Complete positivity of a spin- 1∕2 master equation with memory

We study a non-Markovian spin-1∕2 master equation with exponential memory. We derive the conditions under which the dynamical map describing the reduced system dynamics is completely positive, i.e., the nonunitary evolution of the system is compatible with a description in terms of a closed total spin-reservoir system. Our results show that for a zero-T reservoir, the dynamical map of the model here considered is never completely positive. For moderate- and high-T reservoirs, on the contrary, positivity is a necessary and sufficient condition for complete positivity. We also consider the Shabani-Lidar master equation recently introduced A. Shabani and D.A. Lidar Phys. Rev. A 71 020101 (2005) and we demonstrate that such a master equation is always completely positive.