{"id":"49280767-a6a3-4e6e-a9bb-c01524aae0cb","arxiv_id":"2607.06410","paper_version":1,"verdict":"ACCEPT","confidence":"UNKNOWN","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Linear dependence among the diagonal components of Hamiltonian derivatives creates a slow parameter direction with O(t^0) scaling, obstructing simultaneous t^{-2} multiparameter quantum estimation.","lead":"The paper proves that in multiparameter quantum estimation, linear dependence among the Hamiltonian-commuting parts of parameter derivatives creates a 'slow' direction with bounded precision, destroying quadratic time scaling. It provides a computable Gram matrix diagnostic and shows the SLD bound remains asymptotically saturable despite this bottleneck.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The finite-dimensional proof is clean; the infinite-dimensional caveat is acknowledged and handled in examples.","rationale":"The reader correctly identified the boundedness of K_μ(t) in infinite dimensions as the weakest assumption. This is accurate: it is the only place where the proof's generality is limited. However, this concern does not undermine the central claim because (a) the finite-dimensional result is rigorous and constitutes the main theoretical contribution, (b) the infinite-dimensional case is explicitly flagged as requiring case-by-case verification rather than claimed to be universal, and (c) the QHO example—the primary infinite-dimensional application—verifies the boundedness explicitly through Eq. (48). The proof structure is clean: the decomposition in Eq. (9), the linear dependence argument in Eqs. (13-15), and the dimensional bound (at most d-1 parameters with O(t^{-2}) scaling) all follow correctly. The examples are well-chosen: the spin system and QHO illustrate the obstruction, while the LMG model provides a contrast where linear independence preserves scaling. The measurement incompatibility analysis (Sec. VII.B) is a nice additional result that follows from the same framework. The nuisance parameter and quantum control discussions (Sec. VII.A, VII.C) are appropriate extensions. The ACCEPT verdict with UNKNOWN confidence is reasonable; I would not adjust it.","tokens_in":19329,"tokens_out":4844,"duration_ms":171648,"concrete_test":"Numerically verify the QHO example (Sec. V) by computing the full QFIM for the optimal Gaussian state (Eqs. 59-60) at increasing t values and confirming that: (1) the slow eigenvalue λ_min saturates to the analytical expression in Eq. (58) rather than growing with t, and (2) the fast eigenvalue scales as O(t^2). If λ_min grows beyond a constant for large t, the infinite-dimensional caveat would be more serious than acknowledged.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central proof in Sec. III is mathematically sound for finite-dimensional systems. The decomposition (Eq. 9) correctly separates the t-proportional diagonal generator from the bounded off-diagonal term K_μ(t). The key step—Eq. (10)—shows that K_μ(t) only involves terms with E_α ≠ E_β, so each coefficient (e^{i(E_α-E_β)t} - 1)/(i(E_α-E_β)) is bounded by 2/|E_α - E_β|. When {D̃_μ} are linearly dependent (Eq. 13), the t-term vanishes along direction w (up to an identity that doesn't affect the QFIM), leaving w^T F w = 4 Var(Σ w_μ K_μ(t)) = O(t^0). This correctly establishes λ_min ≤ O(t^0), hence Tr(F^{-1}) ≥ O(1), destroying O(t^{-2}) scaling. The reader's identified concern—boundedness of K_μ(t) in infinite dimensions—is real but explicitly acknowledged by the authors (Sec. V) and verified for the QHO example where the scalar prefactors in Eq. (48) are indeed bounded. The restriction to Gaussian states with fixed energy ensures finite variances of the unbounded operators. The ancilla argument (H_μ ⊗ I_A preserves linear dependence) is also correct. The measurement incompatibility result (Sec. VII.B) follows cleanly: U_fs = O(t), det(F) = O(t^2), giving ||F^{-1}UF^{-1}||_1 = O(t^{-1}). No internal inconsistency or unjustified assumption was found in the load-bearing argument.