Whether the operational quantum description of reality is complete can already be asked at the level of preparations: Can the empirically accessible properties of a finite preparation set be reproduced exactly by an ontological or hidden-variable model, or must every such completion contain an additional hidden structure that is not operationally accessible? We formalize this question through epistemic completeness, a preparation-side notion of classicality requiring exact preservation of empirical preparation properties, defined here as optimal one-way communication-task payoffs, by the corresponding not-fine-tuned ontic quantities obtained by conditioning on the ontic state and allowing all response schemes compatible only with positivity and normalization. For the canonical family of set-distinguishability tasks, we prove a universal equality: in every epistemically complete theory and for every finite preparation set, the average pairwise distinguishability equals the average set distinguishability. Any nonzero deviation therefore certifies epistemic incompleteness and lower-bounds the excess ontic communication power that every ontic completion must conceal. Since unrestricted classical communication models, and more generally finite-dimensional commuting quantum theories, are epistemically complete, every nonzero deviation yields a quantum advantage in a one-way communication task under a natural benchmark-task constraint. The same deviations also furnish dimension-independent witnesses of coherence of the preparation set and of measurement incompatibility in any realization attaining the certifying task values. To characterize the extremal quantum deviations, we formulate dimension-independent semidefinite-programming relaxations and dimension-dependent seesaw lower bounds and show that quantum theory violates the equality in both directions. In particular, the trine and tetrahedral qubit ensembles are numerically certified dimension-independent maximizers of the positive deviation for the $n=3$ and $n=4$ equalities, respectively. These positive-direction violations persist for arbitrarily low positive visibility and arbitrarily large leakage short of complete disclosure in the corresponding visibility and leakage families. We further show that the Kochen–Specker $\psi$-epistemic model exactly saturates the hidden ontic excess implied by the maximal positive violations for $n=3$ and $n=4$, making the witness quantitatively sharp. Finally, our numerical results indicate that the maximal positive deviation increases with the number of preparations, motivating the question of the absolute maximal epistemic incompleteness of quantum theory.

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• Quantum communication
• Quantum formalism
• Quantum foundations