Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

A general proof of the quantum null energy condition

Srivatsan Balakrishnan

Published May 30, 2022• 166 citations
PDF (CC BY)Publisher

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 Tomita-Takesaki 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 Ryu-Takayanagi (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%282019%29020

Related papers

A general proof of the quantum null energy condition

Srivatsan Balakrishnan, Thomas Faulkner, Zuhair U. Khandker +1 more

A bstract 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 $$ \mathcal{O} $$ 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 Tomita-Takesaki 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 Ryu-Takayanagi (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.

arXiv:1706.09432•Journal of High Energy Physics•Sep 1, 2019•120 citations

Proof of the quantum null energy condition for free fermionic field theories

Taha A Malik, Rafael López-Mobilia

The quantum null energy condition (QNEC) is a quantum generalization of the null energy condition, which gives a lower bound on the null energy in terms of the second derivative of the von Neumann entropy or entanglement entropy of some region with respect to a null direction. The QNEC states that $⟨{T}_{kk}{⟩}_{p}\ensuremath{\ge}{\mathrm{lim}}_{A\ensuremath{\rightarrow}0}(\frac{\ensuremath{\hbar}}{2\ensuremath{\pi}A}{S}_{\mathrm{out}}^{\ensuremath{'}\ensuremath{'}})$, where ${S}_{\mathrm{out}}$ is the entanglement entropy restricted to one side of a codimension-2 surface $\mathrm{\ensuremath{\Sigma}}$, which is deformed in the null direction about a neighborhood of point $p$ with area $A$. A proof of QNEC has been given before, which applies to free and super-renormalizable bosonic field theories, and to any points that lie on a stationary null surface. Using similar assumptions and methods, we prove the QNEC for fermionic field theories.

arXiv:1910.07594•Physical review. D/Physical review. D.•Mar 26, 2020•20 citations

Energy and Entropy in Quantum Field Theories

Adam Levine

Energy conditions play an important role in constraining the dynamics of quantum field theories as well as gravitational theories. For example, in semi-classical gravity, the achronal averaged null energy condition (AANEC) can be used to prove that it is always slower to traverse through a wormhole than to travel around via its exterior. Such conditions prevent causality violations that would lead to paradoxes. Recent advances have been made in proving previously conjectured energy conditions directly in quantum field theory (QFT) as well as in uncovering new ones. This thesis will be an exploration of various energy inequalities in conformal field theories (CFTs) as well as semi-classical quantum gravity. At the core of this work lies the recently proved quantum null energy condition (QNEC). The QNEC bounds the null energy flowing past a point by a certain second shape derivative of entanglement entropy. We will demonstrate that the QNEC represents a deep connection between causality, energy and entanglement in quantum field theories. We explore this connection first in the context of holographic CFTs. Calculations in holographic theories will lead us to conjecture that the so-called diagonal QNEC is saturated in all interacting QFTs. We will then provide further, independent evidence that this conjecture holds for all CFTs with a twist gap by explicitly calculating shape derivatives of entanglement entropy using defect CFT techniques.

•Nov 8, 2019•0 citations

The Quantum Null Energy Condition and Entanglement Entropy in Quenches

Márk Mezei, Julio Virrueta

The Quantum Null Energy Condition (QNEC) relates energy to the second variation of entropy in relativistic quantum field theory. We use the QNEC inequality to bound entanglement entropy in quenches. At early times the entanglement entropy grows quadratically in time, and the QNEC provides an upper bound on the prefactor. We demonstrate that the bound is tight, by showing that it is saturated in certain quench protocols: boundary state quenches in conformal field theories in any dimensions. In higher than two dimensions we compute entanglement entropy using AdS/CFT. Our results are the first purely field theoretic applications of the QNEC.

arXiv:1909.00919•Sep 3, 2019•3 citations

Modular Hamiltonians for deformed half-spaces and the averaged null energy condition

Thomas Faulkner, Robert G. Leigh, Onkar Parrikar +1 more

We study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on $$ {\mathrm{\mathbb{R}}}^{1,d-1} $$ . We show that in addition to the usual boost generator, there is a contribution to the modular Hamiltonian at first order in the shape deformation, proportional to the integral of the null components of the stress tensor along the Rindler horizon. We use this fact along with monotonicity of relative entropy to prove the averaged null energy condition in Minkowski space-time. This subsequently gives a new proof of the Hofman-Maldacena bounds on the parameters appearing in CFT three-point functions. Our main technical advance involves adapting newly developed perturbative methods for calculating entanglement entropy to the problem at hand. These methods were recently used to prove certain results on the shape dependence of entanglement in CFTs and here we generalize these results to excited states and real time dynamics. We also discuss the AdS/CFT counterpart of this result, making connection with the recently proposed gravitational dual for modular Hamiltonians in holographic theories.

arXiv:1605.08072•Journal of High Energy Physics•Sep 1, 2016•225 citations

Quantum null energy condition in quenched 2D CFTs

Tanay Kibe, Pratik Roy

The quantum null energy condition (QNEC) is a lower bound on the expectation value of the null-null component of the energy-momentum tensor in terms of null variations of the entanglement entropy. A stronger version of the QNEC (the primary QNEC) is expected to hold in 1 + 1 dimensional conformal field theories (CFT). QNEC has been shown to impose nontrivial quantum thermodynamic restrictions on irreversible entropy production in quenches in 1 + 1 dimensional holographic CFTs. It is therefore natural to study if QNEC imposes similar bounds in other quench setups. In this paper we study QNEC in the Calabrese-Cardy global and local joining quenches using standard CFT techniques. In the global quench we show that the primary QNEC must hold at sufficiently early times and find that it imposes bounds on the four point correlators of twist fields in a boundary state. This is a constraint on the set of boundary states that satisfy the primary QNEC. Furthermore, we find that a violation of the primary QNEC implies a violation of the averaged null energy condition (ANEC) in a conformally transformed frame. In the local quench we find similar bounds on four point correlators from both the primary and the usual QNEC.

