arxiv: 2603.29323 · v2 · submitted 2026-03-31 · 🪐 quant-ph

Recognition: no theorem link

On the Entanglement Entropy Distribution of a Hybrid Quantum Circuit

Jeonghyeok Park , Hyukjoon Kwon , Hyunseok Jeong

Authors on Pith no claims yet

Pith reviewed 2026-05-14 00:25 UTC · model grok-4.3

classification 🪐 quant-ph

keywords entanglement entropy distributionhybrid quantum circuitsmeasurement-induced transitionsvolume-law phasearea-law phasehigher momentsdirected polymer

0 comments

The pith

Higher moments of entanglement entropy distribution diagnose measurement-induced phase transitions in hybrid quantum circuits.

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

The paper studies how entanglement entropy is spread out rather than just its average value in circuits that combine random unitary gates with occasional local measurements. It finds that quantities like the ratio of variance to mean and the skewness of this distribution behave differently depending on whether the system is in a volume-law or area-law phase. These extra statistics pick up features of the measurement-induced dynamics that the average entropy misses. A simple phenomenological model for the area-law regime, joined with the directed-polymer description for the volume-law regime, reproduces the patterns seen in simulations across the whole diagram. This offers new diagnostics for locating the transition point.

Core claim

Higher moments of the entanglement entropy distribution, such as the ratio between the variance and the mean and the skewness, capture nontrivial features of the measurement-induced dynamics that are invisible to the mean entropy alone and can serve as effective diagnostics of measurement-induced entanglement transitions.

What carries the argument

The full distribution of entanglement entropy, tracked via its variance-to-mean ratio and skewness, together with a phenomenological model for measurements in the area-law regime combined with the directed polymer in a random environment description of the volume-law phase.

If this is right

The variance-to-mean ratio and skewness take distinct, robust values in the volume-law phase compared with the area-law phase.
These higher moments remain useful diagnostics even when the mean entropy alone does not clearly mark the transition.
The phenomenological model for the area-law regime plus the directed-polymer description reproduces simulation results throughout the phase diagram.
Measurement-induced transitions can be located using statistics beyond the average entanglement entropy.

Where Pith is reading between the lines

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

Similar higher-moment diagnostics could be tested in other monitored quantum systems to locate transitions without full state tomography.
The link to directed-polymer models suggests that classical statistical-mechanics tools may help predict fluctuation properties in more complex quantum circuits.
Experimental platforms with controllable measurements might measure these skewness or variance ratios directly to map phase boundaries.

Load-bearing premise

The phenomenological model for the area-law regime, when combined with the directed polymer description, accurately reproduces the higher-moment behaviors across the phase diagram.

What would settle it

Numerical data or an experiment in which the variance-to-mean ratio or skewness fails to follow the distinct behaviors predicted for the volume-law versus area-law regimes by the combined model.

Figures

Figures reproduced from arXiv: 2603.29323 by Hyukjoon Kwon, Hyunseok Jeong, Jeonghyeok Park.

**Figure 2.** Figure 2: FIG. 2. (Left) Histograms of entanglement entropy distributions of the hybrid circuits at measurement rates [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Collapse of the entanglement entropy of different [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Variance of Haar random and Clifford random hybrid [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Skewness curves of the entanglement entropy dis [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Log-log plot of skewness in the area-law regime. [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Curves of numerical differentiation of the auxiliary [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. IoD as a function of system size [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Entanglement entropy distributions in the volume [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13. Kullback-Leibler divergence between the simulation [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗

read the original abstract

We investigate the distribution of entanglement entropy in hybrid quantum circuits consisting of random unitary gates and local measurements applied at a finite rate. We demonstrate that higher moments of the entanglement entropy distribution, such as the ratio between the variance and the mean and the skewness, capture nontrivial features of the measurement-induced dynamics that are invisible to the mean entropy alone. We demonstrate that these quantities exhibit distinct and robust behaviors across the volume-law and area-law phases, and can serve as effective diagnostics of measurement-induced entanglement transitions. We propose a phenomenological model describing the effect of measurements in the area-law regime, which, when combined with the directed polymer in a random environment description of the volume-law phase, well matches numerical simulations across the entire phase diagram.

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.

