An operational distinction between quantum entanglement and classical non-separability

Quantum entanglement describes superposition states in multi-dimensional systems, at least two partite, which cannot be factorized and are thus non-separable. Non-separable states exist also in classical theories involving vector spaces. In both cases, it is possible to violate a Bell-like inequality. This has led to controversial discussions, which we resolve by identifying an operational distinction between the classical and quantum cases.
Comment: 12 pages, 4 figures, accepted to the Topical issue of the Philosophical Transactions A: The Quantum Theory of Light, eds. S. Barnett and J. Jeffers, to commemorate Rodney Loudon and the 50th anniversary of the first edition of his seminal book