Universal Statistics of Measurement-Induced Entanglement in Tomonaga-Luttinger liquids

Kabir Khanna; Romain Vasseur

REVIEW 4 major objections 5 minor 4 references

Universal Statistics of Measurement-Induced Entanglement in Tomonaga-Luttinger liquids

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The full statistics of measurement-induced entanglement in one-dimensional Tomonaga-Luttinger liquids are universal: Born-averaging over measurement outcomes is equivalent, at low energy, to averaging over conformal boundary conditions weig

desk verdict First analytic result for the full distribution of measurement-induced entanglement in Tomonaga-Luttinger liquids, with a real but unquantified assumption about non-conformal outcomes that should be pressed in review. read the letter →

arxiv 2512.13809 v2 pith:SHDETNZW submitted 2025-12-15 quant-ph cond-mat.stat-mechcond-mat.str-elhep-th

Kabir Khanna , Romain Vasseur This is my paper

classification quant-phcond-mat.stat-mechcond-mat.str-elhep-th

keywords measurement-inducedentanglementTomonaga-LuttingerliquidconformalfieldtheoryBornaverageboundaryconditionsreplicatrickstatisticsRényientropy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

What this paper tries to establish: after projective measurements on part of a one-dimensional quantum critical system, the Born-averaged post-measurement entanglement—and all its higher cumulants—are universal and conformally invariant. The key mechanism is a replica trick that turns the average over exponentially many measurement outcomes into an average over conformal boundary conditions of the compact free boson, with each boundary condition weighted by its partition function. As a result, all cumulants share a universal scaling in the maximal-separation limit, with a crossover at nl = 1/2. The full distribution is bimodal with heavy tails. The predictions are confirmed numerically on the XX chain.

What carries the argument

The central object is the generalized replica partition function Z_A(k1,k2) and the conformal map from the n-sheeted cylinder to a finite cylinder with length h(ζ). A reflection rotation isolates the measurement dependence in a single replica, producing a 'winding function' that sums over compact-boson windings. Poisson resummation and analytic continuation convert this into the integral representation that becomes the boundary-condition average.

What would settle it

Compute in an interacting gapless chain (e.g., XXZ at Δ≠0) the second and third cumulants of MIE at small cross-ratio and check the predicted scaling ζ^{g/2}/√log(1/ζ) and the crossover at nl=1/2; alternatively, directly compute the contribution of the ferromagnetic outcome to the Born average at small ζ and test whether it vanishes compared with conformal outcomes.

Watch

Extended reading notes

Core claim

The central claim is that the generalized replica partition function Z_A(k1,k2) = (tr ρ^n_{m,A})^{k1} tr ρ_m^{k2} equals, at low energies, an integral over conformal boundary conditions δφ: Z_A ~ ∫ dδφ Z_{C(1),δφ} (Z_{C(n),δφ})^{k1} (Z_{C(1),δφ})^{k2}. Consequently, the l-th cumulant of the post-measurement entanglement equals the l-th cumulant of the forced MIE under p(δφ) ∝ Z_{C(1),δφ}. In the maximal-separation limit ζ→0, cumulants scale as ζ^{g/2}/√log(1/ζ) for n > 1/(2l), as ζ^{g/2} at n = 1/(2l), and as ζ^{2gnl(1−nl)} for n < 1/(2l). The full distribution is bimodal with heavy tails.

Load-bearing premise

Every microscopic measurement outcome can be replaced, at low energies, by a conformally invariant Dirichlet boundary condition, with non-conformal outcomes such as the ferromagnetic state contributing negligibly to the Born average.

Editorial extensions

If this is right

The full statistics of MIE in TLLs are universal: the mean, variance, skewness, and all higher cumulants are fixed by the Luttinger parameter and the cross-ratio.
For large enough Rényi index, every cumulant decays as ζ^{g/2}/√log(1/ζ) when the unmeasured regions are far apart, and the crossover at nl=1/2 is a sharp signature of Born-averaged physics.
The tail near S_m = log 2 implies a finite (if small) probability of generating a Bell pair across the measured region, so a critical state can act as a quantum wire.
Uniform (quenched-disorder) averaging produces different scaling, 1/log(1/ζ), independent of the Rényi index and Luttinger parameter—so Born weighting matters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One could test the recipe in other critical models (e.g., Ising, Potts) where measurements may or may not flow to conformal boundary conditions; the present derivation provides a natural template.
The log-normal tail suggests that experimental single-shot measurements will often find entanglement far from the mean; sample-to-sample fluctuations may be large.
The crossover at nl = 1/2 could be measured by tuning the Rényi index in cold-atom or ion-trap implementations of monitored critical chains.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The new thing here is the full statistics, not just the mean: closed forms for all cumulants of MIE and the full distribution, plus universal ζ→0 scaling forms (ζ^{g/2}/√log with a transition at nl = 1/2) and a distinct prediction for disorder-averaged entanglement (DIE). The two-parameter replica trick and the representation of the Born average as a weighted average over conformal boundary conditions are real advances. There are no fitted parameters, and the numerical checks on the XX chain — second and third cumulants and P(S_m) at ζ = 0.02, L up to 600 — agree well with the theory. This is a serious piece of work, not a routine extension of the authors' earlier mean-MIE paper.

