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.

Figures (5)

• Figure 1: Three preparation-side preservation principles arranged in increasing strength. Left: preparation noncontextuality preserves operational indistinguishability. Here $P_0\simeq P_1$ denotes operational equivalence. If $P_0\simeq P_1$, then the operational two-state distinguishability is trivial, $D_{2,1}^{(\mathcal{O})}(P_0,P_1)=\tfrac{1}{2}$, and preparation noncontextuality sends this to identical epistemic states $\mu_0(\lambda)=\mu_1(\lambda)$ for all $\lambda$, which in turn fixes the ontic distinguishability at $D_{2,1}^{(\Lambda)}(\mu_0,\mu_1)=\tfrac{1}{2}$. Middle: bounded ontological distinctness preserves distinguishability itself for an arbitrary finite preparation set $\mathbf P$, by requiring exact agreement between the operational and ontic quantities $D_{n,1}^{(\mathcal{O})}(\mathbf P)$ and $D_{n,1}^{(\Lambda)}(\boldsymbol\mu)$; see Proposition \ref{['prop:ec-implies-bod']}. Right: epistemic completeness strengthens this further by requiring exact preservation of every empirical preparation-property $S_n^{(\mathcal{O})}(\mathbf P)$ of the form introduced in Eq. \ref{['eq:generic-task']}, through equality with its unrestricted ontic counterpart $S_n^{(\Lambda)}(\boldsymbol\mu)$. The two formula panels at the bottom are written at matched levels. The left panel records the canonical set-distinguishability quantities in their operational and pointwise-solved ontic forms, Eqs. \ref{['eq:Dnm-op']} and \ref{['eq:Dnm-ontic']}. The right panel records the general empirical preparation-property and its pointwise-solved ontic counterpart, Eqs. \ref{['eq:generic-task']} and \ref{['eq:ontic-task-solved']}. In this way, the figure makes visually explicit the strengthening summarized abstractly in Eq. \ref{['eq:hierarchy-core']}: one moves from preserving indistinguishability, to preserving distinguishability, to preserving the full empirical preparation profile of a finite preparation set.
• Figure 2: Operational reading of the positive-deviation side of the equality witness. Alice receives $x\in[n]$ and prepares $P_x$. Bob receives a promised pair $\{i,j\}$ containing $x$ and must guess whether $x=i$ or $x=j$, so the task value is $\overline{D}_n$. The benchmark is imposed on competing classical preparation sets $\bm{C}$ through $\overline{S}_n^{(\mathrm{cl})}(\bm{C})\le p$, with $p$ fixed by the given preparation set as $p=\overline{S}_n^{(\mathcal{O})}(\mathbf P)$. Because unrestricted classical communication obeys the equality witness exactly, any preparation set with $\Delta_n^{(\mathcal{O})}(\mathbf P)>0$ yields a communication advantage in this scenario.
• Figure 3: Operational reading of the negative-deviation side of the equality witness. Alice receives $x\in[n]$ and prepares $P_x$. Bob receives $m\in\{1,\dots,n-1\}$ and must output an $m$-element set containing the true label, so the task value is $\overline{S}_n$. The benchmark is imposed on competing classical preparation sets $\bm{C}$ through $\overline{D}_n^{(\mathrm{cl})}(\bm{C})\le p$, with $p$ fixed by the given preparation set as $p=\overline{D}_n^{(\mathcal{O})}(\mathbf P)$. Because unrestricted classical communication obeys the equality witness exactly, any preparation set with $\Delta_n^{(\mathcal{O})}(\mathbf P)<0$ yields a communication advantage in this dual scenario.
• Figure 4: Three-preparation geometry of the equality witness in the $(\overline{S}_3,\overline{D}_3)$ plane. The diagonal line $\overline{D}_3=\overline{S}_3$ is the exact locus satisfied by every epistemically complete theory. The trine ensemble lies above the line and certifies a positive deviation. The opposite-sign mixed-state qutrit example lies below the line and certifies a negative deviation. The visibility and leakage families introduced in the next section trace explicit positive-direction nonclassical paths starting from the trine point.
• Figure 5: Increasing maximal positive quantum deviation from the equality witness. The values shown are the exact trine and tetrahedral deviations for $n=3$ and $n=4$, together with the numerically certified value displayed for $n=5$. In particular, the $n=3$ maximum is attained exactly by the trine ensemble, while the $n=4$ maximum is numerically certified at the tetrahedral ensemble by hierarchy/see-saw matching. The observed trend suggests that epistemic incompleteness becomes stronger with larger preparation sets and motivates the open question of the absolute maximal epistemic incompleteness of quantum theory.

Theorems & Definitions (47)

• Definition 1: Ontic extension
• Definition 2: Not-fine-tuned ontic task-value
• Lemma 1
• proof
• Definition 3: Epistemically complete ontic extension
• Definition 4: Epistemically complete ontological model
• Definition 5: Epistemically complete operational theory
• Proposition 1
• proof
• Proposition 2