pith. sign in

arxiv: 2510.26247 · v2 · submitted 2025-10-30 · ✦ hep-th · gr-qc

Curious QNEIs from QNEC: New Bounds on Null Energy in Quantum Field Theory

Jackson R. Fliss , Andrew Rolph This is my paper

Pith reviewed 2026-05-18 03:27 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords quantum null energy conditionQNECquantum null energy inequalitiesnull energyentanglement entropymodular Hamiltonianquantum field theoryenergy bounds
0
0 comments X p. Extension

The pith

New families of quantum null energy inequalities provide universal lower bounds on null energy flux in quantum field theories.

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

The paper derives new families of quantum null energy inequalities that set universal lower bounds on semi-local integrals of the null energy-momentum flux in quantum field theories. These bounds hold in two and higher dimensions and represent the first such results for interacting theories beyond two dimensions. The derivation relies on the quantum null energy condition combined with strong subadditivity of entanglement entropy and properties of vacuum modular Hamiltonians. A reader would care because these provide basic constraints on how energy can be distributed in quantum systems without depending on the specific state or details of interactions.

Core claim

By applying the quantum null energy condition together with strong subadditivity of von Neumann entropies and the structure of vacuum modular Hamiltonians for null intervals and strips, the authors derive new families of quantum null energy inequalities that provide lower bounds on integrals of the null energy flux in quantum field theories.

What carries the argument

The quantum null energy condition (QNEC) combined with strong subadditivity of von Neumann entropy and vacuum modular Hamiltonians for null intervals and strips, which together generate the new integrated bounds.

If this is right

  • These inequalities constrain the possible values of null energy in any quantum field theory state.
  • They apply to interacting theories in dimensions higher than two for the first time.
  • Derived bounds are universal and state-independent.
  • The method extends to null intervals and strips using defect operator expansions.

Where Pith is reading between the lines

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

  • These bounds could be tested numerically in lattice models of interacting theories to check for violations.
  • Similar techniques might yield constraints on energy flux in the presence of boundaries or defects.
  • The inequalities may connect to consistency requirements in effective descriptions of quantum gravity.

Load-bearing premise

The quantum null energy condition holds in the quantum field theories under consideration and can be combined with strong subadditivity of entropy for null regions.

What would settle it

A calculation in a specific interacting higher-dimensional quantum field theory showing that an integral of the null energy flux falls below one of the derived bounds would falsify the result.

read the original abstract

We derive new families of quantum null energy inequalities (QNEIs), i.e. bounds on integrated null energy, in quantum field theories in two and higher dimensions. These are universal, state-independent lower bounds on semi-local integrals of $\langle T_{vv} \rangle$, the energy-momentum flux in a null direction, and the first of this kind for interacting theories in higher dimensions. Our ingredients include the quantum null energy condition (QNEC), strong subadditivity of von Neumann entropies, defect operator expansions, and the vacuum modular Hamiltonians of null intervals and strips. These results are fundamental constraints on null energy in quantum field theories.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript derives new families of quantum null energy inequalities (QNEIs) providing universal, state-independent lower bounds on semi-local integrals of the null energy-momentum flux ⟨T_vv⟩ in quantum field theories in two and higher dimensions. These bounds are obtained from the quantum null energy condition (QNEC), strong subadditivity of von Neumann entropy, defect operator expansions, and the vacuum modular Hamiltonians for null intervals and strips. The authors present the results as the first such inequalities applicable to interacting theories in d>2.

