We study Landauers Principle for Repeated Interaction Systems (RIS) consisting of a reference quantum system $mathcal{S}$ in contact with a structured environment $mathcal{E}$ made of a chain of independent quantum probes; $mathcal{S}$ interacts with each probe, for a fixed duration, in sequence. We first adapt Landauers lower bound, which relates the energy variation of the environment $mathcal{E}$ to a decrease of entropy of the system $mathcal{S}$ during the evolution, to the peculiar discrete time dynamics of RIS. Then we consider RIS with a structured environment $mathcal{E}$ displaying small variations of order $T^{-1}$ between the successive probes encountered by $mathcal{S}$, after $nsimeq T$ interactions, in keeping with adiabatic scaling. We establish a discrete time non-unitary adiabatic theorem to approximate the reduced dynamics of $mathcal{S}$ in this regime, in order to tackle the adiabatic limit of Landauers bound. We find that saturation of Landauers bound is equivalent to a detailed balance condition on the repeated interaction system, reflecting the non-equilibrium nature of the repeated interaction system dynamics. This is to be contrasted with the generic saturation of Landauers bound known to hold for continuous time evolution of an open quantum system interacting with a single thermal reservoir in the adiabatic regime.
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