Determine the quantum value of the CGLMP inequalities

Determine the exact quantum value of the CGLMP Bell inequality $I_d$ for arbitrary output dimension $d$, including a closed-form characterization of the optimal quantum strategy and value.

Background

The CGLMP family has a known classical value, but the standard maximally entangled strategy is not generally optimal. The paper notes that already for d=3d=3, a nonmaximally entangled strategy outperforms the maximally entangled one.

The authors therefore use BGD and the NPA hierarchy to obtain lower and upper numerical bounds rather than an exact formula for the quantum optimum. Establishing the value for all dimensions would resolve a central optimization problem for this widely studied family of Bell inequalities.

References

No closed-form expression for the quantum value of this random family is known.

— Efficiently Optimizing the Quantum Value of Bell Inequalities using Batched Gradient Descent  (2610.01699 - Xu et al., 1 Oct 2026) in Section 7, subsection “Random CGLMP network $(n_p,2,n_{\mathrm{out}})$”

We therefore treat CGLMP as a family without a known closed-form expression for $\omega_q$.

— Efficiently Optimizing the Quantum Value of Bell Inequalities using Batched Gradient Descent  (2610.01699 - Xu et al., 1 Oct 2026) in Section 7, subsection “Unsolved families of Bell inequalities,” paragraph “The CGLMP inequalities $(2,2,d)$”