No Finite NPA Level Characterizes the Complete Quantum Set in the Simplest Bell Scenario

Abstract

The Navascués--Pironio--Acín (NPA) hierarchy gives the standard semidefinite outer approximations to quantum behaviors. Whether \emph{any} finite level can already equal the quantum set has remained open even in the bipartite scenario with two binary measurements per party. We demonstrate that \emph{no finite level} is exact. For the symmetric doubly tilted CHSH functional $h_α=A_0B_0+A_0B_1+A_1B_0-A_1B_1+α(A_0+B_0)$, set $T=1-α$. Its quantum maximum satisfies $[ω_{\rm Q}(1-T)-(4-2T)]/T^3\to4/3$, whereas every fixed NPA level satisfies $[ω_L(1-T)-ω_{\rm Q}(1-T)]/T^3\to+\infty$. Under the corresponding boundary rescaling, an explicit expectation of the positive operator $ω_{\rm Q}(1-t^2)I-H_t$ converges to the Motzkin polynomial. A bounded fixed-level error would therefore make the Motzkin polynomial plus a nonnegative constant a sum of squares, which is impossible. Consequently, every standard NPA relaxation based on a fixed finite list of words in the measurement projectors strictly contains the complete quantum set, and its nonquantum behaviors accumulate at a local deterministic behavior. Thus, the finite-level exactness of CHSH and all one-sided tilted CHSH maxima does not extend to an exact finite-level description of the complete quantum set in the minimal scenario.