  Entropy production and the detailed fluctuation theorem are of fundamental
importance for thermodynamic processes. In this paper, we study the multiple
entropy production for multitime quantum processes in a unified framework. For
closed quantum systems and Markovian open quantum systems, the given entropy
productions all satisfy the detailed fluctuation relation. This also shows that
the entropy production rate under these processes is non-negative. For
non-Markovian open quantum systems, the memory effect can lead to a negative
entropy production rate. Thus, in general, the entropy production of the
marginal distribution does not satisfy the detailed FT relation. Our framework
can be applied to a wide range of physical systems and dynamics. It provides a
systematic tool for studying entropy production and its rate under arbitrary
quantum processes.
