\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{holographic entanglement entropy}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{measure_and_probability_theory}{}\paragraph*{{Measure and probability theory}}\label{measure_and_probability_theory}

[[!include measure theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{RyuTakayanagiFormula}{Ryu-Takayanagi formula}\dotfill \pageref*{RyuTakayanagiFormula} \linebreak
\noindent\hyperlink{emergence_of_bulk_spacetime_from_boundary_information_theory}{Emergence of bulk spacetime from boundary information theory}\dotfill \pageref*{emergence_of_bulk_spacetime_from_boundary_information_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Holographic entanglement entropy refers to the expression of [[entanglement entropy]] of [[quantum field theories]] expressed ``holographically'' via a version of [[AdS-CFT duality]] in terms of the [[geometry]] of a higher-dimensional [[bulk]] [[spacetime]]

\hypertarget{RyuTakayanagiFormula}{}\subsubsection*{{Ryu-Takayanagi formula}}\label{RyuTakayanagiFormula}

For [[quantum field theories]] that are exhibited as [[boundary field theories]] on the [[asymptotic boundary]] $A$ of an approximately [[anti de Sitter spacetime]] via some approximation to [[AdS-CFT duality]] (for instance for [[QCD]] via [[AdS-QCD duality]]) their [[entanglement entropy]] of a given [[bounded set|bounded domain]] $B\subset A$ turns out to be proportional to the [[area]] of the minimal-area [[surface]] inside the [[bulk]] spacetime that has the same [[boundary]] $\partial B$ (see \hyperlink{NishiokaRyuTakayanagi09}{Nishioka-Ryu-Takayanagi 09 (3.3)} for review of the formula and \hyperlink{LewkowyczMaldacena13}{Lewkowycz-Maldacena 13} for a conceptual explanation).

\begin{quote}%
graphics grabbed from \hyperlink{NishiokaRyuTakayanagi09}{Nishioka-Ryu-Takayanagi 09}


\end{quote}
This relation is known as the \emph{Ryu-Takayanagi formula} (\hyperlink{RyuTakayanagi06a}{Ryu-Takayanagi 06a}, \hyperlink{RyuTakayanagi06b}{Ryu-Takayanagi 06b}) for holographic computation of entanglement entropy, or \emph{holographic entanglement entropy}, for short.

This is a generalization of the proportionality of [[black hole entropy]] to the area of its [[event horizon]]. Indeed, [[AdS-CFT duality]] applies to the [[near horizon geometry]] of [[black branes]], the higher-dimensional generalizations of [[black holes]] and reduces 4d black holes under suitable [[KK-compactification]] (see also at \emph{[[black holes in string theory]]})

\begin{quote}%
graphics grabbed from \hyperlink{NishiokaRyuTakayanagi09}{Nishioka-Ryu-Takayanagi 09}


\end{quote}
In fact quantum corrections to the [[black hole entropy]] in the presence of matter fields is equal to the [[entanglement entropy]]. (\hyperlink{RyuTakayanagi06a}{Ryu-Takayanagi 06a, p. 13})

Various properties of [[entanglement entropy]] find immediate geometric interpretations this way, for instance subadditivity

\begin{quote}%
graphics grabbed from \hyperlink{NishiokaRyuTakayanagi09}{Nishioka-Ryu-Takayanagi 09}


\end{quote}
\hypertarget{emergence_of_bulk_spacetime_from_boundary_information_theory}{}\subsubsection*{{Emergence of bulk spacetime from boundary information theory}}\label{emergence_of_bulk_spacetime_from_boundary_information_theory}

Further discussion of implications of the Ryu-Takayanagi formula in \hyperlink{vanRaamsdonk10}{van Raamsdonk 10} suggested that the logic may also be turned around: Instead of computing [[entanglement entropy]] of a given [[boundary field theory]] from known [[bulk]] [[geometry]], conversely the [[bulk]] [[spacetime]] may be reconstructed from knowledge of the [[entanglement entropy]] of a boundary field theory.