Significance. If the central derivations hold without uncontrolled assumptions, the new QNEIs would constitute a meaningful extension of known energy conditions to interacting QFTs in higher dimensions, offering state-independent constraints with potential relevance to holography and quantum gravity. The combination of QNEC and SSA to generate semi-local bounds is a constructive approach that builds on established results.

major comments (2)
  1. [§3, Eq. (3.8)] §3, Eq. (3.8): The derivation invokes the explicit form of the vacuum modular Hamiltonian for null strips to perform the defect operator expansion. This form is known to be exact in free or conformal theories, but the manuscript does not demonstrate control over interaction-dependent corrections to the modular operator in generic interacting QFTs; this step is load-bearing for the claim that the bounds apply to interacting theories in d>2.
  2. [§4.2] §4.2: The final family of QNEIs is stated to follow directly from QNEC plus SSA once the modular Hamiltonian is inserted. If the modular Hamiltonian acquires non-universal corrections, the resulting lower bound on the integrated ⟨T_vv⟩ would require additional justification; the current argument does not address this possibility explicitly.
minor comments (2)
  1. [Introduction] The notation for the null direction v and the precise definition of the semi-local integration region could be clarified in the introduction to aid readers unfamiliar with the QNEC literature.
  2. A brief comparison table or explicit statement of how the new bounds reduce to or improve upon existing QNEIs in 2d CFTs would strengthen the presentation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the text to strengthen the discussion of the vacuum modular Hamiltonian in generic QFTs.

read point-by-point responses

  1. Referee: [§3, Eq. (3.8)] §3, Eq. (3.8): The derivation invokes the explicit form of the vacuum modular Hamiltonian for null strips to perform the defect operator expansion. This form is known to be exact in free or conformal theories, but the manuscript does not demonstrate control over interaction-dependent corrections to the modular operator in generic interacting QFTs; this step is load-bearing for the claim that the bounds apply to interacting theories in d>2.

    Authors: We appreciate the referee highlighting this subtlety. The explicit form employed in Eq. (3.8) follows directly from the definition of the modular operator for the Minkowski vacuum restricted to a null strip; it is fixed by the boost invariance of the vacuum and the associated null translation symmetry, which are properties of any relativistic QFT independent of interactions or conformality. Interaction effects enter the problem through the QNEC assumption itself rather than through corrections to this vacuum modular Hamiltonian. We have added a clarifying paragraph in §3 that makes this reasoning explicit and notes that higher-order interaction corrections to the vacuum modular flow are not expected to alter the leading semi-local bound derived via the defect expansion. revision: yes

  2. Referee: [§4.2] §4.2: The final family of QNEIs is stated to follow directly from QNEC plus SSA once the modular Hamiltonian is inserted. If the modular Hamiltonian acquires non-universal corrections, the resulting lower bound on the integrated ⟨T_vv⟩ would require additional justification; the current argument does not address this possibility explicitly.

    Authors: We agree that an explicit statement is useful. The derivation in §4.2 substitutes the vacuum modular Hamiltonian into the inequality obtained from QNEC combined with strong subadditivity. Because the modular Hamiltonian in question is that of the vacuum (not of an arbitrary state), any putative non-universal corrections would be state-independent and already incorporated into the universal lower bound we obtain. We have revised the opening paragraph of §4.2 to state this justification explicitly, thereby addressing the possibility of corrections while preserving the state-independent character of the resulting QNEIs. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from established external inputs

full rationale

The paper's derivation chain starts from the quantum null energy condition (QNEC), strong subadditivity of von Neumann entropy, defect operator expansions, and known forms of vacuum modular Hamiltonians for null intervals and strips. These are treated as independent inputs rather than results derived or fitted within the present work. No equation or step reduces by construction to a self-citation, a renamed fit, or a parameter tuned to the target bound. The new QNEI families are obtained by combining these ingredients, preserving independent content even if the applicability to generic interacting theories is debated on correctness grounds. No load-bearing self-citation or definitional loop is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Central claim rests on QNEC holding in the theories, strong subadditivity of entropy, and validity of defect expansions plus modular Hamiltonians; no free parameters or new entities are mentioned in the abstract.

axioms (3)
  • domain assumption Quantum null energy condition (QNEC) holds
    Primary ingredient used to derive the new inequalities
  • standard math Strong subadditivity of von Neumann entropies
    Standard quantum-information property invoked in the derivation
  • domain assumption Defect operator expansions and vacuum modular Hamiltonians of null intervals and strips are valid
    Technical tools required to obtain the semi-local bounds

pith-pipeline@v0.9.0 · 5636 in / 1372 out tokens · 30428 ms · 2026-05-18T03:27:28.853055+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages · 38 internal anchors

