Duality in quantum information manifolds

Open Systems and Information Dynamics, 11, 71-77, 2004.

ABSTRACT

We consider the information manifold M of density operators on a separable Hilbert space, that lie in some quasinormed Schatten class C(p), for some p less than one. We show that the neighbourhood of a point rho in M can be defined equivalently by states dominated by rho and which dominate rho, or by states such that iI>X(t) is analytic and bounded in the complex circle |t| less than 1/2, where X is the relative Hamiltonian and X(t) is the time evolution of X under the modular automorphism defined by rho. Such functions can be furnished with the norm given by the supremum of the operator norm of the analytic bounded function X(t). With this topology, M becomes a Banach manifold furnished with two flat connections, the quantum analogue of the extreme Amari connections. We show that the connections are dual relative to the Kubo-Mori metric.

REMARKS

I omit the (needed) proof that the norms in different but overlapping balls are equivalent. This proof, while elementary, is interesting, and will be published later.

The article can be found in the archives math-ph/0308037.