Pith. sign in

REVIEW 1 major objections 5 minor 84 references

Detection of a R\'enyi Index Dependent Transition in Entanglement Entropy Scaling

T0 review · 1 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 paper constructs a number-conserving many-body state whose entanglement scaling changes with the Rényi index, and a symmetry-aware bound that detects the change from Rényi-2 data.

desk verdict A solid existence proof for Rényi-index-dependent entanglement scaling in a simple number-conserving state; the main gap is that the α≠1 scalings are asserted from analogy and numerics rather than derived, but that gap is small and fixable. read the letter →

arxiv 2512.24533 v3 pith:SRLVITSI submitted 2025-12-31 cond-mat.str-el

Hatem Barghathi , Adrian Del Maestro This is my paper · ORCID
classification cond-mat.str-el
keywords RényientropyvonNeumannentanglementscalingsymmetry-resolvednumber-conservingstatearealawviolationquantumMonteCarlotransition
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

Experiments and quantum Monte Carlo commonly use the second Rényi entropy as a stand-in for the von Neumann entanglement entropy, assuming the two scale the same way with subsystem size. This paper shows that the assumption can fail qualitatively: it constructs a number-conserving state of two-level sites (fermions or hard-core bosons) in which the leading scaling is S_{alpha>1} ~ ln l, S_{alpha=1} ~ sqrt(l) ln l, and S_{alpha<1} ~ l. The origin is replica biasing: varying alpha reweights the contribution of different charge sectors, and the sectors with the most entanglement are rare enough to be suppressed for alpha>1. To make the effect detectable, the paper proves that the quantity Sbar_alpha = sum_{q_A} P_{q_A} S_alpha(q_A) + H_1({P_{q_A}}) lies between S_alpha and S_1, so observing Sbar_alpha not asymptotically compatible with S_alpha certifies that S_alpha is not asymptotically compatible with S_1.

What carries the argument

The carrying object is the pair-complemented ansatz: each site in the A partition is paired with a site in B whose occupation is its logical NOT, so every n-sector has Schmidt rank C(l,n) and maximal symmetry-resolved entropy ln C(l,n). The second ingredient is the exponentially decaying number distribution P_n = A_l exp(-min{n,l-n}/mu(l)) with mu(l) ~ sqrt(l), whose subextensive width controls which sectors dominate the Rényi sum. The proof machinery is the power-mean inequality applied to the charge-resolved decomposition, yielding the sandwich S_1 >= Sbar_alpha >= S_alpha; Sbar_alpha is built from charge-resolved Rényi entropies plus the Shannon entropy of the charge distribution, and it

What would settle it

Compute, for the explicit Schmidt form with mu(l) ~ sqrt(l) at increasing l up to ~10^5, the exact S_1, S_2, and Sbar_2; the paper predicts S_1 ~ sqrt(l) ln l, S_2 ~ ln l, and Sbar_2/S_2 diverging, so finding S_1 ~ sqrt(l) without the log factor or Sbar_2 ~ S_2 would falsify the central claim.

Watch

Extended reading notes

Core claim

At the center is a half-filled ansatz |Psi> = sum_n sqrt(P_n) |psi_{n,N-n}> with P_n = A_l exp(-min{n,l-n}/mu(l)) and mu(l) ~ sqrt(l), where each projected state |psi_{n,N-n}> is a uniform superposition over configurations whose A-side occupations are the complements of their B-side partners. This makes every symmetry sector maximally entangled, S_alpha(n;l) = ln C(l,n), while the number distribution has a width that grows only as the square root of the subsystem size. The paper shows that these two ingredients produce the full hierarchy of leading scalings: logarithmic for alpha>1, sqrt(l) ln l for alpha=1, and volume-law l for alpha<1. It then proves a general statement for charge-conservi

Load-bearing premise

The load-bearing premise is that the explicit ansatz — realized only at half-filling and not shown to be the ground state of any local Hamiltonian — captures low-energy physics of realistic interacting systems; if no local Hamiltonian can host such a state, the predicted scaling transition remains a construction artifact.

