Markovian semigroup from mixing generalized Weyl dephasing channels

We study the conditions to produce Markovian semigroups under the convex combinations of various generalized Weyl dephasing channels. We show that the convex combinations of generalized Weyl dephasing channels corresponding to two classes of subgroups of $Z_d\times Z_d$ could yield Markovian semigroups. We analyze the (non)invertibility and (non-)Markovianity of generalized Weyl dephasing channels. Surprisingly, we find two results that are quite different from the known conclusions. One is that the noninvertibility and non-Markovianity of generalized Weyl dephasing channels are dependent on the parity of the dimension of the system. The other is that the invertibility and Markovianity of generalized Weyl dephasing channels are equivalent.