2026/07/16 by Anubhav Chaturvedi
#quant-ph
The Navascués--Pironio--Acín (NPA) hierarchy gives the standard semidefinite outer approximations to quantum behaviors. Whether 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 no finite level is exact. For the symmetric doubly tilted CHSH functional hα=A0B0+A0B1+A1B0-A1B1+α(A0+B0), set T=1-α. Its quantum maximum satisfies [ω\rm Q(1-T)-(4-2T)]/T3→4/3, whereas every fixed NPA level satisfies [ωL(1-T)-ω\rm Q(1-T)]/T3→+∞. Under the corresponding boundary rescaling, an explicit expectation of the positive operator ω\rm Q(1-t2)I-Ht 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.