Editorial extensions

If this is right

  • For the constructed state, S_2 ~ ln l while S_1 ~ sqrt(l) ln l: a second-Rényi measurement would miss the dominant von Neumann entanglement by a factor that grows with system size.
  • The bound Sbar_2 is computable with existing replica protocols: histogram the particle number in region A, estimate the sector-resolved swap expectation values, and add the Shannon term H_1.
  • If Sbar_2 and S_2 are found to be asymptotically incompatible, the paper's theorem certifies that S_2 is not asymptotically compatible with S_1, so no extrapolation from S_2 alone can recover the von Neumann scaling.
  • The same inequality gives a rigorous lower bound on S_1 from symmetry-resolved Rényi data even in systems where no transition occurs; regardless of whether Sbar_alpha tracks S_alpha, the bound remains useful.
  • The mechanism identifies a concrete failure mode for tensor-network and QMC entanglement estimates: hidden entanglement can be concentrated in rare charge sectors that are invisible to S_2 but dominant for S_1.

Reading between the lines

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

  • Beyond the paper: the ansatz suggests a general recipe for engineering states with hidden entanglement — take a charge distribution of subextensive width and maximal sector entanglement; tuning the width exponent gamma in mu(l) ~ l^gamma would interpolate the von Neumann scaling between ln l and l while leaving alpha>1 logarithmic.
  • Beyond the paper: if a local Hamiltonian realizing (or approximately realizing) this state is found, the S_2/S_1 mismatch would become a measurable signature of an entanglement phase, and the diagnostic could be used to map it in cold-atom or QMC experiments.
  • Beyond the paper: in single-site-resolved cold-atom experiments, the particle-number histogram P_n is already accessible, so Sbar_2 can likely be extracted without preparing additional copies beyond those used for S_2; a finite-size study at moderate l could test the predicted sqrt(l) ln l form of S_1.
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.

Referee Report

1 major / 5 minor

Summary. The paper claims that a simple U(1)-symmetric, number-conserving state on a bipartite lattice can have a Rényi-index-dependent leading entanglement scaling. For a state defined by particle-number weights P_n = A_ℓ e^{-min{n,ℓ−n}/μ(ℓ)} with μ(ℓ) ∼ √ℓ and maximally entangled charge sectors S_α(n;ℓ)=ln C(ℓ,n), it predicts S_{α>1} ∼ lnℓ, S_{α=1} ∼ √ℓ lnℓ, and S_{α<1} ∼ ℓ. It then introduces S̄_α = Σ P_{q_A} S_α(q_A) + H_1({P_{q_A}}), proves S_1 ≥ S̄_α ≥ S_α for α>1, and shows that if S̄_α and S_α are not asymptotically compatible then S_α and S_1 are not asymptotically compatible, yielding an S_2-based diagnostic. A Tomonaga-Luttinger example is used to show when the diagnostic is inconclusive.

Significance. If fully established, the explicit state is a valuable minimal counterexample to the common assumption that S_2 tracks S_1: the construction uses only two local states per site, and the symmetry-aware bound is clean and operationally meaningful. The paper’s strengths include a self-contained derivation of the inequality chain in Sec. V, a fully specified state with reproducible data and code, and an explicit case where the proposed diagnostic is known to be inconclusive. The main weakness is that the asymptotic scalings for α>1 and α<1 in the explicit state are supported by numerics and analogy rather than by a complete derivation; this is the load-bearing part of the existence claim and needs to be supplied before the result can be regarded as established.

