📝 SummarXiv

Abstract

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 arbitrary-body interactions. We find that estimation of coupling strength of such arbitrary-body encoding Hamiltonians provide 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 also find that 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. 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. 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. 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.