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

TL;DR

This paper introduces exactly solvable tensor network models that mimic key features of the AdS/CFT correspondence, demonstrating how quantum error correction underpins the bulk/boundary relationship in holography.

Contribution

It presents a novel class of tensor network-based toy models for holography that explicitly realize quantum error correction in the AdS/CFT framework.

Findings

Models obey the Ryu-Takayanagi formula

Tripartite information negativity is captured

Bulk logical operators are reconstructible on boundary regions

Abstract

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 Rindler-wedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed by Almheiri et. al in arXiv:1411.7041.