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

by Christian Ecker, Daniel Grumiller, Wilke van der Schee, Shahin Sheikh-Jabbari, Philipp Stanzer

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 6, 036 (2019)

Submission summary

As Contributors: Christian Ecker · Daniel Grumiller
Arxiv Link: (pdf)
Date submitted: 2019-01-30 01:00
Submitted by: Grumiller, Daniel
Submitted to: SciPost Physics
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We consider the Quantum Null Energy Condition (QNEC) for holographic conformal field theories in two spacetime dimensions (CFT$_2$). We show that QNEC saturates for all states dual to vacuum solutions of AdS$_3$ Einstein gravity, including systems that are far from thermal equilibrium. If the Ryu-Takayanagi surface encounters bulk matter QNEC does not need to be saturated, whereby we give both analytical and numerical examples. In particular, for CFT$_2$ with a global quench dual to AdS$_3$-Vaidya geometries we find a curious half-saturation of QNEC for large entangling regions. We also address order one corrections from quantum backreactions of a scalar field in AdS$_3$ dual to a primary operator of dimension $h$ in a large central charge expansion and explicitly compute both, the backreacted Ryu--Takayanagi surface part and the bulk entanglement contribution to EE and QNEC. At leading order for small entangling regions the contribution from bulk EE exactly cancels the contribution from the back-reacted Ryu-Takayanagi surface, but at higher orders in the size of the region the contributions are almost equal while QNEC is not saturated. For a half-space entangling region we find that QNEC is gapped by $h/4$ in the large $h$ expansion.

Ontology / Topics

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Conformal field theory (CFT) Holography
Current status:
Has been resubmitted

Submission & Refereeing History

Published as SciPost Phys. 6, 036 (2019)

Reports on this Submission

Anonymous Report 1 on 2019-3-4 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1901.04499v1, delivered 2019-03-04, doi: 10.21468/SciPost.Report.856


1) Simple and elegant proof of the saturation of the quantum null energy condition for 2d holographic CFTs corresponding to the vacuum solutions of AdS3 gravity.

2) Very detailed (analytical and numerical) analysis of various cases including far-from equilibrium setups, and cases with a bulk scalar and backreactions for which the quantum null energy condition is not saturated.


I don't quite see any definite weaknesses. Some parts of the paper present somewhat speculative statements, such as the concept of "Quantum equilibrium". So, in principle, I would like to see more supports for such statements, which make the manuscript more valuable. However, these type of statements appear only in the last section, and for the bulk of the paper, the authors are quite concrete. So, I see no problem.


In their paper, the authors studied the quantum null energy condition (QNEC) in 2d holographic CFTs. After providing a simple and elegant proof of the saturation of the QNEC for vacuum solutions of AdS3 Einstein gravity, they studied various cases, including far from equilibrium setups and cases with a bulk scalar field. In the latter case, the QNEC needs not to saturate. They also discussed quantum corrections. I found that the manuscript is interesting and sheds light on the QNEC, and hence recommend the paper to be published once the minor concerns in the following are properly addressed (see below).

Requested changes

1) Throughout the paper, "QNEC" is sometimes used to refer actually the RHS of the QNEC in (1.1). For example, there are sentences like "QNEC diverges". These sentences are somewhat strange, as the QNEC per se cannot diverge. I suggest the authors to find a better way of phrasing these sentences.

2) I don't quite see the merit of the numerics presented in Sec. 3.2.
The approximate numerics solution was found by approximating the step function.
Just below the numerical paragraph, however the authors presented
the exact solution (3.6) and (3.7), without approximating the step function.

3) I don't quite understand what the author would mean by the operator interpretation.The LHS of (1.1) is a number (expectation value), and hence the RHS should also be a number. I also failed to see the connection between the twisted Sugawara construction and the identification (6.7). I would hope that the authors could elaborate on this. I guess that it's perhaps simply a matter of putting brackets < ... > in some expressions like (6.7 - 6.9)?

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: good

Author:  Daniel Grumiller  on 2019-03-14  [id 466]

(in reply to Report 1 on 2019-03-04)
reply to objection

We do agree with the referee on the strengths and weaknesses of our work. Regarding the three suggestions for minor changes, we have the following replies.

1) We agree with the referee and have rephrased the few statements where `QNEC' refers only to the right hand side of the QNEC inequality.

2) We have added a brief statement in section 3.2 after Eq. (3.5); while it is true that the analytic discussion of the step-function could be sufficient, we decided to keep the smooth plots in Figure 1 and the corresponding two numerical paragraphs since these plots nicely capture the essence of our statements and corresponding pictures with a theta-function would be harder to decipher.

3) We concede that the referee may have a point and that our presentation was not sufficiently transparent. We have made section 6.3 clearer by introducing bold-faced notation for operators to make it easier to distinguish when we are talking about operators and when we are talking about vacuum expectation values or functions. We adapted the wording after Eq. (6.7) and in footnote 11 accordingly.

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