A general proof of the quantum null energy condition

By S. Balakrishnan, T. Faulkner, Zuhair U. Khandker, Huajia Wang. Journal of High Energy Physics, 2017

179 citations recorded by the source corpus.

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document} 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.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f822f7bcf6d3842086460729bf97cf8deba0249e

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Citations in the corpus

Entanglement entropy: Holography and renormalization group (2018, methodology): “Moreover, the monotonic properties of entanglement measures have been successfully applied to derive nontrivial constraints on energy and entropy in recent studies (Balakrishnan et al., 2017; Bousso et al., 2014, 2015, 2016; Faulkner et al., 2016).”
Recovering the QNEC from the ANEC (2018, methodology): “ilable, since they can lead to their own insights. See [11{16] for some recent progress along these lines. In this paper we will take inspiration from the previous QNEC proof for interacting theories [7] as well as some ideas laid out by Wall [17]. In this way we unify these two seemingly disparate ”
De Sitter Space and the Swampland (2018, background): “D = 10. It would be interesting to ﬁnd out if a bound on |∇V|/V can be obtained with the averaged null energy condition, which is weaker and is known to be satisﬁed by any unitary quantum ﬁeld theory [41, 42]. 3 Accelerating Universe in a Laboratory Suppose a scalar potential V is such that it allow”
Light-ray operators in conformal field theory (2018, background): “ow do they behave under modular ﬂow? Can they appear in OPEs of entangling twist defects? The ANEC can be improved to the quantum null energy condition (QNEC) [104, 105], which was recently proven in [106] together with a higher integer spin generalization. Is there a continuous-spin version of the ”
Notes on Some Entanglement Properties of Quantum Field Theory (2018, background): “It has been used in a number of recent developments, including an attempt to see behind the horizon of a black hole [9], a proof of the quantum null energy condition [10], and too many others to properly cite here.”