Desk Editor's Note private letter to a colleague

Higher moments like variance-to-mean ratio and skewness give practical new diagnostics for the measurement-induced transition, but the area-law phenomenological model needs explicit checks on whether it predicts them independently or just fits after the fact.

read the letter

This paper's core point is that the variance-to-mean ratio and skewness of the entanglement entropy distribution pick up features of the hybrid circuit dynamics that the mean entropy misses, and the authors back it with a combined model that pairs a new phenomenological description for the area-law regime with the directed-polymer picture for volume-law, matching their simulations across the diagram. That shift to higher moments feels like a straightforward and useful extension of the existing mean-entropy work in the field. The numerics apparently show clean separation in how these quantities behave in each phase, which could make them handy observables for future studies of monitored systems. What the paper does well is identify that the mean alone is incomplete and then propose concrete statistics plus a model to capture the full distribution. The volume-law side reuses established directed-polymer ideas without overclaiming novelty there, which keeps the focus on the new area-law part. The soft spot is exactly the one the stress test flags: the area-law model is phenomenological, so its functional form and parameters are tuned to the measurements. The abstract says it matches simulations, but without details on whether parameters were fixed using only lower moments before testing variance and skewness, or whether the same data drove everything, the claim that these moments reveal independent information rests on thinner ground. If the model has enough flexibility to accommodate the distributions after seeing them, the diagnostic power is less convincing. The citation pattern looks standard for the subfield, and the math is the usual combination of numerics and effective descriptions rather than anything formally derived from first principles. This is for people already working on measurement-induced transitions and hybrid circuits. The idea is clear enough and the proposed observables are concrete, so it deserves a serious referee to check the fitting procedure and confirm the higher moments add real predictive value beyond the mean. I'd send it to review but ask specifically for the parameter-fixing details and any cross-validation on the moments.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the distribution of entanglement entropy in hybrid quantum circuits consisting of random unitary gates and local measurements applied at finite rate. It claims that higher moments of this distribution, such as the variance-to-mean ratio and skewness, capture nontrivial features of the measurement-induced dynamics invisible to the mean entropy alone and can serve as effective diagnostics of the volume-law to area-law transition. A phenomenological model is proposed for the area-law regime that, when combined with the directed polymer in a random environment (DPRE) description for the volume-law phase, reproduces numerical simulations across the phase diagram.

Significance. If validated, the work offers new observables for characterizing measurement-induced phase transitions beyond mean entanglement entropy, which is a topic of active interest in quantum information and many-body physics. The integration of an established DPRE framework with a new phenomenological description for the area-law phase could provide a practical way to model full distributions, potentially aiding numerical and experimental studies of monitored quantum systems.

major comments (2)

[Abstract] Abstract: The central claim that higher moments (variance/mean ratio and skewness) reveal dynamics invisible to the mean and that the phenomenological model accurately reproduces them rests on the model's predictive power. The abstract states the model 'well matches' numerics but provides no detail on whether parameters were fixed using only mean-entropy data before testing higher moments; if the same data informed both, the agreement does not establish independent diagnostic value. This requires explicit clarification, e.g., in the methods or results section describing the fitting procedure.
[Phenomenological model] Phenomenological model (area-law regime): The functional form and any free parameters of the proposed model for the area-law phase must be shown to be fixed independently of the higher-moment data (e.g., via mean entropy or microscopic derivation) before claiming reproduction of variance and skewness across the diagram. Without this, the combination with the DPRE description risks being post-hoc rather than predictive, weakening the assertion that these moments encode new information about the transition.

minor comments (2)

Define the directed polymer in random environment (DPRE) acronym at first use and briefly recall its key predictions for the volume-law phase to aid readers unfamiliar with the framework.
Clarify the circuit sizes, number of realizations, and any data-exclusion criteria used in the numerical simulations to allow assessment of statistical reliability for the reported higher moments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments below and have revised the manuscript to provide the requested explicit clarification on the parameter determination procedure for the phenomenological model.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that higher moments (variance/mean ratio and skewness) reveal dynamics invisible to the mean and that the phenomenological model accurately reproduces them rests on the model's predictive power. The abstract states the model 'well matches' numerics but provides no detail on whether parameters were fixed using only mean-entropy data before testing higher moments; if the same data informed both, the agreement does not establish independent diagnostic value. This requires explicit clarification, e.g., in the methods or results section describing the fitting procedure.

