论文标题
量子对称性与非局部对称性
Quantum symmetry vs nonlocal symmetry
论文作者
论文摘要
我们介绍了图$ g $的非本地对称性的概念，该概念定义为$ g $ - automormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormormorphism Game的概念，该量子无法经典地生产。量子组理论和量子信息之间的最新连接表明，该游戏的量子相关性对应于$ c（\ text {qut}（g））$上的奇特状态 - $ g $的量子自动形态群体上函数的代数。这也使我们还可以为任何量子置换组定义非局部对称性。我们研究了这与量子对称性的概念之间的差异和相似性，定义为$ c（\ text {qut}（g））$的非交换性。粗略地说，量子对称性与非本地对称性可以分别视为我们对现实观察的现实模型与非经典性的非经典性。 我们表明，量子对称性是必要的，但不足以进行非局部对称性。特别是，我们表明，五个顶点上的完整图是具有非本地对称性的五个或更少顶点上的唯一连接的图形，尽管其他十几个具有量子对称性。特别是这表明量子对称组在四个点上，$ s_4^+$，没有表现出非本地对称性，回答了文献中的问题。与量子对称性相反，我们表明两个不结合的经典自身形态不能保证非本地对称性。但是，三个截然不同的自动形态确实足够了。我们还为$ |γ| $元素上的有限的Abelian Group $γ$和置换$π$构建的量子排列矩阵结构。计算证据表明，对于增加大小的循环群，几乎所有排列$π$都会导致非本地对称性。令人惊讶的是，当使用$ \ mathbb {z} _2^3 $时，构造永远不会导致非局部对称性。我们还研究在何种条件下采用工会或图形产品时会出现非局部对称性。
We introduce the notion of nonlocal symmetry of a graph $G$, defined as a winning quantum correlation for the $G$-automorphism game that cannot be produced classically. Recent connections between quantum group theory and quantum information show that quantum correlations for this game correspond to tracial states on $C(\text{Qut}(G))$ -- the algebra of functions on the quantum automorphism group of $G$. This allows us to also define nonlocal symmetry for any quantum permutation group. We investigate the differences and similarities between this and the notion of quantum symmetry, defined as non-commutativity of $C(\text{Qut}(G))$. Roughly speaking, quantum symmetry vs nonlocal symmetry can be viewed respectively as non-classicality of our model of reality vs non-classicality of our observation of reality. We show that quantum symmetry is necessary but not sufficient for nonlocal symmetry. In particular, we show that the complete graph on five vertices is the only connected graph on five or fewer vertices with nonlocal symmetry, despite a dozen others having quantum symmetry. In particular this shows that the quantum symmetric group on four points, $S_4^+$, does not exhibit nonlocal symmetry, answering a question from the literature. In contrast to quantum symmetry, we show that two disjoint classical automorphisms do not guarantee nonlocal symmetry. However, three disjoint automorphisms do suffice. We also give a construction of quantum permutation matrices built from a finite abelian group $Γ$ and a permutation $π$ on $|Γ|$ elements. Computational evidence suggests that for cyclic groups of increasing size almost all permutations $π$ result in nonlocal symmetry. Surprisingly, the construction never results in nonlocal symmetry when $\mathbb{Z}_2^3$ is used. We also investigate under what conditions nonlocal symmetry arises when taking unions or products of graphs.
