The exponential Orlicz space in quantum information geometry

TL;DR

This paper develops a quantum exponential Orlicz space within quantum information geometry, establishing a manifold structure with divergence properties and invariance under certain channels, based on von Neumann algebra states.

Contribution

It introduces a quantum exponential Orlicz space and constructs a statistical manifold with divergence and invariance properties, advancing the mathematical framework of quantum information geometry.

Findings

Constructed a quantum exponential Orlicz space.

Established a manifold with a canonical divergence satisfying Pythagoras.

Proved invariance of the manifold under sufficient channels.

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.