Only Classical Parameterized States have Optimal Measurements

When estimating parameters encoded in quantum systems, it is natural to ask if there is an optimal quantum-measurement strategy. This question has been extensively studied when one has an a priori belief about the parameter (the Bayesian regime) or in the limit of infinitely many copies of a state (the asymptotic regime). In this talk, we address this question in a more experimentally realistic scenario. We assume that an experimenter has only finite resources and has no prior knowledge of the unknown parameter. We directly compare measurements without reference to an entropic quantity (e.g the Fisher information) and completely characterise when optimal measurements exist: there is an optimal measurement if and only if the parameterised state is classical. Furthermore, we discuss how one can consider approximately optimal measurements when a strictly optimal measurement does not exist.