  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.
