Pith. sign in

REVIEW 2 major objections 4 minor 44 references

Non-local modular flows across deformed null-cuts

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read The paper derives an explicit non-local modular flow for a massive free scalar across a quadratic null cut and shows that the standard local-boost (Rindler) approximation breaks down at a computable, finite modular time.

desk verdict Exact non-local modular commutators are solid and worth refereeing; the λ~O(1) breakdown horizon is a heuristic the authors themselves flag, not an established result. read the letter →

arxiv 2501.02998 v3 pith:VBUXWIGE submitted 2025-01-06 hep-th

Guan-Cheng Lu , Huajia Wang This is my paper · ORCID
classification hep-th
keywords modularflownon-localHamiltonianhalf-sidedinclusionnull-planeentanglementfreescalarquantumfieldtheoryRindlerapproximationcommutatorstructure
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

Most explicitly understood modular flows in quantum field theory are local, meaning an operator simply moves along a spacetime trajectory. This paper studies a class of genuinely non-local modular flows: the vacuum flow across a null cut whose shape is deformed, $x^+ = \gamma(x_\perp)$, a setting where the half-sided modular inclusion fixes the modular Hamiltonian exactly. For a massive free scalar in $2+1$ dimensions, the modular commutator is computed in closed form; it splits into a local differential part and a non-local integral kernel in the transverse direction. From that commutator the authors derive a differential-integral equation for the finite flow and resum it in the limit of large modular time at fixed $\lambda = \alpha x^- e^{2\pi s}$. The resummation shows that the familiar approximation of the flow by local Rindler boosts breaks down when $\lambda$ reaches order one, thereby making a quantitative prediction for when the local-boost picture of modular flow fails.

What carries the argument

The central machinery is the half-sided modular inclusion, an algebraic property that guarantees the full modular Hamiltonian for a deformed null cut takes the form $\hat{H}_\gamma = \hat{H}_0 - 2\pi P_\gamma$ with $P_\gamma = \int dx^+ dx_\perp\, \gamma(x_\perp) T_{++}$ on the null plane. To compute $[P_\gamma, \phi(x)]$ for a free scalar, the paper rewrites the commutator in terms of vacuum two-point functions and uses the identity $\gamma(z_2)e^{ip_2 z_2} = \gamma((1/i)\partial/\partial p_2)e^{ip_2 z_2}$, so the shape function becomes a differential operator in momentum space; the remaining momentum integrals produce local terms and a transverse convolution kernel $e^{-m|x_2-y_2|}$. For the finite flow, the paper strips off the local evolution with an operator $C(x,\partial_x)$, leaving a differential-integral equation (4.5) whose right-hand side is purely non-local. The series solution is organized in powers of $x^-$, and the resummation of the fastest-growing terms at fixed $\lambda$ relies on the $n$-fold convolution identity (4.27), which yields the explicit kernel (4.30) and the singularities that mark the breakdown of the local-boost approximation.

What would settle it

Compute the coefficients $g^{(n)}(x_2,z^-,z_2)$ from the full recursion (4.23), without dropping the summation, for $n = 3$ through 10, and compare the ratio $g^{(n+1)}/g^{(n)}$ with the estimate $(m/2)\int e^{-m|x_2-y_2|}$ used in the paper: if the ratio grows like $n$ rather than staying bounded, the singularities at $\lambda = 1$ and $\lambda = 2$ in the resummed kernel (4.30) are artifacts of the truncation and the true breakdown time would shift or disappear.