major comments (1)
  1. [Section IV and Appendix A] The central asymptotic claim for the explicit state is not proven for α≠1. In Eq. (4) with P_n from Eq. (13) and S_α(n;ℓ)=ln C(ℓ,n), the α>1 summand is (P_n)^α C(ℓ,n)^{1−α}. This sum is controlled by the edge sectors n=0 and n=ℓ (where C(ℓ,n)=1), not by the sectors n∼μ(ℓ) used in the analogy; that case is not analyzed. For α<1, the sum is controlled by a saddle near n=ℓ/2, and a Laplace-type argument is required. The present text argues by analogy with Sec. III A, where S_α(n)=B min{n,N−n} makes the sums elementary, and by numerical extrapolation (Figs. 2 and 4). Appendix A derives only the α=1 case and does not track the normalization or H_1 term in detail. Because the Rényi-index-dependent scaling trio is the advertised main result, these missing asymptotics are load-bearing. Please supply a direct asymptotic evaluation, or rigorous bounds, for the three cases.
minor comments (5)
  1. [Eq. (13)] The displayed normalization constants appear incorrect for finite N. For even N=2m, the exact normalization is A_N^{-1}=2Σ_{k=0}^{m} e^{−k/μ} − e^{−m/μ}, which does not agree with Eq. (13); for example N=2 and r=e^{−1/μ}=1/2 gives A=1/(2+r), whereas Eq. (13) gives a different value. The asymptotic A≈1/(2μ) used later is unaffected, but the formula should be corrected or labeled as asymptotic.
  2. [Sec. IV and Abstract] The construction in Eq. (18) forces N=ℓ for L=2ℓ: each B-site occupation is the complement of its A-site partner. The abstract and introduction refer generally to 'N particles on L sites' and to 'interacting fermions.' Please restrict the claim to half-filling, or extend the construction, and clarify that no parent Hamiltonian is constructed; Sec. VI already acknowledges this limitation.
  3. [Eq. (29)] The product-of-limits step assumes lim S̄_α/S_1 exists. The ratio bound S̄_α≤S_1 guarantees only boundedness. The conclusion still follows because S_α/S_1 ≤ S_α/S̄_α → 0 when S_α/S̄_α→0, so the proof can be rephrased without the extra limit assumption.
  4. [Appendix A] The appendix evaluates only Σ P_n S_1(n;ℓ). The total von Neumann entropy also contains H_1({P_n}). Please state explicitly that H_1 is subleading for μ∼√ℓ and track the normalization A_ℓ so that the leading coefficient of the √ℓ lnℓ term is unambiguous.
  5. [Notation and text] In Eq. (23), the symbol X_{q_A} is written with a typo ('P_{q_a}'); several names, e.g., 'Korepiny'/'Mukarjhee', appear misspelled. Figures 2 and 4 would be clearer if the text distinguished exact numerical evaluation from fitted asymptotic curves.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the explicit ansatz is evaluated directly and the detection theorem follows from proven inequalities.

full rationale

The derivation chain is self-contained. The many-body state in Eqs. (17)-(18) is an explicit ansatz: P_n is chosen as the exponential distribution of Eq. (13) and the symmetry-resolved entropy is fixed by construction to S_alpha(n;ell) = ln C(ell,n) (Eq. 20). The reported scalings S_{alpha>1} ~ ln ell, S_1 ~ sqrt(ell) ln ell, and S_{alpha<1} ~ ell are then obtained by evaluating the general formulas in Eqs. (4)-(5) with these ingredients; the alpha=1 case is analyzed in Appendix A, and the alpha>1 and alpha<1 cases are supported by the Sec. III A exponential-model analysis and the numerics in Figs. 2-4. There is no fitted parameter being relabeled as a prediction: the state is engineered, not inferred from data, and the entropies are computed from the stated P_n and S_alpha(n). The detection protocol in Sec. V defines Sbar_alpha exactly as the charge-resolved combination in Eq. (27), proves the sandwich bound S_1 >= Sbar_alpha >= S_alpha in Eq. (28) using the standard inequalities (2) and (5), and then derives the incompatibility implication in Eq. (30) as a genuine logical consequence. The definition is chosen to make the bound useful, but the theorem does not presuppose its conclusion. Self-citations such as Refs. [50], [54], and [64] appear only as background references for standard symmetry-resolved identities and prior numerical methods; they are not load-bearing for the central scaling construction or the detection theorem. The substantive limitations are non-circular: the paper does not give a full asymptotic derivation for alpha>1 and alpha<1 for the explicit ln C(ell,n) sector entropy (Appendix A treats only alpha=1), and it explicitly acknowledges that no local Hamiltonian parent state is provided (Sec. VI). These are rigor and physical-relevance gaps, not reductions of the claims to their inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central construction rests on an engineered particle-number distribution and the half-filling complement ansatz; these are legitimate design choices for an existence proof but constitute the main assumptions. No new physical entities are introduced.

