Hamiltonian identifiability assisted by single-probe measurement

We study the Hamiltonian identifiability of a many-body spin-1/2 system assisted by the measure- ment on a single quantum probe based on the eigensystem realization algorithm (ERA) approach employed in [Phys. Rev. Lett. 113, 080401 (2014)]. We demonstrate a potential application of Gr\"obner basis to the identifiability test of the Hamiltonian, and provide the necessary experimental resources, such as the lower bound in the number of the required measurement points, the upper bound in total required evolution time, and thus the total measurement time. Focusing on the examples of the identifiability in the spin chain model with nearest-neighboring interaction, we classify the spin-chain Hamiltonian based on its identifiability, and provide the control protocols to engineer the non-identifiable Hamiltonian to be an identifiable Hamiltonian.