Capacities of the covariant Pauli channel

We study four well-known capacities of a two-parameter family qubit Pauli channels. These are the channels that covariant under $\text{SO}(2)$ group and contain depolarizing channel as special case. find exact expressions for classical capacity entanglement-assisted capacities, analytically determine regions where quantum vanishes. then use flag extension to upper bound private these in entire region parameter space also obtain lower by calculating single-shot numerically. In conjunction with previous results on channels, our result is one step forward determining full channel.