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Local projective measurements that break the U(1) symmetry of a random unitary circuit act as a relevant perturbation at the measurement-induced critical point, driving the system into the same universality class as circuits without the sym

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2026-08-02 07:50 UTC pith:NACJYPCR

load-bearing objection Careful and significant: symmetry-breaking measurements drive the U(1) MIPT to the generic Haar class, with a plausible but unproven bridge between finite charge correlation length and the exact qubit-sector entanglement. the 3 major comments →

arxiv 2607.08589 v2 pith:NACJYPCR submitted 2026-07-09 cond-mat.stat-mech quant-ph

Universality of Measurement-Induced Criticality under Symmetry-Breaking Measurements

Angelo Russotto , Filiberto Ares , Pasquale Calabrese This is my paper
classification cond-mat.stat-mech quant-ph
keywords measurement-induced phase transitionrandom quantum circuitsU(1) symmetrysymmetry-breaking measurementscharge sharpeningpercolation universality classsymmetric simple exclusion processentanglement entropy
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 asks whether the way you measure a random quantum circuit can override the symmetries of its unitary dynamics at a phase transition. Its answer, for circuits with a U(1) charge-conservation law, is yes: any nonzero rate of local projective measurements that break the conservation law is a relevant perturbation of the measurement-induced critical point. At large scales the system therefore behaves like a monitored circuit with no symmetry — in the analytically tractable large-local-Hilbert-space limit, the transition is controlled by two-dimensional percolation, with the entanglement-cut length as the only relevant scale. The mechanism is that each symmetry-breaking measurement acts as a local source–sink defect in a classical stochastic process (a symmetric simple exclusion process) describing charge transport, and the paper proves the resulting charge correlation length is finite for every measurement rate, ruling out a charge-sharpening transition. This matters because it singles out the measurement basis, not the gate symmetry, as the decisive ingredient controlling the universal behavior of these transitions — a conclusion the paper reinforces with finite-size numerics on both Haar and stabilizer circuits.

Core claim

Within the replica approach to random circuits, the averaged entanglement entropy in the large local Hilbert-space limit maps to a two-dimensional classical statistical model whose bond variables carry the conserved charge. In the one-replica limit that charge sector is a symmetric simple exclusion process, and measuring a qubit in the X basis corresponds to resetting the local occupancy to a uniform distribution irrespective of the outcome — a disordered defect that creates or destroys charge. The central technical result is the bound on the averaged connected charge correlation function, E_X[C_X(z,t)] ≤ (1/4)(1−p)^{2t}, which follows because any Brownian-duality path contributing to the co

What carries the argument

The load-bearing object is the large-d statistical-mechanics representation of the replicated circuit: vertices carry permutation degrees of freedom, unmeasured bonds carry binary charge variables, and X-basis measurements cut bonds and reset the local charge to a uniform mixture, independent of the outcome. In the one-replica limit the charge sector becomes a disordered symmetric simple exclusion process (SSEP), and the key identity is the Brownian-duality bound E_X[C_X(z,t)] ≤ (1/4)(1−p)^{2t} = (1/4)e^{−t/ξ_p}, with ξ_p^{−1} = −2 log(1−p). The bound says a charge-correlation 'survival path' must avoid all defects, so after time ξ_p correlations die out. From the finiteness of the typical c

Load-bearing premise

The argument assumes that once the two-point charge correlation length is finite, the charge variables along the minimal entanglement cut can be coarse-grained into independent variables, so the qubit-sector entanglement is controlled solely by the percolation cut length; this factorization is asserted rather than derived, and the rigorous bound in the paper only controls two-point correlations.

What would settle it

In the large-d statistical model, compute the joint distribution p_β of the charges on the minimal cut for a fixed disorder realization at p>0 and test whether it converges to the product 2^{−ℓ_DW}. If multi-point charge correlations survive along the cut, or if S_q/ℓ_DW does not approach log 2 as ℓ_DW/ξ_typ grows, the percolation-universality claim for the qubit sector fails. A second, complementary check: look for a diverging charge correlation length at any finite p in the X-monitored circuit — its presence would contradict the finite-correlation-length theorem.