The main soft spot is exactly the one the authors acknowledge in Sec. 3.1 and then partially wave away. The derivation of the classical winding contribution, Eq. (25), uses an action for a constant boundary-value difference; spatial variations of the measurement outcome m(θ) never enter. So the sum over all microscopic outcomes is effectively replaced by a sum over constant Dirichlet shifts. The paper asserts that non-conformal outcomes such as the ferromagnetic state have small weight, but no bound or RG argument is given. That matters more here than in the mean calculation, because the predicted distribution is heavy-tailed: rare outcomes with S_m near log 2 carry a ζ^{g/2}/√log weight, the same order as the leading cumulant scaling in Eq. (40). If a small fraction of non-conformal outcomes produces near-maximal entanglement with non-negligible Born weight, the higher cumulants could receive leading-order corrections. The numerical agreement at ζ = 0.02 is reassuring but does not probe the ζ→0 regime, which the authors say is beyond accessible system sizes.

The replica-limit analytic continuation in Appendix B follows a known method and is plausible, though not fully rigorous. I do not see that as a deal-breaker; the physics is clear and the structure of the result is sensible. The scaling forms are checked analytically within the paper but not against numerics in the scaling window — a limitation the authors disclose.

Bottom line: this deserves a serious referee. The central claim is conditional on a stated but unquantified assumption, and the paper would be stronger with any quantitative handle on the non-conformal contributions — an RG estimate, a bound, or a cleverer numerical probe. But the derivation is coherent, the results are concrete and falsifiable, and there is no circularity. Send it to peer review, ask the referees to press on the non-conformal outcomes, and expect the authors to have something to say. I would cite this in my own work.

Referee Report

4 major / 5 minor

Summary. The paper studies the statistics of measurement-induced entanglement (MIE) after projective charge measurements on disjoint intervals of a one-dimensional Tomonaga-Luttinger liquid. Using a generalized replica partition function Z_A(k1,k2) = <(tr ρ_{m,A}^n)^{k1} (tr ρ_m)^{k2}>, the authors map the problem to a compact free boson on a cylinder, perform a rotation that isolates the measurement dependence in a single replica, and sum over winding sectors. The central result, Eq. (39), states that the cumulant-generating function of the Born-averaged MIE equals that of the 'forced' MIE averaged over conformal boundary conditions with weight p(δφ) ∝ Z_{C(1),δφ}. From this they derive the ζ→0 scaling Eq. (40), with regimes separated at nl=1/2, and the full distribution P(S_m), which is bimodal with a log-normal tail near S_m=0 and a 1/sqrt(log 2 - S_m) tail near S_m=log 2. They also define disorder-induced entanglement (DIE) and find 1/log(1/ζ) scaling. Numerical checks on the XX chain for κ2, κ3, P(S_m), and DIE show good agreement at moderate sizes and at ζ=0.02.

Significance. If the central equivalence Eq. (39) is correct, this is a substantial advance: it provides parameter-free, conformal-field-theoretic predictions for the full outcome-averaged entanglement statistics in a generic critical 1D system, including heavy tails and a rare Bell-pair-generating outcome. The absence of fitted parameters and the quantitative agreement of several independent numerical observables (Figs. 3a,b, 4, 5) are genuine strengths. The universality claim, however, rests on two non-rigorous steps: the assertion that non-conformal microscopic measurement outcomes are negligible in the Born average, and the analytic continuation of the winding function before the replica limit. The paper explicitly acknowledges the former (Sec. 3.1) but provides no bound, and the numerical verification of the asymptotic scaling Eq. (40) is admittedly beyond the accessible regime. These issues are load-bearing but, in my view, addressable within the manuscript's scope.

major comments (4)