Talking this perspective to the extreme suggests a description of [[bulk]] [[spacetimes]] entirely in terms of [[information theory]]/[[entanglement]]-relations of a boundary [[QFT]] (``[[tensor networks]]'', \hyperlink{Swingle09}{Swingle 09}, \hyperlink{Swingle12}{Swingle 12}, and quantum error correction codes \hyperlink{ADH14}{ADH 14}, \hyperlink{PYHP15}{PYHP 15}, see \hyperlink{Harlow18}{Harlow 18} for review).

\begin{quote}%
graphics grabbed from \hyperlink{Harlow18}{Harlow 18}


\end{quote}
\begin{quote}%
graphics grabbed from \hyperlink{Harlow18}{Harlow 18}


\end{quote}
In this context the Ryu-Takayanagi formula for holographic entanglement entropy (\hyperlink{RyuTakayanagiFormula}{above}) has an exact proof \hyperlink{PYHP15}{PYHP 15, Theorem 2}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item gravitational entropy

\begin{itemize}%
\item [[Bekenstein-Hawking entropy]]


\item [[generalized second law of thermodynamics]]


\item [[black holes in string theory]]



\end{itemize}

\item [[p-adic AdS-CFT]]



\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

The original article are

\begin{itemize}%
\item [[Shinsei Ryu]], [[Tadashi Takayanagi]], \emph{Holographic Derivation of Entanglement Entropy from AdS/CFT}, Phys. Rev. Lett. 96:181602, 2006 (\href{https://arxiv.org/abs/hep-th/0603001}{arXiv:hep-th/0603001})


\item [[Shinsei Ryu]], [[Tadashi Takayanagi]], \emph{Aspects of Holographic Entanglement Entropy}, JHEP 0608:045, 2006 (\href{https://arxiv.org/abs/hep-th/0605073}{arXiv:hep-th/0605073})



\end{itemize}
A proposal for a conceptual explanation is made in

\begin{itemize}%
\item Aitor Lewkowycz, [[Juan Maldacena]], \emph{Generalized gravitational entropy}, J. High Energ. Phys. (2013) 2013: 90 (\href{https://arxiv.org/abs/1304.4926}{arXiv:1304.4926})

\end{itemize}
Review is in

\begin{itemize}%
\item Tatsuma Nishioka, [[Shinsei Ryu]], [[Tadashi Takayanagi]], \emph{Holographic Entanglement Entropy: An Overview}, J.Phys.A42:504008,2009 (\href{https://arxiv.org/abs/0905.0932}{arXiv:0905.0932})


\item Matthew Headrick, \emph{Lectures on entanglement entropy in field theory and holography} (\href{https://arxiv.org/abs/1907.08126}{arXiv:1907.08126})



\end{itemize}
Survey talks include

\begin{itemize}%
\item Meyers, \emph{Holographic entanglement entropy}, (\href{http://www.lpt.ens.fr/IMG/pdf/Myers.pdf}{pdf slides})


\item Shinsei Ryu, \emph{Holographic geometry in Entanglement Renormalization} (\href{http://icmt.illinois.edu/Workshops/JointWorkshopPerimeter/Ryu-PI-ICMT-2012.pdf}{pdf slides})


\item Juan Jottar, \emph{(Entanglement) Entropy in three-dimensional higher spin theories} (\href{http://www.hip.fi/holograv13/talk_folder/Jottar-HologravWorkshop-March2013.pdf}{pdf slides})


\item Matthew Headrick, \emph{Entanglement entropies in holographic field theory} (\href{http://www.ggi.fi.infn.it/talks/talk1853.pdf}{pdf slides})


\item Tadashi Takayanagi, \emph{Entanglement Entropy and Holography (Introductory review)} (\href{http://www.princeton.edu/~jmaciejk/entanglement2012/slides/TakayanagiPCTS2012.pdf}{pdf slides})


\item Tom Hartmann, \emph{Entanglement entropy and geometry}, talk slides, 2014 (\href{http://online.kitp.ucsb.edu/online/qft14/hartman/pdf/Hartman_QFT14_KITP.pdf}{pdf})



\end{itemize}
An influential argument that this relation implies that [[entanglement]] in the boundary theory is what makes spacetime as such appear in the bulk theory is due to

\begin{itemize}%
\item [[Mark Van Raamsdonk]], \emph{Building up spacetime with quantum entanglement}, Gen.Rel.Grav.42:2323-2329,2010; Int.J.Mod.Phys.D19:2429-2435,2010 (\href{https://arxiv.org/abs/1005.3035}{arXiv:1005.3035})


