Bounds for the divisibility-based and distinguishability-based non-Markovianity measures

We derive an upper bound for the distinguishability-based non-Markovianity measure of a two-level system and prove that for certain master equations the exact value of the measure achieves this bound. Furthermore, we obtain an easily calculable lower bound for the divisibility-based non-Markovianity measure of an $n$-level system. We illustrate the calculation of these bounds through examples, considering in detail the spin-boson model. We show that the differences between the two measures in the spin-boson model are caused by the drift vector that is also responsible for the nonunitality of the dynamical map.

Figure 1

(a) The distinguishability measure for the spin-boson model (thick blue line), its upper bound given by Eq. (54) (thin red line), and the approximate value of the bound shown in Eq. (56) (thin green dashed line). (b) The modified divisibility measure defined in Eq. (61) (thick blue line) and its lower bound given in Eq. (62) (thin red line). Note that in both figures the scale of the vertical axis is logarithmic. The strength of the system-environment interaction is $\alpha /{\omega }_A=0.01$.