Watch

Extended reading notes

Core claim

For the quadratic null cut $\gamma(z_2) = (z_2)^2$, the paper obtains the explicit modular commutator $$[P_\gamma, \$\varphi$(x)] = i\left(-|x|^2 \partial_+ + 2x^+ (x\cdot\partial) + x^+\right)\$\varphi$(x) + \frac{i}{2}\, m (x^-)^2 \partial_{x^-} \int dy_2\, $e^{{-m|x_2-y_2|}}$\$\varphi$($x^{0}$,$x^{1}$,y_2).$$ The first line is local and matches the action of a special conformal transformation in the massless limit; the second line is the genuinely non-local part, proportional to $m$ and to the square of the transverse distance $x^-$ to the null plane. From this generator the authors derive the differential-integral equation (4.5) for the finite modular flow, solve it in powers of $x^-$, and in the limit $x^-\to 0$, $s\to\infty$ at fixed $\lambda = \alpha x^- e^{2\pi s}$ resum the series into the explicit kernel (4.30). The resummed kernel has singularities at $\lambda = 1$ (for nearby points, interpreted as formation of caustics) and $\lambda = 2$ (for asymptotically separated points, where the kernel stops decaying), both indicating that perturbation theory and the local-boost approximation break down at $\lambda \sim O(1)$. The paper also checks that the non-local commutator, despite having support outside the right wedge $R_\gamma$, still commutes with all operators in the complementary wedge $L_\gamma$, preserving the von Neumann algebra under modular flow as required by Tomita-Takesaki theory.

Load-bearing premise

The argument's load-bearing premise is that in the large-order recursion (4.23) the first term on the right dominates the sum of the remaining terms enough that the omitted contributions cannot change the singularities; the authors support this only with a crude self-consistency estimate, not a proof, so if the dropped terms contribute at the same order the predicted breakdown at $\lambda \sim O(1)$ could be an artifact.

Editorial extensions

If this is right

  • On the null plane $x^- = 0$ the non-local terms vanish and the modular flow reduces to local null translations, matching the Markov property of the vacuum and the known restriction that non-locality is controlled by distance to the entangling surface.
  • For an operator starting at transverse distance $x^-$ from the boundary, the local-boost (Rindler) approximation is valid only for modular times $s \lesssim -(1/2\pi)\ln(\alpha x^-)$; smaller distance or smaller curvature extends the validity logarithmically.
  • The control parameter $\lambda = \alpha x^- e^{2\pi s}$ does not contain the mass $m$, so even an arbitrarily small mass eventually drives the flow non-local; the mass controls the size of the non-local effects but not the time at which the local-boost picture breaks.
  • For null cuts $\gamma(x_2) = (x_2)^n$ with $n \ge 3$, the modular flow is non-local even in the massless (conformal) limit, so the quadratic case studied here is the minimal example of a family of non-local flows with the same qualitative structure.
  • The commutator $[[\hat{H}_\gamma, \phi(x)], \phi(y)]$ vanishes for $x \in R_\gamma$ and $y \in L_\gamma$ even though the non-local kernel has support outside $R_\gamma$, confirming that the flowed operator remains in the right von Neumann algebra.

Reading between the lines

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

  • The $\lambda$ horizon may be a general scale for modular flows: any small deformation of the entangling surface could seed a long-time breakdown of the Rindler approximation, with the quadratic cut providing the first term of a curvature expansion of the breakdown time.
  • In holographic settings the same $\lambda$ parameter might control the size of the causal shadow, suggesting that entanglement-wedge reconstruction beyond the causal wedge remains possible only for modular times up to the breakdown scale found here.
  • A direct test of the paper's approximation would be to resum the same series for the massless cubic cut $\gamma(x_2) = (x_2)^3$, where non-locality appears already at $m = 0$; if the breakdown still occurs at $\lambda \sim O(1)$ with the same resummation scheme, the phenomenon is generic rather than an artifact of the mass term.
  • The authors' large-$n$ truncation could be replaced by a numerical solution of the full recursion (4.23), which would either confirm the $\lambda = 1, 2$ singularities or locate their true positions; this is a falsifiable, self-contained check.
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

