AI 中文总结
研究在最简单贝尔场景中完整量子集的刻画问题，通过分析对称双倾斜 CHSH 泛函，证明没有有限的 NPA 层级能精确描述该量子集，指出基于固定有限单词列表的 NPA 松弛严格包含量子集，其非量子行为在局部确定性行为处累积。
AI 中文摘要
纳瓦斯克斯 - 皮罗尼奥 - 阿辛（NPA）层级为量子行为提供了标准的半定外逼近。即使在双方各有两个二元测量的二分场景中，是否任何有限层级都能等于量子集仍是未解决的问题。我们证明没有有限层级是精确的。对于对称双倾斜 CHSH 泛函\(h_α = A_0B_0 + A_0B_1 + A_1B_0 - A_1B_1 + α(A_0 + B_0)\)，设\(T = 1 - α\)。其量子最大值满足\([\omega_{\rm Q}(1 - T)-(4 - 2T)]/T^3→4/3\)。通过相应边界重标度，正算子\(\omega_{\rm Q}(1 - t^2)I - H_t\)的显式期望收敛到莫茨金多项式。有界的固定层级误差会使莫茨金多项式加非负常数成为平方和，这是不可能的。所以基于测量投影器中固定有限单词列表的每个标准 NPA 松弛都严格包含完整量子集，且其非量子行为在局部确定性行为处累积。因此，CHSH 及所有单侧倾斜 CHSH 最大值的有限层级精确性并不扩展到最小场景中完整量子集的精确有限层级描述。
英文摘要
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.
CommentsSubmitted to Physical Review Letters on 14 July 2026. Closely overlapping preprints arXiv:2607.13762 and arXiv:2607.13774 appeared on 15 July 2026. This work was completed independently beforehand and gives a distinct fully analytical proof with broader behavior-set consequences
