A PDF file should load here. If you do not see its contents
the file may be temporarily unavailable at the journal website
or you do not have a PDF plug-in installed and enabled in your browser.
Alternatively, you can download the file locally and open with any standalone PDF reader:
https://link.springer.com/content/pdf/10.1007%2FJHEP06%282015%29149.pdf
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)