  When observations must come from incompatible devices and cannot be produced
by compatible devices? This question motivates two integer valued
quantifications of incompatibility, called incompatibility dimension and
compatibility dimension. The first one quantifies how many states are minimally
needed to detect incompatibility if the test states are chosen carefully,
whereas the second one quantifies how many states one may have to use if they
are randomly chosen. With concrete examples we show that these quantities have
unexpected behaviour with respect to noise.
