On the informational completeness of local observables

Understand

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.