A note on Fisher-Helstrom information inequality in pure state models

This paper concerns the design problem of choosing the measurement that provides the maximum Fisher information for the unknown parameter of a quantum system. We show that when the system under investigation is described by a one-parameter $n$-dimensional pure state model an optimal measurement exists, such that Fisher information attains the upper bound constituted by Helstrom information. A characterisation theorem and two strategies of implementations are derived and discussed. These results generalise to $n$-dimensional spaces those obtained for $n=2$ by Barndorff-Nielsen and Gill (2000).