Integral geometry and holography

Czech, Bartlomiej; Lamprou, Lampros; McCandlish, Samuel; Sully, James

doi:10.1007/JHEP10(2015)175

Title: Integral geometry and holography

Abstract

We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS₃/CFT₂ correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS₃ whose kinematic space is two-dimensional de Sitter space.

Authors:

Czech, Bartlomiej ^[1]; Lamprou, Lampros ^[1]; McCandlish, Samuel ^[1]; Sully, James ^[2]

Stanford Univ., Stanford, CA (United States)
SLAC National Accelerator Lab., Menlo Park, CA (United States)

Publication Date:: Tue Oct 27 00:00:00 EDT 2015

Research Org.:: SLAC National Accelerator Laboratory (SLAC), Menlo Park, CA (United States)

Sponsoring Org.:: USDOE Office of Science (SC), High Energy Physics (HEP)

OSTI Identifier:: 1183697

Report Number(s):: SLAC-PUB-16293
Journal ID: ISSN 1029-8479; arXiv:1505.05515

Grant/Contract Number:: AC02-76SF00515

Resource Type:: Accepted Manuscript

Journal Name:: Journal of High Energy Physics (Online)

Additional Journal Information:: Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2015; Journal Issue: 10; Journal ID: ISSN 1029-8479

Publisher:: Springer Berlin

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Theory-HEP; HEPTH; gauge-gravity correspondence; AdS-CFT correspondence

Citation Formats


                    Czech, Bartlomiej, Lamprou, Lampros, McCandlish, Samuel, and Sully, James. Integral geometry and holography.  United States: N. p., 2015. 
Web.  doi:10.1007/JHEP10(2015)175.

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                    Czech, Bartlomiej, Lamprou, Lampros, McCandlish, Samuel, & Sully, James. Integral geometry and holography.  United States.  https://doi.org/10.1007/JHEP10(2015)175

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                    Czech, Bartlomiej, Lamprou, Lampros, McCandlish, Samuel, and Sully, James. Tue .  
"Integral geometry and holography".  United States.  https://doi.org/10.1007/JHEP10(2015)175.  https://www.osti.gov/servlets/purl/1183697.

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@article{osti_1183697,

  title        = {Integral geometry and holography},

  author       = {Czech, Bartlomiej and Lamprou, Lampros and McCandlish, Samuel and Sully, James},

  abstractNote = {We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS3/CFT2 correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volumes, giving a precise entropic interpretation of the length of any bulk curve. We explain how basic geometric concepts -- points, distances and angles -- are reflected in kinematic space, allowing one to reconstruct a large class of spatial bulk geometries from boundary entanglement entropies. In this way, kinematic space translates between information theoretic and geometric descriptions of a CFT state. As an example, we discuss in detail the static slice of AdS3 whose kinematic space is two-dimensional de Sitter space.},

  doi          = {10.1007/JHEP10(2015)175},

  journal      = {Journal of High Energy Physics (Online)},

  number       = 10,

  volume       = 2015,

  place        = {United States},

  year         = {Tue Oct 27 00:00:00 EDT 2015},

  month        = {Tue Oct 27 00:00:00 EDT 2015}

}

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The multiscale entanglement renormalization ansatz (MERA) is a tensor network that provides an efficient way of variationally estimating the ground state of a critical quantum system. The network geometry resembles a discretization of spatial slices of an anti–de Sitter (AdS) spacetime and “geodesics” in the MERA reproduce the Ryu-Takayanagi formula for the entanglement entropy of a boundary region in terms of bulk properties. It has therefore been suggested that there could be an AdS/MERA correspondence, relating states in the Hilbert space of the boundary quantum system to ones defined on the bulk lattice. In this report, we explore this proposalmore »« less
Cited by 56
https://doi.org/10.1103/PhysRevD.91.125036

Full Text Available
Beyond toy models: distilling tensor networks in full AdS/CFT

Journal Article Bao, Ning ; Penington, Geoffrey ; Sorce, Jonathan ; ... - Journal of High Energy Physics (Online)

We present a general procedure for constructing tensor networks that accurately reproduce holographic states in conformal field theories (CFTs). Given a state in a large-N CFT with a static, semiclassical gravitational dual, we build a tensor network by an iterative series of approximations that eliminate redundant degrees of freedom and minimize the bond dimensions of the resulting network. We argue that the bond dimensions of the tensor network will match the areas of the corresponding bulk surfaces. For “tree” tensor networks (i.e., those that are constructed by discretizing spacetime with nonintersecting Ryu-Takayanagi surfaces), our arguments can be made rigorous usingmore »« less
https://doi.org/10.1007/jhep11(2019)069

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$T \bar{T}$ , the entanglement wedge cross section, and the breakdown of the split property

Journal Article Asrat, Meseret ; Kudler-Flam, Jonah - Physical Review D

We consider fine-grained probes of the entanglement structure of two-dimensional conformal field theories deformed by the irrelevant double-trace operatormore »« less
Cited by 21
https://doi.org/10.1103/PhysRevD.102.045009

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Title: Integral geometry and holography

