{"id":"eb67ee1c-1da5-4a9c-b05d-563deb153527","arxiv_id":"2607.07556","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"In random quantum automaton ensembles, the subsystem symmetrization scale depends on the initial state's participation entropy, and the onset of U(1) entanglement asymmetry coincides with the onset of subsystem coherence.","lead":"This paper studies how quantum symmetry breaking spreads in random quantum automaton circuits, finding that the scale at which symmetry restoration occurs depends on the initial state's participation entropy. This matters because it reveals a sharp distinction from standard Haar-random circuits, where the symmetrization scale is universal.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Self-averaging approximation (Eq. 16) is standard but unproven for the QAE; finite-size error is unbounded.","rationale":"The reader's verdict of ACCEPT with HIGH confidence is appropriate. The self-averaging approximation is genuinely standard in the random circuit literature (used identically in [78, 80]), and the paper provides both analytical derivation and numerical cross-checks. The concern is real but bounded: it affects quantitative precision near the onset, not the qualitative physics. The paper clearly flags the heuristic nature of the coherence-asymmetry coincidence argument (Section V.B, 'we do not expect any preference'), and the decoupling inequality (Section V.A) provides independent partial support. The ODE system (Eqs. 46-55) is a genuine technical contribution with a verified stationary solution matching the QAE. No circularity, no post-hoc exclusions. The reader correctly weighted these factors. My only adjustment to the reader's framing: the correctness risk should be explicitly noted as 'moderate' rather than 'unknown' for the finite-size regime, but this does not change the verdict. The paper makes a solid contribution: the L*_A formula, the QAE-QAC equivalence proof, and the coherence connection are novel and well-supported. The self-averaging gap is the right thing to flag but is not severe enough to downgrade from ACCEPT.","tokens_in":30086,"tokens_out":783,"duration_ms":470156,"concrete_test":"Compute E[log Tr ρ_A^2] directly by Monte Carlo sampling over the QAE (not using the replacement) for L=80, θ=π/16, and L_A near L*_A = L - SPE_2. Compare the resulting asymmetry curve point-by-point with Eq. (22). If the deviation near L*_A exceeds the subleading corrections the paper drops (i.e., more than a few percent of log(πL_A)/2), the exactness of the onset formula is questionable at this system size.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader correctly identifies the self-averaging replacement E[log Tr ρ^n] ≈ log E[Tr ρ^n] in Eq. (16) as the weakest link. This step is load-bearing: every analytical Page curve (Eqs. 22, 26, 31) and the coherence-asymmetry coincidence (Section V.B) flow from it. The paper states this 'holds as long as there is a self-averaging property' but provides no proof or error bound for the QAE at finite L. Two specific concerns: (1) The QAE is not Haar-random — it preserves participation entropy, so the distribution of Tr ρ_A^2 has different concentration properties. For highly localized initial states (small θ, where the novel L*_A shift is largest), the number of effectively occupied basis states is small, which could worsen concentration and make the replacement less accurate precisely in the regime where the paper's main new physics appears. (2) The numerical checks (Figs. 2-5) at L=80 show visual agreement but no quantitative comparison of analytical vs. numerical curves near the onset L*_A, where the asymmetry transitions from zero to O(log L_A). This transition region is where self-averaging is most likely to fail, as it involves a competition between exponentially small terms. The concern is not that the approximation is wrong — it is standard in this literature — but that the paper's central quantitative claim (the exact formula L*_A = max(L/2, L - SPE_2)) depends on distinguishing exponentially small terms in Eq. (24), where even small self-averaging errors could shift the onset.","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This paper studies the U(1) entanglement asymmetry in random quantum automaton ensembles (QAE) and 2-local quantum automaton circuits (QAC). The central result is that the symmetrization scale $L_A^* = max(L/2, L - SPE_2(|ψ_0⟩))$ governs the onset of subsystem asymmetry, shifting from the Haar-random value $L/2$ to a participation-entropy-dependent value when the initial state is sufficiently localized. The authors derive this from the QAE second moment (Eq. 20), show that the 2-local circuit ODEs (Eq. 46) converge to the same stationary solution (Eq. 55), and connect the asymmetry onset to the growth of subsystem coherence. The derivations are carried out from first principles using the Choi-Jamiolkowski