Determining system Hamiltonian from eigenstate measurements without correlation functions. (arXiv:1903.06569v1 [quant-ph])

For a local Hamiltonian $H=\sum_i c_i A_i$, with $A_i$s being local
operators, it is known that $H$ could be encoded in a single (non-degenerate)
eigenstate $|\psi\rangle$ in certain cases. One case is that the system
satisfies the Eigenstate Thermalization Hypothesis (ETH), where the local
reduced density matrix asymptotically become equal to the thermal reduced
density matrix [PRX \textbf{8}, 021026 (2018)]. In this case, one can reproduce
$H$ (i.e. $c_i$s) from local measurement results
$\langle\psi|A_i|\psi\rangle=a_i$. Another case is that the two-point
correlation functions $\langle\psi|A_iA_j|\psi\rangle$ are known, one can
reproduce $H$ without satisfying ETH (arXiv: 1712.01850); however, in practice
nonlocal correlation functions $\langle\psi|A_iA_j|\psi\rangle$ are not easy to
obtain. In this work, we develop a method to determine $H$ (i.e., $c_i$s) with
local measurement results $a_i=\langle\psi|A_i|\psi\rangle$ and without the ETH
assumption, by reformulating the task as an unconstrained optimization problem
of certain target function of $c_i$s, with only polynomial number of parameters
in terms of system size when $A_i$s are local operators. Our method applies in
general cases for the known form of $A_i$s, and is tested numerically for both
randomly generated $A_is$ and also the case when $A_i$s are local operators.
Our result shed light on the fundamental question of how a single eigenstate
can encode the full system Hamiltonian, indicating a somewhat surprising answer
that only local measurements are enough without additional assumptions, for
generic cases.

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