Abstract

Citation Formats

Entropy and area journal, August 1993

The Large-N Limit of Superconformal Field Theories and Supergravity journal, January 1999

Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence journal, May 2006

Aspects of holographic entanglement entropy journal, August 2006

A covariant holographic entanglement entropy proposal journal, July 2007

Gravitation from entanglement in holographic CFTs journal, March 2014

Locality of Gravitational Systems from Entanglement of Conformal Field Theories journal, June 2015

Building up spacetime with quantum entanglement journal, June 2010

On the architecture of spacetime geometry journal, October 2014

Cool horizons for entangled black holes journal, August 2013

Entanglement Renormalization journal, November 2007

Holographic geometry of entanglement renormalization in quantum field theories journal, October 2012

Time evolution of entanglement entropy from black hole interiors journal, May 2013

Localized shocks journal, March 2015

Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence journal, June 2015

Boundary Rigidity and Holography journal, January 2004

Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity journal, June 1986

Proof of the strong subadditivity of quantum‐mechanical entropy journal, December 1973

Partial quantum information journal, August 2005

Quantum State Merging and Negative Information journal, October 2006

The information theoretic interpretation of the length of a curve journal, June 2015

Exact Cost of Redistributing Multipartite Quantum States journal, June 2008

A Stress Tensor for Anti-de Sitter Gravity journal, December 1999

Class of Quantum Many-Body States That Can Be Efficiently Simulated journal, September 2008

Theory of minimal updates in holography journal, May 2015

The dS/CFT correspondence journal, October 2001

Notes on de Sitter space and holography journal, November 2002

Entwinement and the emergence of spacetime journal, January 2015

Holographic entanglement entropy for general higher derivative gravity journal, January 2014

Extracting spacetimes using the AdS/CFT conjecture journal, August 2008

The gravity dual of a density matrix journal, July 2012

Bulk locality and quantum error correction in AdS/CFT journal, April 2015

Sur la rigidit� impos�e par la longueur des g�od�siques journal, February 1981

Two dimensional compact simple Riemannian manifolds are boundary distance rigid journal, January 2005

Building up Space–Time with Quantum Entanglement journal, December 2010

A Covariant Holographic Entanglement Entropy Proposal text, January 2007

Extracting spacetimes using the AdS/CFT conjecture text, January 2008

Extracting Spacetimes using the AdS/CFT Conjecture: Part II text, January 2010

Holographic Geometry of Entanglement Renormalization in Quantum Field Theories text, January 2012

On the Architecture of Spacetime Geometry preprint, January 2012

The entropy of a hole in spacetime text, January 2013

Cool horizons for entangled black holes text, January 2013

A theory of minimal updates in holography text, January 2013

Holographic Entanglement Entropy for General Higher Derivative Gravity text, January 2013

Holographic Holes in Higher Dimensions text, January 2014

Holographic Reconstruction of General Bulk Surfaces text, January 2014

Entwinement and the emergence of spacetime text, January 2014

Holographic Holes and Differential Entropy text, January 2014

Entanglement renormalization text, January 2005

Boundary Rigidity and Holography text, January 2003

Aspects of Holographic Entanglement Entropy text, January 2006

A Stress Tensor for Anti-de Sitter Gravity text, January 1999

The exact cost of redistributing multipartite quantum states text, January 2006

Topological Complexity in AdS 3 /CFT 2 journal, June 2018

Kinematic space and wormholes journal, January 2017

Holographic subregion complexity from kinematic space journal, January 2019

Holographic description of boundary gravitons in (3+1) dimensions journal, January 2019

3D holography: from discretum to continuum journal, March 2016

Holographic relations for OPE blocks in excited states journal, March 2019

Holographic complexity in Vaidya spacetimes. Part I journal, June 2018

Entanglement, holography and causal diamonds journal, August 2016

Solving the Schwarzian via the conformal bootstrap journal, August 2017

Kinematic space for conical defects journal, November 2017

On CFT and quantum chaos journal, December 2016

Holographic spacetimes as quantum circuits of path-integrations journal, December 2018

Bit Threads and Holographic Entanglement journal, November 2016

Towards a reconstruction of general bulk metrics journal, December 2016

Losing the IR: a holographic framework for area theorems journal, January 2019

AdS vacuum bubbles, holography and dual RG flows journal, January 2019

Holographic entanglement contour, bit threads, and the entanglement tsunami journal, July 2019

Towards entanglement of purification for conformal field theories journal, June 2018

Berry curvature, horocycles, and scattering states in AdS 3 / CFT 2 journal, December 2019

Complexity Growth Rate in Lovelock Gravity journal, September 2018

Conditional Quantum One-Time Pad journal, February 2020

What’s the point? Hole-ography in Poincaré AdS journal, January 2018

Entanglement wedge reconstruction and entanglement of purification journal, August 2018

Holographic complexity from Crofton’s formula in Lorentzian AdS 3 journal, December 2019

Entropy and area
journal, August 1993

The Large-N Limit of Superconformal Field Theories and Supergravity
journal, January 1999

Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence
journal, May 2006

Aspects of holographic entanglement entropy
journal, August 2006

A covariant holographic entanglement entropy proposal
journal, July 2007

Gravitation from entanglement in holographic CFTs
journal, March 2014

Locality of Gravitational Systems from Entanglement of Conformal Field Theories
journal, June 2015

Building up spacetime with quantum entanglement
journal, June 2010

On the architecture of spacetime geometry
journal, October 2014

Cool horizons for entangled black holes
journal, August 2013

Entanglement Renormalization
journal, November 2007

Holographic geometry of entanglement renormalization in quantum field theories
journal, October 2012

Time evolution of entanglement entropy from black hole interiors
journal, May 2013

Localized shocks
journal, March 2015

Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence
journal, June 2015

Boundary Rigidity and Holography
journal, January 2004

Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity
journal, June 1986

Proof of the strong subadditivity of quantum‐mechanical entropy
journal, December 1973

Partial quantum information
journal, August 2005

Quantum State Merging and Negative Information
journal, October 2006

The information theoretic interpretation of the length of a curve
journal, June 2015

Exact Cost of Redistributing Multipartite Quantum States
journal, June 2008

A Stress Tensor for Anti-de Sitter Gravity
journal, December 1999

Class of Quantum Many-Body States That Can Be Efficiently Simulated
journal, September 2008

Theory of minimal updates in holography
journal, May 2015

The dS/CFT correspondence
journal, October 2001

Notes on de Sitter space and holography
journal, November 2002

Entwinement and the emergence of spacetime
journal, January 2015

Holographic entanglement entropy for general higher derivative gravity
journal, January 2014

Extracting spacetimes using the AdS/CFT conjecture
journal, August 2008

The gravity dual of a density matrix
journal, July 2012

Bulk locality and quantum error correction in AdS/CFT
journal, April 2015

Sur la rigidit� impos�e par la longueur des g�od�siques
journal, February 1981

Two dimensional compact simple Riemannian manifolds are boundary distance rigid
journal, January 2005

Building up Space–Time with Quantum Entanglement
journal, December 2010

A Covariant Holographic Entanglement Entropy Proposal
text, January 2007

Extracting spacetimes using the AdS/CFT conjecture
text, January 2008

Extracting Spacetimes using the AdS/CFT Conjecture: Part II
text, January 2010

Holographic Geometry of Entanglement Renormalization in Quantum Field Theories
text, January 2012

On the Architecture of Spacetime Geometry
preprint, January 2012

The entropy of a hole in spacetime
text, January 2013

Cool horizons for entangled black holes
text, January 2013

A theory of minimal updates in holography
text, January 2013

Holographic Entanglement Entropy for General Higher Derivative Gravity
text, January 2013

Holographic Holes in Higher Dimensions
text, January 2014

Holographic Reconstruction of General Bulk Surfaces
text, January 2014

Entwinement and the emergence of spacetime
text, January 2014

Holographic Holes and Differential Entropy
text, January 2014

Entanglement renormalization
text, January 2005

Boundary Rigidity and Holography
text, January 2003

Aspects of Holographic Entanglement Entropy
text, January 2006

A Stress Tensor for Anti-de Sitter Gravity
text, January 1999

The exact cost of redistributing multipartite quantum states
text, January 2006

Topological Complexity in AdS ₃ /CFT ₂
journal, June 2018

Kinematic space and wormholes
journal, January 2017

Holographic subregion complexity from kinematic space
journal, January 2019

Holographic description of boundary gravitons in (3+1) dimensions
journal, January 2019

3D holography: from discretum to continuum
journal, March 2016

Holographic relations for OPE blocks in excited states
journal, March 2019

Holographic complexity in Vaidya spacetimes. Part I
journal, June 2018

Entanglement, holography and causal diamonds
journal, August 2016

Solving the Schwarzian via the conformal bootstrap
journal, August 2017

Kinematic space for conical defects
journal, November 2017

On CFT and quantum chaos
journal, December 2016

Holographic spacetimes as quantum circuits of path-integrations
journal, December 2018

Bit Threads and Holographic Entanglement
journal, November 2016

Towards a reconstruction of general bulk metrics
journal, December 2016

Losing the IR: a holographic framework for area theorems
journal, January 2019

AdS vacuum bubbles, holography and dual RG flows
journal, January 2019

Holographic entanglement contour, bit threads, and the entanglement tsunami
journal, July 2019

Towards entanglement of purification for conformal field theories
journal, June 2018

Berry curvature, horocycles, and scattering states in ${AdS}_{3} / {CFT}_{2}$
journal, December 2019

Complexity Growth Rate in Lovelock Gravity
journal, September 2018

Conditional Quantum One-Time Pad
journal, February 2020

What’s the point? Hole-ography in Poincaré AdS
journal, January 2018

Entanglement wedge reconstruction and entanglement of purification
journal, August 2018

Holographic complexity from Crofton’s formula in Lorentzian AdS ₃
journal, December 2019

Towards a Fisher-Information Description of Complexity in de Sitter Universe
journal, November 2019

Tensor Networks from Kinematic Space
text, January 2015

Complexity, action, and black holes
text, January 2015