A characterization of the Aerts product of Hilbertian lattices

Let ℋ1 and ℋ2 be complex Hilbert spaces, ℳ1 = P(ℋ1) and ℳ2 = P(ℋ2) the lattices of closed subspaces, and let ℳ be a complete atomistic lattice. We prove under some weak assumptions relating ℳi and ℳ, that if ℳ admits an orthocomplementation, then ℳ is isomorphic to the separated product of ℳ1 and ℳ2 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.