Relating compatibility and divisibility of quantum channels

TL;DR

This paper explores the relationship between compatibility and divisibility of quantum channels, revealing conditions under which these concepts coincide or imply each other, thus advancing understanding of quantum dynamics and Markovianity.

Contribution

It establishes new links between compatibility and divisibility of quantum channels, identifying classes where they coincide or imply each other, and highlights their significance in quantum dynamics.

Findings

Compatibility and divisibility are connected through specific classes of channels.

For self-degradable channels, compatibility and divisibility are equivalent.

Degradable channels with compatibility are divisible; anti-degradable channels with divisibility are compatible.

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 motivate further research on these classes of channels and shed new light on the meaning of these two largely studied notions.