  We consider the quantum information manifold whose underlying set M consists
of density operators rho with the extra property that some fractional power of
rho is of trace class. The topology is defined by defining a neighbourhood of a
point rho to be all density operators that dominate rho and are dominated by
rho. We show that this is the same set as that of all states whose relative
Hamiltonian X in the sense of Araki is bounded, and such that X(t) is
holomorphic in the circle |t| less than 1/2. Here, X(t) is the time evolution
of X determined by the modular automorphism defined by rho. We show that M is a
Banach manifold in Araki's norm, and that both the canonical and the mixture
affine connections can be defined. These are dual relative to the Kubo-Mori
metric, and so generalise Amari's dual theory to quantum theory in infinite
dimensions.
