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An effective method of calculating the non-Markovianity $\mathcal{N}$ for single channel open systems

We propose an effective method which can simplify the optimization of the increase of the trace distance over all pairs of initial states in calculating the non-Markovianity $\mathcal{N}$ for single channel open systems. For the amplitude damping channel, we can unify the results of Breuer $et$ $al$. [Phys. Rev. Lett. \bf 103\rm, 210401 (2009)] in the large-detuning case and the results of Xu $et$ $al$. [Phys. Rev. A \bf 81\rm, 044105 (2010)] in the resonant case; furthermore, for the general off-resonant cases we can obtain a very tight lower bound of $\mathcal{N}$. As another application of our method, we also discuss $\mathcal{N}$ for the non-Markovian depolarizing channel.