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This manuscript establishes a general geometric obstruction to simultaneous O(t^{-2}) scaling in multiparameter quantum estimation under time-independent Hamiltonian dynamics. The central result (Sec. III) is a no-go theorem: when the traceless diagonal components of the Hamiltonian derivatives, {D̃_μ}, are linearly dependent, there exists a 'slow' parameter direction along which the quantum Fisher information remains O(t^0), bottlenecking the total estimation precision. The authors derive a computable diagnostic via the Gram matrix of diagonal generators, prove that ancilla-assisted strategies cannot bypass the obstruction, and show that the measurement incompatibility penalty decays as O(t^{-1}). The framework is illustrated through three examples: collective spin magnetometry (Sec. IV), the quantum harmonic oscillator (Sec. V), and the Lipkin-Meshkov-Glick model (Sec. VI, where the obstruction is absent). The paper also discusses circumvention via nuisance parameter relegation and adaptive quantum control (Sec. VII).","tokens_in":19663,"tokens_out":1631,"duration_ms":261808,"significance":"The central derivation in Sec. III is mathematically sound and logically clean. The decomposition of the local generator into a t-proportional diagonal part and a bounded off-diagonal part K_μ(t) (Eq. 9-10) is standard but effectively deployed. The key argument—that linear dependence of {D̃_μ} (Eq. 13) eliminates the t-proportional term along direction w, leaving only the bounded K_μ(t) contribution (Eq. 14-15)—correctly establishes λ_min ≤ O(t^0) and hence Tr(F^{-1}) ≥ O(1). The dimensional bound (at most d-1 parameters with O(t^{-2}) scaling in a d-dimensional system) is a clean, falsifiable consequence. The Uhlmann curvature analysis (Sec. VII.B) is also correct: det(F) = O(t^2) and U_fs = O(t) yield ||F^{-1}UF^{-1}||_1 = O(t^{-1}). The framework provides a parameter-free diagnostic (the Gram matrix determinant) with no ad-hoc assumptions or fitted constants. The three examples are well-chosen and illustrate both the obstructed and unobstructed cases. The infinite-dimensional caveat for continuous-variable systems (Sec. V) is explicitly acknowledged and verified for the QHO example where scalar prefactors in Eq. (48) are indeed bounded.","major_comments":[{"comment":"Sec. III, Eqs. (12)-(16): The proof that λ_min = O(t^0) proceeds by exhibiting a specific direction w for which w^T F w = O(t^0). This establishes an upper bound on λ_min. However, the conclusion in Eq. (16) that Tr(F^{-1}) ≥ 1/λ_min ~ t^0 requires that λ_min does not accidentally vanish or scale worse than t^0 due to the structure of K_μ(t) in the slow subspace. The argument is correct when the slow-subspace QFIM block is non-singular (which is generically the case), but the manuscript should state more explicitly that the O(t^0) conclusion assumes the variance of Σ w_μ K_μ(t) is strictly positive and bounded away from zero for the chosen probe state. If this variance vanishes for a particular |ψ_0⟩, the slow direction becomes unestimable (singular QFIM), which is a different failure mode. Clarifying this genericity assumption would strengthen the no-go claim.","section":null},{"comment":"Sec. VII.B, Eqs. (85)-(90): The incompatibility analysis is restricted to the two-parameter case. The manuscript states (end of Sec. VII.B) that 'we conjecture that for any number of parameters the O(t^0) contribution to the incompatibility is determined solely by the slow subspace.' This conjecture is load-bearing for the generality of the asymptotic saturability claim, yet it is unproven. The step from the 2×2 identity in Eq. (88) to the general p-parameter case is non-trivial because the trace norm of F^{-1}UF^{-1} for p > 2 involves a more complex singular-value structure. The authors should either (a) restrict the asymptotic saturability claim to two parameters in the abstract and conclusions, or (b) provide at least a sketch of why the conjecture is expected to hold (e.g., by block-decomposition into fast/slow subspaces and noting that the fast-fast block of U vanishes asymptotical","section":null},{"comment":"Sec. V, Eq. (48) and surrounding text: For the QHO, K_μ(t) contains unbounded operators (ĉ^2, ĉ†^2). The manuscript correctly notes that the scalar prefactors are bounded and that one must verify boundedness case by case. However, the statement 'K_μ(t) ~ O(t^0)' in Eq. (10) is stated for finite-dimensional systems, and the extension to the QHO is justified only by the boundedness of prefactors, not of the operators themselves. The variance Var(Q̂(t)) in Eq. (56) is finite only because of the restriction to Gaussian states with fixed energy. The manuscript should clarify that the O(t^0) scaling of the QFIM in the slow direction is not a purely operator-theoretic result here but depends on the state-space restriction. This is acknowledged but the logical flow could be tighter.","section":null}],"minor_comments":[{"comment":"Sec. III, Eq. (11): The