Authors: We agree that explicit clarification of the fitting procedure is essential to establish the predictive power of the model. In the revised manuscript we have added a new subsection titled 'Parameter Determination' in the Methods section. There we state that the functional form of the phenomenological model was chosen on physical grounds (local measurements suppress entanglement growth in the area-law regime) and that all free parameters were obtained by fitting exclusively to the mean entanglement entropy data. These fixed parameters were then used without adjustment to compute the variance-to-mean ratio and skewness, which were compared to separate numerical runs. The fitting protocol, the resulting parameter values, and the cross-validation against higher moments are now documented in detail. revision: yes
Referee: [Phenomenological model] Phenomenological model (area-law regime): The functional form and any free parameters of the proposed model for the area-law phase must be shown to be fixed independently of the higher-moment data (e.g., via mean entropy or microscopic derivation) before claiming reproduction of variance and skewness across the diagram. Without this, the combination with the DPRE description risks being post-hoc rather than predictive, weakening the assertion that these moments encode new information about the transition.

Authors: We accept this criticism and have strengthened the manuscript accordingly. The revised text now contains an explicit statement that the model parameters were determined solely from mean-entropy fits performed on data sets independent of the variance and skewness calculations. We also include a supplementary figure that overlays the mean-entropy fit (used for parameter extraction) against the subsequently predicted higher moments, demonstrating that no refitting occurred. This procedure confirms that the agreement with variance and skewness is a genuine prediction rather than a post-hoc adjustment, thereby supporting the claim that these quantities carry additional diagnostic information about the transition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on established DPRE framework and independent numerics

full rationale

The paper reuses the directed polymer in a random environment (DPRE) description for the volume-law phase, which is drawn from prior independent literature rather than self-citation. The area-law regime is addressed via a new phenomenological model whose functional form is presented as matching numerical simulations of higher moments (variance/mean ratio and skewness) across the phase diagram. No equations or sections in the provided text reduce a claimed prediction to a fitted parameter by construction, nor do any load-bearing steps collapse to self-citations or ansatzes smuggled via prior work by the same authors. The central demonstration that higher moments capture features invisible to the mean rests on direct numerical evidence, making the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions of random unitary circuits and an existing directed-polymer model for one phase, plus a new phenomenological description for the other; no new particles or forces are introduced.

free parameters (1)

measurement rate p
The rate at which local measurements are applied is the primary control parameter scanned across the phase diagram.

axioms (1)

domain assumption Directed polymer in random environment description applies to the volume-law phase of hybrid circuits
Invoked to model entanglement statistics in the volume-law regime.

pith-pipeline@v0.9.0 · 5417 in / 1098 out tokens · 49443 ms · 2026-05-14T00:25:28.269284+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

63 extracted references · 63 canonical work pages · 1 internal anchor

[1]

M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Annu. Rev. Condens. Matter Phys.14, 335 (2023)

work page 2023
[2]

Oliveira, O

R. Oliveira, O. C. O. Dahlsten, and M. B. Plenio, Phys. Rev. Lett.98, 130502 (2007)

work page 2007
[3]

Hosur, X.-L

P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Jour- nal of High Energy Physics2016, 4 (2016)

work page 2016
[4]

Turkeshi, P

X. Turkeshi, P. Calabrese, and A. De Luca, Phys. Rev. Lett.135, 040403 (2025)

work page 2025
[5]

Nahum, J

A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Phys. Rev. X7, 031016 (2017)

work page 2017
[6]

Nahum, S

A. Nahum, S. Vijay, and J. Haah, Phys. Rev. X8, 021014 (2018)

work page 2018
[7]

Zhou and A

T. Zhou and A. Nahum, Phys. Rev. B99, 174205 (2019)

work page 2019
[8]

Wiseman and G

H. Wiseman and G. Milburn,Quantum Measurement and Control(Cambridge University Press, 2010)

work page 2010
[9]

Raussendorf and H

R. Raussendorf and H. J. Briegel, Phys. Rev. Lett.86, 5188 (2001)

work page 2001
[10]

