Exploring Many-body Interactions Through Quantum Fisher Information

The investigation of many-body interactions holds significant importance in both quantum foundations and information. Hamiltonians coupling multiple particles at once, beyond others, can lead to a faster entanglement generation, multiqubit gate implementation and improved error correction. As an increasing number of quantum platforms enable the realization of such physical settings, it becomes interesting to study the verification of many-body interaction resources. In this work, we explore the possibility of higher-order couplings detection through the quantum Fisher information. For a family of symmetric and translationally invariant $k$-body Ising-like Hamiltonians, we derive the bounds on the quantum Fisher information in product states. Due to its ordering with respect to the order of interaction, we demonstrate the possibility of detecting many-body couplings for a given Hamiltonian from the discussed family by observing violations of an appropriate bound. As a possible extension to these observations, we further analyse an example concerning the three-body interaction detection in the XY model.