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

• URL: http://arxiv.org/abs/2608.29734v1
• Date: Sun, 30 Aug 2026 11:48:13 GMT
• Title: The quantum supremum of the $I_{3322}$ Bell inequality is not attained in finite dimension
• Abstract summary: In 2010, Pl and Vértesi found a family of finite-dimensional strategies for the $I_3322$ Bell inequality whose optimized values appeared to converge as the local Hilbert-space dimension grew.<n>We conjectured that this limit is the supremum over all finite-dimensional quantum strategies, but that no finite-dimensional strategy attains it.<n>The proof uses the symmetry of the Bell functional to associate every strategy with a finite matrix of probabilities, one for each pair of spectral subspaces of Alice and Bob.
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• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In 2010, Pál and Vértesi found a family of finite-dimensional strategies for the $I_{3322}$ Bell inequality whose optimized values appeared to converge as the local Hilbert-space dimension grew. They conjectured that this limit is the supremum over all finite-dimensional quantum strategies, but that no finite-dimensional strategy attains it. We prove both claims. The proof uses the symmetry of the Bell functional to associate every strategy with a finite matrix of probabilities, one for each pair of spectral subspaces of Alice and Bob. This matrix gives an upper bound on the Bell value, and finite-dimensional strategies built from the repeating structure found by Pál and Vértesi approach it as the dimension grows. If the bound were attained exactly in finite dimension, the optimality conditions would then require a state that cannot be normalized. Consequently, the set of finite-dimensional quantum correlations is not closed in the $(3,3,2,2)$ scenario, the smallest Bell scenario where this can happen. Moreover, approaching the supremum requires unbounded local dimension. The core of the proof was formalized in Lean~4.

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