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.
"Maximum Fisher information in mixed state quantum systems." Ann. Statist. 32 (4) 1770 - 1779, August 2004. https://doi.org/10.1214/009053604000000436