Single-Shot Holographic Compression from the Area Law

H. Wilming and J. Eisert
Phys. Rev. Lett. 122, 190501 – Published 15 May 2019
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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 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 ϵ by a distribution with support of size exp(S/ϵ), 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.

  • Figure
  • Received 12 October 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.190501

© 2019 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & Technology

Authors & Affiliations

H. Wilming1 and J. Eisert2,3

  • 1Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
  • 2Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, 14195 Berlin, Germany
  • 3Department of Mathematics and Computer Science, Freie Universität Berlin, 14195 Berlin, Germany

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Issue

Vol. 122, Iss. 19 — 17 May 2019

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