Shrinking of Operators in Quantum Error Correction and AdS/CFT
arXiv:1906.05501 · doi:10.1007/JHEP12(2019)128
Abstract
We first show that a class of operators acting on a given bipartite pure state on can shrink its supports on to only or while keeping its mappings. Using this result, we show how to systematically construct the decoders of the quantum error-correcting codes against erasure errors. The implications of the results for the operator dictionary in the AdS/CFT correspondence are also discussed. The "subalgebra code with complementary recovery" introduced in the recent work of Harlow is a quantum error-correcting code that shares many common features with the AdS/CFT correspondence. We consider it under the restriction of the bulk (logical) Hilbert space to a subspace that generally has no tensor factorization into subsystems. In this code, the central operators of the reconstructed algebra on the boundary subregion can emerge as a consequence of the restriction of the bulk Hilbert space. Finally, we show a theorem in this code which implies the validity of not only the entanglement wedge reconstruction but also its converse statement with the central operators.
arXiv comment40 pages, 1 figure, v2: published version
References in corpus (38)
- The Large N Limit of Superconformal Field Theories and Supergravity
- Holographic Derivation of Entanglement Entropy from AdS/CFT
- Quantum secret sharing
- A Covariant Holographic Entanglement Entropy Proposal
- Eternal Black Holes in AdS
- How to share a quantum secret
- Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime
- Quantum corrections to holographic entanglement entropy
- Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
- Bulk Locality and Quantum Error Correction in AdS/CFT
- Time Evolution of Entanglement Entropy from Black Hole Interiors
- Quantum data processing and error correction
- Notes on Some Entanglement Properties of Quantum Field Theory
- Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality
- Relative entropy equals bulk relative entropy
- Holographic representation of local bulk operators
- Holographic duality from random tensor networks
- Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy
- The Gravity Dual of a Density Matrix
- Causality & holographic entanglement entropy
- The Ryu-Takayanagi Formula from Quantum Error Correction
- Codes for the Quantum Erasure Channel
- Holographic Entanglement of Purification
- Remarks on entanglement entropy for gauge fields
- State-Dependent Bulk-Boundary Maps and Black Hole Complementarity
- Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture
- Entanglement of purification: from spin chains to holography
- Causal Holographic Information
- Aspects of Entanglement Entropy for Gauge Theories
- Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography
- Generalization of Quantum Error Correction via the Heisenberg Picture
- Scrambling time from local perturbations of the eternal BTZ black hole
- Bulk-Boundary Duality, Gauge Invariance, and Quantum Error Correction
- Quantum Error Correction of Observables
- Living on the Edge: A Toy Model for Holographic Reconstruction of Algebras with Centers
- Code properties from holographic geometries
- Towards Entanglement of Purification for Conformal Field Theories
- Entanglement with Centers