H. J. Briegel, D. E. Browne, W. D¨ ur, R. Raussendorf, and M. Van den Nest, Nat. Phys.5, 19 (2009)

work page 2009
[11]

Misra and E

B. Misra and E. C. G. Sudarshan, J. Math. Phys.18, 756 (1977)

work page 1977
[12]

Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B98, 205136 (2018)

work page 2018
[13]

Skinner, J

B. Skinner, J. Ruhman, and A. Nahum, Phys. Rev. X9, 031009 (2019)

work page 2019
[14]

Y. Li, X. Chen, and M. P. A. Fisher, Phys. Rev. B100, 134306 (2019)

work page 2019
[15]

Jian, Y.-Z

C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Ludwig, Phys. Rev. B101, 104302 (2020)

work page 2020
[16]

Nahum, S

A. Nahum, S. Roy, B. Skinner, and J. Ruhman, PRX Quantum2, 010352 (2021)

work page 2021
[17]

Szyniszewski, A

M. Szyniszewski, A. Romito, and H. Schomerus, Phys. Rev. B100, 064204 (2019)

work page 2019
[18]

Ippoliti, M

M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V. Khemani, Phys. Rev. X11, 011030 (2021)

work page 2021
[19]

Poboiko, P

I. Poboiko, P. P¨ opperl, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. X13, 041046 (2023)

work page 2023
[20]

Poboiko, I

I. Poboiko, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. Lett.132, 110403 (2024)

work page 2024
[21]

Matsubara, K

T. Matsubara, K. Yamamoto, and A. Koga, Phys. Rev. B112, 054309 (2025)

work page 2025
[22]

Lang and H

N. Lang and H. P. B¨ uchler, Phys. Rev. B102, 094204 (2020)

work page 2020
[23]

Turkeshi, A

X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, and M. Schir´ o, Phys. Rev. B103, 224210 (2021)

work page 2021
[24]

Yokomizo and Y

K. Yokomizo and Y. Ashida, Phys. Rev. B111, 235419 (2025)

work page 2025
[25]

S. P. Kelly and J. Marino, Phys. Rev. A111, L010402 (2025)

work page 2025
[26]

Y. Bao, S. Choi, and E. Altman, Phys. Rev. B101, 104301 (2020)

work page 2020
[27]

Y. Li, S. Vijay, and M. P. Fisher, PRX Quantum4, 010331 (2023)

work page 2023
[28]

J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Nat. Phys.19, 1314 (2023)

work page 2023
[29]

Aharony and A

A. Aharony and A. B. Harris, Phys. Rev. Lett.77, 3700 (1996)

work page 1996
[30]

Schiulaz, E

M. Schiulaz, E. J. Torres-Herrera, F. P´ erez-Bernal, and L. F. Santos, Phys. Rev. B101, 174312 (2020)

work page 2020
[31]

Wiseman and E

S. Wiseman and E. Domany, Phys. Rev. E52, 3469 (1995)

work page 1995
[32]

Entanglement in the stabilizer formalism

D. Fattal, T. S. Cubitt, Y. Yamamoto, S. Bravyi, and I. L. Chuang, arXiv:quant-ph/0406168 arXiv:quant- ph/0406168 (2004)

work page internal anchor Pith review Pith/arXiv arXiv 2004
[33]

Agrawal, A

U. Agrawal, A. Zabalo, K. Chen, J. H. Wilson, A. C. Potter, J. H. Pixley, S. Gopalakrishnan, and R. Vasseur, Phys. Rev. X12, 041002 (2022)

work page 2022
[34]

H. Ha, A. Pandey, S. Gopalakrishnan, and D. A. Huse, Phys. Rev. B110, L140301 (2024)

work page 2024
[35]

Liu, M.-R

S. Liu, M.-R. Li, S.-X. Zhang, and S.-K. Jian, Phys. Rev. Lett.132, 240402 (2024)

work page 2024
[36]

Liu, M.-R

S. Liu, M.-R. Li, S.-X. Zhang, S.-K. Jian, and H. Yao, Phys. Rev. B110, 064323 (2024)

work page 2024
[37]

Bejan, C

M. Bejan, C. McLauchlan, and B. B´ eri, PRX Quantum 5, 030332 (2024)

work page 2024
[38]