2 major / 4 minor

Summary. This paper studies modular flows of the vacuum state across deformed null cuts x⁺ = γ(x⊥) on the null plane x⁻ = 0, for a free (massive or massless) scalar in 2+1 dimensions. Taking the full modular Hamiltonian Ĥ_γ = 2πM₁₀ − 2πP_γ from the half-sided modular inclusion construction of prior work, the authors compute the commutator [P_γ, φ(x)] by mode expansion for monomial profiles γ(x₂) = x₂ⁿ. For γ = (x₂)² and m ≠ 0 they obtain the explicit nonlocal modular commutator Eq. (3.28), with the nonlocal convolution kernel e^{−m|x₂−y₂|}; the massless limit reduces to the action of the special conformal charge. Section 4 derives a differential-integral equation, Eq. (4.5), for the finite modular flow after stripping the local part, solves it to second order in the transverse distance x⁻, and studies the regime x⁻ → 0, s → ∞ at fixed λ = αx⁻e^{2πs}. A partial resummation, Eq. (4.30), suggests the flow kernel develops singularities at λ = 1 and λ = 2, implying breakdown of the local-boost (Rindler) approximation at λ ~ O(1), corresponding to a modular-time horizon s̄ ~ −(1/2π)ln(αx⁻). The paper also verifies that the infinitesimal flow preserves the von Neumann algebra of the region (Appendix 6.2).

Significance. If the results hold, this is a significant addition to the small explicit literature on nonlocal modular flows. The commutator (3.28) is supported by strong internal checks: the linear and quadratic cuts reproduce the known symmetry charges, the massless limit matches the primary-field transformation law, and the null-plane restriction (3.33)–(3.34) reproduces the independent result of [43]. The differential-integral equation (4.5) is a new concrete tool for finite nonlocal modular flow, and Appendix 6.2's explicit vanishing of the flowed commutator for x ∈ R_γ, y ∈ L_γ is a substantive consistency test of the Tomita-Takesaki framework. The paper is commendably transparent about the limits of its resummation (Footnote 4). The principal weakness is that the headline quantitative claim, the validity horizon at λ ~ O(1), rests on a large-n approximation to recursion (4.23) whose accuracy is not established, while Section 5 states the breakdown as a conclusion. A full or numerical treatment of the recursion, or an explicit downgrade of the breakdown claim to conjecture, is needed before the quantitative horizon is established.

