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Symmetry-breaking measurements drive a U(1)-symmetric monitored circuit into the same measurement-induced criticality as a circuit with no symmetry at all.

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2026-07-10 04:41 UTC pith:NACJYPCR

load-bearing objection Clean proof that symmetry-breaking measurements kill charge sharpening and push the MIPT into the ordinary non-symmetric class; numerics back it up. the 2 major comments →

arxiv 2607.08589 v1 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 transitionU(1) symmetrysymmetry-breaking measurementsrandom quantum circuitscharge sharpeningpercolationsymmetric simple exclusion processstabilizer circuits
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.

This paper asks whether local measurements that break a conserved charge can change the critical behavior of a measurement-induced entanglement transition. The authors study random quantum circuits whose unitary gates conserve a U(1) charge, but whose projective measurements deliberately break that charge. They show that the symmetry-breaking measurements act as a relevant perturbation: at large scales the transition falls into the same universality class as ordinary monitored circuits that never had any symmetry. In the analytically tractable limit of large local Hilbert-space dimension the charge sector maps onto a classical exclusion process whose defects create and destroy charge; the resulting charge correlation length stays finite at every nonzero measurement rate, so there is no charge-sharpening transition. Finite-size numerical collapses of tripartite mutual information and Rényi-index dependence for both Haar and stabilizer circuits at finite dimension support the same conclusion. A reader who cares about how symmetries and measurements compete in open quantum dynamics now has a concrete statement of which perturbations matter and which do not.

Core claim

At the measurement-induced phase transition of a U(1)-symmetric random circuit, local projective measurements that break the U(1) symmetry are a relevant perturbation. They drive the entanglement critical point into the same universality class as the corresponding non-symmetric monitored circuit, and they keep the charge correlation length finite at every measurement rate, eliminating the charge-sharpening transition that appears when measurements preserve the symmetry.

What carries the argument

The large-d replica mapping of the trajectory-averaged entanglement entropy onto a two-dimensional classical statistical model whose charge sector is a symmetric simple exclusion process with disordered defects that create and annihilate charge; the finite correlation length of that process is proved by bounding the survival probability of a Brownian particle that is annihilated by every measurement.

Load-bearing premise

That the large-dimension mapping continues to fix the universality class at ordinary qubit dimension, so that the modest system sizes already show the asymptotic non-symmetric exponents rather than a long crossover controlled by a still-large charge correlation length.

What would settle it

A finite-size scaling analysis of the same U(1)-symmetric Haar circuit at d=1, but with measurements only weakly tilted away from the charge-preserving Z basis, that yields a distinct Rényi-index dependence of the critical entanglement coefficient α(n) incompatible with the non-symmetric Haar values, or a clear charge-sharpening signature at intermediate measurement rates.

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

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 / 5 minor

Summary. The manuscript studies measurement-induced phase transitions (MIPTs) in U(1)-symmetric random circuits when local projective measurements explicitly break the conservation law. The central claim is that such measurements are a relevant perturbation: at large scales the entanglement transition belongs to the same universality class as the corresponding non-symmetric monitored circuit, and the charge correlation length remains finite for any measurement rate p>0, ruling out a charge-sharpening transition. In the large-d limit the trajectory-averaged entanglement is mapped via the replica trick and Weingarten calculus to a classical statistical model whose charge sector is a symmetric simple exclusion process with disordered defects; a dual Brownian representation then yields an almost-sure exponential decay of charge correlators. Finite-d support is provided by finite-size scaling of the tripartite mutual information and Rényi-index dependence of the critical entanglement for both U(1)-Haar (L≤24) and U(1)-Clifford (L≤2048) circuits, with critical parameters and scaling functions matching the non-symmetric literature values rather than the symmetry-preserving ones.

Significance. The result cleanly settles a natural question left open by the symmetry-preserving literature (Agrawal et al., Barratt et al.): whether breaking the conservation law only in the measurement channel is enough to restore the generic MIPT universality class. The large-d argument is self-contained and rigorous—the survival-probability bound EX[CX]≤(1/4)(1−p)2t together with Markov and Borel–Cantelli (App. A) is a genuine proof that charge correlations cannot diverge. The Clifford numerics reach L=2048 and produce a clean data collapse onto the non-symmetric scaling function after a single non-universal rescaling, which is strong independent evidence. The observation that U(1)-Clifford circuits have no MIPT under symmetry-preserving measurements but acquire one under symmetry-breaking measurements is a sharp, falsifiable prediction. These strengths make the paper a solid contribution to the theory of monitored circuits.