free parameters (2)
  • μ(N) or μ(ℓ) = √N (γ=1/2) in the explicit example
    Width of the exponential particle-number distribution in Eq. (13). The α-dependent scaling S1≍√ℓ lnℓ, Sα>1≍lnℓ, Sα<1≍ℓ depends on this subextensive width; γ=1/2 is a hand-chosen value, though any 0<γ<1 would preserve the qualitative transition.
  • B = unspecified constant in the toy model
    Coefficient in the illustrative sector entropy Sα(n)=B min{n,N−n} in Sec. III A. It is set by hand in the flat/exponential examples and is not load-bearing for the explicit state, where Sα(n)=ln C(ℓ,n) replaces it.
assumptions (5)
  • ad hoc to paper The constructed state has the prescribed particle-number distribution P_n = A_N e^{-min{n,N−n}/μ(N)} (Eq. 13).
    This distribution is engineered to produce the transition; it is not derived from a Hamiltonian or a dynamical process. Acceptable for an existence proof, but it limits physical generality.
  • domain assumption The reduced density matrix is block diagonal in particle number because the global state conserves total particle number (Sec. II).
    Standard for symmetric states; required for Eqs. (4)-(6) and for the definition of Sbarα.
  • domain assumption The explicit pairing ansatz in Eq. (18) forces half-filling N=ℓ for L=2ℓ.
    Because the B-site occupations are complements nbar_i=1−n_i, the total particle number is always ℓ. The abstract's general 'N particles on L sites' framing is not realized by the construction.
  • standard math Asymptotic estimates use Stirling's approximation and dominance of n≈μ(ℓ) (Appendix A).
    Standard mathematical approximation; the validity conditions (ℓ≫n≫1) are satisfied for n≈√ℓ in the large-ℓ limit.
  • domain assumption The Tomonaga-Luttinger symmetry-resolved relation Eq. (33) from Goldstein-Sela is assumed.
    Used only in the inconclusive-case discussion to show Sbarα≍Sα for a Luttinger liquid; this is prior literature, not derived here.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Detection of a R\'enyi Index Dependent Transition in Entanglement Entropy Scaling." pith.science (2026). https://pith.science/paper/SRLVITSI

@misc{pith2026251224533,
  author       = {Pith},
  title        = {Pith review of: Detection of a R\'enyi Index Dependent Transition in Entanglement Entropy Scaling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SRLVITSI}},
  note         = {Machine review of arXiv:2512.24533}
}
read the original abstract

The scaling of entanglement with subsystem size encodes key information about phases and criticality, but the von Neumann entropy is costly to access in experiments and simulations, often requiring full state tomography. The second R\'enyi entropy is readily measured using two-copy protocols and is often used as a proxy for the von Neumann entanglement entropy, where it is assumed to track its asymptotic scaling. Sugino and Korepiny (Int. J. Mod. Phys. B 32, 1850306 (2018)) revealed that in the ground state of some highly constrained spin models, the scaling of the von Neumann and R\'enyi entropies can differ, varying from power law to logarithmic scaling as a function of the R\'enyi index. Here, we construct a number-conserving many-body state that demonstrates a R\'enyi-index-dependent change in the leading entanglement scaling, generalizing previous results to the case of interacting fermions. We introduce a symmetry-aware lower bound on the von Neumann entropy built from charge-resolved R\'enyi entropies that can provide a protocol for diagnosing anomalous entanglement scaling from experimentally accessible data.

Figures

Figures reproduced from arXiv: 2512.24533 by the authors.

Figure 1
Figure 1. FIG. 1. Particle number distribution [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The entanglement spectrum displays diverse entan [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Particle number distribution [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Different measures of entanglement entropy applied [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The von Neumann entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

84 extracted references · 1 canonical work pages

  1. [1]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.: Theor. Exp.2004, 06002 (2004)

    2004

  2. [2]

    A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Measuring entanglement growth in quench dynamics of bosons in an optical lattice, Phys. Rev. Lett.109, 020505 (2012)

    2012

  3. [3]

    Islam, R

    R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entanglement en- tropy in a quantum many-body system, Nature528, 77 (2015)

    2015

  4. [4]

    M. B. Hastings, I. Gonz´ alez, A. B. Kallin, and R. G. Melko, Measuring Renyi Entanglement Entropy in Quan- tum Monte Carlo Simulations, Phys. Rev. Lett.104, 157201 (2010)

    2010

  5. [5]

    McMinis and N

    J. McMinis and N. M. Tubman, Renyi entropy of the interacting Fermi liquid, Phys. Rev. B87, 081108 (2013)

    2013

  6. [6]