representation, with no free parameters or fitted quantities.","tokens_in":30771,"tokens_out":1282,"duration_ms":371602,"significance":"The paper makes a clean, parameter-free prediction for the symmetrization scale $L_A^*$ and identifies a novel mechanism—interplay between participation-entropy conservation and uniform charge-sector exploration—that is absent in Haar-random circuits. The derivation of the charged partition function (Eq. 20) and the ODE system (Eq. 46) is careful and self-contained. The stationary solution (Eq. 55) is verified by direct substitution, and the numerical integration of the ODE system at L=80 (Figs. 4-5) confirms the analytical Page curves. The connection between asymmetry onset and coherence onset (Section V.B) provides a useful physical interpretation. The results are falsifiable and the framework is extensible to other resource monotones.","major_comments":[{"comment":"§III.A, Eqs. (22)–(25): The self-averaging approximation in Eq. (16), replacing E[log Tr ρ^n] with log E[Tr ρ^n], is load-bearing for all analytical Page curves (Eqs. 22, 26, 31). The paper states this holds 'as long as there is a self-averaging property in the ensemble' but provides no proof or quantitative error bound for the QAE at finite L. This is a standard approximation in the literature, but the concern is specific: the QAE preserves participation entropy, so the distribution of Tr ρ_A^2 has different concentration properties than Haar-random ensembles. For highly localized initial states (small θ, where the novel L*_A shift is largest), the number of effectively occupied basis states is small, which could worsen concentration precisely in the regime where the paper's main new physics appears. Additionally, the numerical checks (Figs. 2–5) at L=80 show visual agreement but no定量比较","section":null},{"comment":"§V.B, Eq. (68) and surrounding text: The claim that the coherence onset coincides with the asymmetry onset relies on the heuristic argument that 'we do not expect any preference in the onset of the off-diagonal entries.' This is plausible given the uniform distribution over permutations and phases, but it is not proven. The decomposition ρ_A = ω_A + χ_A^{in} + χ_A^{out} shows that coherence onset can in principle occur before asymmetry onset (if χ_A^{in} grows before χ_A^{out}). The paper argues this does not happen because the QAE evolution 'does not preserve the U(1) charge,' but this does not by itself rule out different growth rates for χ_A^{in} versus χ_A^{out} at intermediate scales. A more precise argument, or at least an acknowledgment that this is a heuristic supported by numerics, would strengthen the claim.","section":null}],"minor_comments":[{"comment":"Fig. 2 (right panel): The finite-size scaling shows curves for L=10, 20, 100, 1000 but the main text states L=80 for the left panel. Clarify whether the right panel uses different system sizes and, if so, explain the choice.","section":null},{"comment":"Eq. (53): The hypergeometric function $_2F_1(-L_A, 1/2; 1; sin^2(2θ))$ is stated to reproduce a result from [54]. A brief note on the connection (e.g., whether it is the n=2 specialization of the general formula) would help the reader.","section":null},{"comment":"§IV.A, Eq. (40): The notation $|G^α_{n_α, n_-, n_+, n_0}⟩$ introduces four indices whose combinatorial structure is important. A brief comment on the dimension of the state space ($∼L^3$ independent functions) is given later but would be useful already at the definition.","section":null},{"comment":"Appendix B, Eq. (B4): There appears to be a stray 'q' in the last term of the equation (and in Eq. B5), which seems to be the local Hilbert space dimension q=2. If this is intentional, please clarify; if not, it should be removed for consistency with Eqs. (46/47).","section":null},{"comment":"Fig. 1: The schematic is helpful but the axes and quantities shown could be labeled more precisely (e.g., what is plotted on each axis of the sketched curves).","section":null},{"comment":"References: Several arXiv-only references (e.g., [41], [44], [48], [74], [75]) could be updated with published versions where available.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The self-averaging concern raised by the reader is legitimate but, in my assessment, does not rise to the level of a major revision: the approximation is standard in this literature, the numerical checks at L=80 provide reasonable evidence, and the central formula L*_A = max(L/2, L - SPE_2) is derived from the exact second moment without fitting. A brief discussion acknowledging the finite-size limitation and pointing to the numerical agreement would suffice. The coherence-asymmetry coincidence claim is more heuristic and should be flagged as such, but it is a secondary result and does not undermine the main quantitative claim."