论文标题
量子计算和基础中的MERMIN多面体
Mermin polytopes in quantum computation and foundations
论文作者
论文摘要
Mermin Square方案为与国家无关的上下文提供了一个简单的证据。在本文中，我们研究了从Mermin方案获得的polytopes $ \ text {mp}_β$，在一组上下文中由函数$β$参数。为了结合同构，有两种类型的多型$ \ text {mp} _0 $和$ \ text {mp} _1 $，具体取决于$β$的奇偶校验。我们的主要结果是对这两个多面体的顶点的分类。另外，我们描述了与多面体相关的图。 $ \ text {mp} _0 $的所有顶点结果是确定性的。该结果提供了一个新的拓扑证明，证明了在CHSH方案上表征非上下文分布的著名结果。 $ \ text {mp} _1 $可以看作是$λ$ - polytopes的非局部玩具版本，这是用于仿真通用量子计算的一类polytopes。在$ 2 $ qubit的情况下，我们使用$ \ text {mp} _1 $分类$λ$ - polytope的分解，其顶点是分类的，而$（2,3,2）$ bell scemario的非信号polytope则是众所周知的。
Mermin square scenario provides a simple proof for state-independent contextuality. In this paper, we study polytopes $\text{MP}_β$ obtained from the Mermin scenario, parametrized by a function $β$ on the set of contexts. Up to combinatorial isomorphism, there are two types of polytopes $\text{MP}_0$ and $\text{MP}_1$ depending on the parity of $β$. Our main result is the classification of the vertices of these two polytopes. In addition, we describe the graph associated with the polytopes. All the vertices of $\text{MP}_0$ turn out to be deterministic. This result provides a new topological proof of a celebrated result of Fine characterizing noncontextual distributions on the CHSH scenario. $\text{MP}_1$ can be seen as a nonlocal toy version of $Λ$-polytopes, a class of polytopes introduced for the simulation of universal quantum computation. In the $2$-qubit case, we provide a decomposition of the $Λ$-polytope using $\text{MP}_1$, whose vertices are classified, and the nonsignaling polytope of the $(2,3,2)$ Bell scenario, whose vertices are well-known.
