Cooling a quantum system to its ground state is important for the characterization of nontrivial interacting systems and in the context of a variety of quantum information platforms. It can be achieved by employing measurement-based passive steering protocols, where the steering steps are predetermined and are not based on measurement readouts. However, measurements realized by coupling the system to auxiliary quantum degrees of freedom (“detectors”) are rather costly and protocols in which the number of detectors scales with system size will have limited practical applicability. Here, we identify conditions under which measurement-based cooling protocols can be taken to the ultimate dilute limit where the number of detectors is independent of system size. For two examples of frustration-free one-dimensional spin chains, we show that steering on a single link is sufficient to cool these systems into their unique ground states. We corroborate our analytical arguments with finite-size numerical simulations and discuss further applications of dilute cooling.
