  We introduce the notion of compatibility dimension for a set of quantum
measurements: it is the largest dimension of a Hilbert space on which the given
measurements are compatible. In the Schr\"odinger picture, this notion
corresponds to testing compatibility with ensembles of quantum states supported
on a subspace, using the incompatibility witnesses of Carmeli, Heinosaari, and
Toigo. We provide several bounds for the compatibility dimension, using
approximate quantum cloning or algebraic techniques inspired by quantum error
correction. We analyze in detail the case of two orthonormal bases, and, in
particular, that of mutually unbiased bases.
