  The metric underlying the mixed state geometric phase in unitary and
nonunitary evolution [Phys. Rev. Lett. {\bf 85}, 2845 (2000); Phys. Rev. Lett.
{\bf 93}, 080405 (2004)] is delineated. An explicit form for the line element
is derived and shown to be related to an averaged energy dispersion in the case
of unitary evolution. The line element is measurable in interferometry
involving nearby internal states. Explicit geodesics are found in the single
qubit case. It is shown how the Bures line element can be obtained by extending
our approach to arbitrary decompositions of density operators. The proposed
metric is applied to a generic magnetic system in a thermal state.
