Locality of Edge States and Entanglement Spectrum from Strong Subadditivity

TL;DR

This paper demonstrates that for two-dimensional states with an area law, the topological entanglement entropy relates to the minimal relative entropy to local thermal states, revealing boundary locality and entanglement spectrum properties.

Contribution

It establishes a connection between topological entanglement entropy and local thermal states using strong subadditivity, providing insights into boundary locality and entanglement spectrum.

Findings

Topological entanglement entropy equals the minimal relative entropy distance to local thermal states.

States with zero topological entanglement entropy are boundary thermal states of local Hamiltonians.

Entanglement spectrum matches the spectrum of a one-dimensional local thermal state on the boundary.

Abstract

We consider two-dimensional states of matter satisfying an uniform area law for entanglement. We show that the topological entanglement entropy is equal to the minimum relative entropy distance from the reduced state to the set of thermal states of local models. The argument is based on strong subadditivity of quantum entropy. For states with zero topological entanglement entropy, in particular, the formula gives locality of the states at the boundary of a region as thermal states of local Hamiltonians. It also implies that the entanglement spectrum of a two-dimensional region is equal to the spectrum of a one-dimensional local thermal state on the boundary of the region.