What I am going to describe may be called an interplay between concepts of differential geometry and the superposition principle of quantum physics. In particular, it concerns a metrical distance introduced by Bures [14] as a non-commutative version of a construction of Kakutani [24] on the one hand, and on the other hand the purifications of mixed states in physically larger systems, including the problem of geometric phases associated with a distinguished class of such extensions. The Bures distance and the general transition probability [15], [27] are discussed in [lo], [ll], [30]: and further papers. For the sake of clarity, and to avoid technicalities, I will be concerned with finitedimensional objects. Let ‘Ft denote a Hilbert space with complex dimension n,. The set of density operators defined on it is