  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.
