Quantum correlations

Answered Can a finite quantum system reach the largest I₃₃₂₂ violation?

No.

1. The I₃₃₂₂ quantum value is attained spatially but not in finite dimension

   Seth Douglas · version 4.1.0 · 11 September 2026 · 25 pages · arXiv:2609.05555

   No finite-dimensional quantum strategy attains the quantum maximum of the I₃₃₂₂ Bell functional, mixed states and binary measurements included; a strategy on ${\ell }^2(\mathbb{Z})\otimes {\ell }^2(\mathbb{Z})$ does; approaching it needs local dimension $\mathrm{\Theta }(\log (1/\epsilon ))$.

   • Answered Does any finite-dimensional strategy reach the quantum maximum of the I₃₃₂₂ Bell inequality?
   • Answered Can a finite quantum system reach the largest violation of the I₃₃₂₂ Bell inequality?
   • Claimed answered elsewhere What is the exact quantum value of I₃₃₂₂?
   • Open What is the sharp constant of the I₃₃₂₂ dimension rate? posed here

   The paperPDFDOICode

2. The Lovász number is not alone: a continuum of Rényi points in the entanglement-assisted spectrum of graphs

   Nidhal Mghirbi and Seth Douglas · version 1.2.1 · 14 October 2026 · arXiv:2610.20775

   Minimizing Petz–Rényi information radii over classical–quantum channels compatible with a graph gives a continuum of points between known graph parameters; a 1440-vertex graph has $\mathrm{\Sigma }(G)$ strictly below the projective rank of its complement.

   • Answered Do the middle links of the Duan–Winter chain of zero-error quantities ever separate, and what are they on the pentagon?
   • Answered Is the Lovász number the only Rényi point in the entanglement-assisted spectrum of graphs?
   • Open Is the projective rank of a graph’s complement in its entanglement-assisted spectrum? posed here

   The paperPDFDOICode

3. How much dimension does an $\epsilon$-approximation to a non-attained quantum correlation cost? A Diophantine answer for $K_5$

   Seth Douglas · version 1.0.2 · 24 September 2026

   The dimension cost of approximating a non-attained correlation for $K_5$ follows a Diophantine law. Version 1.0.2 shows that an open lemma conjectured in earlier versions is false and settles the drifted case.

   • Answered What does drift in the marginals cost when approximating the non-attained $K_5$ correlation?

   The paperDOICode

4. Exact rational certification of upper and lower quantum bounds for the I₃₃₂₂ Bell inequality

   Nidhal Mghirbi · Zenodo, 11 August 2026; first version 21 July 2026, titled “Proof-carrying exact quantum bounds for the I3322 Bell inequality”

   Finite rational proof objects for both sides of the I₃₃₂₂ quantum value: a Gram certificate on all words of length at most four proves $Q_c\le 0.250875385012$, and an explicit strategy of local dimension 499 with rational projectors has exact value 0.25087538451397653548…, so the two ends differ by less than $5\times {10}^{-10}$. The lower certificate also proves $Q_t>1/4$.

   • Answered How tightly can exact rational arithmetic enclose the I₃₃₂₂ quantum value?

   The paperDOI

5. Critical Bellman–Jacobi dynamics for reversible Bell functionals

   Nidhal Mghirbi · Zenodo, 10 September 2026; first version 12 August 2026

   For a parameterized class of three-setting binary Bell functionals, a strict wall gap and an exactly certifiable endpoint barrier give finite-dimensional nonattainment, attainment by an infinite-dimensional spatial strategy, and nonclosure; for I₃₃₂₂ this proves the Pál–Vértesi nonattainment conjecture. The theory also gives the constructive bound $D(\epsilon )=O(\log (1/\epsilon ))$.

   • Answered Does any finite-dimensional strategy reach the quantum maximum of the I₃₃₂₂ Bell inequality?
   • Answered Which three-setting binary Bell functionals reach their quantum maximum only in infinite dimension?

   The paperDOI

6. Alternating Bellman–Jacobi duality for three-setting binary Bell functionals

   Nidhal Mghirbi · Zenodo, 10 September 2026; first version 10 August 2026

   An exact reduction, for a six-parameter class of three-setting binary Bell functionals that contains I₃₃₂₂ and the Gigena–Kaniewski family, of the operator optimization to an alternating one-dimensional Jacobi problem with a Bellman storage dual. It gives the exact rational bound $\omega (I_{3322})\le 0.250875384626$, a two-sided window of width about $1.12\times {10}^{-10}$ with an independent lower certificate.

   • Answered How tightly can exact rational arithmetic enclose the I₃₃₂₂ quantum value?

   The paperDOI

7. The resource theory of complexness references in real quantum networks

   Seth Douglas · version 1.0.4 · 18 October 2026

   The resource theory in which the shared two-rebit reference of Weilenmann, Gisin and Sekatski is the unit. Version 1.0.4 proves that the unit copies itself exactly and corrects a false remark.

   The paperPDFDOICode

8. The frame bus: exact amplification and catalysis of complexness references in real quantum networks

   Seth Douglas · version 1.0.0 · 13 August 2026

   How real network parties without a shared convention for the complex unit can establish one by consuming pairwise reference states.

   The paperPDFDOICode

9. A repair price for contextuality, and its identity with Min-Rep

   Seth Douglas · version 1.0.0 · 10 August 2026

   The least number of outcome merges that restores a global section; on binary scenarios it is an affine reparametrization of a standard quantity in hardness of approximation.

   • Answered Is the least number of outcome merges that restores a global section a new measure of contextuality?

   The paperDOICode

10. Proof-Carrying Exact Bounds for the I₃₃₂₂ Bell Inequality

    Nidhal Mghirbi · release 2.0.0 · Zenodo, 20 July 2026 · a software record that carries the manuscript

    The first proof-carrying release: a rational certificate on the full word space of length at most four and an explicit dimension-499 strategy prove 0.25087538451 < $Q_t\le Q_c\le 0.2508753855$, an exact window narrower than ${10}^{-9}$, about 265 times narrower than the preceding exact level-three enclosure. The paper above is the later, sharper account.

    The paperDOI

• Answered Does any finite-dimensional strategy reach the quantum maximum of the I₃₃₂₂ Bell inequality?
• Answered Do the middle links of the Duan–Winter chain of zero-error quantities ever separate, and what are they on the pentagon?