四玩家异或博弈中GHZ赤道完备性的严格局部问题阈值
A Sharp Local-Question Threshold for GHZ-Equatorial Completeness in Four-Player XOR Games
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中文总结 AI 辅助
该研究确定四玩家XOR博弈中，每个玩家主动问题数为3时存在GHZ赤道策略，4时不存在，得出4为严格局部问题阈值，证明了相关入射障碍的提升及分离博弈的反驳排除。
中文摘要 AI 辅助
我们确定了四玩家二元异或（XOR）博弈中，当交换算子值为1时，每个玩家所需的最小主动问题数量，此时该博弈不一定存在格林伯格-霍恩-蔡林格（GHZ）赤道实现。此类实现使用四量子比特GHZ态和赤道量子比特可观测量，将完美博弈简化为加性相位方程。我们证明，每个具有交换算子值1且每个玩家最多三个主动问题的四玩家XOR博弈都存在完美GHZ赤道策略。相反，我们构造了一个均匀八字句博弈，每个玩家有四个主动问题，其交换算子值为1但相位方程不一致。因此，四是严格的局部问题阈值。正结果通过将每个整数值入射障碍提升为有序非交换反驳得到，使用了原始电路、森林匹配和三元汉明几何。对于分离博弈，克莱因四群入射关系阻碍了相位系统，而偶子群范式以及一度和二度马格努斯系数排除了任意长度的反驳。
英文摘要
We determine the smallest number of active questions per player at which a four-player binary exclusive-or (XOR) game of commuting-operator value one need not admit a Greenberger--Horne--Zeilinger (GHZ) equatorial realization. Such a realization uses the four-qubit GHZ state and equatorial qubit observables, reducing perfect play to additive phase equations. We prove that every four-player XOR game with commuting-operator value one and at most three active questions per player has a perfect GHZ-equatorial strategy. Conversely, we construct a uniform eight-clause game with four active questions per player whose commuting-operator value is one but whose phase equations are inconsistent. Thus four is the sharp local-question threshold. The positive result follows by lifting every integral incidence obstruction to an ordered noncommutative refutation, using primitive circuits, forest matchings, and ternary Hamming geometry. For the separating game, a Klein four-group incidence relation obstructs the phase system, while an even-subgroup normal form and degree-one and degree-two Magnus coefficients exclude refutations of arbitrary length.
