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

We study the ground state of a gapped quantum many-body system whose entanglement entropy SA can be expressed as SA = a|∂A| − γ , where a,γ are some constants and |∂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 a 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.