arXiv:2503.17448•Physical review. D/Physical review. D.•Jun 3, 2025•2 citations
We prove a conjectured lower bound on ⟨T_ _(x)⟩ ψψψ in any state ψψψ 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 xxx. 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 O\mathcal{O}O in the state ψψψ 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 Tomita-Takesaki 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 Ryu-Takayanagi (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.
A bstract 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 O \mathcal{O} O 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 Tomita-Takesaki 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 Ryu-Takayanagi (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.
The quantum null energy condition (QNEC) is a quantum generalization of the null energy condition, which gives a lower bound on the null energy in terms of the second derivative of the von Neumann entropy or entanglement entropy of some region with respect to a null direction. The QNEC states that $⟨{T}_{kk}{⟩}_{p}\ensuremath{\ge}{\mathrm{lim}}_{A\ensuremath{\rightarrow}0}(\frac{\ensuremath{\hbar}}{2\ensuremath{\pi}A}{S}_{\mathrm{out}}^{\ensuremath{'}\ensuremath{'}})$, where Sout{S}_{\mathrm{out}}Sout​ is the entanglement entropy restricted to one side of a codimension-2 surface $\mathrm{\ensuremath{\Sigma}}$, which is deformed in the null direction about a neighborhood of point ppp with area AAA. A proof of QNEC has been given before, which applies to free and super-renormalizable bosonic field theories, and to any points that lie on a stationary null surface. Using similar assumptions and methods, we prove the QNEC for fermionic field theories.
Energy conditions play an important role in constraining the dynamics of quantum field theories as well as gravitational theories. For example, in semi-classical gravity, the achronal averaged null energy condition (AANEC) can be used to prove that it is always slower to traverse through a wormhole than to travel around via its exterior. Such conditions prevent causality violations that would lead to paradoxes. Recent advances have been made in proving previously conjectured energy conditions directly in quantum field theory (QFT) as well as in uncovering new ones. This thesis will be an exploration of various energy inequalities in conformal field theories (CFTs) as well as semi-classical quantum gravity. At the core of this work lies the recently proved quantum null energy condition (QNEC). The QNEC bounds the null energy flowing past a point by a certain second shape derivative of entanglement entropy. We will demonstrate that the QNEC represents a deep connection between causality, energy and entanglement in quantum field theories. We explore this connection first in the context of holographic CFTs. Calculations in holographic theories will lead us to conjecture that the so-called diagonal QNEC is saturated in all interacting QFTs. We will then provide further, independent evidence that this conjecture holds for all CFTs with a twist gap by explicitly calculating shape derivatives of entanglement entropy using defect CFT techniques.
The Quantum Null Energy Condition (QNEC) relates energy to the second variation of entropy in relativistic quantum field theory. We use the QNEC inequality to bound entanglement entropy in quenches. At early times the entanglement entropy grows quadratically in time, and the QNEC provides an upper bound on the prefactor. We demonstrate that the bound is tight, by showing that it is saturated in certain quench protocols: boundary state quenches in conformal field theories in any dimensions. In higher than two dimensions we compute entanglement entropy using AdS/CFT. Our results are the first purely field theoretic applications of the QNEC.
We study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on R1,d−1 {\mathrm{\mathbb{R}}}^{1,d-1} R1,d−1 . We show that in addition to the usual boost generator, there is a contribution to the modular Hamiltonian at first order in the shape deformation, proportional to the integral of the null components of the stress tensor along the Rindler horizon. We use this fact along with monotonicity of relative entropy to prove the averaged null energy condition in Minkowski space-time. This subsequently gives a new proof of the Hofman-Maldacena bounds on the parameters appearing in CFT three-point functions. Our main technical advance involves adapting newly developed perturbative methods for calculating entanglement entropy to the problem at hand. These methods were recently used to prove certain results on the shape dependence of entanglement in CFTs and here we generalize these results to excited states and real time dynamics. We also discuss the AdS/CFT counterpart of this result, making connection with the recently proposed gravitational dual for modular Hamiltonians in holographic theories.
The quantum null energy condition (QNEC) is a lower bound on the expectation value of the null-null component of the energy-momentum tensor in terms of null variations of the entanglement entropy. A stronger version of the QNEC (the primary QNEC) is expected to hold in 1 + 1 dimensional conformal field theories (CFT). QNEC has been shown to impose nontrivial quantum thermodynamic restrictions on irreversible entropy production in quenches in 1 + 1 dimensional holographic CFTs. It is therefore natural to study if QNEC imposes similar bounds in other quench setups. In this paper we study QNEC in the Calabrese-Cardy global and local joining quenches using standard CFT techniques. In the global quench we show that the primary QNEC must hold at sufficiently early times and find that it imposes bounds on the four point correlators of twist fields in a boundary state. This is a constraint on the set of boundary states that satisfy the primary QNEC. Furthermore, we find that a violation of the primary QNEC implies a violation of the averaged null energy condition (ANEC) in a conformally transformed frame. In the local quench we find similar bounds on four point correlators from both the primary and the usual QNEC.