Canonical Divergence for Flat α-Connections: Classical and Quantum

Abstract

A recent canonical divergence, which is introduced on a smooth manifold M endowed with a general dualistic structure (g, ∇, ∇ * ), is considered for flat α-connections. In the classical setting, we compute such a canonical divergence on the manifold of positive measures and prove that it coincides with the classical α-divergence. In the quantum framework, the recent canonical divergence is evaluated for the quantum α-connections on the manifold of all positive definite Hermitian operators. Also in this case we obtain that the recent canonical divergence is the quantum α-divergence.