On the distinction between distinguishability of states and witness of non-Markovianity of dynamical maps

Non-P-divisibility is the strongest divisibility-based notion of quantum non-Markovianity. The generalized trace distance (GTD) based criterion is known to be an optimal witness of non-P-divisibility of dynamical maps, in the sense that a given map is non-P-divisible if and only if there exists a pair of states that demonstrates increased distinguishability in the GTD sense. This observation forms the basis for associating an information backflow with this type of non-Markovianity. Here we argue that in contrast to the map-level witnessing of non-Markovianity via divisibility, the association of information flow with divisibility must be applicable to individual states or state pairs (in the trace-distance context). In the context of qubit dynamics, we show that this association is generally neither tight nor faithful. We demonstrate this by means of counter-examples: (a) a pair of states whose distinguishability manifestly increases, but the GTD criterion fails to indicate this. (b) manifestly indistinguishable states that are indicated to be GTD distinguishable. In other words, we point out a subtle distinction between indicating state-specific behavior in terms of information backflow or distinguishability and map-level witnessing of non-Markovianity based on the generalized trace distance (GTD). Furthermore, we demonstrate that for qubit unital dynamics, the GTD-based measure provides no advantage over the standard trace distance measure in witnessing non-Markovianity. We determine the class of qubit non-unital channels where the standard trace distance measure is insufficient and the generalized measure is necessary.