major comments (2)
  1. [§4.2, Eqs. (4.23)–(4.31), Footnote 4; §5 Discussion] The central quantitative claim of the paper — that the modular-flow kernel breaks down at λ = αx⁻e^{2πs} ~ O(1) and that this fixes the validity horizon of the local-boost approximation, Eqs. (4.30)–(4.31) — is not established by the analysis of recursion (4.23). The approximation (4.26) drops the n-term sum on the right-hand side of (4.23), and Footnote 4 concedes that this sum “could collectively contribute at the same order.” The self-consistency estimate (4.24)–(4.25) bounds only the two limiting terms of the partial sum Σ_{k<n} g^{(k)} (giving O(√n) and O(1) against O(n)); it does not track the action of the operator (x₂∂_{x₂} + 1/2), which generates extra factors of order m|x₂| and m·sgn(x₂ − y₂) when applied to the n correlated terms of the partial sum, so the subdominance of the neglected term is not demonstrated. Consequently, the singularities at λ = 1 and λ = 2 in (4.30) are not proven to be features of the exact kernel, and the horizon s̄ ~ −(1/2π)ln(αx⁻) is stated in the Discussion bullets with more confidence than the analysis supports. The derivation of the recursion (4.23) itself is also omitted (“we can derive the following recursion equations”), and the preprint does not specify how the leading-λⁿ part of G^{(n)} is separated from the slower-growing terms. I recommend either solving (or numerically resumming) the full recursion, for instance by deriving the exact generating function in λ, or explicitly reformulating the breakdown statements in Section 5 as conjectures.
  2. [§4.1, Eqs. (4.8)–(4.20)] The perturbation theory in small x⁻ is presented without a stated regime of validity. The expansion (4.10) is in powers of x⁻, but the coefficients G^{(n)} and the explicit solutions (4.19)–(4.20) carry powers of m and α, and the paper nowhere states the dimensionless control parameters (e.g., m x⁻ ≪ 1, α x⁻ ≪ 1, and smallness of x⁻ times field gradients) that would justify truncating at order (x⁻)². A heuristic estimate ∫dy₂ e^{−m|x₂−y₂|}φ(y₂) ~ (2/m)φ(x₂) suggests the dominant nonlocal term is actually of order (x⁻)² rather than m(x⁻)², so the truncation may be better behaved than the coefficients suggest, but this should be stated explicitly. Since the λ-series of Section 4.2 is built on the all-orders x⁻ expansion, the paper should identify the control parameter and specify the sense in which the expansion is asymptotic.
minor comments (4)
  1. [§4, Eqs. (4.5) and (4.8)] The argument of the exponential kernel in (4.5) and (4.8) appears as e^{2παs}x⁻x₂, which is inconsistent with the scaling combination λ = αx⁻e^{2πs} used throughout Section 4.2; please correct the typesetting and define the flowed-coordinate variable in the kernel unambiguously.
  2. [Throughout] There are numerous typos and grammatical slips that should be cleaned up: “knwon” and “chosed” in Section 2, “casual wedge” for “causal wedge” in Section 3.3, “suffers a breaks down” in the Section 5 bullet, a stray “in” after the citation [38–41] in Section 1, and “N¨other” and “R´enyi” should be “Noether” and “Rényi”.
  3. [Abstract and Section 1] The abstract states “1 + 2 dimensions” while Section 1 says “2+1 dimensional space-time”; please unify the dimension-counting notation.
  4. [§4.2 after Eq. (4.21)] In (4.21), the symbol “···” denotes terms that grow slower than e^{2πns}; since g^{(n)} is thereby defined only up to slower-growing contributions, the paper should state explicitly that the recursion (4.23) applies to the leading (λⁿ) part of G^{(n)} and verify consistency of this separation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the modular commutator and finite-flow equations are self-contained derivations from the mode expansion, and the prior modular-Hamiltonian inputs are independent published results.

full rationale

Walking the derivation chain: Section 2 adopts the modular Hamiltonian (2.2) from refs. [16,17,43]. Although two of these references share an author with the present paper, they are independent published derivations (perturbation theory in the shape deformation plus half-sided modular inclusion) whose assumptions do not include the modular commutators or flows later computed here. The paper reviews the derivation rather than assuming its own conclusion. Section 3 computes [P_γ, φ(x)] by an explicit mode expansion of P_γ, deriving A and B in equations (3.19)-(3.25) and obtaining the central commutator (3.28). No parameter is fit to the target result, and the local massless limits (γ = 1, x^2, (x^2)^2) reproduce known conformal charges, providing independent consistency checks. Section 4 obtains the differential-integral equation (4.5) by adjoint action on the commutator, not by imposing the answer. The perturbative expansion in x^- and the λ-resummation are explicit calculations. The only load-bearing approximation is (4.26), where the paper explicitly concedes in footnote 4 that the discarded sum 'could collectively contribute at the same order' and that the self-consistency estimate is crude. That is a quantitative rigor/fragility limitation, not a circular reduction: the singularities at λ = 1 and λ = 2 in (4.30) are outputs of the truncated recursion, not inputs assumed in the recursive equation. No equation in the paper equals an input by construction, and no fitted parameter is renamed as a prediction. Therefore the paper receives score 0 for circularity; the skeptic's concern about the validity horizon belongs to correctness risk rather than circularity.

