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

TL;DR

The paper analyzes how non-Markovian memory effects can be witnessed either globally at the level of dynamical maps or locally for pairs of states. It clarifies the relationship between the BLP measure based on trace distance and the generalized GTD-based GBLP measure, highlighting that GTD optimally detects non-P-divisibility for the map as a whole but is not a tight or faithful witness for distinguishability or information backflow on a per-pair basis. For qubit dynamics, unital channels yield equivalence between BLP and GBLP and show no unital non-P-divisible BLP Markovian dynamics, while non-unital channels can exhibit NM detected by GBLP that BLP misses. The results reveal limitations of GTD as a universal pairwise indicator and motivate deeper investigation of the translation component in non-unital dynamics to better capture information flow and state distinguishability.

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.

Figures (3)

• Figure 1: generalized distance between states $\dfrac{1}{2}(\mathbb I + \sigma_z)$ and $\dfrac{1}{2}(\mathbb I + \sigma_x)$ for $p=0.25$ is plotted as a function of time for generalized amplitude damping noise (See Eq. \ref{['Eq: GAD']}). Noise parameters $\gamma$ and $f$ are 0.1 and 4, respectively. Distance and time are in arbitrary units.
• Figure 2: generalized distance is plotted as a function of time for different Bloch radii $r$ with two diametrically opposite points for an isometrically decaying dynamical map (Eq. \ref{['Eq: isometric']}). The decay parameter $\gamma$ and the probability of preparing states $p$ are 0.1 and 0.25, respectively. Distance and time are in arbitrary units.
• Figure 3: Generalized distance is plotted as a function of time for different pairs of diametrically opposite points on the surface of the Bloch sphere for a spin interacting with another spin (Eq. \ref{['Eq: Spin']}). The dynamical parameter $\omega$ and the probability of preparing states $p$ are 1.25 and 0.25, respectively. Distance and time are in arbitrary units.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof