Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence

Journal of High Energy Physics, Jun 2015

We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].

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Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence

Received: April Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence Fernando Pastawski 0 1 Beni Yoshida 0 1 Daniel Harlow 0 1 2 John Preskill 0 1 E. California Blvd. 0 1 Pasadena CA 0 1 U.S.A. 0 1 Open Access, c The Authors. 0 400 Jadwin Hall , Princeton NJ 08540 , U.S.A 1 California Institute of Technology 2 Princeton Center for Theoretical Science, Princeton University We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1]. AdS-CFT Correspondence; Lattice Integrable Models 1 Introduction 4.1 4.2 4.3 2 Isometries and perfect tensors 3 Construction of holographic quantum states and codes 4 Entanglement structure of holographic states Ryu-Takayanagi formula Bipartite entanglement of disconnected regions A map of multipartite entanglement 4.4 Negative tripartite information 5 Quantum error correction in holographic codes AdS-Rindler reconstruction as error correction The physical interpretation of holographic codes Bulk reconstruction from tensor pushing Connected reconstruction and the causal wedge Disconnected reconstruction and the entanglement wedge Holographic stabilizer codes 5.8 Are local gauge constraints enough? 6 Black holes and holography 7 Open problems and outlook A Perfect tensor examples A.1 5-qubit code and 6-qubit state A.2 3 -qutrit code and 4-qutrit state B Proof of RT for negatively curved planar graphs C Counting tensors in the pentagon code C.1 Counting tensors C.2 Connected reconstruction D.1 Analytic bounds D.2 Numerical evaluation D Estimating greedy erasure thresholds Reconstruction from symmetry guarantees E.2 Approximate reconstruction for typical tensors The AdS/CFT correspondence, an exact duality between quantum gravity on a (d+1)dimensional asymptotically-AdS space and a d-dimensional CFT defined on its boundary, has significantly advanced our understanding of quantum gravity, as well as provided a powerful framework for studying strongly-coupled quantum field theories. One aspect of this duality is a remarkable relationship between geometry and entanglement. This notion first appeared in the proposal [2] that two entangled CFT’s have a bulk dual connecting them through a wormhole, and was later quantified by Ryu and Takayanagi via their proposal that entanglement entropy in the CFT is computed by the area of a certain minimal surface in the bulk geometry [3, 4]. This latter proposal, known as the RyuTakayanagi (RT) formula, has led to much further work on sharpening the connection between geometry and entanglement [5–12] In the condensed matter physics community, improved understanding of quantum entanglement has led to significant progress in the numerical simulation of emergent phenomena in strongly-interacting systems. A key ingredient of such algorithms is the use of tensor networks to efficiently represent quantum many-body states [13–15]. Vidal combined this idea with entanglement renormalization to formulate the Multiscale Entanglement Renormalization Ansatz (MERA) [16, 17], a family of tensor networks that efficiently approximate wave functions with long-range entanglement of the type exhibited by ground states of local scale-invariant Hamiltonians [18–20]. The key idea is to represent entanglement at different length scales using tensors in a hierarchical array. In the AdS/CFT correspondence, the emergent radial direction can be regarded as a renormalization scale [21], and spatial slices have a hyperbolic geometry resembling the exponentially growing tensor networks of MERA. This similarity between AdS/CFT and MERA was pointed out by Swingle, who argued that some physics of the AdS/CFT correspondence can be modeled by a MERA-like tensor network where quantum entanglement in the boundary theory is regarded as a building block for the emergent bulk geometry [22, 23]. Recently it has been argued in [1] that the emergence of bulk locality in AdS/CFT can be usefully characterized in the language of quantum error-correcting codes. Certain paradoxical features of (...truncated)


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Fernando Pastawski, Beni Yoshida, Daniel Harlow. Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence, Journal of High Energy Physics, 2015, pp. 149, Volume 2015, Issue 6, DOI: 10.1007/JHEP06(2015)149