Operational Characterization of Divisibility of Dynamical Maps

In this work, we show the operational characterization to the divisibility of dynamical maps in terms of the distinguishability of quantum channels. It is proven that the distinguishability of any pair of quantum channels does not increase under divisible maps, in which the full hierarchy of divisibility is isomorphic to the structure of entanglement between system and environment. This shows that (i) channel distinguishability is the operational quantity signifying (detecting) divisibility (indivisibility) of dynamical maps and (ii) the decision problem for the divisibility of maps is as hard as the separability problem in entanglement theory. We also provide the information-theoretic characterization to the divisibility of maps with conditional min-entropy.