Greenberger-Horne-Zeilinger state is the best probe for multiparameter estimation of independent local fields

We show that estimation of multiple independent field strengths of a local Hamiltonian, i.e., one formed by a sum of single-party terms, necessitates the utility of a highly symmetric genuine multiparty entangled state, viz. the Greenberger-Horne-Zeilinger (GHZ) state, as the input probe. This feature however depends on the choice of the weight matrix considered, to define a figure of merit in the multiparameter estimation. We obtain this result by providing a lower bound on the precision of multiparameter estimation, optimized over input probes, for an arbitrary weight matrix. We find that the bound can be expressed in terms of eigenvalues of the arbitrary multiparty local encoding Hamiltonian. We show that there exists a weight matrix for which this bound is attainable only when the probe is the GHZ state, up to a relative phase. In particular, no pure product state can achieve this lower bound. Indeed, the gap - in precision - between genuinely multiparty entangled and product states acting as probes, increases with increasing number of parties. Finally we also prove that using a probe that is in a mixed state provides a precision lower than that for the GHZ state. To emphasize the importance of the weight matrix considered, we also prove that the choice of identity operator as the same - thereby ignoring the covariances in the precision matrix - does not require the use of genuine multiparty entanglement in input probes for attaining the best precision, and the optimal probe can be a pure product.
Comment: 11 pages