Building bulk geometry from the tensor Radon transform
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Springer
Received: September 9, 2020 Accepted: October 29, 2020 Published: December 4, 2020
ChunJun Cao,a Xiao-Liang Qi,b Brian Swinglea,c and Eugene Tangd a
Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, 20742, U.S.A. b Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA, 94305, U.S.A. c Condensed Matter Theory Center, University of Maryland, College Park, MD, 20742, U.S.A. d Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, 91125, U.S.A.
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: Using the tensor Radon transform and related numerical methods, we study how bulk geometries can be explicitly reconstructed from boundary entanglement entropies in the specific case of AdS3 /CFT2 . We find that, given the boundary entanglement entropies of a 2d CFT, this framework provides a quantitative measure that detects whether the bulk dual is geometric in the perturbative (near AdS) limit. In the case where a welldefined bulk geometry exists, we explicitly reconstruct the unique bulk metric tensor once a gauge choice is made. We then examine the emergent bulk geometries for static and dynamical scenarios in holography and in many-body systems. Apart from the physics results, our work demonstrates that numerical methods are feasible and effective in the study of bulk reconstruction in AdS/CFT. Keywords: AdS-CFT Correspondence, Conformal Field Theory, Models of Quantum Gravity ArXiv ePrint: 2007.00004
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2020)033
JHEP12(2020)033
Building bulk geometry from the tensor Radon transform
Contents 1 Introduction
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2 Boundary rigidity and bulk metric reconstruction
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3 Numerical methods for reconstruction 3.1 Discretization and optimization procedures
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5 Geometry detection
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6 Discussion
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A The tensor Radon transform A.1 General definitions A.2 s-injectivity
27 27 28
B The Radon transform on the Poincare disk
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C The holomorphic gauge
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D The numerical (inverse) Radon transform
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E Accuracy of the numerical reconstruction E.1 Constrained optimization E.2 Interpolation and regularization
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Introduction
Recent progress [1–7] in quantum gravity has shown that spacetime geometry can emerge from quantum entanglement. This emergence provides appealing explanations for many intuitive properties of the physical world, including the existence of gravity [8–11], conditions on the allowed distribution of energy and matter [12, 13], and the unitarity of black hole dynamics [14, 15]. Most of these developments have taken place in the context of
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JHEP12(2020)033
4 Reconstructed geometries 4.1 Holographic reconstructions 4.1.1 Mixture of thermal states 4.2 1D free fermion 4.2.1 Local deformations 4.2.2 Global deformations 4.3 Random disorder
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More generally, but less explicitly, there are also proposals for bulk reco
