Canonical divergence for flat $\alpha$-connections: Classical and Quantum

A recently introduced canonical divergence for a dualistic structure $(\mathrm{g},\nabla,\nabla^*)$ on a smooth manifold $\mathrm{M}$ is considered for flat $\alpha$-connections. In the classical setting, we compute the recent canonical divergence on the manifold of positive measures and prove that it coincides with the classical $\alpha$-divergence. In the quantum framework, such a divergence is evaluated for the quantum $\alpha$-connections on the manifold of positive definite matrices. Also, in this case we obtain that the recent canonical divergence is the quantum $\alpha$-divergence.