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful reading and constructive report. The referee raises two major comments: (1) the self-averaging approximation E[log Tr ρ^n] ≈ log E[Tr ρ^n] lacks a proof or quantitative error bound for the QAE at finite L, with particular concern about the regime of highly localized initial states; and (2) the claim that coherence onset coincides with asymmetry onset relies on a heuristic argument rather than a rigorous proof. Both comments are well-taken. We will address them by adding a quantitative finite-size scaling analysis of the self-averaging property (including in the small-θ regime) and by revising the language around the coherence-asymmetry coincidence to explicitly acknowledge it as a heuristic supported by numerics, while providing additional analytical backing from the structure of the QAE second moment.","responses":[{"response":"The referee correctly identifies that the self-averaging approximation in Eq. (16) is load-bearing and that its justification for the QAE at finite L—particularly in the small-θ regime—is insufficient in the current manuscript. We agree and will address this in two ways. First, we will add a quantitative finite-size scaling analysis: for homogeneous product states with θ = π/16 (the most localized case shown in our figures), we will compute the ratio Var[Tr ρ_A^2] / (E[Tr ρ_A^2])^2 as a function of L for several values of L_A, including L_A near L_A^*, and show that it decays with system size. This directly tests concentration in the regime of concern. Second, we note that the referee's concern about small effective support is partially mitigated by the structure of the QAE second moment: although the initial state may occupy few basis states, the random permutation spreads these over D basis states, and the charged partition function E[Z_A^(2)(α)] in Eq. (20) depends on I_2(|ψ_0⟩) and D_A f(α)^{L_A} in a way that is analytically controlled. The effective number of occupied basis states after permutation is min(K, D_A) where K = 2^{SPE_2} is the effective support, and for L_A < L_A^* the relevant observables are dominated by the extensive factor 2^{L-L_A} rather than by K. Nevertheless, we agree that a quantitative bound or at least a numerical demonstration of concentration is needed, and we will add it. We will also add quantitative error bars or residuals to the numerical comparisons in Figs. 2–5, replacing the purely visual agreement with explicit measures of deviation between the ODE integration and the analytical Page curves.","revision_made":"yes","referee_comment":"§III.A, Eqs. (22)–(25): The self-averaging approximation in Eq. (16), replacing E[log Tr ρ^n] with log E[Tr ρ^n], is load-bearing for all analytical Page curves (Eqs. 22, 26, 31). The paper states this holds 'as long as there is a self-averaging property in the ensemble' but provides no proof or quantitative error bound for the QAE at finite L. This is a standard approximation in the literature, but the concern is specific: the QAE preserves participation entropy, so the distribution of Tr ρ_A^2 has different concentration properties than Haar-random ensembles. For highly localized initial states (small θ, where the novel L*_A shift is largest), the number of effectively occupied basis states is small, which could worsen concentration precisely in the regime where the paper's main new physics appears. Additionally, the numerical checks (Figs. 2–5) at L=80 show visual agreement but no定量比较"},{"response":"The referee is correct that the argument in §V.B is heuristic and that the statement 'we do not expect any preference in the onset of the off-diagonal entries' is not a proof. We acknowledge that the decomposition ρ_A = ω_A + χ_A^{in} + χ_A^{out} does not, by itself, rule out different growth rates for χ_A^{in} and χ_A^{out} at intermediate scales. We will revise the manuscript to explicitly state that the coincidence of coherence and asymmetry onset is a heuristic argument supported by two ingredients: (1) the analytical computation showing that both onset conditions reduce to the same inequality Eq. (25)/Eq. (75) in the thermodynamic limit, and (2) the numerical evidence from Figs. 4–7 showing simultaneous onset. We will also add a brief discussion of why the QAE structure makes different growth rates unlikely: the second moment E_QAE[U*⊗U⊗U*⊗U] in Eq. (11) treats the |I^0⟩ component (which controls χ_A^{in} via the dephased purity) and the |I^-⟩ component (which controls χ_A^{out} via the charged moments) through the same permutation average, so there is no structural mechanism in the second moment that would favor one over the other. However, we agree this falls short of a rigorous proof, and we will state this limitation clearly. We will also note that a rigorous proof would require control of higher moments of the charged partition function, which is beyond the scope of this work.","revision_made":"partial","referee_comment":"§V.B, Eq. (68) and surrounding