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Symmetry restoration in quantum automata depends on the initial state

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2026-07-09 06:58 UTC pith:NAQDVR7B

load-bearing objection Participation-entropy-dependent symmetrization scale is a clean new result for quantum automaton ensembles the 2 major comments →

arxiv 2607.07556 v1 pith:NAQDVR7B submitted 2026-07-08 cond-mat.stat-mech quant-ph

Entanglement Asymmetry in Random Quantum Automata

Olalla A. Castro-Alvaredo , D\'avid Sz\'asz-Schagrin , Michele Mazzoni This is my paper
classification cond-mat.stat-mech quant-ph
keywords subsystemasymmetryensembleaveragedifferententanglementgrowthhaar-random
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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.

What carries the argument

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.

Load-bearing premise

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-

What would settle it

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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

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.

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 (2)
  1. §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
  2. §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.
minor comments (6)
  1. 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.
  2. 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.
  3. §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.
  4. 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).
  5. 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).
  6. References: Several arXiv-only references (e.g., [41], [44], [48], [74], [75]) could be updated with published versions where available.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: §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

    Authors: 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: yes

  2. Referee: §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.

    Authors: 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: partial

Circularity Check

1 steps flagged

No significant circularity; central result derived from first-principles computation with standard self-citations to prior methodology

specific steps
  1. self citation load bearing [Section IV.A, Eqs. (37) and (41), and Appendix B]
    "The expectation value U_{j,k} can be obtained in the very same way as the corresponding expectation value in the global ensemble QAE (see Appendix A), with the only difference that now U_{j,k} is an operator acting non-trivially only on the replica Hilbert space (H_j ⊗ H_k)^{⊗4}. Thus, by replacing D = 2^L in Eq. (11) with D = 4, we have: [Eq. 37]... The following relations can be found in [32]: [Eq. 41]"

    The 2-local circuit ODEs (Eq. 46) and the two-body gate expectation value (Eq. 37) are derived by direct analogy to Ref. [32] (co-authored by present authors Szász-Schagrin and Mazzoni). However, this is a methodological self-citation, not a circular derivation. The key computation — the second moment E_QAE[U*⊗U⊗U*⊗U] in Eq. (11) — is derived from first principles in Appendix A.2 (Eqs. A20-A29) by explicitly averaging over random permutations and phases. The replacement D→4 for the two-body gate is a dimension substitution, not a fitting procedure. The stationary solution (Eq. 55) is verified by direct substitution into the ODEs (Eq. 46), which is a mathematical check, not a circular assumption. The self-citation provides methodological scaffolding but the load-bearing computation is self-

full rationale

The central result L*_A = max(L/2, L - SPE_2(|ψ_0⟩)) in Eq. (25) is derived analytically from the QAE second moment (Eq. 11), which is computed from first principles in Appendix A.2 without fitting any parameters. The participation entropy SPE_2 is a property of the initial state (Eq. 3), not a fitted parameter. The derivation chain is: (1) compute E[U*⊗U⊗U*⊗U] by explicit averaging over permutations and phases (Appendix A.2, Eqs. A20-A29), (2) compute the charged partition function Z_A^(2)(α) (Eq. 20), (3) integrate over α to get E[Tr ρ_A^2,Q] (Eq. 22), (4) analyze the asymptotic behavior of the ratio r (Eq. 24) to find the onset scale L*_A (Eq. 25). Each step introduces new computation. The self-averaging approximation (Eq. 16) is a standard assumption in this literature, not a circular step — it does not define the output in terms of the input. The 2-local circuit results (Section IV) derive the ODEs from the same QAE structure with D=4 (Eq. 37), and the stationary solution (Eq. 55) is verified by direct substitution. The coherence-asymmetry coincidence (Section V.B) is shown by computing E[Tr ω_A^2] (Eq. 72) and E[Tr ρ_A^2 - Tr ω_A^2] (Eq. 73) independently and showing they yield the same onset condition (Eq. 75). The self-citations to [32] are for methodology (the ODE framework) and are not load-bearing for the central claim, which is independently derived. Score 2 reflects minor self-citation that is not circular.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters are fitted to data. The tilting angle θ and Dicke parameter k are initial-state specifications, not fitting parameters. No new physical entities, particles, or forces are introduced. The axioms are standard results from random circuit theory (self-averaging, Markovian limit, permutation invariance) except for the heuristic argument that coherence and asymmetry onsets coincide, which is ad hoc to this paper.

axioms (5)
  • domain assumption Self-averaging property: E[log Tr ρ^n] ≃ log E[Tr ρ^n] for the QAE and QAC ensembles
    Invoked in Eq. (16). Standard in random circuit literature but stated without proof for the QAE. All analytical Page curves depend on this.
  • domain assumption The QAE measure is the composition of the flat measure on S_D and uniform measure on D copies of U(1)
    Stated in Section II.A. This defines the ensemble; the second moment is derived from it in Appendix A.2.
  • standard math The 2-local QAC continuous-time limit yields a Markovian master equation (Eq. 35)
    Derived in Ref. 100 (Piroli-Sunderhauf-Qi) and used here. The Lindbladian form follows from the continuous-time limit of discrete random two-body gates.
  • standard math Permutation invariance of the initial state implies permutation invariance of the averaged state at all t>0
    Used in Section IV.A to close the ODE system. Follows from the site-permutation invariance of the Lindbladian L (Eq. 36).
  • ad hoc to paper Coherence onset coincides with asymmetry onset because 'we do not expect any preference in the onset of the off-diagonal entries'
    Invoked in Section V.B to argue L*_A is the same for both quantities. Plausible given the uniform measure over permutations and phases, but not proven rigorously.

pith-pipeline@v1.1.0-glm · 32639 in / 3094 out tokens · 459840 ms · 2026-07-09T06:58:46.871330+00:00 · methodology

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read the original abstract

We investigate the subsystem entanglement asymmetry in random quantum automaton ensembles, which are generated by permuting the basis states in the Hilbert space and applying global phase shifts. We compute the ensemble average of the $U(1)$ subsystem asymmetry in different connectivity geometries, showing that the late-time limit of the ensemble associated to a 2-local circuit geometry coincides with the all-to-all ensemble average. By focusing on different subsystem sizes, we demonstrate that, similarly to Haar-random circuits, the system locally symmetrizes. However, in sharp contrast to the Haar-random setting, the scale at which symmetrization happens depends on the initial state, a phenomenon we associate with the interplay of conservation of the participation entropy and the uniform exploration of charge sectors. Additionally, we connect the growth of the subsystem asymmetry to the subsystem coherence and show that their growth is characterized by the same symmetrization scale.

Figures

Figures reproduced from arXiv: 2607.07556 by D\'avid Sz\'asz-Schagrin, Michele Mazzoni, Olalla A. Castro-Alvaredo.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic depiction of our main results. We apply a global random permutation of basis states followed by a random [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The entanglement asymmetry Page curve ( [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The averaged entanglement asymmetry under 2-local random quantum automaton circuit evolution starting from an [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The entanglement asymmetry Page curve averaged over the 2-local quantum automaton ensemble (markers) starting [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The coherence [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The averaged coherence [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

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