Quantum description of reality is epistemically incomplete

Abstract

We ask whether the operational quantum description is complete at the level of preparations: can the empirically accessible properties of a finite preparation set be reproduced exactly by a hidden-variable description, or must every such completion contain additional structure that is not operationally accessible? We formalize this through epistemic completeness, a preparation-side notion of classicality requiring exact preservation of empirical preparation-properties by the corresponding ontic quantities obtained by conditioning on the ontic state and allowing all response schemes compatible with positivity and normalization. For the canonical family of set-distinguishability tasks, we prove that every epistemically complete theory satisfies an equality: for every finite preparation set, the average pairwise distinguishability equals the average set-distinguishability. Any nonzero deviation certifies epistemic incompleteness and lower-bounds the excess ontic communication power that every ontic completion must conceal. Because unrestricted classical communication models, and more generally commuting quantum theories, are epistemically complete, every nonzero deviation also yields a quantum communication advantage and witnesses of coherence and measurement incompatibility. We formulate semidefinite-programming relaxations and see-saw lower bounds, and show that quantum theory violates the equality in both directions. The trine and tetrahedral qubit ensembles are numerically certified maximizers for the n=3 and n=4 equalities; their violations persist for arbitrarily low positive visibility and arbitrarily large leakage short of complete disclosure; and the Kochen--Specker ψ-epistemic model exactly saturates the hidden ontic excess for these maximal positive violations. Numerics suggest that the maximal positive deviation increases with the number of preparations.