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

TL;DR

This paper critically examines the effectiveness of the generalized trace distance criterion as a witness of non-Markovianity, revealing its limitations in detecting distinguishability and information backflow in quantum dynamics.

Contribution

It identifies the shortcomings of the GTD criterion as a universal witness of non-Markovianity and clarifies when the standard trace distance suffices versus when the generalized measure is needed.

Findings

GTD criterion is not a tight witness of distinguishability for individual state pairs.

GTD can falsely indicate distinguishability for indistinguishable states.

For qubit unital dynamics, GTD offers no advantage over trace distance.

Abstract

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. Nevertheless, we point out that the criterion is not a tight witness of distinguishability when applied to individual pairs of states; specifically, there can exist a pair of states whose distinguishability manifests in an increase, but the GTD criterion fails to indicate this. Note that this is beside the fact that pairs of states can exist whose distinguishability doesn't evolve under a non-P-divisible map. Worse, the GTD criterion is not a faithful witness in that manifestly indistinguishable states that are indicated to be GTD distinguishable can exist. In other words, the case of GTD shows that a witness of non-Markovianity is not necessarily a universally applicable witness of distinguishability or information backflow across the entire state space. 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.