TY - JOUR

T1 - Single-Shot Holographic Compression from the Area Law

AU - Wilming, H.

AU - Eisert, J.

N1 - Funding Information: H. W. would like to thank Renato Renner and Joseph M. Renes for interesting discussions regarding Lemma 2. J. E. thanks the ERC (TAQ), the DFG (CRC 183 project B01, EI 519/14-1, EI 519/15-1, EI 519/7-1), and the Templeton Foundation for funding. This work has also received funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 817482 (PASQuanS). H. W. further acknowledges support from the Swiss National Science Foundation through SNSF Project No. 200020_165843 and through the National Center of Competence in Research Quantum Science and Technology (QSIT). This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research and Innovation.

PY - 2019/5/15

Y1 - 2019/5/15

N2 - 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 state that fulfills an area law, the reduced quantum state of a region of space can be unitarily compressed into a thickened boundary 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 boundary. The thickness of the boundary scales inversely proportional to the error for arbitrary spin systems and logarithmically with the error for quasifree 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 any probability distribution with entropy S can be approximated to error Ïby a distribution with support of size exp(S/Ï), 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.

AB - 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 state that fulfills an area law, the reduced quantum state of a region of space can be unitarily compressed into a thickened boundary 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 boundary. The thickness of the boundary scales inversely proportional to the error for arbitrary spin systems and logarithmically with the error for quasifree 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 any probability distribution with entropy S can be approximated to error Ïby a distribution with support of size exp(S/Ï), 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.

UR - http://www.scopus.com/inward/record.url?scp=85065819624&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.122.190501

DO - 10.1103/PhysRevLett.122.190501

M3 - Article

C2 - 31144922

AN - SCOPUS:85065819624

VL - 122

JO - Physical review letters

JF - Physical review letters

SN - 0031-9007

IS - 19

M1 - 190501

ER -