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 $|\psi\rangle,|\phi\rangle$ could be ontologically indistinct (there are ontological states which fail to distinguish between these quantum states). A similar method proves that if $|\langle\phi|\psi\rangle|^{2}\in(0,\frac{1}{4})$ then $|\psi\rangle,|\phi\rangle$ must approach ontological distinctness as $d\rightarrow\infty$. The robustness of these results to small experimental error is also discussed.