A characterization of the Aerts product of Hilbertian lattices

Abstract

Let H_1 and H_2 be complex Hilbert spaces, L_1=P(H_1) and L_2=P(H_2) the lattices of closed subspaces, and let L be a complete atomistic lattice. We prove under some weak assumptions relating L_i and L, that if L admits an orthocomplementation, then L is isomorphic to the separated product of L_1 and L_2 defined by Aerts. Our assumptions are minimal requirements for L 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 L.

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