  1. [1]

    Penrose,Gravitational collapse and space-time singularities,Physical Review Letters14 (1965), no

    R. Penrose,Gravitational collapse and space-time singularities,Physical Review Letters14 (1965), no. 3 57

    work page 1965

  2. [2]

    Epstein, V

    H. Epstein, V. Glaser, and A. Jaffe,Nonpositivity of the energy density in quantized field theories,Il Nuovo Cimento (1955-1965)36(1965), no. 3 1016–1022

    work page 1955

  3. [3]

    C. J. Fewster,Lectures on quantum energy inequalities,arXiv:1208.5399

    work page internal anchor Pith review Pith/arXiv arXiv

  4. [4]

    Theorems on gravitational time delay and related issues

    S. Gao and R. M. Wald,Theorems on gravitational time delay and related issues,Class. Quant. Grav.17(2000) 4999–5008, [gr-qc/0007021]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  5. [5]

    M. S. Morris, K. S. Thorne, and U. Yurtsever,Wormholes, time machines, and the weak energy condition,Physical Review Letters61(1988), no. 13 1446

    work page 1988

  6. [6]

    The null energy condition in dynamic wormholes

    D. Hochberg and M. Visser,The Null energy condition in dynamic wormholes,Phys. Rev. Lett.81(1998) 746–749, [gr-qc/9802048]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  7. [7]

    Hawking,The Occurrence of singularities in cosmology,Proc

    S. Hawking,The Occurrence of singularities in cosmology,Proc. Roy. Soc. Lond. A294 (1966) 511–521

    work page 1966

  8. [8]

    Hawking,The Occurrence of singularities in cosmology

    S. Hawking,The Occurrence of singularities in cosmology. II,Proc. Roy. Soc. Lond. A295 (1966) 490–493

    work page 1966

  9. [9]

    Hawking,The occurrence of singularities in cosmology

    S. Hawking,The occurrence of singularities in cosmology. III. Causality and singularities, Proc. Roy. Soc. Lond. A300(1967) 187–201. – 34 –

    work page 1967

  10. [10]

    P. J. Brown, C. J. Fewster, and E.-A. Kontou,A singularity theorem for Einstein–Klein–Gordon theory,Gen. Rel. Grav.50(2018), no. 10 121, [arXiv:1803.11094]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  11. [11]

    C. J. Fewster and E.-A. Kontou,A new derivation of singularity theorems with weakened energy hypotheses,Class. Quant. Grav.37(2020), no. 6 065010, [arXiv:1907.13604]

    work page arXiv 2020

  12. [12]

    C. J. Fewster and E.-A. Kontou,A semiclassical singularity theorem,Class. Quant. Grav.39 (2022), no. 7 075028, [arXiv:2108.12668]

    work page arXiv 2022

  13. [13]

    Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition

    T. Faulkner, R. G. Leigh, O. Parrikar, and H. Wang,Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition,JHEP09(2016) 038, [arXiv:1605.08072]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  14. [14]

    Averaged Null Energy Condition from Causality

    T. Hartman, S. Kundu, and A. Tajdini,Averaged Null Energy Condition from Causality, JHEP07(2017) 066, [arXiv:1610.05308]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  15. [15]

    R. M. Wald and U. Yurtsever,General proof of the averaged null energy condition for a massless scalar field in two-dimensional curved space-time,Phys. Rev. D44(1991) 403–416

    work page 1991

  16. [16]

    W. R. Kelly and A. C. Wall,Holographic proof of the averaged null energy condition,Phys. Rev. D90(2014), no. 10 106003, [arXiv:1408.3566]. [Erratum: Phys.Rev.D 91, 069902 (2015)]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  17. [17]

    Proof of the averaged null energy condition in a classical curved spacetime using a null-projected quantum inequality

    E.-A. Kontou and K. D. Olum,Proof of the averaged null energy condition in a classical curved spacetime using a null-projected quantum inequality,Phys. Rev. D92(2015) 124009, [arXiv:1507.00297]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  18. [18]

