A characterization of the Aerts product of Hilbertian lattices

Let ℋ ₁ and ℋ ₂ be complex Hilbert spaces, ℳ ₁ = P(ℋ ₁) and ℳ ₂ = P(ℋ ₂) the lattices of closed subspaces, and let ℳ be a complete atomistic lattice. We prove under some weak assumptions relating ℳ _i and ℳ, that if ℳ admits an orthocomplementation, then ℳ is isomorphic to the separated product of ℳ ₁ and ℳ ₂ defined by Aerts. Our assumptions are minimal requirements for ℳ to describe the experimental propositions concerning a compound system consisting of so-called separated quantum systems. The proof does not require any assumption on the orthocomplementation of ℳ.