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We formulate multiparameter quantum estimation in the parametric andsemiparametric setting. While the Holevo Cram\'er-Rao bound (CRB) requires nosubstantial modifications in moving from the former to the latter, wegeneralize the Helstrom CRB appropriately. We show that the Holevo CRB cannotbe greater than twice the generalized Helstrom CRB. We also present a tighter,intermediate, bound. Finally, we show that for parameters encoded in the firstmoments of a Gaussian state there always exists a Gaussian measurement thatgives a classical Fisher information matrix that is one-half of the quantumFisher information matrix.

Upper bounds on the Holevo Cram\'er-Rao bound for multiparameter quantum parametric and semiparametric estimation