The exponential Orlicz space in quantum information geometry

Abstract

We review the construction of a quantum version of the exponential statistical manifold over the set of all faithful normal positive functionals on a von Neumann algebra. The construction is based on the relative entropy approach to state perturbation. We construct a quantum version of the exponential Orlicz space and discuss the properties of this space and its dual with respect to Kosaki $L_p$-spaces. We show that the constructed manifold admits a canonical divergence satisfying a Pythagorean relation. We also prove that the manifold structure is invariant under sufficient channels.

18 pages,to be published in the special issue Half a Century of Information Geometery of the journal Information Geometry

References in corpus (2)

• Rényi relative entropies and noncommutative $L_p$-spaces
• Rényi relative entropies and noncommutative $L_p$-spaces II