Duality in quantum information manifolds

Abstract

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.