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We investigate the Weyl channels being covariant with respect to the maximumcommutative group of unitary operators. This class includes the quantumdepolarizing channel and the "two-Pauli" channel as well. Then, we show thatour estimation of the output entropy for a tensor product of the phase dampingchannel and the identity channel based upon the decreasing property of therelative entropy allows to prove the additivity conjecture for the minimaloutput entropy for the quantum depolarizing channel in any prime dimesnsion andfor the "two Pauli" channel in the qubit case.Comment: A completely revised version, 20 page

On Weyl channels being covariant with respect to the maximum commutative group of unitaries