Resourcefulness vs indivisibility in quantum channels

Open quantum dynamics can be categorized in several ways, including according to their divisibility. For any quantum channel acting for any finite time period, we propose measures of P-indivisibility and CP-indivisibility, where P and CP stand for positivity and complete positivity respectively. Subsequently, we also propose two quantities to measure the resourcefulness - with respect to an arbitrary quantum resource - of any quantum channel within any finite time interval. Moreover, we find a bridge between these two classes of metrics, viz. those quantifying divisibilities of quantum channels and those gauging their resourcefulness, by identifying two separate relations between elements of one class with those of the other. Lastly, we verify the two relations using quantum non-Markovianity as a channel resource.
Comment: 7 pages, 3 figures