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The I3322 Bell inequality requires infinite dimensions

  • Andrea Coladangelo
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Abstract

The I3322 inequality is one of the simplest bipartite Bell inequalities, with three possible questions and two possible answers per party. Yet, despite its simplicity, it has long been conjectured by Pál and Vértesi (Physical Review A, 2010) to possess the following quirk: no finite-dimensional quantum strategy can attain its maximal violation, but an infinite-dimensional strategy can. A Bell correlation, with five questions and three answers per party, that provably possesses the same property has since been discovered by Coladangelo and Stark (Nature Communications, 2020). However, proving that the same property holds for I3322, which lives in the simplest Bell scenario in which such a phenomenon could occur, has remained elusive. Here, we provide a proof of this conjecture. An approximate proof, containing all the main ideas, was produced by GPT 5.5 Pro after various rounds of interaction. The presentation of this proof was revised substantially by the author for completeness, correctness, and clarity.

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Cite

BibTeXRIS
BibTeX
@misc{coladangelo2026i3322,
  title = {{The $I_3322$ Bell inequality requires infinite dimensions}},
  author = {Andrea Coladangelo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.06038},
  eprint = {2609.06038},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06038},
}
RIS
TY  - UNPB
TI  - The I_3322 Bell inequality requires infinite dimensions
AU  - Andrea Coladangelo
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06038
UR  - https://arxiv.org/abs/2609.06038
AB  - The I_3322 inequality is one of the simplest bipartite Bell inequalities, with three possible questions and two possible answers per party. Yet, despite its simplicity, it has long been conjectured by Pál and Vértesi (Physical Review A, 2010) to possess the following quirk: no finite-dimensional quantum strategy can attain its maximal violation, but an infinite-dimensional strategy can. A Bell correlation, with five questions and three answers per party, that provably possesses the same property has since been discovered by Coladangelo and Stark (Nature Communications, 2020). However, proving that the same property holds for I_3322, which lives in the simplest Bell scenario in which such a phenomenon could occur, has remained elusive. Here, we provide a proof of this conjecture. An approximate proof, containing all the main ideas, was produced by GPT 5.5 Pro after various rounds of interaction. The presentation of this proof was revised substantially by the author for completeness, correctness, and clarity.
ER  -