The quantum supremum of the I3322 Bell inequality is not attained in finite dimension
Abstract
In 2010, Pál and Vértesi found a family of finite-dimensional strategies for the I3322 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.
Related
- The I3322 quantum value is attained spatially but not in finite dimension
- A sharp gradient-corrected Lieb--Thirring inequality in one dimension with applications to density functional theory
- Complex coordinates forced by finite orthogonality hypergraphs in dimension three
- The Quantum KKL Inequality
- Equality in the Bosonic Quantum Entropy Power Inequality
BibTeX
@misc{pauwels2026quantum,
title = {{The quantum supremum of the $I_3322$ Bell inequality is not attained in finite dimension}},
author = {Jef Pauwels},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29734},
eprint = {2608.29734},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29734},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The quantum supremum of the I_3322 Bell inequality is not attained in finite dimension
AU - Jef Pauwels
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29734
UR - https://arxiv.org/abs/2608.29734
AB - 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.
ER -