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arXiv:2609.06038

The $I_{3322}$ Bell inequality requires infinite dimensions

Andrea Coladangelo

quant-phSubmitted Sep 5, 2026• Announced Sep 8, 2026• 0 citations• Revised Oct 5, 2026 (v2)
v1 · Sep 5, 2026v2 · Oct 5, 2026
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Abstract

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.

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79 pages

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The I3322I_{3322}I3322​ Bell inequality requires infinite dimensions
The I3322I_{3322}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 I3322I_{3322}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.
The quantum supremum of the I3322I_{3322}I3322​ Bell inequality is not attained in finite dimension
In 2010, Pál and Vértesi found a family of finite-dimensional strategies for the I3322I_{3322}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)(3,3,2,2)(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.
We introduce a three-parameter family of Bell functionals that extends those studied previously [Kaniewski, Phys. Rev. Res. 2, 033420 (2020)] by including a marginal contribution. An analysis of their largest value achievable by quantum realizations naturally splits the family into two branches, and for the first of them we show that this value is given by a simple function of the parameters defining the functionals. In this case we completely characterize the realizations attaining the optimal value and show that these functionals can be used to self-test any partially entangled state of two qubits. The optimal measurements, however, are not unique and form a one-parameter family of qubit measurements. The second branch, which includes the well-known I3322{I}_{3322}I3322​ functional, is studied numerically. We identify the region in the parameter space where the optimal value can be attained with two-dimensional quantum systems and characterize the state and measurements attaining this value. Finally, we show that the set of realizations introduced by P\'al and V\'ertesi [Phys. Rev. A 82, 022116 (2010)] to obtain the maximal violation of the I3322{I}_{3322}I3322​ inequality succeeds in approaching the optimal value for a large subset of the functionals in this branch. In these cases we analyze and discuss the main features of the optimal realizations.
Let SSS be the common tensor-product and commuting-operator supremum of the I3322I_{3322}I3322​ Bell functional in the Collins-Gisin normalization. Starting from a certified value window and Bellman/path equivalence, we prove that no finite-dimensional quantum strategy attains SSS, whereas a spatial strategy on ℓ2(Z)⊗ℓ2(Z)\ell^2(\mathbb{Z})\otimes\ell^2(\mathbb{Z})ℓ2(Z)⊗ℓ2(Z) does. The finite-dimensional statement includes mixed states and binary POVMs. Consequently Cq(3,3;2,2)C_q(3,3;2,2)Cq​(3,3;2,2) is not closed and Cqs(3,3;2,2)∖Cq(3,3;2,2)C_{qs}(3,3;2,2)\setminus C_q(3,3;2,2)Cqs​(3,3;2,2)∖Cq​(3,3;2,2) is nonempty. We also establish the dimension complexity D(ε)=Θ(log⁡(1/ε))D(ε)=Θ(\log(1/ε))D(ε)=Θ(log(1/ε)): approaching SSS requires and suffices local dimension logarithmic in inverse error. The constructive upper bound is D(ε)≤23.9010650log⁡(1/ε)D(ε)\le 23.9010650\log(1/ε)D(ε)≤23.9010650log(1/ε) for all sufficiently small εεε, with natural logarithms; the lower constants are existential. The proofs use critical Bellman storage, spectral transport and normalizable orbit measures. A finite weighted-flow argument supplies the unrestricted quantitative lower bound without identifying distinct joint spectral components. This revision replaces an unsupported step in the earlier lower argument and records additional proof corrections. Prior numerical certification and independent concurrent attainment results are credited. Exact arithmetic and Lean 4 check specified numerical, scalar and finite accounting facts; the complete analytic proof is not formalized.
In quantum information, asymmetry, i.e., the lack of symmetry, is a resource allowing one to accomplish certain tasks that are otherwise impossible. Similarly, in a Bell test using any given Bell inequality, the maximum violation achievable using quantum strategies respecting or disregarding a certain symmetry can be different. In this work, we focus on the symmetry involved in the exchange of parties and explore when we have to trade this symmetry for a lower-dimensional quantum strategy in achieving the maximal violation of given Bell inequalities. For the family of symmetric Collins-Gisin-Linden-Massar-Popescu inequalities, we provide evidence showing that there is no such trade-off. However, for several other Bell inequalities with a small number of dichotomic measurement settings, we show that symmetric quantum strategies in the minimal Hilbert space dimension can only lead to a suboptimal Bell violation. In other words, there exist symmetric Bell inequalities that can only be maximally violated by asymmetric quantum strategies of minimal dimension. In contrast, one can also find examples of asymmetric Bell inequalities that are maximally violated by symmetric correlations. The implications of these findings on the geometry of the set of quantum correlations and the possibility of performing self-testing therefrom are briefly discussed.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
Completely determining the relationship between quantum correlation sets is a long-standing open problem, known as Tsirelson's problem. Following recent progress by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)] only two instances of the problem remain open. One of them is the question of whether the set of finite-dimensional quantum correlations is strictly contained in the set of infinite-dimensional ones (i.e. whether Cq≠Cqs\mathcal C_{q} \neq \mathcal C_{qs}Cq​=Cqs​). The usual formulation of the question assumes finite question and answer sets. In this work, we show that, when one allows for either infinite answer sets (and finite question sets) or infinite question sets (and finite answer sets), there exist correlations that are achievable using an infinite-dimensional quantum strategy, but not a finite-dimensional one. For the former case, our proof exploits a recent result [Nat. Comm. 8, 15485 (2017)], which shows self-testing of any pure bipartite entangled state of arbitrary local dimension ddd, using question sets of size 3 and 4 and answer sets of size ddd. For the latter case, a key step in our proof is to show a novel self-test, inspired by [Nat. Comm. 8, 15485 (2017)], of all bipartite entangled states of any local dimension d, using question sets of size O(d)O(d)O(d), and answer sets of size 4 and 3 respectively.