 One can show that common cause completeness of such quantum probability spaces can be characterized in terms of measure theoretic atomicity of \(({\cal P} ({\cal N}),\phi)\) in complete analogy with the classical case: \(({\cal P} ({\cal N}),\phi)\) is common cause complete if it contains at most one measure theoretic atom (Kitajima 2008; Gyenis &amp; Rédei 2014; Kitajima &amp; Rédei 2015). This entails that measure theoretically purely non-atomic quantum probability spaces are common cause complete. Specifically, quantum probability spaces \(({\cal P} ({\cal N}),\phi)\) with a type III or type II von Neumann algebra \({\cal N}\) are common cause complete because the quantum probability spaces defined by these types of von Neumann algebras are purely measure theoretically non-atomic. The quantum probability space \(({\cal P} ({\cal H}),\phi)\) is not purely nonatomic, it contains a lot of measure theoretic atoms: all the rank-one projections (equivalently: one dimensional linear spaces spanned by vectors in \({\cal H}\)) are atoms of the Hilbert lattice \({\cal P} ({\cal H})\), hence all rank-one projections that are assigned non-zero probability by state \(\phi\) are also measure theoretic atoms in \(({\cal P} ({\cal N}),\phi)\). Thus \(({\cal P} ({\cal H}),\phi)\) is not common cause complete. Whether common cause incomplete quantum probability spaces are common cause completable, is not known; results are available on common cause extendability of quantum probability spaces only: Every quantum probability space \(({\cal P} ({\cal N}),\phi)\) is embeddable into a larger quantum probability space \(({\cal P}({\cal N}'),\phi')\) in such a way that every correlation in \(({\cal P} ({\cal N}),\phi)\) has a common cause in \(({\cal P}({\cal N}'),\phi')\) (Hofer-Szabó, Rédei, &amp; Szabó 1999; 2013: 62, Proposition 6.3).