  Motivated by the recent efforts to describe the gravitational interaction as
a classical channel arising from continuous quantum measurements, we study what
types of dynamics can emerge from a collisional model of repeated interactions
between a system and a set of ancillae. We show that contingent on the model
parameters the resulting dynamics ranges from exact unitarity to arbitrarily
fast decoherence (quantum Zeno effect). For a series of measurements the
effective dynamics includes feedback-control, which for a composite system
yields effective interactions between the subsystems. We quantify the amount of
decoherence accompanying such induced interactions, generalizing the lower
bound found for the gravitational example. However, by allowing multipartite
measurements, we show that interactions can be induced with arbitrarily low
decoherence. These results have implications for gravity-inspired decoherence
models. Moreover, we show how the framework can include terms beyond the usual
second-order approximation, which can spark new quantum control or simulation
protocols. Finally, within our simple approach we re-derive the quantum
filtering equations for the different regimes of effective dynamics, which can
facilitate new connections between different formulations of open systems.
