A general proof of the quantum null energy condition
Abstract
We prove a conjectured lower bound on ⟨T_ _(x)⟩ $ψ$ in any state $ψ$ of a CFT on Minkowski space, dubbed the Quantum Null Energy Condition (QNEC). The bound is given by the second order shape deformation, in the null direction, of the geometric entanglement entropy of an entangling cut passing through $x$. Our proof involves a combination of the two independent methods that were used recently to prove the weaker Averaged Null Energy Condition (ANEC). In particular the properties of modular Hamiltonians under shape deformations for the state ψ play an important role, as do causality considerations. We study the two point function of a “probe” operator $$\mathcal{O}$$ in the state $ψ$ and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider causality on modular time for the modular evolved probe operators, which we constrain using TomitaTakesaki theory as well as certain generalizations pertaining to the theory of modular inclusions. The QNEC follows from very similar considerations to the derivation of the chaos bound and the causality sum rule. We use a kind of defect Operator Product Expansion to apply the replica trick to these modular flow computations, and the displacement operator plays an important role. We argue that the proof extends to more general relativistic QFT with an interacting UV fixed point and also prove a higher spin version of the QNEC. Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the RyuTakayanagi (RT) surface near the boundary of AdS, combined with the requirement of entanglement wedge nesting. Our methods were, as such, designed as a precise probe of the RT surface close to the boundary of a putative gravitational/stringy dual of any QFT with an interacting UV fixed point.
 Authors:

 Univ. of Illinois, Urbana, IL (United States)
 Univ. of Illinois, Urbana, IL (United States); Univ. of California, Santa Barbara, CA (United States)
 Publication Date:
 Research Org.:
 Univ. of Illinois at UrbanaChampaign, IL (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC); National Science Foundation (NSF); Defense Advanced Research Projects Agency (DARPA)
 OSTI Identifier:
 1612303
 Grant/Contract Number:
 SC0015655; PHY1607611; D15AP00108
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Journal of High Energy Physics (Online)
 Additional Journal Information:
 Journal Volume: 2019; Journal Issue: 9; Journal ID: ISSN 10298479
 Publisher:
 Springer Berlin
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Physics; AdSCFT Correspondence; Conformal Field Theory
Citation Formats
Balakrishnan, Srivatsan, Faulkner, Thomas, Khandker, Zuhair U., and Wang, Huajia. A general proof of the quantum null energy condition. United States: N. p., 2019.
Web. doi:10.1007/jhep09(2019)020.
Balakrishnan, Srivatsan, Faulkner, Thomas, Khandker, Zuhair U., & Wang, Huajia. A general proof of the quantum null energy condition. United States. doi:10.1007/jhep09(2019)020.
Balakrishnan, Srivatsan, Faulkner, Thomas, Khandker, Zuhair U., and Wang, Huajia. Tue .
"A general proof of the quantum null energy condition". United States. doi:10.1007/jhep09(2019)020. https://www.osti.gov/servlets/purl/1612303.
@article{osti_1612303,
title = {A general proof of the quantum null energy condition},
author = {Balakrishnan, Srivatsan and Faulkner, Thomas and Khandker, Zuhair U. and Wang, Huajia},
abstractNote = {We prove a conjectured lower bound on ⟨T_ _(x)⟩ $ψ$ in any state $ψ$ of a CFT on Minkowski space, dubbed the Quantum Null Energy Condition (QNEC). The bound is given by the second order shape deformation, in the null direction, of the geometric entanglement entropy of an entangling cut passing through $x$. Our proof involves a combination of the two independent methods that were used recently to prove the weaker Averaged Null Energy Condition (ANEC). In particular the properties of modular Hamiltonians under shape deformations for the state ψ play an important role, as do causality considerations. We study the two point function of a “probe” operator $\mathcal{O}$ in the state $ψ$ and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider causality on modular time for the modular evolved probe operators, which we constrain using TomitaTakesaki theory as well as certain generalizations pertaining to the theory of modular inclusions. The QNEC follows from very similar considerations to the derivation of the chaos bound and the causality sum rule. We use a kind of defect Operator Product Expansion to apply the replica trick to these modular flow computations, and the displacement operator plays an important role. We argue that the proof extends to more general relativistic QFT with an interacting UV fixed point and also prove a higher spin version of the QNEC. Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the RyuTakayanagi (RT) surface near the boundary of AdS, combined with the requirement of entanglement wedge nesting. Our methods were, as such, designed as a precise probe of the RT surface close to the boundary of a putative gravitational/stringy dual of any QFT with an interacting UV fixed point.},
doi = {10.1007/jhep09(2019)020},
journal = {Journal of High Energy Physics (Online)},
issn = {10298479},
number = 9,
volume = 2019,
place = {United States},
year = {2019},
month = {9}
}
Web of Science
Works referenced in this record:
Topological censorship
journal, September 1993
 Friedman, John L.; Schleich, Kristin; Witt, Donald M.
