Two quantum sample spaces (projective decompositions) \(\{P_j\}\) and \(\{Q_k\}\) associated with the same Hilbert space are said to be compatible provided all the projectors in one decomposition commute with all the projectors in the other decomposition: \(P_jQ_k = Q_k P_j\) for all \(j\) and \(k\). Otherwise they are incompatible. Note that all projectors in the two event algebras commute if and only if this is true for the sample spaces, so it makes sense to talk about compatible and incompatible frameworks. When the frameworks are compatible one can combine them. In particular, the coarsest common refinement is given by the sample space or decomposition formed by all nonzero products of the form \(P_jQ_k\), and the corresponding event algebra contains all the projectors in the original algebras along with the additional ones constructed from their products.