Non-Markovian dephasing and depolarizing channels

We introduce a method to construct non-Markovian variants of completely positive (CP) dynamical maps, particularly, qubit Pauli channels. We identify non-Markovianity with the breakdown in CP divisibility of the map, i.e., appearance of a not-completely positive intermediate map. In particular, we consider the case of non-Markovian dephasing in detail. The eigenvalues of the Choi matrix of the intermediate map crossover at a point which corresponds to a singularity in the canonical decoherence rate of the corresponding master equation and thus to a momentary noninvertibility of the map. Thereafter, the rate becomes negative, indicating non-Markovianity. We quantify the non-Markovianity by two methods, one based on CP divisibility Hall et al., Phys. Rev. A 89, 042120 (2014), which does not require optimization but requires normalization to handle the singularity, and another method, based on distinguishability Breuer et al. Phys. Rev. Lett. 103, 210401 (2009), which requires optimization but is insensitive to the singularity.