Relating Compatibility and Divisibility of Quantum Channels

Abstract

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

Experimental Communication Through Superposition of Quantum Channels

“…where any channel description ordered lower on the partial ordering can be extracted from a channel description ordered higher on the partial ordering, either by taking a subspace or post-selecting on the channel higher on the ordering. More broadly speaking, our partial ordering here is a specific kind of channel divisibility [20,21] that involves dimensional reduction. With this partial ordering in mind, we compare the maximum coherent information of our depolarizing channels under different channel models and implementations.…”