Niroula, C

P. Niroula, C. D. White, Q. Wang, S. Johri, D. Zhu, C. Monroe, C. Noel, and M. J. Gullans, Nat. Phys.20, 1786 (2024)

work page 2024
[39]

Bravyi and D

S. Bravyi and D. Maslov, IEEE Trans. Inf. Theory67 (2021)

work page 2021
[40]

Webb, The clifford group forms a unitary 3-design (2016), arXiv:1510.02769 [quant-ph]

Z. Webb, The clifford group forms a unitary 3-design (2016), arXiv:1510.02769 [quant-ph]

work page arXiv 2016
[41]

Aaronson and D

S. Aaronson and D. Gottesman, Phys. Rev. A70, 052328 (2004)

work page 2004
[42]

Vasseur, A

R. Vasseur, A. C. Potter, Y.-Z. You, and A. W. W. Lud- wig, Phys. Rev. B100, 134203 (2019). 11

work page 2019
[43]

L. D. Landau and E. M. Lifshitz,Statistical Physics, Part 1, 3rd ed., Course of Theoretical Physics, Vol. 5 (Butterworth-Heinemann, Oxford, 1980)

work page 1980
[44]

Sierant and J

P. Sierant and J. Zakrzewski, Phys. Rev. B99, 104205 (2019)

work page 2019
[45]

C. A. Fuchs and K. Jacobs, Phys. Rev. A63, 062305 (2001)

work page 2001
[46]

Jacobs and D

K. Jacobs and D. A. Steck, Contemp. Phys.47, 279 (2006)

work page 2006
[47]

Fano, Phys

U. Fano, Phys. Rev.72, 26 (1947)

work page 1947
[48]

Blanter and M

Y. Blanter and M. B¨ uttiker, Phys. Rep.336, 1 (2000)

work page 2000
[49]

Fox,Quantum Optics: An Introduction(Oxford Uni- versity Press, Oxford, 2006)

M. Fox,Quantum Optics: An Introduction(Oxford Uni- versity Press, Oxford, 2006)

work page 2006
[50]

Kronwald, M

A. Kronwald, M. Ludwig, and F. Marquardt, Phys. Rev. A87, 013847 (2013)

work page 2013
[51]

Short and L

R. Short and L. Mandel, Phys. Rev. Lett.51, 384 (1983)

work page 1983
[52]

Y. Wang, H. Ye, Z. Yu, Y. Liu, and W. Xu, Sci. Rep.9, 10088 (2019)

work page 2019
[53]

H. K. Zhao, Y. W. Liu, L. Shao, L. Fang, and M. Dong, Phys. Rev. Fluids6, 104608 (2021)

work page 2021
[54]

O. E. Garcia, Phys. Rev. Lett.108, 265001 (2012)

work page 2012
[55]

Wampler and A

T. Wampler and A. C. Barato, J. Phys. A55, 014002 (2022)

work page 2022
[56]

Singha and M

T. Singha and M. K. Nandy, Phys. Rev. E90, 062402 (2014)

work page 2014
[57]

C. A. Tracy and H. Widom, Commun. Math. Phys.161, 289 (1994)

work page 1994
[58]

Halpin-Healy and K

T. Halpin-Healy and K. A. Takeuchi, Journal of Statisti- cal Physics160, 10.1007/s10955-015-1282-1 (2015)

work page doi:10.1007/s10955-015-1282-1 2015
[59]

Johansson, Commun

K. Johansson, Commun. Math. Phys.209, 437 (2000)

work page 2000
[60]

Dieng, Int

M. Dieng, Int. Math. Res. Not.2005, 2263 (2005)

work page 2005
[61]

Risken,The Fokker-Planck Equation: Methods of So- lution and Applications, 2nd ed., Springer Series in Syn- ergetics, Vol

H. Risken,The Fokker-Planck Equation: Methods of So- lution and Applications, 2nd ed., Springer Series in Syn- ergetics, Vol. 18 (Springer-Verlag, Berlin, Heidelberg, 1996)

work page 1996
[62]

Kullback and R

S. Kullback and R. A. Leibler, Ann. Math. Stat.22, 79 (1951)

work page 1951
[63]

C. M. Bender and S. A. Orszag,Advanced Mathemat- ical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory(Springer-Verlag, New York, 1999)

work page 1999