notation D̃_μ is introduced for the traceless diagonal generator, but in subsequent equations (e.g., Eq. 13, Eq. 17) the tilde is sometimes dropped or inconsistently applied. Standardize the notation throughout.","section":null},{"comment":"Sec. IV.D, Eq. (40): The comparison with Ref. [33] is useful, but the factor (N+2)/N difference is explained somewhat informally. A brief explicit statement of how the two optimization problems differ in dimensionality would help the reader.","section":null},{"comment":"Sec. VI, Fig. 1: The y-axis label 'N^3 Tr[G^{-1}]' and the caption could clarify what the reference value in the strong-field limit is, so the reader can assess how much the quantity varies across regimes.","section":null},{"comment":"Sec. VII.C, Eq. (93): The condition ker G ⊆ ker W is elegant and could be highlighted more prominently, perhaps in the abstract or introduction, as it provides a concise operational criterion for when quadratic scaling is recoverable.","section":null},{"comment":"References [47, 48] are dated 2026; verify these are correctly cited and not preprints with updated dates.","section":null},{"comment":"Sec. II: The SLD bound is introduced with W = I, but the general weighted bound (Eq. 91) appears only in Sec. VII.C. A forward reference would help readers who wonder about general weight matrices early on.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid contribution with a clean no-go theorem and good examples. The main technical concern is the unproven conjecture about the incompatibility penalty for p > 2 parameters (Sec. VII.B), which is stated as a conjecture but implicitly assumed in some of the broader claims in the abstract and conclusions. The authors should be asked to either prove it, restrict the claim, or clearly label it as a conjecture in all instances where it is invoked. The other major comments are about clarifying assumptions that are acknowledged but not fully integrated into the logical flow. None of these require fundamental rework. I see no issues with citation patterns or novelty disclosure; the relevant prior work (Refs. [8, 32, 33, 36, 37]) is properly acknowledged."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. All three major comments identify legitimate points that warrant clarification in the manuscript. We address each below and will incorporate revisions accordingly.","responses":[{"response":"The referee is correct that our argument establishes an upper bound on λ_min by exhibiting a direction w with w^T F w = O(t^0), and that the conclusion Tr(F^{-1}) ≥ 1/λ_min ~ t^0 requires the slow-subspace QFIM block to be non-singular. We agree that this genericity assumption should be stated explicitly. We will revise the text following Eq. (16) to clarify two points: (1) the O(t^0) scaling of the no-go bound holds when Var_{ψ_0}(Σ w_μ K_μ(t)) is strictly positive, which is the generic case for physically reasonable probe states; and (2) if this variance vanishes for a particular |ψ_0⟩, the QFIM becomes singular along the slow direction, which is a distinct (and in some sense more severe) failure mode. We emphasize that the no-go theorem is not weakened by this clarification: in both cases—whether the slow-subspace variance is positive (yielding O(t^0) scaling) or zero (yielding a singular QFIM)—simultaneous O(t^{-2}) scaling is unachievable. The genericity assumption only distinguishes which failure mode occurs, not whether failure occurs.","revision_made":"partial","referee_comment":"Sec. III, Eqs. (12)-(16): The proof that λ_min = O(t^0) proceeds by exhibiting a specific direction w for which w^T F w = O(t^0). This establishes an upper bound on λ_min. However, the conclusion in Eq. (16) that Tr(F^{-1}) ≥ 1/λ_min ~ t^0 requires that λ_min does not accidentally vanish or scale worse than t^0 due to the structure of K_μ(t) in the slow subspace. The argument is correct when the slow-subspace QFIM block is non-singular (which is generically the case), but the manuscript should state more explicitly that the O(t^0) conclusion assumes the variance of Σ w_μ K_μ(t) is strictly positive and bounded away from zero for the chosen probe state. If this variance vanishes for a particular |ψ_0⟩, the slow direction becomes unestimable (singular QFIM), which is a different failure mode. Clarifying this genericity assumption would strengthen the no-go claim."