The Cohomology of Non-Locality and Contextuality
The Cohomology of Non-Locality and Contextuality
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DOI：
10.4204/eptcs.95.1
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发表时间：
2012-01-01
影响因子：
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通讯作者：
Barbosa, Rui Soares
中科院分区：
文献类型：
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作者：
Abramsky, Samson;Mansfield, Shane;Barbosa, Rui Soares
在亚当·勃兰登堡（Adam Brandenburger）之前的一篇论文中，我们使用层理论来分析非定域性和上下文性的结构。在此基础上，我们证明了非定域性和上下文性现象可以用全局截面存在的障碍来精确刻画，我们的目标是在此基础上，利用层上同调这一强有力的工具来研究非定域性和上下文性的结构.我们使用切赫上同调的阿贝尔预层来自支持的概率模型，被视为一个兼容的家庭的分布，为了定义一个上同调障碍的家庭作为一定的上同调类。如果族具有全局截面，则此类将消失。因此，障碍物的不消失提供了足够的（但不是必要的）条件的模型是上下文。我们表明，对于一些突出的例子，包括PR盒，GHZ状态，Peres-Mermin幻方，和18矢量配置由于Cabello等人。在四维中给出了Kochen-Specker定理的证明，障碍不会消失，从而产生上下文的上同调见证。
In a previous paper with Adam Brandenburger, we used sheaf theory to analyze the structure of non-locality and contextuality. Moreover, on the basis of this formulation, we showed that the phenomena of non-locality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections.Our aim in the present work is to build on these results, and to use the powerful tools of sheaf cohomology to study the structure of non-locality and contextuality. We use the Cech cohomology on an abelian presheaf derived from the support of a probabilistic model, viewed as a compatible family of distributions, in order to define a cohomological obstruction for the family as a certain cohomology class. This class vanishes if the family has a global section. Thus the non-vanishing of the obstruction provides a sufficient (but not necessary) condition for the model to be contextual.We show that for a number of salient examples, including PR boxes, GHZ states, the Peres-Mermin magic square, and the 18-vector configuration due to Cabello et al. giving a proof of the Kochen-Specker theorem in four dimensions, the obstruction does not vanish, thus yielding cohomological witnesses for contextuality.