Universal Structure of Nonlocal Operators for Deterministic Navigation and Geometric Locking

We establish a universal geometric framework that transforms the search for optimal nonlocal operators from a combinatorial black box into a deterministic predict-verify operation. We discover that the principal eigenvalue governing nonlocality is rigorously dictated by a low-dimensional manifold parameterized by merely two fundamental angular variables, $θ$ and $ϕ$, whose symmetry leads to further simplification. This geometric distillation establishes a precise mapping connecting external control parameters directly to optimal measurement configurations. Crucially, a comparative analysis of the geometric angles against the principal eigenvalue spectrum, including its magnitude, susceptibility, and nonlocal gap, reveals a fundamental dichotomy in quantum criticality. While transitions involving symmetry sector rotation manifest as geometric criticality with drastic operator reorientation, transitions dominated by strong anisotropy exhibit geometric locking, where the optimal basis remains robust despite clear signatures of phase transitions in the spectral indicators. This distinction offers a novel structural classification of quantum phase transitions and provides a precision navigation chart for Bell experiments.

💡 Research Summary

This paper introduces a universal geometric framework that transforms the notoriously hard problem of optimizing non‑local operators (NLOs) for Bell‑type inequalities into a deterministic predict‑verify procedure. The authors show that the principal eigenvalue |λ₁|, which quantifies the strength of non‑locality, is completely determined by a low‑dimensional manifold parameterized by only two spherical angles, θ and ϕ. By mapping each local measurement operator onto a unit vector on the Bloch sphere, the normalization constraints are automatically satisfied and intrinsic symmetries (parity, global rotations, etc.) impose simple relations between operator pairs. For a unit cell of size u=1 the optimal pair reduces to a₁=