Composite systems: uniqueness

Composite systems are of central importance in quantum physics, for modeling entanglement. But, are there essentially different ways of constructing a composite system for two independent physical systems? We show that the answer is “No”. For quantum physics and real quantum mechanics, we prove that the standard composite system, the Segre embedding, is unique, in the sense that any composite system candidate is essentially the Segre embedding, with possibly time reversals on the component state spaces. Consequently, there are no other types of composite systems that could potentially expose new properties of quantum mechanics. We, thus, are considering two well-known state space families: Projective linear subspaces \(\mathfrak {S}\subset \mathbb {P}(V)\) $\mathfrak{S}\subset \mathbb{P}(V)$ , where V is a separable Hilbert space over the reals or complexes (quantum physics). A candidate composite system for state spaces \(\mathfrak {S}_1\) ${\mathfrak{S}}_1$ and \(\mathfrak {S}_2\) ${\mathfrak{S}}_2$ is a state space, \(\mathfrak {S}\) $\mathfrak{S}$ , with an isometric embedding of \(\mathfrak {S}_1 \times \mathfrak {S}_2\) ${\mathfrak{S}}_1\times {\mathfrak{S}}_2$ into \(\mathfrak {S}\) $\mathfrak{S}$ , where, on the Cartesian product, the probability transition form is the product of the transition forms on the two factors. In essence, this article classifies the isometric embeddings of \(\mathbb {P}(V_1)\times \mathbb {P}(V_2)\) $\mathbb{P}(V_1)\times \mathbb{P}(V_2)$ into \(\mathbb {P}(V)\) $\mathbb{P}(V)$ , as Wigner’s Theorem classifies the isometric embeddings of \(\mathbb {P}(V_1)\) $\mathbb{P}(V_1)$ into \(\mathbb {P}(V_2)\) $\mathbb{P}(V_2)$ .