[Secs. 3.1–3.2, Eq. (25)] The derivation reduces every lattice measurement outcome to a constant Dirichlet shift. The classical action used in Eq. (25), S_C(1)[φ_cl,m] = (g/2)(δm)^2/h, depends only on a constant boundary-value difference, not on the spatial profile m(θ). The sum over m in Eq. (24) is therefore effectively a sum over constant boundary conditions only. The paper acknowledges in Sec. 3.1 that outcomes such as |↑↑↑…⟩ do not flow to conformal boundary conditions, but only states that their contributions are 'small in practice'; no RG estimate, operator-content argument, or numerical bound is supplied. This is load-bearing because the predicted P(S_m) and the ζ→0 scaling Eq. (40) are controlled by rare near-maximal outcomes; if a small Born-weight fraction of non-conformal outcomes yields S_m near log 2, the leading-order cumulant scaling could be modified. The numerical checks at ζ=0.02 and L≤600 do n
[Sec. 3.2 and Appendix B] The analytic continuation of the winding function W from Eq. (31) to Eq. (32) is not justified. The derivation in Appendix B uses Poisson resummation, completing the square, and a Dirac-delta Fourier representation for positive integers k1 and k2; the final expression involves fractional powers of sums (the k1 and k2 powers of theta functions), and the replica limit k→0 is then taken after this continuation. This is a standard replica-trick heuristic, but it is not a derivation unless a uniqueness or continuity argument is supplied. Because the replica limit is the entire content of the paper, this step needs either a clearer justification (e.g., via the theta-function representation being entire in the exponent) or an explicit check: expand both sides of Eq. (31)/(32) to order k^2 along k1=k, k2=-nk and verify against the stated Eq. (37). At minimum, the paper should flag this as a work
[Sec. 4.1, Eq. (37)] The statement that 'taking the replica limit in (17) with the derived generalized replica partition function (30) reproduces precisely (37)' is not demonstrated. Eq. (37) is the first nontrivial test of the central equivalence and the basis for the general cumulant formula Eq. (39), so the intermediate algebra should be shown (or sketched in an appendix). This is not purely cosmetic: the prefactors and normalization in Eqs. (30) and (32) can contribute to derivatives along k1=k, k2=-nk, and the reader cannot verify that the κ2 formula is free of such contributions.
[Sec. 4.2 and Appendix A] The derivation of Eq. (40) computes the scaling of moments and then asserts the same scaling for cumulants. For a heavy-tailed distribution this is plausible, but it should be justified: cumulants are linear combinations of products of lower moments, and one must check that no cancellation removes the leading term. A small-ζ numerical check of the moment/cumulant ratio, even at the smallest reachable ζ, would strengthen the claim. The paper's own statement (Sec. 4.2) that the scaling window is 'beyond the numerical window accessible in this work' is an important caveat that should be stated more prominently; Fig. 3(c) is a theoretical extension, not a numerical verification.

minor comments (5)

[Sec. 1] Typo: 'indciates' should be 'indicates'. Also 'à priori' should be italicized and without a space in French ('a priori').
[Eq. (25)] The symbol δm is not defined before use. Please define it as the difference of the (constant) boundary values on the two cylinder boundaries, and explicitly note that the spatial-profile dependence has been dropped at this stage; this will make the approximation in the main text transparent.
[Eq. (32)] The prefactor sqrt((nk1+k2+1)g/(2πh)) in Eq. (32) is not derived in the main text. The naive completion-of-square calculation in Appendix B gives different powers of sqrt(2πg/h) and sqrt(2πgn/h); please clarify how these are absorbed and whether they affect the replica limit.
[Sec. 4.4.2, Eq. (59)] There appears to be a typo in the expression involving 'ε g π logζ' around Eq. (59); the sign and placement of π should be checked against Eq. (58).
[Fig. 3 caption] The caption says 'we see the scaling Eq. (40)' for the theoretical curves in panel (c). Rephrase to make clear that this is a prediction extended into the asymptotic regime, not a numerical observation.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the higher-cumulant predictions are derived from a replica/CFT calculation and benchmarked numerically; self-citations are supporting rather than load-bearing.

