  We ask whether the knowledge of a single eigenstate of a local Hamiltonian is
sufficient to uniquely determine the Hamiltonian. We present evidence that the
answer is "yes" for generic local Hamiltonians, given either the ground state
or an excited eigenstate. In fact, knowing only the two-point equal-time
correlation functions of local observables with respect to the eigenstate
should generically be sufficient to exactly recover the Hamiltonian for
finite-size systems, with numerical algorithms that run in a time that is
polynomial in the system size. We also investigate the large-system limit, the
sensitivity of the reconstruction to error, and the case when correlation
functions are only known for observables on a fixed sub-region. Numerical
demonstrations support the results for finite one-dimensional spin chains
(though caution must be taken when extrapolating to infinite-size systems in
higher dimensions). For the purpose of our analysis, we define the
"$k$-correlation spectrum" of a state, which reveals properties of local
correlations in the state and may be of independent interest.
