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Journal of High Energy Physics

, 2018:83 | Cite as

Modular flow as a disentangler

  • Yiming Chen
  • Xi DongEmail author
  • Aitor Lewkowycz
  • Xiao-Liang Qi
Open Access
Regular Article - Theoretical Physics

Abstract

In holographic duality, the entanglement entropy of a boundary region is proposed to be dual to the area of an extremal codimension-2 surface that is homologous to the boundary region, known as the Hubeny-Rangamani-Takayanagi (HRT) surface. In this paper, we study when the HRT surfaces of two boundary subregions R, A are in the same Cauchy slice. This condition is necessary for the subregion-subregion mapping to be local for both subregions and for states to have a tensor network description. To quantify this, we study the area of a surface that is homologous to A and is extremal except at possible intersections with the HRT surface of R (minimizing over all such possible surfaces), which we call the constrained area. We give a boundary proposal for an upper bound of this quantity, a bound which is saturated when the constrained surface intersects the HRT surface of R at a constant angle. This boundary quantity is the minimum entropy of region A in a modular evolved state — a state that has been evolved unitarily with the modular Hamiltonian of R. We can prove this formula in two boundary dimensions or when the modular Hamiltonian is local. This modular minimal entropy is a boundary quantity that probes bulk causality and, from this quantity, we can extract whether two HRT surfaces are in the future or past of each other. These entropies satisfy some inequalities reminiscent of strong subadditivity and can be used to remove certain corner divergences.

Keywords

AdS-CFT Correspondence Gauge-gravity correspondence Black Holes 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of PhysicsTsinghua UniversityBeijingChina
  2. 2.Department of PhysicsUniversity of CaliforniaSanta BarbaraU.S.A.
  3. 3.Stanford Institute for Theoretical Physics, Department of PhysicsStanford UniversityStanfordU.S.A.
  4. 4.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.

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