A characterization of the Aerts product of Hilbertian lattices

TL;DR

This paper characterizes the Aerts product of Hilbertian lattices, showing that under minimal assumptions, a complete atomistic lattice with orthocomplementation is isomorphic to the separated product of two Hilbert space lattices, relevant for quantum systems.

Contribution

It provides a minimal assumption framework to identify when a lattice models separated quantum systems as an Aerts product.

Findings

Lattice with orthocomplementation is isomorphic to the Aerts separated product.

Minimal assumptions are sufficient for the characterization.

Proof does not rely on orthocomplementation of the lattice.

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.