Determining a local Hamiltonian from a single eigenstate

TL;DR

This paper demonstrates that a single eigenstate of a local Hamiltonian, along with two-point correlation functions, can uniquely determine the Hamiltonian in finite systems, with potential applications in quantum physics.

Contribution

It shows that local Hamiltonians can be reconstructed from a single eigenstate and its two-point correlations, introducing the concept of the '$k$-correlation spectrum' for analysis.

Findings

Reconstruction is possible for finite-size systems using polynomial-time algorithms.

Single eigenstates typically suffice to determine the Hamiltonian uniquely.

Numerical tests on 1D spin chains support the theoretical results.

Abstract

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.