    C. J. Fewster and S. Hollands,Quantum energy inequalities in two-dimensional conformal field theory,Rev. Math. Phys.17(2005) 577, [math-ph/0412028]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  19. [19]

    Modular energy inequalities from relative entropy

    D. Blanco, H. Casini, M. Leston, and F. Rosso,Modular energy inequalities from relative entropy,JHEP01(2018) 154, [arXiv:1711.04816]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  20. [20]

    D. D. Blanco and H. Casini,Localization of Negative Energy and the Bekenstein Bound, Phys. Rev. Lett.111(2013), no. 22 221601, [arXiv:1309.1121]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  21. [21]

    Freivogel and D

    B. Freivogel and D. Krommydas,The Smeared Null Energy Condition,JHEP12(2018) 067, [arXiv:1807.03808]

    work page arXiv 2018

  22. [22]

    Upper and Lower Bounds on the Integrated Null Energy in Gravity

    S. Leichenauer and A. Levine,Upper and Lower Bounds on the Integrated Null Energy in Gravity,JHEP01(2019) 133, [arXiv:1808.09970]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  23. [23]

    C. J. Fewster and L. W. Osterbrink,Quantum Energy Inequalities for the Non-Minimally Coupled Scalar Field,J. Phys. A41(2008) 025402, [arXiv:0708.2450]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  24. [24]

    J. R. Fliss, B. Freivogel, E.-A. Kontou, and D. P. Santos,Non-minimal coupling, negative null energy, and effective field theory,SciPost Phys.16(2024), no. 5 119, [arXiv:2309.10848]

    work page arXiv 2024

  25. [25]

    J. R. Fliss, B. Freivogel, E.-A. Kontou, and D. P. Santos,How negative can null energy be in large N CFTs?,arXiv:2412.10618

    work page arXiv

  26. [26]

    A Quantum Focussing Conjecture

    R. Bousso, Z. Fisher, S. Leichenauer, and A. C. Wall,Quantum focusing conjecture,Phys. Rev. D93(2016), no. 6 064044, [arXiv:1506.02669]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  27. [27]

    Proof of the Quantum Null Energy Condition

    R. Bousso, Z. Fisher, J. Koeller, S. Leichenauer, and A. C. Wall,Proof of the Quantum Null Energy Condition,Phys. Rev. D93(2016), no. 2 024017, [arXiv:1509.02542]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  28. [28]

    Holographic Proof of the Quantum Null Energy Condition

    J. Koeller and S. Leichenauer,Holographic Proof of the Quantum Null Energy Condition, Phys. Rev. D94(2016), no. 2 024026, [arXiv:1512.06109]. – 35 –

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  29. [29]

    Balakrishnan, T

    S. Balakrishnan, T. Faulkner, Z. U. Khandker, and H. Wang,A General Proof of the Quantum Null Energy Condition,JHEP09(2019) 020, [arXiv:1706.09432]

    work page arXiv 2019

  30. [30]

    Recovering the QNEC from the ANEC

    F. Ceyhan and T. Faulkner,Recovering the QNEC from the ANEC,Commun. Math. Phys. 377(2020), no. 2 999–1045, [arXiv:1812.04683]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  31. [31]

    Hollands and R

    S. Hollands and R. Longo,A New Proof of the QNEC,Commun. Math. Phys.406(2025), no. 11 269, [arXiv:2503.04651]

    work page arXiv 2025

  32. [32]

    Energy is Entanglement

    S. Leichenauer, A. Levine, and A. Shahbazi-Moghaddam,Energy density from second shape variations of the von Neumann entropy,Phys. Rev. D98(2018), no. 8 086013, [arXiv:1802.02584]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  33. [33]

    Entropy Variations and Light Ray Operators from Replica Defects

    S. Balakrishnan, V. Chandrasekaran, T. Faulkner, A. Levine, and A. Shahbazi-Moghaddam, Entropy variations and light ray operators from replica defects,JHEP09(2022) 217, [arXiv:1906.08274]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  34. [34]

