  In this paper we reconsider the single parameter quantum Fisher information
(QFI) and compare it with the two-parameter one. We find simple relations
connecting the single parameter QFI (both in the asymmetric and symmetric phase
shift cases) to the two parameter Fisher matrix coefficients. Following some
clarifications about the role of an external phase [Phys. Rev. A 85, 011801(R)
(2012)], the single-parameter QFI and its over-optimistic predictions have been
disregarded in the literature. We show in this paper that both the single- and
two-parameter QFI have physical meaning and their predicted quantum
Cram\'er-Rao bounds are often attainable with the appropriate experimental
setup. Moreover, we give practical situations of interest in quantum metrology,
where the phase sensitivities of a number of input states approach the quantum
Cram\'er-Rao bound induced by the single-parameter QFI, outperforming the
two-parameter QFI.
