Distribution of non-Markovian intervals for open qubit systems

We study the non-Markovianity of open qubit systems using the measure \mathscrN proposed by Breuer, Laine and Piilo Phys. Rev. Lett. 103 210401 (2009). We find that for the three types of quantum noises, amplitude-damping, dephasing and depolarizing noises, there exist some non-Markovian time intervals whose distribution is independent of the selection of the pair of initial states. Therefore, the maximization in the definition of measure \mathscrN can be actually removed without influencing the detection of non-Markovianity.