full rationale

The paper's new results (Eqs. 39-40 and Eq. 50) are obtained by analytic continuation of the winding function W^{(n)}_{k1,k2} (Eqs. 31-32) within a replica calculation. This is a derivation, not a fit: no parameter is adjusted to the cumulant data, and the predictions are independently benchmarked against free-fermion numerics for κ2, κ3, and P(S_m) (Figs. 3-4). The dependence on the authors' previous work [69] is for the mean-MIE replica evaluation and for the earlier 'Born-averaging over boundary conditions' interpretation; the two-parameter generalized replica partition function and all higher-cumulant expressions are new, so the self-citation is supporting rather than load-bearing. The paper explicitly acknowledges the main physical assumption in Sec. 3.1: microscopic outcomes such as the ferromagnetic |↑↑↑…⟩ need not flow to conformal boundary conditions, and are asserted to be 'small in practice'. Eq. (25) likewise keeps only the constant boundary-value difference δm. This is a low-energy physical approximation/limitation (a correctness risk), not a circular reduction: the predicted scaling and distribution are non-trivial consequences of the replica/CFT calculation, not restatements of the assumption. The small-ζ regime of Eq. (40) is beyond the accessible numerical window and is presented as an analytic prediction rather than as a fit. Honest finding: no significant circularity; score 2 reflects only the minor, non-load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper has no free parameters fitted to data; g is the Luttinger parameter of the model. It relies on the standard CFT description of TLLs and on the ad hoc assumption that every measurement outcome can be represented by a conformal Dirichlet boundary condition, with non-conformal outcomes assumed negligible. The replica trick and its analytic continuation are taken as standard.

assumptions (6)

standard math Low-energy description of the gapless XXZ chain by the compact free-boson (Luttinger liquid) CFT with action S = g/(4π)∫((∂_x φ)^2+(∂_τ φ)^2).
Section 2; standard bosonization result used to model the critical state.
ad hoc to paper Projective measurement of the local charge (σ^z) maps to imposing Dirichlet boundary conditions on the bosonic field φ in the measured region.
Section 2; the mapping is motivated by symmetry and previous numerical agreement, not proven for all outcomes. The paper itself notes that some lattice states (e.g., |↑↑↑…⟩) do not flow to conformal Dirichlet BCs.
ad hoc to paper Non-conformal measurement outcomes contribute negligibly to Born-averaged observables.
Section 3.1; asserted because symmetry-breaking is 'only true near the boundary' and previous numerics show conformal invariance. No quantitative bound is given.
domain assumption The replica trick and the analytic continuation in k (taking k→0 after continuation) are valid for these partition functions.
Section 3 and Appendix B; standard but non-rigorous in this context; the winding function is analytically continued via Poisson resummation and a Gaussian integral.
domain assumption The geometric contribution to the free energy cancels in the replica limit for the cylinder target manifold (F_geom^n = n F_geom^1).
Section 3.1; proven in previous work [69], not re-derived here; essential for dropping the geometric factor.
ad hoc to paper The sum over all measurement outcomes m can be replaced by an integral over the single parameter δφ = φ|_B1 − φ|_B2 labeling conformal boundary conditions.
Section 3.2/4.1; the outcome of the replica calculation is that the full sum depends only on δφ, but this relies on the earlier assumptions about boundary conditions and the analytic continuation.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Universal Statistics of Measurement-Induced Entanglement in Tomonaga-Luttinger liquids." pith.science (2026). https://pith.science/paper/SHDETNZW

@misc{pith2026251213809,
  author       = {Pith},
  title        = {Pith review of: Universal Statistics of Measurement-Induced Entanglement in Tomonaga-Luttinger liquids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SHDETNZW}},
  note         = {Machine review of arXiv:2512.13809}
}

read the original abstract

We study the statistics of measurement-induced entanglement (MIE) after partial measurement on a class of one-dimensional quantum critical states described by Tomonaga-Luttinger liquids at low energies. Using a replica trick to average over measurement outcomes in the charge basis and tools from conformal field theory (CFT), we derive closed-form expressions for the cumulants of MIE. We show that exact Born-averaging over microscopic measurement outcomes becomes equivalent at low energy to averaging over conformal boundary conditions weighted by their corresponding partition functions. Our results yield distinctive critical behavior across all cumulants in the regime where the unmeasured parts of the system are maximally separated. We also obtain the full distribution of MIE, finding that it is generically bimodal and exhibits fat-tails. We corroborate our analytical predictions by numerical calculations and find good agreement between them.

Figures

Figures reproduced from arXiv: 2512.13809 by the authors.

