One from many: Scalar estimation in a multiparameter context

Achievable sensitivity bounds are difficult to formulate for quantum multiparameter estimation. We consider a specialized case: many parameters of a Hamiltonian are unknown and one seeks an estimate for a scalar function of the Hamiltonian. This problem exhibits genuine multiparameter behavior, though it is superficially similar to single-parameter estimation. By uniting saturable single-parameter quantum bounds with geometric reasoning we prove the conditions, necessary and sufficient, for saturating the fundamental and attainable bound in this context.