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In this paper, we analyze the evolution of the generalized Pauli channelsgoverned by the memory kernel master equation. We provide necessary andsufficient conditions for the memory kernel to give rise to the legitimate(completely positive and trace-preserving) quantum evolution. In particular, weanalyze a class of kernels generating the quantum semi-Markov evolution, whichis a natural generalization of the Markovian semigroup. Interestingly, theconvex combination of Markovian semigroups goes beyond the semi-Markov case.Our analysis is illustrated with several examples.

Memory kernel approach to generalized Pauli channels: Markovian, semi-Markov, and beyond