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arXiv:2407.19252

Resourcefulness vs indivisibility in quantum channels

Priya Ghosh, Soumyajit Pal, Ujjwal Sen

Published Jul 27, 2024• 0 citations
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Abstract

Open quantum dynamics can be categorized in several ways, including according to their divisibility. For any quantum channel acting for any finite time period, we propose measures of P-indivisibility and CP-indivisibility, where P and CP stand for positivity and complete positivity respectively. Subsequently, we also propose two quantities to measure the resourcefulness - with respect to an arbitrary quantum resource - of any quantum channel within any finite time interval. Moreover, we find a bridge between these two classes of metrics, viz. those quantifying divisibilities of quantum channels and those gauging their resourcefulness, by identifying two separate relations between elements of one class with those of the other. Lastly, we verify the two relations using quantum non-Markovianity as a channel resource.

DOI
10.48550/arxiv.2407.19252

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Open quantum dynamics can be categorized in several ways, including according to their divisibility. For any quantum channel acting for any finite time period, we propose measures of P-indivisibility and CP-indivisibility, where P and CP stand for positivity and complete positivity respectively. Subsequently, we also propose two quantities to measure the resourcefulness - with respect to an arbitrary quantum resource - of any quantum channel within any finite time interval. Moreover, we find a bridge between these two classes of metrics, viz. those quantifying divisibilities of quantum channels and those gauging their resourcefulness, by identifying two separate relations between elements of one class with those of the other. Lastly, we verify the two relations using quantum non-Markovianity as a channel resource.
The role of completely positive--indivisibility (CP-indivisibility) and incompatibility as valuable resources for various information-theoretic tasks is widely acknowledged. This study delves into the intricate relationship between CP-divisibility and channel compatibility. Our investigation focuses on the behavior of incompatibility robustness of quantum channels for a pair of generic dynamical maps. We show that the incompatibility robustness of channels is monotonically nonincreasing for a pair of generic CP-divisible dynamical maps. Further, our explicit study of the behavior of incompatibility robustness with time for some specific dynamical maps reveals nonmonotonic behavior in the CP-indivisible regime. Additionally, we propose a measure of CP-indivisibility based on the incompatibility robustness of quantum channels. Our investigation provides valuable insights into the nature of quantum dynamical maps and their relevance in information-theoretic applications.
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We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘ indivisible ’ channels which can not be written as non-trivial products of other channels and study the set of ‘ infinitesimal divisible ’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.