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Entanglement, Gravity, and Quantum Error Correction Xi Dong Institute for Advanced Study July 12, 2016 [XD, arXiv: 1601.06788 to appear in Nature Commun.] [XD, Phys. Rev. Lett. 116, 251602 (2016)] [XD, Harlow, Wall, Phys. Rev. Lett. 117, 021601 (2016)] [XD, Miao, JHEP 1512, 100 (2015)] [Almheiri, XD, Harlow, JHEP 1504, 163 (2015)] [XD, JHEP 1401, 044 (2014)] 21st International Conference on General Relativity and Gravitation, Columbia University Gravity Entanglement July 12, 2016 Quantum Error Correction Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Bekenstein-Hawking entropy for black holes ππ 3 Area Horizon π= 4πΊπ β β’ Led to much progress in understanding quantum gravity, e.g. holographic principle. July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Anti-de Sitter/Conformal Field Theory Correspondence [Maldacena β97] Quantum gravity in AdSd+1 (bulk) Holographic CFTs on πAdSd+1 (boundary) Isometry group π(π, 2) Conformal group π(π, 2) Black hole states Thermal states Gauge symmetry Global symmetry States and operators States and operators β’ Best-understood model of emergent spacetime/gravity β’ Framework for generalizing area law beyond black holes July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Holographic Entanglement Entropy A simple and powerful prescription for entanglement [Ryu & Takayanagi β06] entropy: Area Minimal Surface π= Spacetime βSpooky action 4πΊπ geometry at a distanceβ Recall the definition: π β βTr Οπ΄ ln Οπ΄ This is only the tip of the iceberg! July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Holographic Renyi Entropy A simple and powerful prescription for Renyi entropy: 2 π ππ πβ1 Area Cosmic Braneπ ππ = π 4πΊπ [XD 1601.06788] Has tension and backreacts on ambient geometry by creating conical deficit angle πβ1 2π . 1 ππ β ln TrΟππ΄ 1βπ More general measures of entanglement π β’ Gravity dual of Renyi entropy is a cosmic brane! β’ As π β 1: probe brane settles at minimal surface. β’ One-parameter generation of Ryu-Takayanagi. July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Example: Renyi entropy for two disks in holographic CFT 2 π ππ πβ1 Area Cosmic Braneπ ππ = π 4πΊπ β’ For one disk, it was calculated by exploiting a symmetry and finding hyperbolic black hole solutions. [Hung, Myers, Smolkin & Yale β11] β’ Area-law prescription is more powerful: applies to arbitrary regions. β’ For two disks, we study mutual Renyi information: πΌπ π΄1 , π΄2 β ππ π΄1 + ππ π΄2 β ππ (π΄1 βͺ π΄2 ) β’ To linear order in πΏπ = π β 1, brane backreaction is weak: 23βπ π π+1 πΆπ πΏπ 2βπ₯ πΌπ = 2 π₯ π΅ πβ1 π π2 β 1 Ξ 2 July 12, 2016 π₯ 2βπ₯ 2 π+1 2βπ ; , + π(πΏπ2 ) 2 2 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) [XD 1601.06788] So far: We used the area law to understand structure of quantum entanglement and to efficiently study Renyi entropies. Rest of the talk: How to use it to understand quantum gravity? July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) AdS/CFT: our best-understood model of quantum gravity [Maldacena β97] Quantum gravity in AdSd+1 Holographic CFTs on πAdSd+1 Isometry group π(π, 2) Conformal group π(π, 2) Black hole states Thermal states Gauge symmetry Global symmetry States and operators States and operators lim π Ξ π π, π₯ = π(π₯) πββ π π, π₯ = ? β’ What operator in CFT represents a local bulk operator? β’ Answering this question helps us reconstruct the bulk. July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Global AdS reconstruction [Hamilton, Kabat, Lifschytz, Lowe β06] July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) AdS-Rindler reconstruction for disk π΄ July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) What region of the dual spacetime is described by a general subregion in a holographic CFT? July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Reconstruction conjecture for entanglement wedge β’ Entanglement wedge is defined as a bulk region bounded by the Ryu-Takayanagi minimal surface. β’ It may change discontinuously. β’ Conjecture: Any bulk operator in entanglement wedge of π΄ may be represented as a CFT operator on π΄. [Czech, Karczmarek, Nogueira & Van Raamsdonk β12] [Wall β12] [Headrick, Hubeny, Lawrence & Rangamani β14] July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Conjecture: Any bulk operator in entanglement wedge of π΄ may be represented as a CFT operator on π΄. Ingredients for proving the conjecture: οHolography as a quantum error correcting code οCFT relative entropy = bulk relative entropy β Reconstruction theorem for entanglement wedge July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Holography as a quantum error [Almheiri, XD, Harlow β14] correcting code β’ π(π₯) can be represented on π΄ βͺ π΅, π΅ βͺ πΆ, or π΄ βͺ πΆ. β’ Obviously they cannot be the same CFT operator. β’ This redundancy is a defining feature for quantum error correction. July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Relative entropy is bulk relative entropy β’ Quantum corrections to the area law: Area Cosmic Braneπ ππ = + ππ ,bulk 4πΊπ β’ Provides (alternative) derivation of similar statements about modular Hamiltonian πΎ β β ln Οπ΄ Area πΎ= + πΎbulk 4πΊπ β’ and relative entropy π π π β Tr π ln π β Tr π ln π π(π|π) = πbulk (π|π) July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) [Jafferis, Lewkowycz, Maldacena & Suh 1512.06431] Reconstruction theorem for [XD, Harlow & Wall entanglement wedge 1601.05416] Any bulk operator ππ in the entanglement wedge π of π΄ may be reconstructed as a CFT operator ππ΄ on π΄ via quantum error correction, as long as the relative entropy of any two bulk states satisfies π(ππ΄ |ππ΄ ) = π bulk (ππ |ππ ) π Intuitive proof: β’ WLG assume ππ is Hermitian. β’ Take any bulk state π, let π be π ππππ acting on π. β’ Use π ππ΄ ππ΄ = πbulk ππ ππ = 0 to conclude ππ΄ = ππ΄ . β’ No measurements on π΄ distinguish the two states β ππ , ππ΄ = 0 under state π for any ππ΄ . β’ ππ must have a CFT realization as ππ΄ on π΄. Via theorem in [Almheiri, XD, Harlow β14] July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) What We Learned β’ Area law is universal in quantum gravity and is not restricted to black hole or entanglement entropy. β’ It is a powerful statement for Renyi entropy, generalizing the Ryu-Takayanagi prescription. β’ Quantum information theory enables us to understand the basic dictionary of quantum gravity. β’ Viewing holography as a quantum error correcting code, we can make progress on how to βbuild spacetime from entanglementβ. July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong) Thank you! July 12, 2016 Entanglement, Gravity, and Quantum Error Correction (Xi Dong)