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

A recent canonical divergence, which is introduced on a smooth manifold $mathrm{M}$ endowed with a general dualistic structure $(mathrm{g}, abla, abla^*)$, is considered for flat $alpha$-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 $alpha$-divergence. In the quantum framework, the recent canonical divergence is evaluated for the quantum $alpha$-connections on the manifold of all positive definite Hermitian operators. Also in this case we obtain that the recent canonical divergence is the quantum $alpha$-divergence.