AI 中文总结
本文针对n量子比特泡利群，引入非contextuality属性概念，证明两量子比特时其存在非contextuality属性，而n≥2时稳定子态上布尔值框架函数为常数，明确其contextuality源于辛极空间几何。
AI 中文摘要
n量子比特泡利群是大多数量子应用的核心要素，涵盖计算、纠错、基准测试及模拟等领域。尽管仅由一组离散算子构成，它却展现出量子理论的诸多典型特征，包括 contextuality，该特征已被证实是各类量子优势的关键资源。本文拓展了相关分析，引入「非 contextuality 属性」的概念，其不存在性可证明科亨-施佩克尔定理，与基于估值不存在性的常见论证类似且对其进行了推广。我们将该概念与布尔值框架函数的存在性建立形式关联，并针对n量子比特泡利群刻画了所有此类框架函数。对于两量子比特，我们证明泡利群存在非 contextuality 属性，即便其不存在估值；进一步证明这是n≥2时唯一非平凡的此类情况，方法是证明稳定子态上的任意布尔值框架函数在量子比特数超过2时为常数。我们还对n量子比特泡利群背后的辛理论开展类似分析，该理论中所有n值均存在非恒定布尔值框架函数，但仅以受限形式存在。相比之下，这表明n量子比特泡利群中的 contextuality 不仅是泡利群作为其底层辛向量空间表示的投影性质的结果，更是辛极空间本身几何的结果。从几何角度看，本文结果确定了二元仿射-辛空间中所有极大迷向平层的卡梅伦-利布勒集合。
英文摘要
The $n$-qubit Pauli group is an essential ingredient to most quantum applications, from computing and error correction to benchmarking and simulation. Despite comprising merely a discrete set of operators, it exhibits many quintessential features of quantum theory, including contextuality, which has been identified as a key resource to quantum advantage in a variety of different flavours. Here, we extend this analysis, introducing the notion of a `noncontextual property' whose nonexistence proves the Kochen-Specker theorem, similarly to and generalising common arguments based on the nonexistence of valuations. We relate this notion formally to the existence of Boolean-valued frame functions, and characterise all such frame functions in the case of the $n$-qubit Pauli group. For two qubits, we show that the Pauli group admits noncontextual properties, despite admitting no valuations. We then establish this as the only nontrivial such case with $n\geq 2$, by proving that any Boolean-valued frame function on stabiliser states is constant for more than two qubits. We also perform a similar analysis for the symplectic theory underlying the $n$-qubit Pauli group, for which nonconstant Boolean-valued frame functions exist for all $n$, yet only in restricted form. By comparison, this shows that contextuality in the $n$-qubit Pauli group is not only a consequence of the projective nature of the Pauli group as a representation of its underlying symplectic vector space, but of the geometry of symplectic polar spaces itself. In geometric terms, our result determines all Cameron-Liebler sets of maximal totally isotropic flats in the binary affine-symplectic space.
Comments4+19 pages, 1 figure, 2 tables; comments welcome
