 In quantum theory one uses non-classical (quantum) probability spaces to describe quantum physical systems. A general quantum probability space is a pair \(({\cal P} ({\cal N}),\phi)\), where \({\cal P} ({\cal N})\) is the orthocomplemented, orthomodular lattice of projections of a non-commutative von Neumann algebra \({\cal N}\) and \(\phi\) is a countably additive probability measure on \({\cal P} ({\cal N})\), which is the restriction to \({\cal P} ({\cal N})\) of a normal state on \({\cal N}\). Rédei &amp; Summers (2007a) provides a brief description of non-commutative probability theory in terms of von Neumann algebras; Landsman (2017) contains an encyclopedic treatment of the algebraic structures relevant for quantum theory; the SEP entry on  quantum theory and mathematical rigor  gives a very brief informal introduction to some basic facts on von Neumann algebras, including their Murray-von Neumann classification, which is relevant from the perspective of the Common Cause Principle—see below). A special example of a quantum probability space is obtained by taking the set of all bounded operators \({\cal B} ({\cal H} )\) on a (possibly infinite dimensional) Hilbert space \({\cal H}\) as the von Neumann algebra and \(\phi\) a quantum state given by a density matrix. The resulting specific quantum probability space \(({\cal P}({\cal B}({\cal H})),\phi)\) is known as the “Hilbert space formalism” of quantum mechanics; this space describes quantum systems of finite degrees of freedom. The orthomodular lattice \({\cal P}({\cal B}({\cal H}))\), frequently denoted as \({\cal P} ({\cal H})\), is the set of all projections on \({\cal H}\), this lattice is also called “Hilbert lattice”.