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Non-Markovian Toy Quantum Chain

We propose a simple structure for stationary non-Markovian quantum chains in the framework of collisional dynamics of open quantum systems. To this end, we modify the microscopic Markovian system--reservoir model, consider multiple collisions with each of the molecules with an overlap between the collisional time intervals. We show how the equivalent Markovian quantum chain can be constructed with the addition of satellite quantum memory to the system. We distinguish quantum from classical non-Markovianity. Moreover, we define the counts of non-Markovianity by the required number of satellite qubits and bits, respectively. As the particular measure of quantum non-Markovianity the discord of the satellite w.r.t. the system is suggested. Simplest qubit realizations are discussed, and the significance for real system--environment dynamics is also pointed out.