Collapse dynamics are diffusive

Non-interferometric experiments have been successfully employed to constrain models of spontaneous wave function collapse, which predict a violation of the quantum superposition principle for large systems. These experiments are grounded on the fact that, according to these models, the dynamics is driven by a noise that, besides collapsing the wave function in space, generates a diffusive motion with characteristic signatures, which, though small, can be tested. The non-interferometric approach might seem applicable only to those models which implement the collapse through a noisy dynamics, not to any model, which collapses the wave function in space. Here we show that this is not the case: under reasonable assumptions, any collapse dynamics (in space) is diffusive. Specifically, we prove that any space-translation invariant dynamics which complies with the no-signaling constraint, if collapsing the wave function in space, must change the average momentum of the system, and/or its spread.