Hierarchy of non-Markovianity and k-divisibility phase diagram of quantum processes in open systems
Hong-Bin Chen1*, Jiun-Yi Lien1, Guang-Yin Chen2, Yueh-Nan Chen1,3
1Department of Physics, National Cheng Kung University, Tainan, Taiwan
2Department of Physics, National Chung Hsing University, Taichung, Taiwan
3Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan
* presenting author:Hong-Bin Chen, email:hongbinchen@phys.ncku.edu.tw
In recent years, many efforts have been devoted to the construction of a proper measure of quantum non-Markovianity. However, those proposed measures are shown to be at variance in different situations. In this work, we utilize the theory of k-positive maps to generalize a hierarchy of k divisibility and develop a powerful tool, called k-divisibility phase diagram, to show the landscape of non-Markovianity. This allows one to study different measures in a unified framework and provides a deeper insight into the nature of quantum non-Markovianity. By exploring the phase diagram with several paradigms, we can explain the origin of the discrepancy between three frequently used measures and find the condition under which these measures coincide with each other.

Keywords: complete positivity, k-positivity, k-divisibility, non-Markovianity, quantum process