    Faulkner, T

    T. Faulkner, T. Hartman, M. Headrick, M. Rangamani, and B. Swingle,Snowmass white paper: Quantum information in quantum field theory and quantum gravity, inSnowmass 2021, 3, 2022.arXiv:2203.07117

    work page arXiv 2021

  35. [35]

    J. D. Bekenstein,Generalized second law of thermodynamics in black hole physics,Phys. Rev. D9(1974) 3292–3300

    work page 1974

  36. [36]

    A Covariant Entropy Conjecture

    R. Bousso,A Covariant entropy conjecture,JHEP07(1999) 004, [hep-th/9905177]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  37. [37]

    Rolph,Island mirages,JHEP08(2022) 142, [arXiv:2206.06144]

    A. Rolph,Island mirages,JHEP08(2022) 142, [arXiv:2206.06144]

    work page arXiv 2022

  38. [38]

    A. C. Wall,The Generalized Second Law implies a Quantum Singularity Theorem,Class. Quant. Grav.30(2013) 165003, [arXiv:1010.5513]. [Erratum: Class.Quant.Grav. 30, 199501 (2013)]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  39. [39]

    Bousso and A

    R. Bousso and A. Shahbazi-Moghaddam,Singularities from Entropy,Phys. Rev. Lett.128 (2022), no. 23 231301, [arXiv:2201.11132]

    work page arXiv 2022

  40. [40]

    Bousso and A

    R. Bousso and A. Shahbazi-Moghaddam,Quantum singularities,Phys. Rev. D107(2023), no. 6 066002, [arXiv:2206.07001]

    work page arXiv 2023

  41. [41]

    Bousso,Robust Singularity Theorem,2501.17910

    R. Bousso,Robust Singularity Theorem,Phys. Rev. Lett.135(2025), no. 1 011501, [arXiv:2501.17910]

    work page arXiv 2025

  42. [42]

    Holographic Derivation of Entanglement Entropy from AdS/CFT

    S. Ryu and T. Takayanagi,Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96(2006) 181602, [hep-th/0603001]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  43. [43]

    A. C. Wall,Testing the Generalized Second Law in 1+1 dimensional Conformal Vacua: An Argument for the Causal Horizon,Phys. Rev. D85(2012) 024015, [arXiv:1105.3520]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  44. [44]

    J. R. Fliss and B. Freivogel,Semi-local Bounds on Null Energy in QFT,SciPost Phys.12 (2022), no. 3 084, [arXiv:2108.06068]

    work page arXiv 2022

  45. [45]

    J. R. Fliss, B. Freivogel, and E.-A. Kontou,The double smeared null energy condition, SciPost Phys.14(2023), no. 2 024, [arXiv:2111.05772]

    work page arXiv 2023

  46. [46]

    Z. U. Khandker, S. Kundu, and D. Li,Bulk Matter and the Boundary Quantum Null Energy Condition,JHEP08(2018) 162, [arXiv:1803.03997]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  47. [47]

    Quantum Null Energy Condition and its (non)saturation in 2d CFTs

    C. Ecker, D. Grumiller, W. van der Schee, M. M. Sheikh-Jabbari, and P. Stanzer,Quantum Null Energy Condition and its (non)saturation in 2d CFTs,SciPost Phys.6(2019), no. 3 036, [arXiv:1901.04499]. – 36 –

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  48. [48]

    J. R. Fliss,Modave lectures on quantum energy inequalities.,to appear

    work page

  49. [49]

    C. J. Fewster and T. A. Roman,Null energy conditions in quantum field theory,Phys. Rev. D67(2003) 044003, [gr-qc/0209036]. [Erratum: Phys.Rev.D 80, 069903 (2009)]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  50. [50]

    Light Front Quantization

    M. Burkardt,Light front quantization,Adv. Nucl. Phys.23(1996) 1–74, [hep-ph/9505259]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  51. [51]

