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We investigate the relation between two approaches to the characterisation ofquantum Markovianity, divisibility and lack of information backflow. We showthat a bijective dynamical map is completely-positive-divisible if and only ifa monotonic non-increase of distinguishability is observed for two equiprobablestates of the evolving system and an ancilla. Moreover our proof isconstructive: given any such map that is not completely-positive-divisible, wegive an explicit construction of two states that, when taken with the same apriori probability, exhibit information back-flow. Finally, while an ancilla isnecessary for the equivalence to hold in general, we show that it is alwayspossible to witness the non-Markovianity of bijective maps without using anyentanglement between system and ancilla.

Constructive method for detecting the information backflow of bijective non-completely-positive-divisible dynamics