The role of CP-indivisibility and incompatibility as valuable resources for
various information-theoretic tasks is widely acknowledged. This study delves
into the intricate relationship between CP-divisibility and channel
compatibility. Our investigation focuses on the behaviour of incompatibility
robustness of quantum channels for a pair of generic dynamical maps. We show
that the incompatibility robustness of channels is monotonically non-increasing
for a pair of generic CP-divisible dynamical maps. We have explicitly studied
the behaviour of incompatibility robustness with time for some specific
dynamical maps and found non-monotonic behaviour in the CP-indivisible regime.
Additionally, we propose a measure of CP-indivisibility based on the
incompatibility robustness of quantum channels. Our investigation provides
valuable insights into the nature of quantum dynamical maps and their relevance
in information-theoretic applications.Comment: 10 pages, 5 figure