major comments (2)
  1. Sec. 4 and Fig. 4: the strongest finite-d diagnostic is the Rényi-index dependence α(n)=a(1+1/n)+b, which matches the non-symmetric Haar values of Zabalo et al. rather than the U(1)-preserving values of Agrawal et al. With only L≤24 the logarithmic window is short; the manuscript should report the systematic uncertainty on a and b under variation of Lmin (analogous to the App. C analysis already performed for pc and ν) so that the reader can judge how robust the discrimination between the two universality classes remains.
  2. Sec. 3.4 and App. A.2: the analytic bound on the typical charge correlation length is derived for the large-d SSEP. The manuscript correctly notes that a parametrically large ξ(θ) for nearly symmetry-preserving measurements can produce long crossovers at finite d. A short quantitative estimate (even a rough scaling argument) of the system size needed to exit that crossover for the Haar d=1 data would strengthen the claim that L=24 already probes the asymptotic non-symmetric fixed point rather than an intermediate regime.
minor comments (5)
  1. Fig. 1 caption and Sec. 2: the local Hilbert space is C2⊗Cd; it would help the reader if the figure explicitly labels which factor is measured in the X basis and which is measured in a fixed qudit basis.
  2. Eq. (24) and the surrounding text: the measurement operator M is written without the conventional probability-conserving normalization; a one-sentence reminder that the 2^{NX} factor is restored when computing expectation values would avoid confusion for readers less familiar with the disordered SSEP literature.
  3. App. C: the two fitting procedures (polynomial collapse and cost-function minimization) are carefully documented; adding the numerical value of the minimal cost S(p∗c,ν∗) for the Clifford data would make the quality of the collapse more transparent.
  4. References: the recent works on noise as a symmetry-breaking field (e.g. Dias et al., Liu et al.) are cited in the discussion of random measurement bases; a brief cross-reference in Sec. 3.4 would better situate App. B within that literature.
  5. Typographical: “themeasurement-inducedphasetransition” (abstract) and a few missing spaces after commas in the large-d section should be cleaned up.

Circularity Check

0 steps flagged

No significant circularity: large-d charge-correlation bound is derived from the SSEP transfer matrix and measurement operator; finite-d exponents are compared to independent external literature values.

full rationale

The central analytic claim (finite charge correlation length for any p>0, ruling out charge sharpening) follows by direct calculation from the one-replica SSEP transfer matrix (Eq. 23/38) and the X-measurement operator (Eq. 24/39). The dual Brownian representation yields the average bound EX[CX(z,t)]≤(1/4)(1-p)2t (Eq. 29/A.54) and, via Markov + Borel–Cantelli, the almost-sure liminf bound (Eqs. A.55–A.58). These steps do not invoke fitted parameters, self-referential definitions, or load-bearing self-citations; they are first-principles consequences of the model constructed in Sec. 3. At finite d the extracted ν and α(n) coefficients are compared to published non-symmetric Haar/Clifford values (Zabalo et al., Gullans–Huse) rather than being forced by construction from the present data. Self-citations are limited to standard technical tools (Weingarten calculus, Stim) that are externally verified and not used to close a logical loop. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The analytic claim rests on standard large-d Weingarten calculus, the replica trick, and the known mapping of the charge sector to SSEP; the only free parameters are the non-universal critical rates and exponents extracted from finite-size collapses. No new particles or forces are postulated.

free parameters (4)
  • pc (Haar d=1) = 0.143(3)
    Critical measurement rate extracted from tripartite-mutual-information crossings and polynomial/cost-function collapses; non-universal and fitted to the L=16–24 data.
  • ν (Haar d=1) = 1.3(2)
    Correlation-length exponent from the same finite-size scaling; compatible with literature but still a fit.
  • pc (U(1) Clifford) = 0.0851(1)
    Critical rate for the stabilizer circuit, fitted from L≥768 data.
  • ν (U(1) Clifford) = 1.27(3)
    Correlation-length exponent for the stabilizer circuit.
axioms (4)
  • domain assumption In the d→∞ limit the averaged replicated Haar gate reduces to the projector given by Weingarten calculus (Eq. 13), and unmeasured bonds force identical permutations.
    Standard large-d technique used throughout the MIPT literature; invoked in Sec. 3.2.
  • domain assumption The replica limit k→0 of the Rényi entropy can be taken inside the statistical model, yielding a Q=1 classical partition function.
    Replica trick as employed in Refs. [50,51,58]; Sec. 3.1–3.3.
  • domain assumption The dynamical exponent of the MIPT is z=1, so that steady-state data can be taken at t∝L.
    Standard assumption for brickwork MIPTs; used for all finite-size scaling (Secs. 4–5).
  • standard math Two-dimensional bond percolation has pc=1/2 and ν=4/3.
    Classical result used to identify the large-d entanglement transition (Sec. 3.3).

pith-pipeline@v1.1.0-grok45 · 47608 in / 2815 out tokens · 30221 ms · 2026-07-10T04:41:26.986288+00:00 · methodology

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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 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 ↗

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