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The status of the quantum state is perhaps the most controversial issue inthe foundations of quantum theory. Is it an epistemic state (state ofknowledge) or an ontic state (state of reality)? In realist models of quantumtheory, the epistemic view asserts that nonorthogonal quantum states correspondto overlapping probability measures over the true ontic states. This naturallyaccounts for a large number of otherwise puzzling quantum phenomena. Forexample, the indistinguishability of nonorthogonal states is explained by thefact that the ontic state sometimes lies in the overlap region, in which casethere is nothing in reality that could distinguish the two states. For this towork, the amount of overlap of the probability measures should be comparable tothe indistinguishability of the quantum states. In this letter, I exhibit afamily of states for which the ratio of these two quantities must be $\leq2de^{-cd}$ in Hilbert spaces of dimension $d$ that are divisible by $4$. Thisimplies that, for large Hilbert space dimension, the epistemic explanation ofindistinguishability becomes implausible at an exponential rate as the Hilbertspace dimension increases.

$\psi$-epistemic models are exponentially bad at explaining the distinguishability of quantum states