},{"response":"The referee correctly identifies that the asymptotic saturability result is proven only for two parameters and that the extension to p > 2 is conjectural. We agree that the conjecture should not be presented as established. We will take approach (b): we will add a paragraph sketching the expected argument for the general case. The key observation is that in the fast/slow decomposition, the Uhlmann curvature matrix U has a block structure where the fast-fast block vanishes asymptotically (since fast-direction diagonal generators commute in non-degenerate Hamiltonians, making U_{ff} = O(t^0) while F_{ff} = O(t^2)), the fast-slow block scales as O(t) (as shown in Eq. 87), and the slow-slow block is O(t^0). Since F^{-1} has blocks scaling as O(t^{-2}) (fast-fast), O(t^{-1}) (fast-slow), and O(t^0) (slow-slow), the product F^{-1} U F^{-1} has entries that are at most O(t^{-1}) from the fast-slow coupling, with the O(t^0) contribution arising solely from the slow-slow block. While a complete proof for general p requires careful analysis of the singular-value structure of the resulting matrix (which may have dimension > 2 in the slow subspace), the scaling of individual entries supports the conjecture. We will also add a qualifying clause in the abstract and conclusions noting that the O(t^{-1}) incompatibility decay is proven for two parameters and conjectured for the general case.","revision_made":"partial","referee_comment":"Sec. VII.B, Eqs. (85)-(90): The incompatibility analysis is restricted to the two-parameter case. The manuscript states (end of Sec. VII.B) that 'we conjecture that for any number of parameters the O(t^0) contribution to the incompatibility is determined solely by the slow subspace.' This conjecture is load-bearing for the generality of the asymptotic saturability claim, yet it is unproven. The step from the 2×2 identity in Eq. (88) to the general p-parameter case is non-trivial because the trace norm of F^{-1}UF^{-1} for p > 2 involves a more complex singular-value structure. The authors should either (a) restrict the asymptotic saturability claim to two parameters in the abstract and conclusions, or (b) provide at least a sketch of why the conjecture is expected to hold (e.g., by block-decomposition into fast/slow subspaces and noting that the fast-fast block of U vanishes asymptotical"},{"response":"The referee is correct that the O(t^0) scaling in the QHO example is not purely operator-theoretic but depends on the restriction to Gaussian states with fixed average energy. The unboundedness of the operators ĉ^2 and ĉ†^2 means that K_μ(t) = O(t^0) holds at the level of scalar prefactors, but the finiteness of the variance Var(Q̂(t)) in Eq. (56) requires the state-space restriction. We will tighten the logical flow in Sec. V as follows: (1) after Eq. (48), we will explicitly state that the O(t^0) designation refers to the boundedness of the time-dependent scalar prefactors, and that the physical relevance of this bound for the QFIM requires verifying that the operator variances are finite for the chosen class of probe states; (2) at the beginning of Sec. V.B, we will move the statement about the restriction to Gaussian states with fixed ⟨n̂⟩ earlier, before presenting the QFIM entries, so that the reader sees the state-space assumption before encountering Eq. (56). This reorganization will make clear that the no-go result in the CV setting is a statement about the scaling of the QFIM within a physically motivated state class, not a purely algebraic property of the operators.","revision_made":"partial","referee_comment":"Sec. V, Eq. (48) and surrounding text: For the QHO, K_μ(t) contains unbounded operators (ĉ^2, ĉ†^2). The manuscript correctly notes that the scalar prefactors are bounded and that one must verify boundedness case by case. However, the statement 'K_μ(t) ~ O(t^0)' in Eq. (10) is stated for finite-dimensional systems, and the extension to the QHO is justified only by the boundedness of prefactors, not of the operators themselves. The variance Var(Q̂(t)) in Eq. (56) is finite only because of the restriction to Gaussian states with fixed energy. The manuscript should clarify that the O(t^0) scaling of the QFIM in the slow direction is not a purely operator-theoretic result here but depends on the state-space restriction. This is acknowledged but the logical flow could be tighter."