  A typical bipartite quantum protocol, such as EPR-steering, relies on two
quantum features, entanglement of states and incompatibility of measurements.
Noise can delete both of these quantum features. In this work we study the
behavior of incompatibility under noisy quantum channels. The starting point
for our investigation is the observation that compatible measurements cannot
become incompatible by the action of any channel. We focus our attention to
channels which completely destroy the incompatibility of various relevant sets
of measurements. We call such channels incompatibility breaking, in analogy to
the concept of entanglement breaking channels. This notion is relevant
especially for the understanding of noise-robustness of the local measurement
resources for steering.
