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arxiv: 2606.24928 · v1 · pith:7GVBVOJOnew · submitted 2026-06-21 · 🪐 quant-ph · hep-th

Reading Weakly, Acting Strongly: A Static Parity Horizon and its Dynamical Bypass in the Monitored Lipkin-Meshkov-Glick Model

Stavros Mouslopoulos This is my paper

Pith reviewed 2026-06-26 10:30 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords lipkin-meshkov-glick modelparity doubletinstanton actionstatic parity horizonmonitored dynamicsj_z measurementdistinguishability measureshomodyne monitoring
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The pith

The instanton action that splits the LMG parity doublet also sets the leading exponent for parity information in static J_z magnetisation readouts.

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

The paper examines the broken-symmetry phase of the Lipkin-Meshkov-Glick model, where the two lowest states form a near-degenerate parity doublet split by tunnelling. It shows that the same instanton action S_inst controlling the doublet splitting also governs how much parity information a static J_z magnetisation measurement can extract, even though J_z primarily distinguishes the two wells and remains nearly blind to their relative sign. WKB barrier arguments combined with exact diagonalisation up to N=4500 demonstrate that the spectral gap, total-variation distance, Jensen-Shannon divergence, and Chernoff information all share one instanton exponent rather than the doubled exponent expected from a naive small-deviation expansion. This separation between reading and disturbing the parity label defines the static parity horizon for the idealised static channel. Continuous-monitoring simulations of 1.48 million trajectories show that time-resolved homodyne records can extract additional parity information over a finite window of system sizes organised by the ratio xi of coherent doublet rotation to measurement-induced dephasing.

Core claim

Although J_z measures magnetisation rather than parity, the spectral gap, the total-variation distance, and the nonlinear distinguishability measures (Jensen-Shannon divergence and Chernoff information) share a single instanton exponent set by the tunnelling action S_inst, rather than the doubled exponent a naive small-deviation expansion in the lobes would suggest; exact diagonalisation confirms this common leading exponent lies within a few percent of the WKB value, while the off-diagonal coupling grows as |J_01| -> N m_*/2, so the bath disturbs the parity label far more strongly than it can read it from a frozen histogram, establishing the static parity horizon as a benchmark for the idea

What carries the argument

The static parity horizon: the separation in the idealised static J_z channel where the off-diagonal coupling grows linearly with N and disturbs parity more than it reads it from a histogram.

If this is right

  • The spectral gap, total-variation distance, Jensen-Shannon divergence and Chernoff information all share the same single instanton exponent.
  • The off-diagonal element |J_01| grows linearly with system size as N m_*/2, allowing stronger disturbance than readout of the parity label.
  • Time-resolved homodyne records extract parity information hidden from the single-shot histogram over a finite window of system sizes organised by xi = omega_01/Gamma_01.
  • The extraction window closes again under strong measurement.

Where Pith is reading between the lines

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

  • The organising ratio xi may allow experimental tuning of information extraction windows in other monitored double-well systems by adjusting measurement strength relative to coherent splitting.
  • The separation between static and dynamical extraction could appear in monitored versions of related spin or bosonic models with broken symmetry.
  • If size-dependent corrections remain negligible, the common-exponent behaviour should persist to larger N where WKB becomes more accurate.

Load-bearing premise

The instanton action extracted from the spectral gap directly governs the distinguishability measures without additional system-size-dependent corrections that would alter the leading exponent.

What would settle it

Exact diagonalisation at N larger than 4500 in which the fitted exponents for total-variation distance or Jensen-Shannon divergence deviate from the WKB instanton value by more than a few percent.

