  We use the relative modular operator to define a generalized relative entropy
for any convex operator function g on the positive real line satisfying g(1) =
0. We show that these convex operator functions can be partitioned into convex
subsets each of which defines a unique symmetrized relative entropy, a unique
family (parameterized by density matrices) of continuous monotone Riemannian
metrics, a unique geodesic distance on the space of density matrices, and a
unique monotone operator function satisfying certain symmetry and normalization
conditions. We describe these objects explicitly in several important special
cases, including the familiar logarithmic relative entropy. The relative
entropies, Riemannian metrics, and geodesic distances obtained by our procedure
all contract under completely positive, trace-preserving maps. We then define
and study the maximal contraction associated with these quantities.