    Grover, Entanglement of Interacting Fermions in Quantum Monte Carlo Calculations, Phys

    T. Grover, Entanglement of Interacting Fermions in Quantum Monte Carlo Calculations, Phys. Rev. Lett. 111, 130402 (2013)

    2013

  7. [7]

    Pollmann and J

    F. Pollmann and J. E. Moore, Entanglement spectra of critical and near-critical systems in one dimension, New Journal of Physics12, 025006 (2010)

    2010

  8. [8]

    Piˇ zorn, F

    I. Piˇ zorn, F. Verstraete, and R. M. Konik, Tree tensor 10 networks and entanglement spectra, Phys. Rev. B88, 195102 (2013)

    2013

Show all 84 references
  1. [9]

    Deng and L

    X. Deng and L. Santos, Entanglement spectrum of one- dimensional extended bose-hubbard models, Phys. Rev. B84, 085138 (2011)

    2011

  2. [10]

    Srednicki, Entropy and area, Phys

    M. Srednicki, Entropy and area, Phys. Rev. Lett.71, 666 (1993)

    1993

  3. [11]

    Eisert, M

    J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010)

    2010

  4. [12]

    M. B. Hastings, An area law for one-dimensional quan- tum systems, Journal of Statistical Mechanics: Theory and Experiment2007, P08024 (2007)

    2007

  5. [13]

    Anshu, I

    A. Anshu, I. Arad, and D. Gosset, An area law for 2d frustration-free spin systems (Association for Computing Machinery, New York, NY, USA, 2022) p. 12–18

    2022

  6. [14]

    The notation≍is standard in asymptotic analysis and analytic number theory, denoting equivalence up to a fi- nite multiplicative constant

  7. [15]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009)

    2009

  8. [16]

    Vidal, J

    G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entan- glement in quantum critical phenomena, Phys. Rev. Lett. 90, 227902 (2003)

    2003

  9. [17]

    Holzhey, F

    C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nuclear Physics B424, 443 (1994)

    1994

  10. [18]

    V. E. Korepin, Universality of entropy scaling in one di- mensional gapless models, Phys. Rev. Lett.92, 096402 (2004)

    2004

  11. [19]

    M. M. Wolf, Violation of the entropic area law for fermions, Phys. Rev. Lett.96, 010404 (2006)

    2006

  12. [20]

    Gioev and I

    D. Gioev and I. Klich, Entanglement entropy of fermions in any dimension and the widom conjecture, Phys. Rev. Lett.96, 100503 (2006)

    2006

  13. [21]

    Simon, Natural entanglement in bose-einstein conden- sates, Phys

    C. Simon, Natural entanglement in bose-einstein conden- sates, Phys. Rev. A66, 052323 (2002)

    2002

  14. [22]

    C. M. Herdman, S. Inglis, P. N. Roy, R. G. Melko, and A. Del Maestro, Path-integral Monte Carlo method for R´ enyi entanglement entropies, Phys. Rev. E90, 013308 (2014)

    2014

  15. [23]

    Leschke, A

    H. Leschke, A. V. Sobolev, and W. Spitzer, Scaling of r´ enyi entanglement entropies of the free fermi-gas ground state: A rigorous proof, Phys. Rev. Lett.112, 160403 (2014)

    2014

  16. [24]

    Swingle, Entanglement renormalization and hologra- phy, Phys

    B. Swingle, Entanglement renormalization and hologra- phy, Phys. Rev. D86, 065007 (2012)

    2012

  17. [25]

    Huijse and B

    L. Huijse and B. Swingle, Area law violations in a super- symmetric model, Phys. Rev. B87, 035108 (2013)

    2013

  18. [26]

    H. Ma, A. T. Schmitz, S. A. Parameswaran, M. Hermele, and R. M. Nandkishore, Topological entanglement en- tropy of fracton stabilizer codes, Phys. Rev. B97, 125101 (2018)

    2018

  19. [27]

    M. G. Nezhadhaghighi and M. A. Rajabpour, Quantum entanglement entropy and classical mutual information in long-range harmonic oscillators, Phys. Rev. B88, 045426 (2013)

    2013

  20. [28]

    Chakraborty and N

    D. Chakraborty and N. Angelinos, Entanglement entropy in ground states of long-range fermionic systems, Phys. Rev. A110, 042408 (2024)

