  For a general multipartite quantum state, we formulate a locally checkable
condition, under which the expectation values of certain nonlocal observables
are completely determined by the expectation values of some local observables.
The condition is satisfied by ground states of gapped quantum many-body systems
in two spatial dimensions, assuming a widely conjectured form of area law is
correct. Its implications on the studies of gapped quantum many-body systems,
quantum state tomography, and quantum state verification are discussed. These
results are based on a partial characterization of states with small yet
nonzero conditional mutual information, which may be of independent interest.
