Canonical Divergence for Flat α-Connections: Classical and Quantum

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. In this case as well, we obtain that the recent canonical divergence is the quantum α -divergence.