the abstract. As written, the parity result is conditional, not definitive. The L=3 vs L=4 comparison also mixes parity with a change in interaction graph (all-to-all vs ring), as the authors note; the L=5/6 and L=7/8 pairs are the cleaner test.\n\nTwo smaller points. The M=200 perturbation averages lack error bars, and no significance tests are given for the parity gaps; a few standard errors would help. And there is no quantitative comparison with OTOC, Krylov complexity, or other chaos measures—fine as future work, but it limits how strongly one can claim the diagnostics are complementary.\n\nWho benefits: people working on finite-size quantum chaos, scrambling, and open-system complexity. It is a within-subfield contribution, not a paradigm shift. I’d engage with it, and I’d urge the editor to send it out. The basis robustness check is the one substantive revision I’d demand.\n\nRecommendation: peer review yes, with a request for the basis test and error bars.","headline":"Useful new geometric diagnostics for open-system complexity, honestly presented, but the parity claim rests on a basis-dependent representation and needs a robustness check.","tokens_in":25692,"tokens_out":4843,"would_cite":true,"duration_ms":42489,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q50","49Q22"],"pacs":["03.65.Yz","05.45.Mt"],"model":"deepseek-v4-flash","headline":"This paper claims that complexity in open quantum systems can be measured geometrically, by tracking how a subsystem's quantum state spreads over the Bloch sphere, with two Wasserstein-distance diagnostics that grow with interaction…","keywords":["dynamical complexity","geometric quantum states","Wasserstein distance","complex projective Hilbert space","quantum kicked top","open quantum systems","state-space coverage","parity symmetry"],"falsifier":"Recompute $\\Gamma$ and $S_1$ for the same kicked-top initial states and coupling strengths $\\kappa=0.5$ and $\\kappa=2.5$, but condition the global wavefunction on a different environment basis, for instance a basis obtained by applying a generic product or Haar-random unitary to the environment qubits before the decomposition, or the Schmidt basis of the global state. If the even-$L$ versus odd-$L$ ordering of the two diagnostics reverses, disappears, or becomes non-monotonic under this change, then the parity-symmetry claim is an artifact of the computational-basis conditioning rather than a property of the open-system dynamics.","tokens_in":24651,"feed_emoji":"⚛️","tokens_out":8399,"duration_ms":61956,"temperature":0.7,"pith_summary":"This paper tries to establish that dynamical complexity in finite-size open quantum systems can be read directly from how a subsystem's state spreads over the geometry of pure states. The authors represent the reduced state of a qubit not as a density matrix but as a probability measure on the Bloch sphere ($\\mathbb{CP}^1$), obtained by conditioning the global wavefunction on the environment's computational basis. They introduce two Wasserstein-based diagnostics, a distinguishability measure $\\Gamma$ that plays the role of a Lyapunov exponent for ensembles and a state-space coverage index $S_1$ that measures how much of the sphere the dynamics explore over long times. Applied to the quantum kicked top, both diagnostics grow with interaction strength, and their dependence on environment size is organized by parity: integer-spin (even-qubit) systems typically show greater sensitivity and coverage than half-integer-spin (odd-qubit) systems. If these claims hold, complexity in the deep quantum regime becomes a geometric, state-resolved phenomenon visible at experimentally relevant finite sizes.","feed_headline":"Parity splits how open quantum systems grow complex","feed_subtitle":"Bloch-sphere spreading, read with Wasserstein distance, turns sensitivity and coverage into two clean diagnostics.","key_machinery":"The carrying object is the geometric quantum state (GQS): an environment-conditioned probability measure $Q_S(Z,t)=\\sum_j \\lambda^E_j(t)\\,\\delta_{Z^S_j(t)}$ on the subsystem's projective Hilbert space $\\mathbb{CP}^{d_S-1}$, built by decomposing the global wavefunction in the environment's computational basis (the conditional pure states need not be orthogonal, unlike a Schmidt decomposition). On top of it sit two diagnostics: $\\Gamma$, the ensemble Lyapunov-type measure defined by the time-averaged logarithmic growth of the Wasserstein distance $W_1$ (with Fubini–Study ground metric) between nearby GQSs, and $S_1 = 1 - W_1(\\nu_S,\\sigma)/W_1(\\delta_{Z_0},\\sigma)$, where $\\nu_S$ is the time-aggregated GQS and $\\sigma$ the uniform measure on $\\mathbb{CP}^1$. The