\item [[Mark Van Raamsdonk]], \emph{Building up spacetime with quantum entanglement II: It from BC-bit} (\href{https://arxiv.org/abs/1809.01197}{arXiv:1809.01197})



\end{itemize}
reviewed in

\begin{itemize}%
\item [[Mark Van Raamsdonk]], \emph{Lectures on Gravity and Entanglement}, chapter 5 in New Frontiers in Fields and Strings TASI 2015 Proceedings of the 2015 Theoretical Advanced Study Institute in Elementary Particle Physics 2015 Theoretical Advanced Study Institute in Elementary Particle Physics (\href{https://arxiv.org/abs/1609.00026}{arXiv:1609.00026})

\end{itemize}
Relation to [[renormalization]] of [[entanglement]] and [[tensor networks]] is due to

\begin{itemize}%
\item Brian Swingle, \emph{Entanglement Renormalization and Holography} (\href{https://arxiv.org/abs/0905.1317}{arXiv:0905.1317})


\item Brian Swingle, \emph{Constructing holographic spacetimes using entanglement renormalization} (\href{https://arxiv.org/abs/1209.3304}{arXiv:1209.3304})



\end{itemize}
and further in terms of quantum error correcting codes due to

\begin{itemize}%
\item [[Ahmed Almheiri]], Xi Dong, [[Daniel Harlow]], \emph{Bulk Locality and Quantum Error Correction in AdS/CFT}, JHEP 1504:163,2015 (\href{https://arxiv.org/abs/1411.7041}{arXiv:1411.7041})


\item Fernando Pastawski, Beni Yoshida, [[Daniel Harlow]], John Preskill, \emph{Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence}, JHEP 06 (2015) 149 (\href{https://arxiv.org/abs/1503.06237}{arXiv:1503.06237})



\end{itemize}
reviewed in

\begin{itemize}%
\item [[Daniel Harlow]], \emph{TASI Lectures on the Emergence of Bulk Physics in AdS/CFT} (\href{https://arxiv.org/abs/1802.01040}{arXiv:1802.01040})

\end{itemize}
See also

\begin{itemize}%
\item Felix M. Haehl, Eric Mintun, Jason Pollack, Antony J. Speranza, [[Mark Van Raamsdonk]], \emph{Nonlocal multi-trace sources and bulk entanglement in holographic conformal field theories}, J. High Energ. Phys. (2019) 2019: 005 (\href{https://arxiv.org/abs/1904.01584}{arxiv:1904.01584}, \href{https://youtu.be/kRCwzyliJ1M}{talk recording})

\end{itemize}
Further development of these tensor networks in

\begin{itemize}%
\item Ning Bao, Geoffrey Penington, Jonathan Sorce, Aron C. Wall, \emph{Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT} (\href{https://arxiv.org/abs/1812.01171}{arXiv:1812.01171})


\item Ning Bao, Geoffrey Penington, Jonathan Sorce, Aron C. Wall, \emph{Holographic Tensor Networks in Full AdS/CFT} (\href{https://arxiv.org/abs/1902.10157}{arXiv:1902.10157})



\end{itemize}
Computation of [[black hole entropy]] in 4d via [[AdS4-CFT3 duality]] from [[holographic entanglement entropy]] in the [[ABJM theory]] for the [[M2-brane]] is discussed in

\begin{itemize}%
\item Jun Nian, Xinyu Zhang, \emph{Entanglement Entropy of ABJM Theory and Entropy of Topological Black Hole} (\href{https://arxiv.org/abs/1705.01896}{arXiv:1705.01896})

\end{itemize}
Discussion in terms of [[DHR superselection theory]]:

\begin{itemize}%
\item Horacio Casini, Marina Huerta, Javier M. Magan, Diego Pontello, \emph{Entanglement entropy and superselection sectors I. Global symmetries} (\href{https://arxiv.org/abs/1905.10487}{arXiv:1905.10487})

\end{itemize}
[[!redirects Ryu-Takayanagi formula]]



\end{document}