Bures geometry and related concepts

The closely related concepts Bures metric, Bures distance, and fidelity (transition probability) can be defined equally well as a ``functor'' within the hierarchy of quantum systems or intrinsically within any quantum system. Some of these possibilities will be discussed in connection with inequalities, the use of ``amplitudes'', and symmetries. We further point to a parallel transport and its natural gauge theory which is related to the Bures geometry.