Measurements satisfying the quantum Cramer-Rao equality

The situation where the quantum Cramer-Rao inequality for a general measurement becomes equality is analyzed in some detail in the case of a family of pure states. In particular, it turns out that under some natural assumptions, the measurement in question is simple, and the states must have a special form. This fact in turn allows us to obtain in the two-dimensional case a characterization of the pure states for which the quantum Cramer-Rao equality holds.