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

No fitted constants: α is the shape parameter of the null cut and m is the physical mass. The load-bearing assumptions are the previously derived modular Hamiltonian and the two approximations in the finite-flow section. No new entities are postulated.

assumptions (5)
  • domain assumption The full modular Hamiltonian for a deformed null cut is H_hat_gamma = M10 - 2π P_gamma with P_gamma = ∫ d^{d-2}x⊥ dx+ γ(x⊥) T++ (Eq. 2.2, 2.10).
    Taken from prior work [16,17,43] using half-sided modular inclusion and the Markov property of the vacuum on null planes; not rederived. The paper notes the proof requires γ positive definite and cites [43,44] for arbitrary γ.
  • standard math Half-sided modular inclusion (R0, R_gamma, |Ω⟩) holds, giving the algebraic relations (2.5)-(2.6).
    Theoretical input from Tomita-Takesaki theory and Borchers; assumed, not proven in this paper.
  • domain assumption Free scalar field mode expansion and canonical commutation relations are valid, and P_gamma can be normal-ordered using T++ = ∂+φ∂+φ (or the traceless tilde T++ for m=0).
    Standard free QFT; the equivalence of T++ and tilde T++ in P_gamma is argued in Section 3.1 via a vanishing boundary term.
  • domain assumption The ansatz (4.7): the stripped field tilde φ(s,x) is supported on the null surface y^+ = x^+ and expandable as an integral kernel G over (y^-, y2).
    The evolution equation (4.5) has no derivative or integral along x^+, so support on fixed x^+ is consistent; the expansion in local operators on the null surface uses the free-field basis.
  • ad hoc to paper In the resummation (4.23)-(4.30), at large n the first term on the RHS dominates and the sum over k contributes subdominantly (O(√n) vs O(n)).
    Explicitly acknowledged approximation with a crude self-consistency check (4.24)-(4.25); no rigorous bound. The breakdown singularities at λ=1 and λ=2 depend on this assumption.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Non-local modular flows across deformed null-cuts." pith.science (2026). https://pith.science/paper/VBUXWIGE

@misc{pith2026250102998,
  author       = {Pith},
  title        = {Pith review of: Non-local modular flows across deformed null-cuts},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VBUXWIGE}},
  note         = {Machine review of arXiv:2501.02998}
}
abstract

Modular flows probe important aspects of the entanglement structures, especially those of QFTs, in a dynamical framework. Despite the expected non-local nature in the general cases, the majority of explicitly understood examples feature local space-time trajectories under modular flows. In this work, we study a particular class of non-local modular flows. They are associated with the relativistic vacuum state and sub-regions whose boundaries lie on a planar null-surface. They satisfy a remarkable algebraic property known as the half-sided modular inclusion, and as a result the modular Hamiltonians are exactly known in terms of the stress tensor operators. To be explicit, we focus on the simplest QFT of a massive or massless free scalar in $2+1$ dimensions. We obtain explicit expressions for the generators. They can be separated into a sum of local and non-local terms showing certain universal pattern. The preservation of von-Neumann algebra under modular flow works in a subtle way for the non-local terms. We derive a differential-integral equation for the finite modular flow, which can be analyzed in perturbation theory of small distance deviating from the entanglement boundary, and re-summation can be performed in appropriate limits. Comparison with the general expectation of modular flows in such limits are discussed.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

44 extracted references · 31 canonical work pages

  1. [43]

    Casini, E

    H. Casini, E. Test´ e and G. Torroba,Modular hamiltonians on the null plane and the markov property of the vacuum state , Journal of Physics A: Mathematical and Theoretical 50 (2017) 364001

    work page 2017

  2. [1]

    J. M. Maldacena, The Large N limit of superconformal field theories and supergravity , Adv. Theor. Math. Phys. 2 (1998) 231 [ hep-th/9711200]

    arXiv 1998

  3. [2]

    Maldacena, The large-N limit of superconformal field theories and supergravity , International Journal of Theoretical Physics 38 (1999) 1113

    J. Maldacena, The large-N limit of superconformal field theories and supergravity , International Journal of Theoretical Physics 38 (1999) 1113

    work page 1999

  4. [3]

    Witten, Anti-de Sitter space and holography , Adv

    E. Witten, Anti-de Sitter space and holography , Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150]. – 30 –

    arXiv 1998

  5. [4]

