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 (NCP) 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 non-invertibility 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., PRA 89, 042120, 2014), which doesn't require optimization but requires normalization to handle the singularity, and another method, based on distinguishability (Breuer et al. PRL 103, 210401, 2009), which requires optimization but is insensitive to the singularity.