Determining the structure of real-space entanglement spectrum from approximate conditional independence

TL;DR

This paper demonstrates that in gapped quantum systems with area-law entanglement entropy, the real-space entanglement spectrum exhibits a structure that is weakly correlated with local operators, indicating a form of approximate conditional independence.

Contribution

It introduces a framework linking the structure of the real-space entanglement spectrum to the area law and ground state properties using an operator extension of strong subadditivity.

Findings

Certain linear combinations of the entanglement spectrum have minimal correlation with local operators.

The entanglement spectrum structure is inherited from the ground state properties.

The locality of the entanglement spectrum can be explained by the area law of entanglement entropy.

Abstract

We study the ground state of a gapped quantum many-body system whose entanglement entropy $S_A$ can be expressed as $S_A = a|\partial A| - \gamma$, where $a, \gamma$ are some constants and $|\partial A|$ is an area of the subsystem $A$. By using a recently proved operator extension of strong subadditivity of entropy,[I. H. Kim, J. Math. Phys. 53, 122204 (2012)] we show that certain linear combination of the real-space entanglement spectrum has a small correlation with almost any local operator. Our result implies that there exists a structure relating the real-space entanglement spectrum over different subsystems. Further, this structure is inherited from the generic property of the ground state alone, suggesting that the locality of the entanglement spectrum may be attributed to the area law of entanglement entropy.