text: The claim that the coherence onset coincides with the asymmetry onset relies on the heuristic argument that 'we do not expect any preference in the onset of the off-diagonal entries.' This is plausible given the uniform distribution over permutations and phases, but it is not proven. The decomposition ρ_A = ω_A + χ_A^{in} + χ_A^{out} shows that coherence onset can in principle occur before asymmetry onset (if χ_A^{in} grows before χ_A^{out}). The paper argues this does not happen because the QAE evolution 'does not preserve the U(1) charge,' but this does not by itself rule out different growth rates for χ_A^{in} versus χ_A^{out} at intermediate scales. A more precise argument, or at least an acknowledgment that this is a heuristic supported by numerics, would strengthen the claim."}],"tokens_in":29849,"tokens_out":1324,"duration_ms":221579,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper identifies that the U(1) entanglement asymmetry symmetrization scale in quantum automaton ensembles shifts from the Haar value L/2 to max(L/2, L - SPE_2) depending on the initial state's participation entropy. That is a genuinely new result, and the connection between asymmetry onset and coherence onset at the same scale is a nice second finding that gives the physics a clean interpretation. The QAE second moment (Eq. 11) was already computed in Ref. 32 for purity, so the technical machinery is not from scratch — but applying it to charged partition functions and extracting the shifted onset formula (Eq. 25) is new work. The derivation from Eq. 20 through Eq. 26 is clean and each step is justified. The ODE system for the 2-local circuit (Eq. 46) is derived carefully in Appendix B, and the stationary solution (Eq. 55) is verified by direct substitution, which proves the QAE and QAC(∞) averages coincide at finite size. That is a solid analytical result. The decoupling inequality (Eq. 63) and the coherence-onset argument (Section V.B) together give a coherent physical picture for why the shift happens. The soft spot is the self-averaging approximation in Eq. 16 — replacing E[log Tr ρ^n] with log E[Tr ρ^n]. The stress-test note flags this as load-bearing and unproven for the QAE at finite L, and specifically worries that for highly localized initial states (small θ), where the novel shift is largest, concentration could be worse. I think the concern is real but proportionate: the approximation is standard in this literature (same step appears in Refs. 78, 80), and the numerical checks at L=80 show visual agreement. However, there is no quantitative comparison of analytical vs. numerical curves near the onset L*_A, which is exactly where self-averaging is most likely to strain. A finite-size error bound or a sharper numerical comparison near the transition would strengthen the claim. The heuristic argument in Section V.B that coherence and asymmetry onsets coincide — 'we do not expect any preference in the onset of the off-diagonal entries' — is plausible but not proven. The paper flags it as an expectation, which is honest, but a reader should treat it as a conjecture supported by numerics rather than a theorem. No code or data is shipped, but the methods are specified in enough detail for reimplementation. No circularity, no post-hoc exclusions. This paper is for researchers working on quantum many-body dynamics, random circuits, and quantum resource theories. It deserves a serious referee who can check the asymptotic analysis in Eqs. 23-26 and push on the self-averaging question. I lean positive — the core result is new, the derivation is sound, and the soft spots are the kind that a good revision can address.","headline":"Participation-entropy-dependent symmetrization scale is a clean new result for quantum automaton ensembles","tokens_in":31050,"tokens_out":681,"would_cite":true,"duration_ms":172234,"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":"Symmetry restoration in quantum automata depends on the initial state","keywords":[],"falsifier":"If the self-averaging approximation fails at finite system sizes, the analytical Page curves (Eqs. 22, 26, 31) would deviate from numerical QAE sampling for moderate L, and the sharp onset at L*_A would be smeared rather than abrupt.","tokens_in":30047,"feed_emoji":"🎲","tokens_out":1283,"duration_ms":69527,"temperature":0.7,"pith_summary":"The paper studies how U(1) charge symmetry is restored in subsystems of random quantum automaton ensembles — circuits that randomly permute computational basis states and apply random phases. In standard Haar-random circuits, any subsystem smaller than half the total system symmetrizes (loses its asymmetry) at late times, with the threshold fixed at L/2. The authors show that in quantum automaton ensembles this threshold shifts to L*_A = max(L/2, L - SPE_2(|ψ_0⟩)), where SPE_2 is the participation entropy of the initial state — a measure of how spread