Single-shot holographic compression from the area law
arXiv:1809.10156 · doi:10.1103/PhysRevLett.122.190501
Abstract
The area law conjecture states that the entanglement entropy of a region of space in the ground state of a gapped, local Hamiltonian only grows like the surface area of the region. We show that, for any quantum state that fulfills an area law, the reduced quantum state of a region of space can be unitarily compressed into a thickened surface of the region. If the interior of the region is lost after this compression, the full quantum state can be recovered to high precision by a quantum channel only acting on the thickened surface. The thickness of the boundary scales inversely proportional to the error for arbitrary spin systems and logarithmically with the error for quasi-free bosonic systems. Our results can be interpreted as a single-shot operational interpretation of the area law. The result for spin systems follows from a simple inequality showing that probability distributions with low entropy can be approximated by distributions with small support, which we believe to be of independent interest. We also discuss an emergent approximate correspondence between bulk and boundary operators and the relation of our results to tensor network states.
arXiv comment4+5 pages, 1 figure; v2: improved presentation, removed erroneous lemma from appendix (results unchanged), close to published version
References in corpus (38)
- Holographic Derivation of Entanglement Entropy from AdS/CFT
- Gaussian Quantum Information
- Entanglement Entropy and Quantum Field Theory
- Concentrating Partial Entanglement by Local Operations
- Area laws for the entanglement entropy - a review
- A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
- Entropy and Area
- Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
- Geometric and Renormalized Entropy in Conformal Field Theory
- The holographic principle
- Average Entropy of a Subsystem
- Entanglement renormalization
- An Area Law for One Dimensional Quantum Systems
- Entanglement Renormalization and Holography
- Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
- Bulk Locality and Quantum Error Correction in AdS/CFT
- Notes on Some Entanglement Properties of Quantum Field Theory
- Holographic duality from random tensor networks
- Violation of the entropic area law for Fermions
- Entropy scaling and simulability by Matrix Product States
- Entropy, entanglement, and area: analytical results for harmonic lattice systems
- The Ryu-Takayanagi Formula from Quantum Error Correction
- An entanglement-area law for general bosonic harmonic lattice systems
- Entanglement Entropy and the Fermi Surface
- An area law for entanglement from exponential decay of correlations
- An area law for the entropy of low-energy states
- Single-copy entanglement in critical spin chains
- Bidirectional holographic codes and sub-AdS locality
- Correlations, spectral gap, and entanglement in harmonic quantum systems on generic lattices
- Statistics dependence of the entanglement entropy
- Entanglement renormalization and wavelets
- Area law and its violation: A microscopic inspection into the structure of entanglement and fluctuations
- On symplectic eigenvalues of positive definite matrices
- Code properties from holographic geometries
- Gaussian quantum marginal problem
- Exact holographic mapping in free fermion systems
- Real-space renormalization yields finite correlations
- Entanglement area law from specific heat capacity