Wasserstein distance with the Fubini–Study metric is what restores a geometric notion of trajectory separation that the unitary global evolution and the density-matrix description both lose.","core_discovery":"On the paper's own terms, the central discovery is that the transition from periodic, coherent subsystem motion to interaction-driven complexity can be tracked as a reorganization of probability mass on $\\mathbb{CP}^{d-1}$, and that two optimal-transport quantities capture this reorganization: $\\Gamma$, the average logarithmic growth of the Wasserstein distance between a reference geometric quantum state and its perturbed copies (an open-system analogue of the maximal Lyapunov exponent), and $S_1$, the Wasserstein proximity of the time-aggregated measure to the uniform distribution on $\\mathbb{CP}^{d-1}$. In the three-qubit kicked top, $\\Gamma$ rises with the interaction strength $\\kappa$, and its spatial pattern over initial spin-coherent states mirrors the classical kicked top's Lyapunov map. Increasing the environment size $L-1$ at fixed single-qubit subsystem, the two diagnostics show a finite-size parity effect: even-$L$ (integer-spin) systems generally achieve larger $\\Gamma$ and $S_1$ than odd-$L$ (half-integer-spin) systems, with the separation growing more pronounced at strong coupling and vanishing in the classical limit $j\\to\\infty$. In recurrent regimes, recurrence alone does not imply simplicity: at $\\kappa=\\pi j$ the dynamics recur but $\\Gamma>0$, whereas at $\\kappa=2\\pi j$ distinguishability is preserved and both diagnostics stay small.","pith_inferences":["Editorial inference: because the GQS is built by conditioning in a fixed environment basis, the two diagnostics are basis-dependent by construction; the central claim would be strengthened if the parity ordering of $\\Gamma$ and $S_1$ were shown to persist under generic rotations of the environment basis, not just the computational one.","Editorial inference: the resemblance of the quantum $\\Gamma$ and $S_1$ phase-space maps to the classical kicked-top Lyapunov map suggests a testable correspondence, comparing these quantum diagnostics against the classical Lyapunov exponent and information dimension on the same $(\\theta,\\phi)$ grids across $L$, which could sharpen the analogy into a quantitative semiclassical relation.","Editorial inference: one could probe the framework's stability by applying it to a dissipative or noisy driven qubit beyond the kicked top, where the GQS measure would be replaced by an ensemble generated by quantum trajectories, testing whether environment-conditioned spreading survives when the environment is not a closed computational basis.","Editorial inference: the claim that integer-spin systems show greater sensitivity could be checked on existing kicked-top experimental platforms by measuring the two diagnostics on identical initial states, which would settle whether the parity effect is observable or a numerical artifact."],"forward_implications":["For a single-qubit subsystem of the kicked top, complexity becomes a quantitative, two-dimensional diagnosis, sensitivity ($\\Gamma$) and spread ($S_1$), that grows with interaction strength $\\kappa$ and sharply distinguishes the noninteracting periodic regime ($\\Gamma=0$) from interacting regimes ($\\Gamma>0$).","Environment size controls complexity in a parity-structured way: for the systems studied, integer-spin (even-$L$) kicked tops generally show larger $\\Gamma$ and $S_1$ than half-integer-spin (odd-$L$) tops, an effect attributed to the Floquet operator's parity symmetry and expected to vanish in the classical limit.","Recurrent quantum dynamics can still be dynamically complex: at $\\kappa=\\pi j$ the subsystem recurs but $\\Gamma>0$, while at $\\kappa=2\\pi j$ both diagnostics vanish, so recurrence alone does not certify simplicity.","The framework extends to larger subsystems ($L_S>1$) and provides a state-based geometric complement to operator diagnostics such as out-of-time-ordered correlators and Loschmidt echoes, useful when observables remain stable while the underlying state structure changes."],"supporting_citations":[{"why":"Anza and Crutchfield, Beyond density matrices: Geometric quantum states, supplies the GQS representation of mixed states as probability measures on projective Hilbert space.","marker":"[24]"},{"why":"Villani, Optimal transport: old and new, provides the Wasserstein-metric and optimal-transport machinery used for all distances between GQSs.","marker":"[28]"},{"why":"Lombardi and Matzkin, Entanglement and chaos in the kicked top, motivates the kicked top as the model system connecting entanglement and