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arXiv:2609.35399

Heuristic lower bounds on real Grothendieck constants of finite and infinite order

Erika Bene, Tamás Vértesi

quant-phSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

We propose candidate lower bounds on the real Grothendieck constants $K_{\mathrm G}(d)$ and their infinite-dimensional limit $K_{\mathrm G}$ using rotationally invariant cubic kernels. For the continuous construction, we identify an explicit threshold above which hemispherical binary strategies are unstable under cubic boundary deformations. Supported by numerical evidence, we conjecture that hemispherical strategies are globally optimal at this threshold. This conjecture would imply $K_{\mathrm G}\geq9π/16\simeq1.767146$, improving the rigorous lower bound $6π/11\simeq1.713596$. We also construct finite symmetric coefficient matrices from unit vectors in dimensions up to $24$. Large-scale see-saw optimization supports the continuous predictions and suggests improved lower bounds on $K_{\mathrm G}(d)$ in the studied dimensions from $4$ to $24$, conditional on the conjectured binary optima being exact. Selected matrices also yield finite-setting candidate dimension witnesses when interpreted as bipartite correlation Bell functionals. If the corresponding binary upper bounds hold, these witnesses would certify a local Hilbert-space dimension of at least five for each party.

Comments
21 pages, 2 figures, 3 tables

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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.
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In this paper, we prove that KG(3)<KG(4){K}_{G}(3)<{K}_{G}(4)KG​(3)<KG​(4), where KG(d){K}_{G}(d)KG​(d) denotes the Grothendieck constant of order ddd. To this end, we use a branch-and-bound algorithm commonly used in the solution of NP-hard problems. It has recently been proven that ${K}_{G}(3)\ensuremath{\le}1.4644$. Here we prove that ${K}_{G}(4)\ensuremath{\ge}1.4841$, which has implications for device-independent witnessing dimensions greater than two. Furthermore, the algorithm with some modifications may find applications in various black-box quantum information tasks with large number of inputs and outputs.
1. Exact planar GHZ nonlocality threshold for all n ≥ 4. We determine the sharp supremum of violations of full-correlation Bell inequalities by the n-qubit GHZ state under planar (XY) measurements, over arbitrary finite numbers of measurement settings. The result is the exact formula κₙ = (π/2)ⁿ ζ(n) / [2(1 + 2⁻ⁿ) ζ(2n)], n ≥ 4. In particular, κ₄ = 105/34, corresponding to the critical visibility v₄ = 34/105. The result replaces bounds for particular finite measurement configurations with an exact analytic expression for the stated class. 2. Substantially improved rigorous bounds for the open three-party GHZ problem. We establish 2.0349681535… ≤ κ₃ ≤ 105 ζ(3)/(2π³) = 2.0353294291… . Both bounds improve the previously published 2024 interval. The lower bound comes from a finite-harmonic Bell witness supported by 272 reproducible exact inequalities; the upper bound follows from an explicit square-wave/Möbius local model. The conjecture that the upper bound is attained remains open. The paper discusses an obstruction involving the Möbius function and zeros of the Riemann zeta function. 3. Sharp covariance inequalities under strong mixing. We establish the optimal coefficient 4√2 for bounded complex-valued observations, improving the classical bound 2π. More generally, the optimal Hilbert-space coefficient is 4K_G(k), where K_G(k) is the real Grothendieck constant of order k. For real scalar observations, the sharp coefficient is 4. 4. Sharp multilinear dependence inequalities. For n-linear forms that are real-valued on real inputs, we identify the optimal real and complex comparison factors 2ⁿ and 2ⁿ κₙ. These results strengthen classical estimates in a natural restricted class. They do not settle the more general problem for arbitrary complex multilinear forms considered by Bradley and Bryc (1985). 5. Asymptotically sharp growth of higher-order interaction measures. For the centred Fourier–Lancaster interaction functional, we derive explicit finite-dimensional bounds and establish L_d = 2ᵈ/(ed) · (1 + O(d⁻¹)). Thus, unlike the universal characteristic-function factorization defect, this higher-order interaction functional admits no dimension-independent bound. 6. Further structural results. The paper also develops restrictions to discrete phase groups, centred-interaction comparisons and quantitative links between universal Fourier factorization ceilings, strong mixing and multilinear dependence. Updated version, 18 October 2026 — two-PDF proof supplement. This record contains the original 14-page P04 main preprint (P04_CF_V6_S8.pdf), reproduced without alteration, and a 47-page English mathematical proof supplement (P04_K2_K7_R13_PROOF_SUPPLEMENT_V2.pdf). The supplement expands the analytical proof of the exact planar GHZ formula for n ≥ 4 and develops the three-party K7 lower bound using geometric reductions and 272 reproducible exact-rational inequalities. The exact three-party constant remains open.
Suppose that Alice and Bob make local two-outcome measurements on a shared entangled quantum state. We show that, for all positive integers d, there exist correlations that can only be reproduced if the local Hilbert-space dimension is at least d. This resolves a conjecture of Brunner et al. [Phys. Rev. Lett. 100, 210503 (2008)] and establishes that the amount of entanglement required to maximally violate a Bell inequality must depend on the number of measurement settings, not just the number of measurement outcomes. We prove this result by establishing a lower bound on a new generalization of Grothendieck’s constant.