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

If this is right

  • For every measurement rate p>0, charge correlations in the large-d model have finite correlation length; no charge-sharpening transition exists when measurements break the symmetry, in contrast to the symmetry-preserving case.
  • The entanglement transition lies in the 2D percolation universality class — the same as non-symmetric Haar circuits — with the minimal-cut length ℓ_DW as the only diverging scale.
  • At finite local Hilbert-space dimension, the critical exponents and Rényi entanglement-growth coefficients of the monitored U(1) Haar and stabilizer circuits match the corresponding non-symmetric circuits, supporting the same universality class there.
  • For symmetric stabilizer circuits, where symmetry-preserving measurements produce no transition at finite p, X-basis measurements restore a volume-law phase at small p and a transition in the non-symmetric Clifford universality class.
  • Any symmetry-breaking direction in the measurement basis (not only X) produces a finite correlation length ξ_p(θ) for p>0, so even a slightly tilted measurement should drive the same universal behavior after a possibly long crossover.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the factorization along the minimal cut is as strong as the paper's argument implies, then in the large-d limit the entire charge distribution p_β should become asymptotically i.i.d. Bernoulli, making S_q = ℓ_DW log 2 exactly at large scales — a stronger statement than the paper proves and a direct target for numerical verification in the SSEP.
  • The Brownian-duality argument that controls two-point functions likely extends to higher-order charge correlations; checking whether k-point functions decay with rate k/ξ_p would settle whether the multi-point correlations omitted in Sec. 3.4 are truly harmless.
  • The same 'measurement basis selects the universality class' picture suggests a testable prediction for monitored free-fermion circuits: symmetry-breaking measurements should destroy the distinct criticality found for charge-preserving monitoring and drive the transition toward the generic percolation-like class at large scales.
  • One might use the finite correlation length ξ_p(θ) to engineer crossovers: by tuning the tilt θ toward the symmetry-preserving axis, experiments on near-term quantum processors could observe effective exponents interpolating between the U(1)-symmetric and generic universality classes as system size grows.

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

3 major / 5 minor

Summary. The paper studies monitored random quantum circuits with U(1)-symmetric Haar gates and local projective measurements that break the U(1) symmetry (X-basis measurements on the qubit sector). The central claim is that symmetry-breaking measurements are a relevant perturbation at the measurement-induced critical point, so that at large scales the transition belongs to the same universality class as the generic (non-symmetric) monitored Haar circuit. In the large-d limit the authors map the problem to a 2D statistical mechanics model in which the charge sector is a symmetric simple exclusion process (SSEP) with random space-time defects representing the measurements. They prove that the averaged and typical two-point charge correlations decay exponentially in time for any measurement rate p>0, implying the absence of a charge-sharpening transition. They then argue that the qubit-sector entanglement contribution is controlled solely by the percolation minimal-cut length, so the whole transition is in the 2D percolation universality class. Finite-d numerical results for both U(1)-symmetric Haar circuits and U(1)-symmetric stabilizer circuits are presented and compared with the corresponding non-symmetric circuits.

Significance. If the central claim holds, the paper settles an important question: local measurements that break the symmetry of the unitary gates act as a relevant perturbation at the MIPT and drive the system to the universality class of the corresponding symmetry-free monitored circuit. The large-d mapping is explicit and the derivation of the outcome-independent Boltzmann weight for X measurements is clean. The proof that the X-averaged two-point charge correlation decays exponentially, together with the Borel-Cantelli argument for almost-sure decay, is a genuine analytic result and is stronger than a numerical observation. The numerical study is also substantial: exact Haar simulation up to L=24 and stabilizer simulations up to L=2048, with a comparison of the scaling function to the non-symmetric stabilizer circuit. However, the analytic bridge from finite two-point charge correlation length to the universality of the qubit-sector entanglement is not fully derived; the manuscript itself presents this step as an argument. Because this bridge is load-bearing for the central claim, the paper needs either a proof of the missing step or a careful reframing of the claim as a conjecture supported

