Geometry from divergence functions and complex structures

Abstract

Motivated by the geometrical structures of quantum mechanics, we introduce an almost complex structure [Formula: see text] on the product [Formula: see text] of any parallelizable statistical manifold [Formula: see text]. Then, we use [Formula: see text] to extract a pre-symplectic form and a metric-like tensor on [Formula: see text] from a divergence function. These tensors may be pulled back to [Formula: see text], and we compute them in the case of an N-dimensional symplex with respect to the Kullback–Leibler relative entropy, and in the case of (a suitable unfolding space of) the manifold of faithful density operators with respect to the von Neumann–Umegaki relative entropy.