out the state is in the Hilbert space. The mechanism is that quantum automaton operations preserve the participation entropy, so a highly localized initial state cannot generate enough off-diagonal structure in small subsystems to break the symmetry. The paper proves this for the global all-to-all ensemble (QAE) and shows numerically that the infinite-depth limit of a 2-local circuit (QAC) converges to the same stationary asymmetry profile. A second result ties the asymmetry onset to subsystem coherence: the Rényi-2 relative entropy of coherence begins growing at exactly the same subsystem size L*_A, because both phenomena are governed by whether the off-diagonal entries of the reduced density matrix can become comparable in norm to the diagonal ones.","feed_headline":"Symmetry threshold in quantum automata shifts with initial state","feed_subtitle":"Random permutation circuits preserve a state's localization, pushing the onset of charge-symmetry breaking past the Haar-random value of L/2","key_machinery":"Choi-Jamiolkowski vectorization in four-replica Hilbert space; charged partition functions Z^(2)_A(α) = Tr[e^{iαQ_A} ρ_A e^{-iαQ_A} ρ_A]; permutation-invariant states |G^α_{n_α,n_-,n_+,n_0}⟩ enabling a closed system of O(L^3) linear ODEs for the 2-local circuit dynamics; decoupling inequality bounding the trace distance between ρ_A and the maximally mixed state.","core_discovery":"The central object is the symmetrization scale L*_A = max(L/2, L - SPE_2(|ψ_0⟩)), which determines the subsystem size at which U(1) entanglement asymmetry and subsystem coherence simultaneously begin to grow in random quantum automaton ensembles. Unlike Haar-random circuits where this scale is universally L/2, here it depends on the initial state's participation entropy, which is conserved by the automaton dynamics. The paper derives this scale analytically for the global ensemble, confirms it numerically for the 2-local circuit at infinite depth, and shows that both the asymmetry Page curve and the coherence Page curve share the same onset.","pith_inferences":[],"forward_implications":["For initial states with participation entropy growing linearly in L (e.g., homogeneous product states with θ near π/4), the threshold L*_A approaches L/2 and the QAE reproduces Haar-random behavior; for localized states the threshold can shift dramatically, requiring subsystems nearly as large as the full system before asymmetry appears.","For Dicke states |D_k⟩, the critical k below which the threshold shifts from L/2 scales as k < 0.11L, giving a concrete phase boundary in the (k, L) plane.","The total system's Rényi-2 coherence equals its participation entropy and is exactly conserved, making the initial state's localization a permanent constraint on all downstream resource dynamics.","The same ODE system that governs asymmetry dynamics also yields coherence dynamics for free, suggesting that other quantum resource monotones (nonstabilizerness, non-Gaussianity) might be accessible through similar replica-trick extensions."],"fun_headline_variants":["Initial state dictates symmetry onset in quantum automata","State-dependent symmetrization in random quantum automata","Quantum automata symmetrization scale relies on initial state","Initial localization sets entanglement asymmetry onset in automata","Random quantum automata symmetry threshold varies by initial state"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The analytical results rely on replacing the average of a logarithm with the logarithm of an average (the self-averaging approximation in Eq. 16). The authors state this is valid for the quantities of interest but do not provide a proof or finite-size error bound. Additionally, the claim that the coherence onset coincides exactly with the asymmetry onset rests on a heuristic argument that the random phases and permutations do not preferentially populate intra-sector versus跨跨-","fun_headline_variants_meta":{"raw":{"variants":["Initial state dictates symmetry onset in quantum automata","State-dependent symmetrization in random quantum automata","Quantum automata symmetrization scale relies on initial state","Initial localization sets entanglement asymmetry onset in automata","Random quantum automata symmetry threshold varies by initial state"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1055,"prompt_tokens":489,"completion_tokens":566,"prompt_tokens_details":null},"tokens_in":489,"tokens_out":566,"duration_ms":26548,"temperature":1.0,"reasoning_tokens":568,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T06:58:46.871330+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If the self-averaging approximation fails at finite system sizes, the analytical Page curves (Eqs. 22, 26, 31) would deviate from numerical QAE sampling for moderate L, and the sharp onset at L*_A would be smeared rather than abrupt.","supporting_citations":[],"review_version":1}