  In recent times, there has been a growing scholarly focus on investigating
the intricacies of quantum channel mixing. It has been commonly believed, based
on intuition in the literature, that every generalized Pauli channel with
dimensionality $d$ could be represented as a convex combination of $(d+1)$
generalized Pauli dephasing channels (see [Phys. Rev. A 103, 022605 (2021)] as
a reference). To our surprise, our findings indicate the inaccuracy of this
intuitive perspective. This has stimulated our interest in exploring the
properties of convex combinations of generalized Pauli channels, beyond the
restriction to just $(d+1)$ generalized Pauli dephasing channels. We
demonstrate that many previously established properties still hold within this
broader context. For instance, any mixture of invertible generalized Pauli
channels retains its invertibility. It's worth noting that this property
doesn't hold when considering the Weyl channels setting. Additionally, we
demonstrate that every Pauli channel (for the case of $d=2$) can be represented
as a mixture of $(d+1)$ Pauli dephasing channels, but this generalization
doesn't apply to higher dimensions. This highlights a fundamental distinction
between qubit and general qudit cases. In contrast to prior understanding, we
show that non-invertibility of mixed channels is not a prerequisite for the
resulting mapping to constitute a Markovian semigroup.
