A $\psi$-Ontology Result without the Cartesian Product Assumption

We introduce a weakening of the Preparation Independence Postulate of Pusey, Barrett, and Rudolph that does not presuppose that the space of ontic states resulting from a product state preparation can be represented by the Cartesian product of subsystem state spaces. On the basis of this weakened assumption, it is shown that, in any model that reproduces the quantum probabilities, any pair of pure quantum states $\mathrm{\mid }\psi ⟩$, $\mathrm{\mid }\varphi ⟩$ with $\bkt{\phi}{\psi} \leq 1/\sqrt{2}$ must be ontologically distinct.