Figures

Figures reproduced from arXiv: 2606.24928 by Stavros Mouslopoulos.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
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Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
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Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
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Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p040_4.png] view at source ↗
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Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p041_5.png] view at source ↗
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Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p043_6.png] view at source ↗
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Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p046_7.png] view at source ↗
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Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p067_8.png] view at source ↗
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Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p078_9.png] view at source ↗
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Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p080_10.png] view at source ↗
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Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p082_11.png] view at source ↗
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Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p085_12.png] view at source ↗
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Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p087_13.png] view at source ↗
read the original abstract

We study the broken-symmetry phase of the Lipkin-Meshkov-Glick (LMG) model, whose two lowest states form a near-degenerate parity doublet split by tunnelling. We show that the same instanton action S_inst that sets the doublet splitting also controls how much parity information a static J_z magnetisation readout can extract. Although J_z measures magnetisation rather than parity - and so distinguishes the two wells easily while remaining almost blind to their relative sign - WKB barrier arguments together with exact diagonalisation show that the spectral gap, the total-variation distance, and the nonlinear distinguishability measures (Jensen-Shannon divergence and Chernoff information) share a single instanton exponent, rather than the doubled exponent a naive small-deviation expansion in the lobes would suggest. Exact diagonalisation up to N = 4500 supports a common leading exponent for all four quantities, with fitted values within a few percent of the WKB instanton value in the largest reliable windows. The same coupling acts strongly inside the doublet: its off-diagonal element grows as |J_01| -> N m_*/2, so the bath can disturb the parity label far more strongly than it can read it from a frozen histogram. We call this separation the static parity horizon - a benchmark for the idealised static J_z channel, not a universal bound on time-resolved monitoring. Restoring the full monitored dynamics, continuous-monitoring simulations (1.48 million full-LMG trajectories with matched QND controls across 77 independent settings) show that a time-resolved homodyne record extracts parity information hidden from the single-shot histogram, over a finite window of system sizes organised by the ratio xi = omega_01/Gamma_01 of coherent doublet rotation to measurement-induced dephasing, and closing again under strong measurement.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript examines the broken-symmetry phase of the Lipkin-Meshkov-Glick model, where the two lowest states form a near-degenerate parity doublet split by tunneling. It claims that the same WKB instanton action S_inst governing the spectral gap also sets the leading exponential scaling of parity information extractable from static J_z magnetisation measurements, with the total-variation distance, Jensen-Shannon divergence, and Chernoff information all sharing this single exponent (rather than a doubled one from naive lobe expansions). This is supported by exact diagonalization up to N=4500 and 1.48 million continuous-monitoring trajectories, leading to the introduction of a 'static parity horizon' as a benchmark for the idealised static channel, with dynamical bypass possible via time-resolved homodyne records over a finite window of system sizes controlled by xi = omega_01/Gamma_01.

Significance. If the shared-exponent claim holds, the work offers a concrete benchmark for how static versus dynamical readouts interact with parity symmetry in monitored spin systems, highlighting a separation between readout strength and disturbance within the doublet. The scale of the exact-diagonalization (N=4500) and trajectory ensemble (1.48 million) constitutes a clear computational strength that allows direct numerical comparison to WKB. The absence of an independent analytic derivation for the distinguishability exponents, however, keeps the result primarily numerical and limits its generality beyond the LMG model.

major comments (2)
  1. [Abstract] Abstract: The statement that fitted exponents for TV distance, JS divergence, and Chernoff information lie 'within a few percent of the WKB instanton value in the largest reliable windows' provides no details on the fitting procedure, reported uncertainties, data-exclusion rules, or explicit tests for subleading corrections (log N, constant, or 1/N terms). Because the central claim of a shared leading exponent (rather than doubled) rests entirely on these fits, the lack of such controls leaves open the possibility that quantity-dependent finite-N corrections produce an apparent collapse.
  2. [Abstract] Abstract and numerical sections: The assumption that the instanton action extracted from the spectral gap directly transfers to the distinguishability measures without additional system-size-dependent corrections that would alter the leading coefficient is invoked via WKB+ED comparison but is not independently verified; for quantities scaling as exp(-S N + o(N)), inclusion or exclusion of subleading terms in the fit window can shift the extracted S by an amount comparable to the reported tolerance, undermining the claim of exact identity.
minor comments (1)
  1. [Abstract] The introduction of the term 'static parity horizon' would benefit from an explicit operational definition (e.g., a precise inequality involving readout fidelity versus disturbance) at first use rather than after the numerical results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our numerical analysis. The points raised about fitting procedures and the transfer of the instanton exponent are valid and will be addressed by expanding the documentation in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The statement that fitted exponents for TV distance, JS divergence, and Chernoff information lie 'within a few percent of the WKB instanton value in the largest reliable windows' provides no details on the fitting procedure, reported uncertainties, data-exclusion rules, or explicit tests for subleading corrections (log N, constant, or 1/N terms). Because the central claim of a shared leading exponent (rather than doubled) rests entirely on these fits, the lack of such controls leaves open the possibility that quantity-dependent finite-N corrections produce an apparent collapse.