    Levin and X.-G

    M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function , Physical Review Letters 96 (2006)

    work page 2006

  6. [5]

    Kitaev and J

    A. Kitaev and J. Preskill, Topological entanglement entropy, Physical Review Letters 96 (2006)

    work page 2006

  7. [6]

    Ryu and T

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from the anti–de Sitter space/conformal field theory correspondence , Phys. Rev. Lett. 96 (2006) 181602

    work page 2006

  8. [7]

    V. E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, Journal of High Energy Physics 2007 (2007) 062

    work page 2007

Show all 44 references
  1. [8]

    Faulkner, A

    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, Journal of High Energy Physics 2013 (2013)

    2013

  2. [9]

    Lewkowycz and J

    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, Journal of High Energy Physics 2013 (2013)

    2013

  3. [10]

    Engelhardt and A

    N. Engelhardt and A. C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime , Journal of High Energy Physics 2015 (2015)

    2015

  4. [11]

    R. C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions , Journal of High Energy Physics 2011 (2011) 125

    2011

  5. [12]

    Casini, M

    H. Casini, M. Huerta, R. C. Myers and A. Yale, Mutual information and the F-theorem, JHEP 10 (2015) 003 [ 1506.06195]

    2015 arXiv

  6. [13]

    Casini, E

    H. Casini, E. Test´ e and G. Torroba,Markov Property of the Conformal Field Theory Vacuum and the a Theorem , Phys. Rev. Lett. 118 (2017) 261602 [ 1704.01870]

    2017 arXiv

  7. [14]

    Casini, I

    H. Casini, I. S. Landea and G. Torroba, Entropic g theorem in general spacetime dimensions, Physical Review Letters 130 (2023)

    2023

  8. [15]

    Bousso, Z

    R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer and A. C. Wall, Proof of the quantum null energy condition , Phys. Rev. D 93 (2016) 024017

    2016

  9. [16]

    Faulkner, R

    T. Faulkner, R. G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for deformed half-spaces and the averaged null energy condition , Journal of High Energy Physics 2016 (2016) 38

    2016

  10. [17]

    Balakrishnan, T

    S. Balakrishnan, T. Faulkner, Z. U. Khandker and H. Wang, A general proof of the quantum null energy condition , Journal of High Energy Physics 2019 (2019) 20

    2019

  11. [18]

    H. J. Borchers, On revolutionizing quantum field theory with tomita’s modular theory , Journal of mathematical Physics 41 (2000) 3604

    2000

  12. [19]

    Sorce, Notes on the type classification of von neumann algebras , Reviews in Mathematical Physics 36 (2023)

    J. Sorce, Notes on the type classification of von neumann algebras , Reviews in Mathematical Physics 36 (2023) . – 31 –

    2023

  13. [20]

    Witten, Aps medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory , Reviews of Modern Physics 90 (2018)

    E. Witten, Aps medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory , Reviews of Modern Physics 90 (2018)

    2018

  14. [21]

    Witten, Why does quantum field theory in curved spacetime make sense? And what happens to the algebra of observables in the thermodynamic limit? 2022

    E. Witten, Why does quantum field theory in curved spacetime make sense? And what happens to the algebra of observables in the thermodynamic limit? 2022. 2112.11614. 10.1007/978-3-031-17523-7˙11

    2022 arXiv

  15. [22]

    Leutheusser and H

    S. Leutheusser and H. Liu, Causal connectability between quantum systems and the black hole interior in holographic duality , Phys. Rev. D 108 (2023) 086019 [2110.05497]

    2023 arXiv

  16. [23]

    S. A. W. Leutheusser and H. Liu, Emergent Times in Holographic Duality , Phys. Rev. D 108 (2023) 086020 [ 2112.12156]

    2023 arXiv

  17. [24]

    Leutheusser and H

    S. Leutheusser and H. Liu, Subregion-subalgebra duality: emergence of space and time in holography, 2024

    2024

  18. [25]

