Hierarchy of non-Markovianity and k -divisibility phase diagram of quantum processes in open systems

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.