    2024

  21. [29]

    J. Zhao, N. Laflorencie, and Z. Y. Meng, Unconventional scalings of quantum entropies in long-range heisenberg chains, Phys. Rev. Lett.134, 016707 (2025)

    2025

  22. [30]

    Irani, Ground state entanglement in one-dimensional translationally invariant quantum systems, Journal of Mathematical Physics51, 022101 (2010)

    S. Irani, Ground state entanglement in one-dimensional translationally invariant quantum systems, Journal of Mathematical Physics51, 022101 (2010)

    2010

  23. [31]

    Vitagliano, A

    G. Vitagliano, A. Riera, and J. I. Latorre, Volume-law scaling for the entanglement entropy in spin-1/2 chains, New Journal of Physics12, 113049 (2010)

    2010

  24. [32]

    Movassagh and P

    R. Movassagh and P. W. Shor, Supercritical entangle- ment in local systems: Counterexample to the area law for quantum matter, Proceedings of the National Academy of Sciences113, 13278 (2016)

    2016

  25. [33]

    Dell’Anna, O

    L. Dell’Anna, O. Salberger, L. Barbiero, A. Trombettoni, and V. E. Korepin, Violation of cluster decomposition and absence of light cones in local integer and half-integer spin chains, Phys. Rev. B94, 155140 (2016)

    2016

  26. [34]

    Menon, A

    V. Menon, A. Gu, and R. Movassagh, Symmetries, cor- relation functions, and entanglement of general quantum motzkin spin-chains (2024), arXiv:2408.16070 [quant-ph]

    2024 arXiv

  27. [35]

    J. Wang, Z. Liu, Z. Yan, and C. Wu, Singularity and universality from von neumann to r´ enyi entanglement entropy and disorder operator in motzkin chains, Phys. Rev. B112, 075128 (2025)

    2025

  28. [36]

    Sugino and V

    F. Sugino and V. Korepin, R´ enyi entropy of highly entan- gled spin chains, International Journal of Modern Physics B32, 1850306 (2018)

    2018

  29. [37]

    Here Tr| ˆO|= Tr √ O†Ois the trace norm of operator ˆO

  30. [38]

    J. L. W. V. Jensen, Sur les fonctions convexes et les in´ egalit´ es entre les valeurs moyennes, Acta Mathematica 30, 175–193 (1906)

    1906

  31. [39]

    R. V. L. Hartley, Transmission of information, The Bell System Technical Journal7, 535 (1928)

    1928

  32. [40]

    H. M. Wiseman and J. A. Vaccaro, Entanglement of indistinguishable particles shared between two parties, Phys. Rev. Lett.91, 097902 (2003)

    2003

  33. [41]

    Goldstein and E

    M. Goldstein and E. Sela, Symmetry-resolved entan- glement in many-body systems, Phys. Rev. Lett.120, 200602 (2018)

    2018

  34. [42]

    D. X. Horv´ ath and P. Calabrese, Symmetry resolved en- tanglement in integrable field theories via form factor bootstrap, J. High Energy Phys.2020(11), 131

    2020

  35. [43]

    Murciano, R

    S. Murciano, R. Bonsignori, and P. Calabrese, Symme- try decomposition of negativity of massless free fermions, SciPost Phys.10, 111 (2021)

    2021

  36. [44]

    D. X. Horvath, P. Calabrese, and O. A. Castro-Alvaredo, Branch Point Twist Field Form Factors in the sine- Gordon Model II: Composite Twist Fields and Symmetry Resolved Entanglement, SciPost Phys.12, 088 (2022)

    2022

  37. [45]

    M. J. Arildsen, V. Cr´ epel, N. Regnault, and B. Esti- enne, Symmetry resolved entanglement entropy in a non-abelian fractional quantum hall state (2025), arXiv:2508.05494 [cond-mat.str-el]

    2025 arXiv

  38. [46]

    Monkman and J

    K. Monkman and J. Sirker, Symmetry-resolved entan- glement: general considerations, calculation from cor- relation functions, and bounds for symmetry-protected topological phases, Journal of Physics A: Mathematical and Theoretical56, 495001 (2023)

    2023

  39. [47]

    Monkman and J

    K. Monkman and J. Sirker, Operational entanglement of symmetry-protected topological edge states, Phys. Rev. Res.2, 043191 (2020)

