The quantum supremum of the $I_{3322}$ Bell inequality is not attained in finite dimension

Technical Summary: The Quantum Supremum of the I3322 Bell Inequality is Not Attained in Finite Dimension

Problem Statement
The paper addresses a fundamental question in quantum nonlocality: Does every Bell inequality possess an optimal quantum strategy within a finite-dimensional Hilbert space? While the set of quantum correlations for a fixed finite dimension is compact (ensuring a maximum exists), the union of these sets over all finite dimensions need not be closed. Consequently, a sequence of strategies with increasing dimensions might approach a supremum that no single finite-dimensional strategy can attain.

The specific focus is the I3322 Bell inequality, defined in the $(3, 3, 2, 2)$ scenario (three measurement settings per party, two binary outcomes). This scenario is significant because, based on Beigi's result that scenarios with only two settings per party are closed, it represents the smallest binary-outcome setting in which non-closure of the set of finite-dimensional correlations ($C_q$) is theoretically possible.

In 2010, Pál and Vértesi (PV) identified a family of finite-dimensional strategies for I3322 utilizing a repeating structure of Schmidt coefficients and spectral labels. Their numerical optimization suggested these values converge to a specific limit ($\approx 0.25087538$) as the dimension grows, but never reach it. They conjectured that this limit is the true supremum over all finite-dimensional strategies and that no finite-dimensional strategy attains it.

Methodology
The proof proceeds by establishing a rigorous correspondence between arbitrary finite-dimensional quantum strategies and a specific variational family (the PV family), followed by an analysis of the optimality conditions.

1. Symmetrization and Reduction:
   The author first reduces an arbitrary finite-dimensional strategy to a symmetric form. Using a direct-sum doubling technique, they construct an equivalent strategy where Bob's measurements are outcome-flipped, conjugated copies of Alice's. This symmetry allows the third measurement (which enters the Bell functional linearly) to be optimized analytically.

2. Spectral-Weight Matrix Construction:
   Applying Jordan's lemma to the remaining pair of measurements, the strategy is reduced to a finite set of spectral subspaces. The author defines a spectral-weight matrix $\theta$, representing the joint probability distribution of spectral labels for Alice and Bob. They derive an upper-bound functional, $\Phi(\theta)$, which bounds the Bell value of any strategy $S$ such that $I_{3322}(S) \leq \Phi(\theta)$.

3. Equivalence to the PV Family (Theorem 1):
   The paper proves that the supremum of the functional $\Phi(\theta)$ over all valid spectral-weight matrices is exactly equal to the supremum of the Pál–Vértesi family ($\beta_{PV}$). This is achieved by showing that any spectral-weight matrix can be approximated by a weighted average of finite PV chains. As the chain length $N \to \infty$, the weighted average converges to $\Phi(\theta)$. Thus, no strategy outside the PV family can exceed the PV supremum.

4. Proof of Non-Attainment (Theorem 2):
   To prove that the supremum is not attained in finite dimensions, the author assumes a finite-dimensional strategy exists that achieves the supremum. This implies the associated spectral-weight matrix $\theta$ satisfies $\Phi(\theta) = \beta_{PV}$.

   • By analyzing the first-order optimality conditions of $\Phi$, they derive a recurrence relation for a sequence of auxiliary coefficients $u_i$ and spectral labels $c_i$ extracted from $\theta$.
   • These coefficients must satisfy a three-term recurrence relation resembling an eigenvalue equation for a Jacobi operator.
   • The proof demonstrates a contradiction: for the value to be maximal, the auxiliary coefficients must grow geometrically in one direction (implying an infinite, non-normalizable state) while simultaneously being required to be square-summable (normalizable) for a finite-dimensional strategy. This incompatibility proves that no finite-dimensional strategy can attain the supremum.

Key Results

• Theorem 1: The finite-dimensional quantum supremum of I3322 equals the supremum of the Pál–Vértesi family ($I^*_{3322} = \beta_{PV}$).
• Theorem 2: No finite-dimensional quantum strategy attains the value $I^*_{3322}$. Every finite-dimensional strategy yields a value strictly below the supremum.
• Corollary 1 (Non-closure): The set of finite-dimensional quantum correlations $C_q(3, 3, 2, 2)$ is not closed. Its closure $C_{qa}$ contains correlations that cannot be generated by any finite-dimensional tensor-product strategy.
• Corollary 2 (Dimension Witness): Approaching the supremum requires unbounded local dimension. For any finite dimension $k$, the maximum value $\beta_k$ is strictly less than $I^*_{3322}$.

Significance and Claims
The paper establishes that the $(3, 3, 2, 2)$ scenario is the smallest binary-outcome Bell scenario where the set of finite-dimensional correlations is not closed, confirming that non-closure is possible in this minimal setting. This resolves the specific attainment question for the I3322 inequality, confirming the 2010 conjecture by Pál and Vértesi.

The author explicitly states that their results apply to finite-dimensional strategies. The question of whether an infinite-dimensional tensor-product strategy (in the set $C_{qs}$) attains the supremum remains open. If such a strategy exists, it would require infinite-dimensional entanglement; if not, it would imply a separation between tensor-product and commuting-operator correlations ($C_{qs} \neq C_{qc}$) at this specific scenario.

The core mathematical proof was formalized in Lean 4, ensuring rigorous verification of the logical steps. The paper also notes that the value of the supremum is known numerically to high precision but lacks a known closed-form analytic expression.

Note on Authorship and AI
The author, Jef Pauwels, includes a statement regarding the role of AI in the discovery process. They note that the initial proof sketch was generated by an AI model (OpenAI Codex/GPT-5.6-sol) after being prompted to persist with the conjecture. The subsequent work involved human translation of the AI's "non-English" mathematical jargon into standard quantum information terminology, verification, and formalization in Lean 4. The author takes full responsibility for the final claims and the "proof digestion" required to make the machine-generated argument accessible and verifiable.