Optimal Distillation of Non-Markovianity: Bounds, Multi-Copy Gain, and the Weak-to-Essential Transition

Abstract

Quantum channels generally reduce the distinguishability of quantum states, limiting information transmission and processing. Previous work introduced a protocol capable of increasing the distinguishability of states after the action of a specific quantum channel. Here we show how to systematically determine the maximal distinguishability gain achievable by this method. We develop an algorithm that identifies the optimal implementation of the protocol and applies to arbitrary quantum channels in a straightforward manner. Using this approach, we demonstrate that a weakly non-Markovian channel can effectively be converted into an essentially non-Markovian one through a distillation-like process. We further analyze the quantitative features of the optimized protocol, characterizing the conditions under which the enhancement is most pronounced. Our results provide a general framework to assess and optimize distinguishability recovery in open quantum systems.

Figures (6)

• Figure 1: Schematic representation of the non-Markovianity distillation protocol. AZEVEDO
• Figure 2: Change in distinguishability as a function of the parameter $\varepsilon$, comparing the original (undistilled) dynamics with the optimized distilled dynamics. The latter exhibits a positive $\Delta D'_n$ for $n=2$ and $n=3$, signaling a transition from weak to essential non-Markovianity. As well as increase in change of distinguishability with the increase of number of copies
• Figure 3: Dependence of the General Bound $\|X-Y\|_1$ on the number of copies $n$. In saturating pairs, increasing $n$ does not significantly improve the bound, while for non-saturating states, the bound increases notably from $n=2$ to $n=3$, indicating that collective processing can amplify distinguishability. Larger $n$ becomes computationally costly without further substantial improvement.
• Figure 4: Comparison between the distilled and undistilled change in distinguishability and its corresponding bounds for sampled values of $\varepsilon$, considering two copies of the channel. saturating states fail to certify the regime change in the weakly non-Markovian region. Moreover, in the strongly non-Markovian region, the initial change in distinguishability already saturates the general bound, leaving no room for further improvement.
• Figure 5: Comparison of the effect of changing the number of copies from $n=2$ to $n=3$ for both choices of states for sampled values of $\varepsilon$. Saturating states now witness a change in the regime of non-Markovianity at three-copy level, even though in the region $\varepsilon \in[0.25,\ 0.5]$ the increase in change of distinguishability from $n=2$ is quite small, in contrast to what is observed for the non-saturating states.

Theorems & Definitions (17)

• Definition 2.1: Markovian dynamics
• Definition 2.2: Weakly non-Markovian dynamics
• Theorem 3.1
• Theorem 3.2: Stinespring-type dilation Nadja-proof
• Lemma A.1: Norm duality watroustheory
• Theorem A.2
• proof
• Corollary A.3
• proof
• Lemma A.4