major comments (3)
  1. [Sec. 3.4, Eqs. (21)-(22) and Eq. (28)] The claim that finiteness of the typical charge correlation length implies that p_beta can be coarse-grained into O(ell_DW/xi_typ) effectively independent variables along the minimal cut is asserted, not derived. The rigorous content is Eq. (29) and App. A.1: for the uniform initial state, the X-averaged two-point, time-separated correlation decays exponentially, and with probability one the decay is exponential for fixed spacetime pairs. This does not control multi-point correlations of the conditioned distribution p_beta on the X-dependent minimal cut. Since S_q in Eq. (22) is (1/(1-n)) log sum_beta p_beta^n, an O(ell_DW) multi-point connected contribution with a p-dependent coefficient would change the coefficient of ell_DW and hence the universal coefficient alpha(n) in Eq. (34). The paper explicitly calls this step an 'argument' rather than a proof; as written, the large-d universal
  2. [Sec. 3.4, Eq. (29); Sec. 6] The bound in Eq. (29) is a temporal two-point correlation, C_X(z,t)=<n(z,t)n(0,0)>_c, measured between time t and time 0. The absence of a charge-sharpening transition is, however, a statement about charge fluctuations and the equal-time charge correlation length in the steady state. The manuscript does not supply the relation between the temporal decay proved in Eq. (29) and the equal-time correlations that define charge sharpening in Refs. [62,63]. For a diffusive SSEP the two are plausibly related, but the implication is not demonstrated. Thus the statement in the abstract and Sec. 6 that the result 'rules out a charge-sharpening transition' goes beyond what is strictly proved.
  3. [Sec. 3.4 and App. A.2] The same multi-point gap affects the tilted-basis analysis. Equation (65) gives an exponential bound on the two-point correlation for any trajectory and initial state, and Eq. (61) correctly tracks the cos^2 theta survival factor. However, the conclusion that 'all of the results derived in the previous section apply' (last paragraph of App. A.2) again requires the unproven factorization of p_beta along the minimal cut. The paper should either prove the relevant multi-point decay or explicitly label this part of the argument as a conjecture. Given that the central claim—'the only possible diverging length scale is the percolation length scale'—rests on this step, the current wording overstates the strength of the analytic result.
minor comments (5)
  1. [Eq. (13)] The notation for the vectorized states |gamma;sigma>_j |delta;sigma>_{j+1} <alpha;sigma|_j <beta;sigma|_{j+1} is compact but could confuse a reader who is not familiar with the Choi-Jamiolkowski convention. A brief note indicating the ordering of the bras and kets in the doubled Hilbert space would improve readability.
  2. [App. A.2] There is a typo: 'Boltzmann weigth' should be 'Boltzmann weight'.
  3. [Sec. 4, Fig. 3 inset] The red cross marking the universal value from Ref. [38] is visually useful, but the numerical value of I_H(0) used for comparison is not stated in the text. Please include it explicitly so the reader can compare it with the reported E[I_3,1(p_c)] = -0.45(5).
  4. [Sec. 4, Eq. (31)] The quantity I_3,n is called 'topological entanglement entropy' in one place, but it is the tripartite mutual information and is not a topological invariant. Using 'tripartite mutual information' consistently would avoid confusion.
  5. [Sec. 3.4, below Eq. (29)] The definition of the typical correlation length C_typ(z,t) = exp(E_X[log C_X(z,t)]) conditions on non-zero values of C_X(z,t). Please make the conditioning explicit in the definition, since C_X(z,t) can vanish for many realizations.

Circularity Check

0 steps flagged

No significant circularity: the central large-d result is derived from an independent statistical-mechanics mapping and compared against external percolation benchmarks.

full rationale

The paper's central large-d claim is that symmetry-breaking X measurements render the charge correlation length finite for every p>0, so the entanglement transition is governed by the percolation minimal cut. This is not circular. The finite-correlation-length statement is derived in App. A.1 from an explicit survival-probability bound for SSEP trajectories, Eq. (29), with the Borel-Cantelli argument giving the almost-sure decay; it does not assume the entropy-conclusion it is used to support. The statistical-mechanics mapping follows Ref. [62], which is by a different group, and the percolation identification and universal exponents come from external, machine-checkable or standard results ([7,38,47,90-92]); the paper compares, rather than fits, its values against those external benchmarks. The numerical finite-size scaling fits p_c, nu, and alpha(n) from the present data and then compares them with the independently determined values of Refs. [38] and [47]; no fitted parameter is renamed as a prediction. The only weak analytic link is the assertion in Sec. 3.4 that finiteness of the two-point typical charge correlation length implies p_beta can be coarse-grained into O(ell_DW/xi_typ) independent variables along the minimal cut, and hence that S_q is controlled solely by ell_DW. This is an unproven bridge and a genuine limitation, but it is not a circular reduction: the paper does not define ell_DW in terms of S_q, and the percolation universality for the qudit sector is already external. The authors also flag their own remaining finite-size uncertainties. Thus there is no step in which a prediction is equivalent by construction to an input, and no load-bearing self-citation chain.

Axiom & Free-Parameter Ledger

6 free parameters · 8 axioms · 0 invented entities

The analytic core is parameter-free: no constant is fitted in the large-d derivation—the correlation bound E[C] <= (1-p)^(2t)/4 (Eq. (29)) and the almost-sure decay (Eq. (58)) contain no adjustable parameters. All listed free parameters reside in the finite-d numerical FSS analyses and are outputs used for comparison with external benchmarks ([38,47]), not inputs to the central derivation. The main axiomatic burden sits on the replica trick, the large-d limit, and the coarse-graining step asserting that finite two-point charge correlations make S_q track the percolation cut. No invented physical entities are introduced: the measurement defects in the SSEP are exact images of the X-basis projectors under the statistical-mechanics mapping (Eqs. (23)-(24)), not new postulates, and they carry no falsifiable handle outside the mapping itself.

free parameters (6)
  • pc (Haar circuit) = 0.143(3)
    Critical measurement rate fitted to the trajectory-averaged tripartite mutual information of the d=1 U(1)-Haar circuit (Sec. 4, Eq. (33)). Output of the analysis, not an input to the analytic claim; used to anchor the data collapse and the universal-value comparisons.
  • nu (Haar circuit) = 1.3(2)
    Correlation-length exponent fitted in the same FSS analysis (Eq. (33)); degenerate between the two candidate universality classes within errors, as the paper acknowledges.
  • pc (stabilizer circuit) = 0.0851(1)
    Fitted to the U(1)-Clifford circuit FSS (Sec. 5, Eq. (36)) using only L>=768 data.
  • nu (stabilizer circuit) = 1.27(3)
    Fitted in the same stabilizer FSS; mildly L_min-dependent (Fig. 9), which the paper states dominates its uncertainty.
  • a, b (Renyi coefficient fit) = a=0.98(3), b=-0.27(4)
    Fitted to alpha(n)=a(1+1/n)+b for the Haar critical entanglement entropy (Sec. 4); compared with the external Haar-MIPT values a_H=1.01, b_H=-0.31 of Ref. [38].
  • non-universal rescaling a~0.94 = 0.94
    Horizontal rescaling factor applied to the non-symmetric Clifford data in Fig. 5 (right panel) to display the collapse; a hand-chosen fit parameter in the comparison plot.
axioms (8)
  • domain assumption Replica trick: Renyi entropies obtained by analytic continuation k->0 (Q->1), exchanging the replica limit with the large-d limit (Eqs. (7)-(10))
    Standard in the MIPT literature [53,54,62] but not a mathematically rigorous interchange of limits; used throughout Sec. 3.
  • standard math In the d->infinity limit only minimal-Cayley-distance permutations survive in the bond weight B (Eq. (15))
    Large-d dominance of the diagonal Weingarten term in Eq. (13); standard ingredient of the Haar random circuit mapping.
  • standard math SSEP self-duality: the charge two-point function equals the survival probability of a single Brownian particle (Eqs. (48)-(52))
    Known duality result for the symmetric exclusion process, cited to Ref. [124]; the backbone of the correlation bound (29).
  • standard math 2D percolation criticality of the averaged minimal cut: pc=1/2, nu=4/3
    Inherited from the non-symmetric monitored circuit mapping, Refs. [7,53,54,90-92]; the qudit sector reduces to this exactly (Eq. (18)).
  • ad hoc to paper Finite two-point charge correlation length implies p_beta factorizes into ~ell_DW/xi_typ independent variables, so S_q is controlled by ell_DW
    The coarse-graining step in Sec. 3.4 bridging the proven correlation bound to the universality class of the qubit sector; asserted, not derived. Identified here as the weakest assumption.
  • domain assumption Large-d conclusions carry over to finite local Hilbert-space dimension
    The finite-d Haar and Clifford numerics (Secs. 4-5) support this transfer but do not prove it; the paper notes percolation universality is only strictly the large-d limit [45,53].
  • domain assumption Fixed dynamical exponent z=1 in the finite-size scaling ansatz (Eq. (32))
    Quoted: 'As generally observed for MIPTs, we assume the dynamical exponent z=1 and thus fix the time proportional to the system size' (Sec. 4).
  • domain assumption Born-probability absorption via one extra replica (Q=kn+1) commutes with the k->0 limit
    Standard replica-trick device inherited from Ref. [62]; unproven interchange of limits (Sec. 3.1).

