  We deal with the maximization of classical Fisher information in a quantum
system depending on an unknown parameter. This problem has been raised by
physicists, who defined [Helstrom (1967) Phys. Lett. A 25 101-102] a quantum
counterpart of classical Fisher information, which has been found to constitute
an upper bound for classical information itself [Braunstein and Caves (1994)
Phys. Rev. Lett. 72 3439-3443]. It has then become of relevant interest among
statisticians, who investigated the relations between classical and quantum
information and derived a condition for equality in the particular case of
two-dimensional pure state systems [Barndorff-Nielsen and Gill (2000) J. Phys.
A 33 4481-4490]. In this paper we show that this condition holds even in the
more general setting of two-dimensional mixed state systems. We also derive the
expression of the maximum Fisher information achievable and its relation with
that attainable in pure states.