**Figure 1.** Setup. Left: Schematic illustration of a TLL modeled as a spin-1/2 chain. Red (green) sites denote the measured (unmeasured) sites. Green shaded curves illustrate the entanglement between the unmeasured spins. Right: Manifolds M1 and C(1), and the conformal mapping between them. The intervals A = [x1 , x2 ] and C = [x3 , x4 ] on the cylinder M1 denote unmeasured regions, while B = [x1 , x4 ] ∪ [x2 , x3 ] = B1 ∪ B2 … view at source ↗

**Figure 2.** n-sheeted Riemann surface. Illustration of the gluing condition in region A from Eq. (10), following Refs. [88, 89]. For simplicity, the constituent manifolds are drawn as infinite planes rather than cylinders, and region C is omitted. The measurement region B1 (B2 ) is depicted in green (blue). The resulting manifold is the n-sheeted Riemann cylinder, which we denote by Mn (see an illustration of the infinite plane… view at source ↗

**Figure 3.** Cumulants of MIE versus cross ratio. Markers show numerical results for cumulants of MIE the XX chain for the setup in [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Distribution of MIE. Left: Comparison of the theory from Eq. (50) and the numerical distribution P(Sm) for the XX chain at a fixed value of the cross-ratio ζ = 0.02 and L = 600 (see [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

**Figure 5.** Figure 5: Disorder Induced Entanglement (DIE). Markers show numerical results and solid curves are theoretical predictions. References [1] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W. K. Wootters, Teleporting an unknown quantum state via dual classical and …

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

4 extracted references · 2 linked inside Pith

[1]

[1]C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W . K. Wootters,Teleport- ing an unknown quantum state via dual classical and einstein-podolsky-rosen channels, Physical review letters70(13), 1895 (1993). [2]P . W . Shor,Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995), doi:10.1103/PhysRevA.52.R249...

arXiv 1993
[2]

Johansen, M

Gaebler, J. Johansen, M. Mills, S. A. Moses, J. M. Pino, A. Ransfordet al.,Non-Abelian topological order and anyons on a trapped-ion processor, Nature626(7999), 505 (2024), doi:10.1038/s41586-023-06934-4. [12]D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T . Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P . Bonilla Ataides, N....

doi:10.1038/s41586-023-06934-4 2024
[3]

Huang, A

Mark, H.-Y. Huang, A. Kale, H. Pichler, F . G. S. L. Brandãoet al.,Preparing random states and benchmarking with many-body quantum chaos, Nature613(7944), 468 (2023), doi:10.1038/s41586-022-05442-1. [31]J. S. Cotler, D. K. Mark, H.-Y. Huang, F . Hernández, J. Choi, A. L. Shaw, M. Endres and S. Choi,Emergent quantum state designs from individual many-body ...

arXiv 2023
[4]

Problems, Progress of Theoretical Physics5(4), 544 (1950), doi:10.1143/ptp/5.4.544. [71]J. M. Luttinger,An Exactly Soluble Model of a Many-Fermion System, Journal of Mathe- matical Physics4(9), 1154 (1963), doi:10.1063/1.1704046. [72]F . D. M. Haldane,’luttinger liquid theory’ of one-dimensional quantum fluids. i. properties of the luttinger model and the...

arXiv 1950

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

[1]C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres and W . K. Wootters,Teleport- ing an unknown quantum state via dual classical and einstein-podolsky-rosen channels, Physical review letters70(13), 1895 (1993). [2]P . W . Shor,Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493 (1995), doi:10.1103/PhysRevA.52.R249...

arXiv 1993

[2] [2]

Johansen, M

Gaebler, J. Johansen, M. Mills, S. A. Moses, J. M. Pino, A. Ransfordet al.,Non-Abelian topological order and anyons on a trapped-ion processor, Nature626(7999), 505 (2024), doi:10.1038/s41586-023-06934-4. [12]D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T . Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P . Bonilla Ataides, N....

doi:10.1038/s41586-023-06934-4 2024

[3] [3]

Huang, A

Mark, H.-Y. Huang, A. Kale, H. Pichler, F . G. S. L. Brandãoet al.,Preparing random states and benchmarking with many-body quantum chaos, Nature613(7944), 468 (2023), doi:10.1038/s41586-022-05442-1. [31]J. S. Cotler, D. K. Mark, H.-Y. Huang, F . Hernández, J. Choi, A. L. Shaw, M. Endres and S. Choi,Emergent quantum state designs from individual many-body ...

arXiv 2023

[4] [4]

Problems, Progress of Theoretical Physics5(4), 544 (1950), doi:10.1143/ptp/5.4.544. [71]J. M. Luttinger,An Exactly Soluble Model of a Many-Fermion System, Journal of Mathe- matical Physics4(9), 1154 (1963), doi:10.1063/1.1704046. [72]F . D. M. Haldane,’luttinger liquid theory’ of one-dimensional quantum fluids. i. properties of the luttinger model and the...

arXiv 1950