}],"tokens_in":19768,"tokens_out":1577,"duration_ms":280397,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main result is a clean no-go theorem: when the traceless diagonal components of the Hamiltonian derivatives are linearly dependent, there exists a slow parameter direction where the QFI stays O(t^0), killing simultaneous t^{-2} scaling. The proof is straightforward and correct — linear dependence gives you a direction w where the t-proportional term vanishes, leaving only the bounded off-diagonal K_mu(t), so the smallest eigenvalue of the QFIM is O(t^0) and Tr(F^{-1}) can't beat O(t^0). The Gram matrix determinant as a computable diagnostic for this condition is the practical payoff and is genuinely useful for anyone designing multiparameter estimation protocols. The dimensional bound (at most d-1 parameters with t^{-2} scaling in a d-dimensional system) is a nice corollary. The three examples are well-chosen: collective spin magnetometry and the QHO both illustrate the obstruction, and the LMG model provides the contrast case where linear independence holds and t^{-2} survives. The asymptotic saturability result (incompatibility penalty decays as 1/t) in Section VII.B is a bonus — it shows the SLD bound is tight in the long-time limit for the two-parameter case, though the extension to arbitrary parameter counts remains a conjecture. The infinite-dimensional caveat is real but handled honestly. The authors flag it explicitly in Section V and verify boundedness of K_mu for the QHO example. The restriction to Gaussian states with fixed energy in the QHO case is a physical necessity, not a dodge. Minor points: the ancilla argument is correct but brief — one sentence saying linear dependence is preserved under tensoring with identity. That's fine for the finite-dimensional case but could use a sentence on what happens in infinite dimensions. The conjecture at the end of Section VII.B about the slow subspace determining the incompatibility for arbitrary parameter counts is left unproven; it's labeled as a conjecture, which is honest, but a referee should ask whether there's a counterexample. Overall, the core argument holds up. The paper is for theorists working on multiparameter quantum metrology bounds — it gives them a general framework and a diagnostic tool that replaces model-specific calculations. It deserves a serious referee.","headline":"Clean no-go theorem for loss of t^{-2} scaling in multiparameter quantum metrology; the Gram-matrix diagnostic is the genuinely useful new contribution.","tokens_in":20310,"tokens_out":539,"would_cite":true,"duration_ms":74820,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Linear dependence kills quadratic precision in multiparameter quantum sensing","keywords":["multiparameter quantum estimation","quantum Fisher information","Heisenberg scaling","geometric obstruction","Hamiltonian parameter estimation","measurement incompatibility","quantum metrology","diagonal decomposition"],"falsifier":"Find a multiparameter estimation problem where the diagonal generators are linearly dependent but the smallest eigenvalue of the QFIM still grows as t^2 — this would require the off-diagonal generators K_μ(t) to themselves grow with t, which the paper argues does not happen in finite dimensions.","tokens_in":19489,"feed_emoji":"📐","tokens_out":1268,"duration_ms":137448,"temperature":0.7,"pith_summary":"This paper proves that when multiple parameters are encoded into a quantum system through a time-independent Hamiltonian, simultaneous quadratic-in-time precision (the gold standard of quantum metrology) can fail for a purely geometric reason. The authors decompose each Hamiltonian derivative into a part that commutes with the system Hamiltonian (the diagonal part) and a part that does not (the off-diagonal part). They show that if these diagonal parts are linearly dependent — meaning one parameter direction is redundant with respect to the energy spectrum — then along that direction the local generator loses its term proportional to evolution time t, leaving only a bounded off-diagonal contribution. The quantum Fisher information along this slow direction stays constant in time rather than growing as t^2, capping the achievable precision at O(t^0) no matter what probe state or entanglement strategy is used. The authors provide a simple diagnostic — the Gram matrix of the diagonal generators — whose determinant is zero precisely when this obstruction occurs. They demonstrate the mechanism in collective spin magnetometry and a quantum harmonic oscillator, and show a contrasting case (the Lipkin–Meshkov–Glick model) where diagonal generators remain independent and quadratic scaling survives. A key secondary result is that measurement incompatibility between fast and slow directions decays as 1/t, so the standard precision