chaos.","marker":"[33]"},{"why":"Dogra, Madhok, and Lakshminarayan, Quantum signatures of chaos, thermalization, and tunneling in the exactly solvable few-body kicked top, supplies the parity-symmetry structure of even- and odd-qubit kicked tops.","marker":"[37]"},{"why":"Anand, Davis, and Ghose, Quantum recurrences in the kicked top, provides the recurrence times used to select the $\\kappa=\\pi j$ and $\\kappa=2\\pi j$ regimes.","marker":"[40]"},{"why":"Haake, Kuś, and Scharf, Classical and quantum chaos for a kicked top, underpins the classical limit $j\\to\\infty$ and the classical kicked-top Lyapunov behavior.","marker":"[41]"},{"why":"Flamary et al., POT: Python Optimal Transport, supplies the numerical earth-mover-distance solver used to evaluate the Wasserstein distances.","marker":"[51]"}],"fun_headline_variants":["Wasserstein distance reveals spin-parity split in open qubit complexity","Integer spin boosts sensitivity and coverage in open systems","Parity splits open-system complexity into even vs odd spin","Recurring but complex: spin parity shapes open-system diagnostics","Two Wasserstein diagnostics track how open systems grow complex"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All reported values of the two diagnostics are computed after conditioning the global wavefunction on one fixed choice, the environment's computational basis, and the paper does not test whether the parity ordering of $\\Gamma$ and $S_1$ survives conditioning in any other basis.","fun_headline_variants_meta":{"raw":{"variants":["Wasserstein distance reveals spin-parity split in open qubit complexity","Integer spin boosts sensitivity and coverage in open systems","Parity splits open-system complexity into even vs odd spin","Recurring but complex: spin parity shapes open-system diagnostics","Two Wasserstein diagnostics track how open systems grow complex"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002017,"raw_usage":{"total_tokens":7886,"prompt_tokens":991,"completion_tokens":6895,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":6813}},"tokens_in":607,"tokens_out":6895,"duration_ms":50541,"temperature":1.0,"reasoning_tokens":6813,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:20:12.856287+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute $\\Gamma$ and $S_1$ for the same kicked-top initial states and coupling strengths $\\kappa=0.5$ and $\\kappa=2.5$, but condition the global wavefunction on a different environment basis, for instance a basis obtained by applying a generic product or Haar-random unitary to the environment qubits before the decomposition, or the Schmidt basis of the global state. If the even-$L$ versus odd-$L$ ordering of the two diagnostics reverses, disappears, or becomes non-monotonic under this change, then the parity-symmetry claim is an artifact of the computational-basis conditioning rather than a property of the open-system dynamics.","supporting_citations":[{"cited_title":"Quantum Chaos on Complexity Geometry","cited_arxiv_id":"2004.03501","evidence_quote":"Anza and Crutchfield, Beyond density matrices: Geometric quantum states, supplies the GQS representation of mixed states as probability measures on projective Hilbert space."},{"cited_title":"Rabinovici, A","cited_arxiv_id":null,"evidence_quote":"Villani, Optimal transport: old and new, provides the Wasserstein-metric and optimal-transport machinery used for all distances between GQSs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Lombardi and Matzkin, Entanglement and chaos in the kicked top, motivates the kicked top as the model system connecting entanglement and chaos."},{"cited_title":"The quantum Wasserstein distance of order 1.IEEE Transac- tions on Information Theory, 67(10):6627–6643, 2021","cited_arxiv_id":null,"evidence_quote":"Dogra, Madhok, and Lakshminarayan, Quantum signatures of chaos, thermalization, and tunneling in the exactly solvable few-body kicked top, supplies the parity-symmetry structure of even- and odd-qubit kicked tops."},{"cited_title":"Chaudhury, A","cited_arxiv_id":null,"evidence_quote":"Anand, Davis, and Ghose, Quantum recurrences in the kicked top, provides the recurrence times used to select the $\\kappa=\\pi j$ and $\\kappa=2\\pi j$ regimes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Haake, Kuś, and Scharf, Classical and quantum chaos for a kicked top, underpins the classical limit $j\\to\\infty$ and the classical kicked-top Lyapunov behavior."},{"cited_title":"Braumüller, A","cited_arxiv_id":null,"evidence_quote":"Flamary et al., POT: Python Optimal Transport, supplies the numerical earth-mover-distance solver used to evaluate the Wasserstein distances."}],"review_version":1}