    2020

  40. [48]

    O. A. Castro-Alvaredo and L. Santamar ´ ıa-Sanz, Symmetry-resolved measures in quantum field theory: A short review, Modern Physics Letters B39, 2430002 (2025). 11

    2025

  41. [49]

    Parez, Symmetry-resolved r´ enyi fidelities and quan- tum phase transitions, Phys

    G. Parez, Symmetry-resolved r´ enyi fidelities and quan- tum phase transitions, Phys. Rev. B106, 235101 (2022)

    2022

  42. [50]

    Barghathi, C

    H. Barghathi, C. M. Herdman, and A. Del Maestro, R´ enyi generalization of the accessible entanglement en- tropy, Phys. Rev. Lett.121, 150501 (2018)

    2018

  43. [51]

    R. G. Melko, C. M. Herdman, D. Iouchtchenko, P.-N. Roy, and A. Del Maestro, Entangling qubit registers via many-body states of ultracold atoms, Phys. Rev. A93, 042336 (2016)

    2016

  44. [52]

    Monkman and J

    K. Monkman and J. Sirker, Symmetry-resolved entangle- ment ofC 2-symmetric topological insulators, Phys. Rev. B107, 125108 (2023)

    2023

  45. [53]

    Klich and L

    I. Klich and L. S. Levitov, Scaling of entanglement en- tropy and superselection rules (2008), arXiv:0812.0006 [quant-ph]

    2008 arXiv

  46. [54]

    Barghathi, E

    H. Barghathi, E. Casiano-Diaz, and A. Del Maestro, Op- erationally accessible entanglement of one-dimensional spinless fermions, Phys. Rev. A100, 022324 (2019)

    2019

  47. [55]

    Kiefer-Emmanouilidis, R

    M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer, and J. Sirker, Evidence for unbounded growth of the number entropy in many-body localized phases, Phys. Rev. Lett.124, 243601 (2020)

    2020

  48. [56]

    Kiefer-Emmanouilidis, R

    M. Kiefer-Emmanouilidis, R. Unanyan, J. Sirker, and M. Fleischhauer, Bounds on the entanglement entropy by the number entropy in non-interacting fermionic sys- tems, SciPost Phys.8, 083 (2020)

    2020

  49. [57]

    A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Quantum ther- malization through entanglement in an isolated many- body system, Science353, 794 (2016)

    2016

  50. [58]

    Pichler, G

    H. Pichler, G. Zhu, A. Seif, P. Zoller, and M. Hafezi, Measurement Protocol for the Entanglement Spectrum of Cold Atoms, Phys. Rev. X6, 041033 (2016)

    2016

  51. [59]

    Lukin, M

    A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kauf- man, S. Choi, V. Khemani, J. L´ eonard, and M. Greiner, Probing entanglement in a many-body–localized system, Science364, 256 (2019)

    2019

  52. [60]

    Brydges, A

    T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Probing R´ enyi entanglement entropy via random- ized measurements, Science364, 260 (2019)

    2019

  53. [61]

    S. V. Isakov, M. B. Hastings, and R. G. Melko, Topologi- cal entanglement entropy of a Bose–Hubbard spin liquid, Nature Phys.7, 772 (2011)

    2011

  54. [62]

    Helmes and S

    J. Helmes and S. Wessel, Entanglement entropy scaling in the bilayer heisenberg spin system, Phys. Rev. B89, 245120 (2014)

    2014

  55. [63]

    Wang and M

    L. Wang and M. Troyer, Renyi Entanglement Entropy of Interacting Fermions Calculated Using the Continuous- Time Quantum Monte Carlo Method, Phys. Rev. Lett. 113, 110401 (2014)

    2014

  56. [64]

    C. M. Herdman, P.-N. Roy, R. G. Melko, and A. Del Maestro, Entanglement area law in superfluid 4He, Na- ture Phys.13, 556 (2017)

    2017

  57. [65]

    Wildeboer, A

    J. Wildeboer, A. Seidel, and R. G. Melko, Entanglement entropy and topological order in resonating valence-bond quantum spin liquids, Phys. Rev. B95, 100402 (2017)

    2017

  58. [66]

