Characterize the quantum correlation set

Characterize the set of quantum behaviors realizable over arbitrary finite-dimensional local Hilbert spaces by deriving a tractable exact formulation for that set and, consequently, an exact general method for computing the quantum value of a Bell inequality.

Background

The paper defines the quantum value as the supremum of a Bell functional over behaviors generated by finite-dimensional quantum strategies. Numerical methods such as the see-saw method and batched gradient descent provide lower bounds, while the NPA hierarchy provides outer upper bounds.

A general tractable exact characterization of the quantum behavior set would resolve the fundamental optimization difficulty underlying the numerical methods discussed in the paper. The problem is especially significant because the quantum value may be uncomputable in general and the NPA hierarchy converges to the commuting-operator value rather than necessarily to the finite-dimensional tensor-product value.

References

These random instances have no known closed-form classical or quantum value.

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

The difficulty is that $\mathcal Q$ admits no known tractable characterization, so no tractable exact convex formulation is known in general.

— Efficiently Optimizing the Quantum Value of Bell Inequalities using Batched Gradient Descent  (2610.01699 - Xu et al., 1 Oct 2026) in Section 2, subsection “Existing optimizers for the quantum value”