 Physical Review Letters, Vol. 71, Issue 10
The gravity dual of a density matrix
journal, July 2012
 Czech, Bartłomiej; Karczmarek, Joanna L.; Nogueira, Fernando
 Classical and Quantum Gravity, Vol. 29, Issue 15
Gravitational Collapse and SpaceTime Singularities
journal, January 1965
 Penrose, Roger
 Physical Review Letters, Vol. 14, Issue 3
AdS dual of the critical O(N) vector model
journal, December 2002
 Klebanov, I. R.; Polyakov, A. M.
 Physics Letters B, Vol. 550, Issue 34
Relative entropy and the Bekenstein bound
journal, September 2008
 Casini, H.
 Classical and Quantum Gravity, Vol. 25, Issue 20
Conformal field theories near a boundary in general dimensions
journal, September 1995
 McAvity, D. M.; Osborn, H.
 Nuclear Physics B, Vol. 455, Issue 3
Entanglement entropy of two disjoint intervals in conformal field theory: II
journal, January 2011
 Calabrese, Pasquale; Cardy, John; Tonni, Erik
 Journal of Statistical Mechanics: Theory and Experiment, Vol. 2011, Issue 01
Extension of the Structure Theorem of Borchers and its Application to HalfSided Modular Inclusions
journal, June 2005
 Araki, Huzihiro; ZsidÓ, LÁSzlÓ
 Reviews in Mathematical Physics, Vol. 17, Issue 05
Some remarks about the localization of states in a quantum field theory
journal, December 1965
 Schlieder, Siegfried
 Communications in Mathematical Physics, Vol. 1, Issue 4
Gravitational Radiation from Colliding Black Holes
journal, May 1971
 Hawking, S. W.
 Physical Review Letters, Vol. 26, Issue 21
The generalized second law implies a quantum singularity theorem
journal, July 2013
 Wall, Aron C.
 Classical and Quantum Gravity, Vol. 30, Issue 16
HalfSided modular inclusions of vonNeumannAlgebras
journal, October 1993
 Wiesbrock, HansWerner
 Communications in Mathematical Physics, Vol. 157, Issue 1
Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence
journal, May 2006
 Ryu, Shinsei; Takayanagi, Tadashi
 Physical Review Letters, Vol. 96, Issue 18
Causality & holographic entanglement entropy
journal, December 2014
 Headrick, Matthew; Hubeny, Veronika E.; Lawrence, Albion
 Journal of High Energy Physics, Vol. 2014, Issue 12
A covariant entropy conjecture
journal, July 1999
 Bousso, Raphael
 Journal of High Energy Physics, Vol. 1999, Issue 07
Halfsided modular inclusion and the construction of the Poincaré group
journal, September 1996
 Borchers, H. J.
 Communications in Mathematical Physics, Vol. 179, Issue 3
A stereoscopic look into the bulk
journal, July 2016
 Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel
 Journal of High Energy Physics, Vol. 2016, Issue 7
Causality constraints on corrections to the graviton threepoint coupling
journal, February 2016
 Camanho, Xián O.; Edelstein, José D.; Maldacena, Juan
 Journal of High Energy Physics, Vol. 2016, Issue 2
A covariant holographic entanglement entropy proposal
journal, July 2007
 Hubeny, Veronika E.; Rangamani, Mukund; Takayanagi, Tadashi
 Journal of High Energy Physics, Vol. 2007, Issue 07
On the duality condition for quantum fields
journal, January 1976
 Bisognano, Joseph J.
 Journal of Mathematical Physics, Vol. 17, Issue 3
The CPTtheorem in twodimensional theories of local observables
journal, January 1992
 Borchers, H. J.
 Communications in Mathematical Physics, Vol. 143, Issue 2
Conformal collider physics: energy and charge correlations
journal, May 2008
 Hofman, Diego M.; Maldacena, Juan
 Journal of High Energy Physics, Vol. 2008, Issue 05
Geodesic focusing, energy conditions and singularities
journal, March 1987
 Borde, A.
 Classical and Quantum Gravity, Vol. 4, Issue 2
On geometric entropy
journal, July 1994
 Callan, Curtis; Wilczek, Frank
 Physics Letters B, Vol. 333, Issue 12
Nuclear maps and modular structures. I. General properties
journal, February 1990
 Buchholz, Detlev; D'Antoni, Claudio; Longo, Roberto
 Journal of Functional Analysis, Vol. 88, Issue 2