Quantum Fisher-Bures information of two-level systems and a three-level extension

Braunstein and Caves have recently demonstrated that the Bums metric on the mixed quantum states is equivalent--up to a proportionality factor of four--to the statistical distinguishability or quantum Fisher information metric. The volume element of these metrics can then--adapting a fundamental Bayesian principle of Jeffreys to the quantum context--serve as a reparametrization-invariant prior measure over the quantum states. The implications of this line of reasoning for the two-level systems, in general, and an embedding of them into a certain set of three-level systems are investigated.