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A Markovian process of a system is defined classically as a process in whichthe future state of the system is fully determined by only its present state,not by its previous history. There have been several measures ofnon-Markovianity to quantify the degrees of non-Markovian effect in a processof an open quantum system based on information backflow from the environment tothe system. However, the condition for the witness of the system informationbackflow does not coincide with the classical definition of a Markovianprocess. Recently, a new measure with a condition that coincides with theclassical definition in the relevant limit has been proposed. Here, we focus onthe new definition (measure) for quantum non-Markovian processes, andcharacterize the Markovian condition as a quantum process that has noinformation backflow through the reduced environment state (IBTRES) and nosystem-environment correlation effect (SECE). The action of IBTRES producesnon-Markovian effects by flowing the information of quantum operationsperformed by an experimenter at earlier times back to the system through theenvironment, while the SECE can produce non-Markovian effect without carryingany earlier quantum operation information. We give the necessary and sufficientconditions for no IBTRES and no SECE, respectively, and show that a process isMarkovian if and only if it has no IBTRES and no SECE. The quantitativemeasures and algorithms for calculating non-Markovianity, IBTRES and soly-SECEare explicitly presented.

Non-Markovianity, information backflow and system-environment correlation for open-quantum-system processes