    Witten, Gravity and the crossed product , Journal of High Energy Physics 2022 (2022)

    E. Witten, Gravity and the crossed product , Journal of High Energy Physics 2022 (2022)

    2022

  19. [26]

    Chandrasekaran, R

    V. Chandrasekaran, R. Longo, G. Penington and E. Witten, An algebra of observables for de sitter space , Journal of High Energy Physics 2023 (2023)

    2023

  20. [27]

    Chandrasekaran, G

    V. Chandrasekaran, G. Penington and E. Witten, Large n algebras and generalized entropy, Journal of High Energy Physics 2023 (2023)

    2023

  21. [28]

    Kudler-Flam, S

    J. Kudler-Flam, S. Leutheusser and G. Satishchandran, Generalized Black Hole Entropy is von Neumann Entropy , 2309.15897

    arXiv

  22. [29]

    Jensen, J

    K. Jensen, J. Sorce and A. J. Speranza, Generalized entropy for general subregions in quantum gravity, JHEP 12 (2023) 020 [ 2306.01837]

    2023 arXiv

  23. [30]

    Vidal and R

    G. Vidal and R. F. Werner, Computable measure of entanglement , Phys. Rev. A 65 (2002) 032314

    2002

  24. [31]

    Svetlichny, Distinguishing three-body from two-body nonseparability by a bell-type inequality, Phys

    G. Svetlichny, Distinguishing three-body from two-body nonseparability by a bell-type inequality, Phys. Rev. D 35 (1987) 3066

    1987

  25. [32]

    I. H. Kim, B. Shi, K. Kato and V. V. Albert, Chiral central charge from a single bulk wave function , Phys. Rev. Lett. 128 (2022) 176402

    2022

  26. [33]

    R. Fan, R. Sahay and A. Vishwanath, Extracting the quantum hall conductance from a single bulk wave function , Phys. Rev. Lett. 131 (2023) 186301

    2023

  27. [34]

    X. Dong, D. Harlow and A. C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality , Physical Review Letters 117 (2016)

    2016

  28. [35]

    Faulkner and A

    T. Faulkner and A. Lewkowycz, Bulk locality from modular flow , JHEP 07 (2017) 151 [1704.05464]. – 32 –

    2017 arXiv

  29. [36]

    Sorce, Analyticity and the unruh effect: a study of local modular flow , Journal of High Energy Physics 2024 (2024)

    J. Sorce, Analyticity and the unruh effect: a study of local modular flow , Journal of High Energy Physics 2024 (2024)

    2024

  30. [37]

    Chen and H

    L. Chen and H. Wang, Causal shadow and non-local modular flow: from degeneracy to perturbative genesis by correlation , Journal of High Energy Physics 2023 (2023)

    2023

  31. [38]

    Casini and M

    H. Casini and M. Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Classical and Quantum Gravity 26 (2009) 185005

    2009

  32. [39]

    Cardy and E

    J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 1612 (2016) 123103 [ 1608.01283]

    2016 arXiv

  33. [40]

    R. E. Arias, H. Casini, M. Huerta and D. Pontello, Entropy and modular hamiltonian for a free chiral scalar in two intervals , Physical Review D 98 (2018)

    2018

  34. [41]

    Erdmenger, P

    J. Erdmenger, P. Fries, I. A. Reyes and C. P. Simon, Resolving modular flow: a toolkit for free fermions , JHEP 12 (2020) 126 [ 2008.07532]

    2020 arXiv

  35. [42]

    Lashkari, H

    N. Lashkari, H. Liu and S. Rajagopal, Modular flow of excited states , Journal of High Energy Physics 2021 (2021)

    2021

  36. [44]

    Casini, E

    H. Casini, E. Teste and G. Torroba, All the entropies on the light-cone , JHEP 05 (2018) 005 [ 1802.04278]. – 33 –

    2018 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.