    Zhao, Y.-C

    J. Zhao, Y.-C. Wang, Z. Yan, M. Cheng, and Z. Y. Meng, Scaling of Entanglement Entropy at Deconfined Quan- tum Criticality, Phys. Rev. Lett.128, 010601 (2022)

    2022

  59. [67]

    D’Emidio and A

    J. D’Emidio and A. W. Sandvik, Entanglement Entropy and Deconfined Criticality: Emergent So(5) Symmetry and Proper Lattice Bipartition, Phys. Rev. Lett.133, 166702 (2024)

    2024

  60. [68]

    Laflorencie, Quantum entanglement in condensed matter systems, Physics Reports646, 1 (2016), quantum entanglement in condensed matter systems

    N. Laflorencie, Quantum entanglement in condensed matter systems, Physics Reports646, 1 (2016), quantum entanglement in condensed matter systems

    2016

  61. [69]

    Casiano-Diaz, C

    E. Casiano-Diaz, C. M. Herdman, and A. D. Maestro, A path integral ground state Monte Carlo algorithm for entanglement of lattice bosons, SciPost Physics14, 054 (2023), 2207.11301

    2023 arXiv

  62. [70]

    For any two positive functionsf(x) andg(x),f=O(g) is defined as:∃x 0 and a constantc >0 such that∀x > x0, f(x)≤cg(x)

  63. [71]

    Giamarchi,Quantum Physics in One Dimension (Clarendon Press, Oxford, U.K., 2004)

    T. Giamarchi,Quantum Physics in One Dimension (Clarendon Press, Oxford, U.K., 2004)

    2004

  64. [72]

    Elben, B

    A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, and P. Zoller, R´ enyi Entropies from Random Quenches in Atomic Hubbard and Spin Models, Phys. Rev. Lett.120, 050406 (2018)

    2018

  65. [73]

    Elben, B

    A. Elben, B. Vermersch, C. F. Roos, and P. Zoller, Sta- tistical correlations between locally randomized measure- ments: A toolbox for probing entanglement in many- body quantum states, Phys. Rev. A99, 052323 (2019)

    2019

  66. [74]

    Raussendorf and H

    R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett.86, 5188 (2001)

    2001

  67. [75]

    H. J. Briegel, D. E. Browne, W. D¨ ur, R. Raussendorf, and M. Van den Nest, Measurement-based quantum compu- tation, Nature Physics5, 19 (2009)

    2009

  68. [76]

    Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys

    G. Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys. Rev. Lett.91, 147902 (2003)

    2003

  69. [77]

    Van den Nest, W

    M. Van den Nest, W. D¨ ur, A. Miyake, and H. J. Briegel, Universal resources for measurement-based quantum computation, Phys. Rev. Lett.97, 150504 (2006)

    2006

  70. [78]

    Gross, S

    D. Gross, S. T. Flammia, and J. Eisert, Most quantum states are too entangled to be useful as computational resources, Phys. Rev. Lett.102, 190501 (2009)

    2009

  71. [79]

    G. K. Brennen and A. Miyake, Measurement-based quan- tum computer in the gapped ground state of a two-body hamiltonian, Phys. Rev. Lett.101, 010502 (2008)

    2008

  72. [80]

    Raussendorf and J

    R. Raussendorf and J. Harrington, Fault-tolerant quan- tum computation with high threshold in two dimensions, Phys. Rev. Lett.98, 190504 (2007)

    2007

  73. [81]

    D. V. Else, I. Schwarz, S. D. Bartlett, and A. C. Doherty, Symmetry-protected phases for measurement- based quantum computation, Phys. Rev. Lett.108, 240505 (2012)

    2012

  74. [82]

    Calabrese, J

    P. Calabrese, J. Dubail, and S. Murciano, Symmetry- resolved entanglement entropy in wess-zumino-witten models, J. High Energy Phys.10, 067

  75. [83]

    Bianchi, P

    E. Bianchi, P. Dona, and R. Kumar, Non-Abelian symmetry-resolved entanglement entropy, SciPost Physics17, 127 (2024)

    2024

  76. [84]

    Barghathi and A

    H. Barghathi and A. Del Maestro, Github repos- itory,https://github.com/DelMaestroGroup/ papers-code-EntanglementScalingTransition 10.5281/zenodo.18118762 (2025)

    2025 doi

Pith tools

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