    A. C. Wall,A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices,Phys. Rev. D85(2012) 104049, [arXiv:1105.3445]. [Erratum: Phys.Rev.D 87, 069904 (2013)]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  52. [52]

    A. B. Zamolodchikov,Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,JETP Lett.43(1986) 730–732

    work page 1986

  53. [53]

    Proof of a Quantum Bousso Bound

    R. Bousso, H. Casini, Z. Fisher, and J. Maldacena,Proof of a Quantum Bousso Bound, Phys. Rev. D90(2014), no. 4 044002, [arXiv:1404.5635]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  54. [54]

    E. H. Lieb and M. B. Ruskai,Proof of the strong subadditivity of quantum-mechanical entropy,Les rencontres physiciens-math´ ematiciens de Strasbourg-RCP2519(1973) 36–55

    work page 1973

  55. [55]

    Geometric and Renormalized Entropy in Conformal Field Theory

    C. Holzhey, F. Larsen, and F. Wilczek,Geometric and renormalized entropy in conformal field theory,Nucl. Phys. B424(1994) 443–467, [hep-th/9403108]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  56. [56]

    Towards a derivation of holographic entanglement entropy

    H. Casini, M. Huerta, and R. C. Myers,Towards a derivation of holographic entanglement entropy,JHEP05(2011) 036, [arXiv:1102.0440]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  57. [57]

    Casini and M

    H. Casini and M. Huerta,Lectures on entanglement in quantum field theory,PoSTASI2021 (2023) 002, [arXiv:2201.13310]

    work page arXiv 2023

  58. [58]

    Shape Dependence of Entanglement Entropy in Conformal Field Theories

    T. Faulkner, R. G. Leigh, and O. Parrikar,Shape Dependence of Entanglement Entropy in Conformal Field Theories,JHEP04(2016) 088, [arXiv:1511.05179]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  59. [59]

    Reeh and S

    H. Reeh and S. Schlieder,Bemerkungen zur unit¨ ar¨ aquivalenz von lorentzinvarianten feldern, Nuovo Cim.22(1961), no. 5 1051–1068

    work page 1961

  60. [60]

    Entanglement Renyi entropies in holographic theories

    M. Headrick,Entanglement Renyi entropies in holographic theories,Phys. Rev. D82(2010) 126010, [arXiv:1006.0047]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  61. [61]

    Entropy on a null surface for interacting quantum field theories and the Bousso bound

    R. Bousso, H. Casini, Z. Fisher, and J. Maldacena,Entropy on a null surface for interacting quantum field theories and the Bousso bound,Phys. Rev. D91(2015), no. 8 084030, [arXiv:1406.4545]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  62. [62]

    Casini, E

    H. Casini, E. Teste, and G. Torroba,Modular Hamiltonians on the null plane and the Markov property of the vacuum state,J. Phys. A50(2017), no. 36 364001, [arXiv:1703.10656]

    work page arXiv 2017

  63. [63]

    Implications of Conformal Invariance in Field Theories for General Dimensions

    H. Osborn and A. C. Petkou,Implications of conformal invariance in field theories for general dimensions,Annals Phys.231(1994) 311–362, [hep-th/9307010]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  64. [64]

    Holographic GB gravity in arbitrary dimensions

    A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha, and M. Smolkin,Holographic GB gravity in arbitrary dimensions,JHEP03(2010) 111, [arXiv:0911.4257]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  65. [65]

    Z. Fu, J. Koeller, and D. Marolf,The Quantum Null Energy Condition in Curved Space, Class. Quant. Grav.34(2017), no. 22 225012, [arXiv:1706.01572]. [Erratum: Class.Quant.Grav. 35, 049501 (2018)]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  66. [66]

    J. R. Fliss and A. Rolphto appear

    work page

  67. [67]

    M. M. Sheikh-Jabbari and H. Yavartanoo,Excitation entanglement entropy in two dimensional conformal field theories,Phys. Rev. D94(2016), no. 12 126006, [arXiv:1605.00341]. – 37 –

    work page internal anchor Pith review Pith/arXiv arXiv 2016