We consider networks featuring preparation, transformation, and measurement
devices, in which devices exchange communication via mediating physical
systems. We investigate the problem of testing the dimension of the mediating
systems in the device-independent scenario, that is, based on observable data
alone. A general framework for tackling this problem is presented, considering
both classical and quantum systems. These methods can then also be used to
certify the non-classicality of the mediating systems, given an upper bound on
their dimension. Several case studies are reported, which illustrate the
relevance of the framework. These examples also show that, for fixed dimension,
quantum systems largely outperform classical ones. Moreover, the use of a
transformation device considerably improves noise tolerance when compared to
simple prepare-and-measure networks. These results suggest that the classical
simulation of quantum systems becomes costly in terms of dimension, even for
simple networks.Comment: 11 pages, published versio