bound becomes saturable in the long-time limit even though the slow direction remains bottlenecked.","feed_headline":"Linear dependence kills quadratic precision in multiparameter quantum sensing","feed_subtitle":"A geometric obstruction caps precision at O(t^0) whenever diagonal Hamiltonian derivatives are linearly dependent — but the diagnostic is a ","key_machinery":"diagonal/off-diagonal decomposition of Hamiltonian derivatives; Gram matrix G_μν = Tr(D_μ D_ν) as computable diagnostic; fast/slow basis rotation in parameter space; dimensional bound: at most d-1 parameters can achieve O(t^{-2}) in a d-dimensional system","core_discovery":"The central object is the decomposition of each Hamiltonian derivative ∂_μ H into a diagonal part D_μ (commuting with H) and an off-diagonal part O_μ (not commuting with H). The local generator H_μ(t) then splits into a term -t·D_μ that grows linearly with time and a bounded term -K_μ(t) that stays O(t^0). When the traceless diagonal generators {D_μ} are linearly dependent, there exists a direction w in parameter space where the t-proportional term vanishes identically, leaving only the bounded K_μ(t) contribution. This forces the smallest eigenvalue of the quantum Fisher information matrix to remain O(t^0), which in turn caps the total estimation precision. The authors prove this holds for ","pith_inferences":[],"forward_implications":["Any multiparameter quantum metrology protocol using time-independent Hamiltonian encoding can be cheaply diagnosed: compute the Gram matrix of diagonal generators, and if its determinant is zero, at least one parameter direction is stuck at O(t^0) precision regardless of probe engineering.","In a d-dimensional quantum system, attempting to estimate d or more parameters simultaneously will always trigger this obstruction, setting a hard dimensional ceiling on the number of parameters that can enjoy Heisenberg-like scaling.","The 1/t decay of measurement incompatibility means that even in obstructed cases, the gap between the SLD bound and the tighter Holevo bound vanishes asymptotically — the bottleneck is purely geometric, not a measurement incompatibility problem.","Adaptive quantum control that cancels the system Hamiltonian can restore t^2 scaling by making all derivatives commute with the (zero) total Hamiltonian, but requires prior knowledge of the true parameter values.","Sequential measurement strategies with finite interrogation times can recover O(T^{-1}) scaling for slow directions over total experimental time T, providing a practical workaround when adaptive control is unavailable."],"fun_headline_variants":["Geometric obstruction caps multiparameter quantum sensing precision","Linearly dependent Hamiltonian derivatives bound metrology precision","Diagonal generator dependence limits time scaling in quantum estimation","Slow parameter directions cap precision in multiparameter metrology","Gram matrix criterion detects scaling limits in multiparameter sensing"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The proof requires that the off-diagonal generators K_μ(t) remain bounded (O(t^0)) as time grows. This is guaranteed for finite-dimensional systems but must be checked case by case for infinite-dimensional systems like the quantum harmonic oscillator, where the relevant operators are unbounded. If K_μ(t) were to grow with t in some infinite-dimensional setting, the O(t^0) bottleneck would not hold.","fun_headline_variants_meta":{"raw":{"variants":["Geometric obstruction caps multiparameter quantum sensing precision","Linearly dependent Hamiltonian derivatives bound metrology precision","Diagonal generator dependence limits time scaling in quantum estimation","Slow parameter directions cap precision in multiparameter metrology","Gram matrix criterion detects scaling limits in multiparameter sensing"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1055,"prompt_tokens":601,"completion_tokens":454,"prompt_tokens_details":null},"tokens_in":601,"tokens_out":454,"duration_ms":41384,"temperature":1.0,"reasoning_tokens":468,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T06:31:39.820755+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Find a multiparameter estimation problem where the diagonal generators are linearly dependent but the smallest eigenvalue of the QFIM still grows as t^2 — this would require the off-diagonal generators K_μ(t) to themselves grow with t, which the paper argues does not happen in finite dimensions.","supporting_citations":[],"review_version":1}