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Aspects of Holography and Quantum Error Correction by Pratik Rath a Dissertation Submitted in Partial Satisfaction of the Requir Aspects of Holography And Quantum Error Correction by Pratik Rath A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Yasunori Nomura, Chair Professor Raphael Bousso Professor Richard Borcherds Summer 2020 Aspects of Holography And Quantum Error Correction Copyright 2020 by Pratik Rath 1 Abstract Aspects of Holography And Quantum Error Correction by Pratik Rath Doctor of Philosophy in Physics University of California, Berkeley Professor Yasunori Nomura, Chair The holographic principle has been a central theme in most of the progress in the field of quantum gravity in recent years. Our understanding of the AdS/CFT duality, the best known embodiment of the holographic principle, has taken a quantum leap in the last decade. A key role in the elucidation of how the holographic duality functions has been played by ideas from quantum information theory. In particular, the modern understanding of the holographic dictionary is that it works as a quantum error correcting code. In this dissertation, we focus on a two-pronged approach to developing a deeper insight into the framework of quantum gravity. Firstly, despite the fact that we have learnt a lot about quantum gravity from AdS/CFT, it is not directly applicable to our universe which is an accelerating cosmological spacetime. Taking inspiration from the holographic principle and formulating ideas from AdS/CFT in the abstract language of quantum error correction, we take some preliminary steps in freeing ourselves from the crutches of AdS spacetimes and understanding features of holography in a wider class of spacetimes. We develop a framework for holography in general spacetimes using the Ryu-Takayanagi formula as a postulate and discuss conditions for bulk reconstruction, the existence of a bulk dual and qualitative features of putative holographic theories in arbitrary spacetimes. Secondly, the holographic dictionary is not completely understood even within the realm of AdS/CFT. We clarify some aspects and propose novel entries to the AdS/CFT dictionary which shed light on how a gravitational description of a quantum mechanical system emerges holographically. In particular, this includes an understanding of how the holographic com- putation of Renyi entropy arises from a general feature of quantum error correction, supple- mented by the special property that gravitational states have maximally mixed edge modes. Further, we resolve a long standing conjecture about the nature of tripartite entanglement of holographic states. Finally, we propose novel holographic duals to the reflected entropy in the presence of entanglement islands, and the Connes cocycle flow. i To my parents, Bratati Dash and Pradyot Kumar Rath, for their limitless trust in my passion and eternal support in my endeavour, and to my wife, Shoan Jain, for her vital role in this often arduous process. ii Contents Contents ii List of Figures v 1 Introduction 1 1.1 The Holographic Principle . 1 1.2 AdS/CFT and Quantum Error Correction . 2 1.3 Holography in General Spacetimes . 3 1.4 Holographic Dictionary . 5 I Holography in General Spacetimes 8 2 Classical Spacetimes as Amplified Information in Holographic Quantum Theories 9 2.1 Introduction . 9 2.2 Framework . 13 2.3 Classicalization and Spacetime . 14 2.4 Reconstructing Spacetime . 17 2.5 Spacetime Is Non-Generic . 26 2.6 Black Hole Interior . 34 3 Spacetime from Unentanglement 37 3.1 Introduction . 37 3.2 Maximally Entropic States Have No Spacetime . 39 3.3 Spacetime Emerges through Deviations from Maximal Entropy . 55 3.4 Holographic Hilbert Spaces . 61 3.5 Conclusion . 64 4 Pulling the Boundary into the Bulk 71 4.1 Introduction . 71 4.2 Motivation . 73 4.3 Holographic Slice . 74 iii 4.4 Examples . 80 4.5 Interpretation and Applications . 89 4.6 Relationship to Renormalization . 96 4.7 Discussion . 99 II Holographic Dictionary 101 5 Comments on Holographic Entanglement Entropy in TT Deformed CFTs102 5.1 Introduction . 102 5.2 Field Theory Calculation . 103 5.3 Bulk Calculation . 107 5.4 Discussion . 108 6 Holographic Renyi Entropy from Quantum Error Correction 111 6.1 Introduction . 111 6.2 Operator-algebra Quantum Error Correction . 114 6.3 Interpretation of OQEC . 117 6.4 Cosmic Brane Prescription in OQEC . 122 6.5 Tensor Networks . 126 6.6 Discussion . 127 7 Entanglement Wedge Cross Sections Require Tripartite Entanglement 131 7.1 Introduction . 131 7.2 SR Conjecture vs Bipartite Entanglement . 135 7.3 EP Conjecture vs Bipartite Entanglement . 144 7.4 Discussion . 147 8 Islands for Reflected Entropy 150 8.1 Introduction . 