    Authors: We agree that additional documentation of the fitting procedure is required to support the shared-exponent claim. In the revised manuscript we will add a dedicated subsection (or appendix) that specifies: (i) the exact N-ranges used for each fit (typically 2000–4500 for the largest reliable windows), (ii) the functional forms tested, including pure exponential, exponential plus log N, and exponential plus 1/N corrections, (iii) uncertainty estimation via bootstrap resampling of the ED data and trajectory ensembles, (iv) explicit data-exclusion rules based on convergence thresholds for the spectral gap and distinguishability measures, and (v) side-by-side comparisons of extracted exponents with and without subleading terms to demonstrate stability of the leading coefficient. These additions will allow direct assessment of whether quantity-dependent finite-N effects could mimic the reported collapse. revision: yes

  2. Referee: [Abstract] Abstract and numerical sections: The assumption that the instanton action extracted from the spectral gap directly transfers to the distinguishability measures without additional system-size-dependent corrections that would alter the leading coefficient is invoked via WKB+ED comparison but is not independently verified; for quantities scaling as exp(-S N + o(N)), inclusion or exclusion of subleading terms in the fit window can shift the extracted S by an amount comparable to the reported tolerance, undermining the claim of exact identity.

    Authors: We acknowledge that the evidence for identical leading exponents is numerical (WKB for the gap combined with ED for the distinguishability measures) rather than supported by an independent analytic derivation of the TV/JS/Chernoff exponents. Such an analytic derivation would be valuable but lies outside the scope of the present work, which centers on establishing the numerical coincidence and its physical interpretation through the static parity horizon. The expanded fitting analysis described above will include explicit sensitivity tests across multiple windows and correction terms, showing that the leading coefficient remains consistent to within the stated tolerance. This strengthens the numerical support without claiming an analytic identity beyond what the WKB+ED comparison demonstrates. revision: partial

Circularity Check

0 steps flagged

No circularity: WKB + ED comparison is independent numerical verification

full rationale

The paper derives the instanton action S_inst for the spectral gap via standard WKB barrier arguments, then uses exact diagonalization (N up to 4500) to numerically extract leading exponents for total-variation distance, Jensen-Shannon divergence and Chernoff information, reporting that fitted values lie within a few percent of the WKB result in chosen windows. This constitutes an empirical check of shared scaling rather than any reduction of one quantity to another by definition, self-citation, or renaming of a fitted parameter as an independent prediction. No load-bearing self-citation, ansatz smuggling, or uniqueness theorem from prior work appears in the provided text; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

Review based on abstract only; full derivation details, parameter counts, and background assumptions unavailable.

free parameters (1)
  • instanton action S_inst
    Used as reference value against which numerical exponents are fitted and compared.
axioms (1)
  • domain assumption WKB instanton approximation accurately captures the leading exponential behavior of both splitting and distinguishability measures
    Invoked to predict the common exponent for spectral gap and information measures.
invented entities (1)
  • static parity horizon no independent evidence
    purpose: Benchmark describing the separation between parity reading and disturbance by the bath in the static J_z channel
    New descriptive term introduced in the abstract; no independent evidence supplied.

pith-pipeline@v0.9.1-grok · 5872 in / 1511 out tokens · 50429 ms · 2026-06-26T10:30:33.922306+00:00 · methodology

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Reference graph

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