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read the original abstract

We study the critical properties of random quantum circuits with a $U(1)$ symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored $U(1)$-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored $U(1)$-symmetric Haar and stabilizer random circuits.

Figures

Figures reproduced from arXiv: 2607.08589 by Angelo Russotto, Filiberto Ares, Pasquale Calabrese.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Two dimensional statistical model describing the average Rényi-n entropy of the monitored circuit of Sec. 2 in the limit of large local Hilbert space dimension. The lattice vertices correspond to the averaged replicated unitary gates and bonds rep￾resent the sites of the quantum system. Broken bonds correspond to local projective measurements. On each vertex is defined a permutation degree of freedom σ ∈ S… view at source ↗
Figure 3
Figure 3. Figure 3: Trajectory-averaged tripartite mutual information E[I3,1], for n = 1, in the stationary state at t ∼ 2L of the U(1)-symmetric Haar random circuit with measure￾ments in the X basis, as a function of the measurement rate p and for increasing system size L. We average over ∼ 2 × 103 quantum trajectories. The errorbars correspond to one standard deviation of the mean. For L ≥ 16, the data develop a clear cross… view at source ↗
Figure 4
Figure 4. Figure 4: Half-system Rényi-n entanglement entropy E[Sn] averaged over ∼ 103 quan￾tum trajectories at the estimated critical point pc = 0.143 in the U(1)-symmetric Haar random circuit with symmetry-breaking measurements in the X basis. Left panel: Log￾arithmic growth of E[Sn(pc)] as a function of the system size L for various Rényi indices. The symbols are the exact numerical results, while dashed lines represent th… view at source ↗
Figure 5
Figure 5. Figure 5: Left panel: Tripartite mutual information I3,1, for n = 1, averaged over ∼ 6 × 103 quantum trajectories, in the stationary state at t ∼ 2L, of the symmetric random stabilizer circuit with projective measurements in the X basis, as a function of the measurement rate p and different system sizes L. The errorbars are of the order of the symbol size and correspond to one standard deviation of the mean. Inset: … view at source ↗
Figure 6
Figure 6. Figure 6: Estimates of the critical measurement rate pc (upper panels) and critical exponent ν (lower panels) for the X-monitored U(1) Haar-random circuit, obtained from a non-linear fit of the function in Eq. (74) to numerical simulation data. Left panels: we vary the degree M of the fitting function for different minimum system sizes Lmin included in the dataset. Right panels: the same data are shown explicitly as… view at source ↗
Figure 6
Figure 6. Figure 6: Logarithmic growth of the half-system entanglement entropy SL/2 averaged over ∼ 104 quantum trajectories in the stationary state at t ∼ 2L of the symmetric random stabilizer circuit with projective measurements in the X basis at the estimated critical point p C c in Eq. (36). Symbols correspond to the numerical data, while errorbars are smaller than the symbol size. The dashed line is the result of a linea… view at source ↗
Figure 7
Figure 7. Figure 7: Heatmap of the cost function S(pc, ν) defined in Eq. (75), for the numerical simulation data of the X-monitored U(1) Haar-random circuit. The symbols indicate the location of the global minimum by varying the minimum system size Lmin included in the dataset: red circle for Lmin = 16 and orange square for Lmin = 20. The dashed lines delimit the region S(pc, ν) = 1.3 S(p ∗ c , ν∗ ), which provides an estimat… view at source ↗
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Same as [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Same as [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗

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