150 8.2 Islands Formula . 152 8.3 Path integral Argument . 155 8.4 Phase transitions . 159 8.5 Discussion . 170 9 Gravity Dual of Connes Coycle Flow 175 9.1 Introduction . 175 9.2 Connes Cocycle Flow . 177 9.3 Kink Transform . 184 9.4 Bulk Kink Transform = Boundary CC Flow . 189 9.5 Predictions . 196 9.6 Discussion . 198 iv A Appendix 209 A.1 Reconstruction from a Single Leaf . 209 A.2 Reconstructability of Two-sided Black Holes and Complementarity . 210 A.3 Calculations for the Schwarzschild-AdS Spacetime . 213 A.4 Calculations for the de Sitter Limit of FRW Universes . 217 A.5 Intersection of Domains of Dependence . 224 A.6 Uniqueness of the Holographic Slice . 226 A.7 Convexity of Renormalized Leaves . 229 A.8 Flat Renyi Spectrum . 231 A.9 Inequalities of Entanglement of Purification . 232 A.10 Null Limit of the Kink Transform . 232 Bibliography 235 v List of Figures 2.1 If the HRT surface E(Γ) behaves continuously under a change of Γ, we can reconstruct the entire spacetime region inside the holographic screen, M, despite the fact that d families of HRT surfaces all anchored on a single leaf σ(0) do not in general span the same hypersurface. 21 2.2 A point p in an entanglement shadow S can be reconstructed as an intersec- tion of entanglement wedges associated with spatial regions on leaves if all the future-directed and past-directed light rays emanating from p reach outside the entanglement shadow early enough. Here we see that all past-directed light rays escape the shadow before the first of them intersects the holographic screen. 22 2.3 A point p in an entanglement shadow may be reconstructed as the intersection of a finite number of entanglement wedges if it is on caustics of these entanglement wedges (denoted by the dotted lines). 24 3.1 The volume V (r+;R) of the Schwarzschild-AdS spacetime that can be recon- structed from the boundary theory, normalized by the corresponding volume V (R) in empty AdS space: f = V (r+;R)=V (R). Here, R is the infrared cut- off of (d + 1)-dimensional AdS space, and r+ is the horizon radius of the black hole. 42 3.2 The Penrose diagram of de Sitter space. The spacetime region covered by the flat- slicing coordinates is shaded, and constant time slices in this coordinate system are drawn. The codimension-1 null hypersurface Σ0 is the cosmological horizon for an observer at r = 0, to which the holographic screen of the FRW universe asymptotes in the future. 45 3.3 Constant time slices and the spacetime region covered by the coordinates in static slicing of de Sitter space. Here, Σ is the τ = 0 hypersurface, and Ξ is the bifurcation surface, given by ρ = α with finite τ................... 46 3.4 The spacetime volume of the reconstructable region in (2 + 1)-dimensional flat FRW universes for w 2 (−0:9; −1), normalized by the reconstructable volume for w = −0:9........................................ 49 vi 3.5 Reconstructable spacetime regions for various values of w in (3 + 1)-dimensional flat FRW universes. The horizontal axis is the distance from the center, nor- malized by that to the leaf. The vertical axis is the difference in conformal time from the leaf, normalized such that null ray from the leaf would reach 1. The full reconstructable region for each leaf would be the gray region between the two lines rotated about the vertical axis. 50 3.6 Diagrams representing the achronal surface Σ in which two HRRT surfaces, m(ABC) and m(B), live. m(AB)Σ and m(BC)Σ are the representatives of m(AB) and m(BC), respectively. They are shown to be intersecting at p. On a spacelike Σ, one could deform around this intersection to create two new surfaces with smaller areas. 51 3.7 This depicts how one can scan across the representative m(BC)Σ by bipartitioning BC on the achronal surface Σ. At each of these intersections, p(xi), θu = θv = 0 if the state on the leaf is maximally entropic and Σ is null and non-expanding. 53 3.8 A schematic depiction of the entanglement entropy in the Schwarzschild-AdS spacetime, normalized by the maximal value of entropy in the subregion, QA = SA=SA;max, and depicted as a function of the size L of subregion A; see Eq. (3.20). The scales of the axes are arbitrary. As the mass of the black hole is lowered (the temperature T of the holographic theory is reduced from the cutoff Λ), QA deviates from 1 in a specific manner. 57 3.9 The entanglement entropy in the holographic theory of flat FRW spacetimes nor- malized by the maximal value of entropy in the subregion, Qw( ) = Sw( )=Smax( ), as a function of the size of the subregion, a half opening angle . As the equa- tion of state parameter w is increased from −1, Qw( ) deviates from 1 in a way different from the Schwarzschild-AdS case.
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