AI 中文总结
本文解决多体GHZ-Mermin关联的忠实局域确定性模型的最小测量依赖问题，得到不同粒子数下的精确依赖阈值，其服从闭合定律，最优隐变量密度有闭合形式，可用于设备无关证书领域。
AI 中文摘要
Bell推导基于定域性、确定性和测量独立性三个假设。Hall[Phys. Rev. Lett. 105, 250404 (2010)]针对单态精确量化了第三个假设的代价，而在Kochen-Specker分析[Phys. Rev. A 84, 022102 (2011)]中，针对三体Mermin完美关联器的代价为1/3的放弃分数，留下了Mermin态本身最优模型的问题未解决。本文解决了该问题的忠实版本——即重现所有完整关联器且所有真子集边缘为零——并将其扩展到13个粒子。一个归约定理表明，忠实性约束是自由的，因此Hall仅基于关联器的阈值被提升为忠实值F(3)=1/3；通过整数算术在已证明的下界与显式构造之间挤压，精确验证的线性规划最优解给出F(5)=2/5、F(7)=4/9、F(9)=8/17、F(11)=16/33和F(13)=32/65，n=11之前的每个值在后续偶数大小中重复。所有计算点均服从闭合定律F=R/[2(R+1)]，其中R=2^floor((n-1)/2)为Mermin违背比，已证明的组合下界在每个计算核心上是紧的，且通用上限F≤1/2表明，在任何大小下，统计都不要求完全放弃测量独立性。最优隐变量密度具有闭合物理形式：制备态的受挫稳定子哈密顿量的语境基态上的均匀测度，该结构在三类纠缠的cluster态上得到了样本外验证。这些下界构成了多体设备无关证书的伪造阈值，以及量子关联的任何测量依赖解释必须满足的精确需求曲线。所有定理的完整证明均在正文和附录中给出。
英文摘要
Bell derivations rest on locality, determinism, and measurement independence. Hall [Phys. Rev. Lett. 105, 250404 (2010)] priced the third assumption exactly for the singlet state, and in the Kochen-Specker analysis of Phys. Rev. A 84, 022102 (2011) priced the four tripartite Mermin perfect correlators at a surrendered fraction of 1/3, leaving open the problem of an optimal model for the Mermin state itself. This paper solves the faithful version of that problem -- every full correlator reproduced and every proper-subset marginal vanishing -- and extends it to thirteen parties. A reduction theorem shows the faithfulness constraints are free, so Hall's correlator-only threshold is promoted to the faithful value, F(3) = 1/3; linear-programming optima, certified exactly by an integer-arithmetic squeeze between a proven lower bound and an explicit construction, then give F(5) = 2/5, F(7) = 4/9, F(9) = 8/17, F(11) = 16/33, and F(13) = 32/65, each value through n = 11 repeated at the following even size. All computed points obey the closed law F = R/[2(R+1)] with R = 2^floor((n-1)/2) the Mermin violation ratio, a proven combinatorial lower bound is tight on every computed core, and a universal ceiling F <= 1/2 shows the statistics never require total abandonment of measurement independence at any size. The optimal hidden-variable densities have a closed physical form: uniform measures on the contextual ground states of the prepared state's frustrated stabilizer Hamiltonian, a structure confirmed out of sample on cluster states in three entanglement classes. The floors constitute counterfeiting thresholds for multipartite device-independent certificates and an exact demand curve that any measurement-dependent account of quantum correlations must fund. Complete proofs of all theorems are given in the main text and appendices.
Comments9 pages, 1 table; all exact values machine-verifiable from archived code (DOI in paper)
