Quantum Superpositions Cannot be Epistemic

Quantum superposition states are behind many of the curious phenomena exhibited by quantum systems, including Bell non-locality, quantum interference, quantum computational speed-up, and the measurement problem. At the same time, many qualitative properties of quantum superpositions can also be observed in classical probability distributions leading to a suspicion that superpositions may be explicable as probability distributions over less problematic states, that is, a suspicion that superpositions are \emph{epistemic}. Here, it is proved that, for any quantum system of dimension $d>3$, this cannot be the case for almost all superpositions. Equivalently, any underlying ontology must contain ontic superposition states. A related question concerns the more general possibility that some pairs of non-orthogonal quantum states $\mathrm{\mid }\psi ⟩,\mathrm{\mid }\varphi ⟩$ could be ontologically indistinct (there are ontological states which fail to distinguish between these quantum states). A similar method proves that if $\mathrm{\mid }⟨\varphi \mathrm{\mid }\psi ⟩{\mathrm{\mid }}^2\in (0,\frac{1}{4})$ then $\mathrm{\mid }\psi ⟩,\mathrm{\mid }\varphi ⟩$ must approach ontological distinctness as $d\to \mathrm{\infty }$. The robustness of these results to small experimental error is also discussed.