Geometry of quantum states: dual connections and divergence functions

In a finite quantum state space with a monotone metric, a family of torsion-free affine connections is introduced, in analogy with the classical α-connections defined by Amari. The dual connections with respect to the metric are found and it is shown that they are, in general, not torsion-free. The torsion and the Riemannian curvature are computed and the existence of efficient estimators is treated. Finally, geodesics are used to define a divergence function.