  A recent canonical divergence, which is introduced on a smooth manifold
$\mathrm{M}$ endowed with a general dualistic structure
$(\mathrm{g},\nabla,\nabla^*)$, 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.
