We analyze the scaling of quantum Fisher information with the number of system particles in the limit of large number of particles, as a function of the number of parties interacting with each other, for encoding Hamiltonians having $k$-body interactions, where $k$ is arbitrary. We find that estimation of coupling strength of such arbitrary-body encoding Hamiltonians provides a super-Heisenberg scaling that increases monotonically with an increase in the number of interacting particles, in the limit of large number of system particles. Moreover, we ask if genuine multiparty entanglement is indispensable in attaining the best metrological precision if we employ nonlocal terms in the Hamiltonian. We identify a dichotomy in the answer. Specifically, we find that Hamiltonians having odd-body interactions necessarily require genuine multipartite entanglement in probes to attain the best metrological precision, but the situation is opposite in the case of Hamiltonians with even-body interactions. The optimal probes corresponding to Hamiltonians that contain even-body interaction terms may be entangled, but certainly not so in all bipartitions, and particularly it is possible to attain optimal precision using asymmetric probes. Asymmetry, which therefore is a resource in this scenario rather than genuine multiparty entanglement, refers to the disparity between states of local parts of the global system. Thereby we find a complementarity in the requirement of asymmetry and genuine entanglement in optimal probes for estimating strength of odd- and even-body interactions respectively. Additionally, we provide an upper bound on the number of parties up to which one can always obtain an asymmetric product state that gives the best metrological precision for even-body interactions. En route, we find the quantum Fisher information in closed form for two- and three-body interactions for arbitrary number of parties. Additionally, we analyze the finite-size scaling of the QFI and examine the impact of local dephasing noise during the encoding process. We also provide an analysis of the case when the Hamiltonian contains local fields and up to $k$-body interaction terms, where the strength of interaction gradually decreases with an increase in the number of parties interacting with each other. Interestingly, we find a similar dichotomy in the nature of the optimal probe in this case as well, i.e., for encoding Hamiltonians with up to even- and odd-body interactions. Further, we identify conditions on the local component of the Hamiltonian for which this dichotomy is still shown to exist for two- and three-body encoding Hamiltonians with arbitrary local dimensions.

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• Quantum communication, protocols & technology
• Quantum entanglement
• Quantum information theory
• Quantum metrology
• Quantum sensing