Single-shot holographic compression from the area law

TL;DR

This paper demonstrates that quantum states satisfying the area law can be efficiently compressed into a boundary surface, enabling high-precision recovery of the full state from the boundary alone, with implications for quantum information and tensor networks.

Contribution

It introduces a single-shot holographic compression scheme for area law states, linking entanglement entropy to boundary-based quantum state compression.

Findings

Quantum states with area law can be unitarily compressed into a boundary surface.

Recovery of the full state is possible from the boundary with high precision.

The boundary thickness scales inversely or logarithmically with the error depending on the system.

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.