Relating Compatibility and Divisibility of Quantum Channels

We connect two key concepts in quantum information: compatibility and divisibility of channels. Two channels are compatible if they can be both obtained via marginalization from a third channel. A channel divides another it reproduces its action by sequential composition with (In)compatibility is central importance for studying the difference between classical dynamics. The relevance stands close relationship onset Markovianity. emphasize simulability character divisibility, and, despite their structural difference, we find set -- self-degradable which notions coincide. also show that, degradable channels, implies anti-degradable compatibility. These results motivate further research on these classes shed new light meaning largely studied notions.