On the informational completeness of local observables

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.