同步相关阵与Connes嵌入猜想
Synchronous correlation matrices and Connes' embedding conjecture
摘要：
在最近的一篇论文中，引入了同步量子相关阵的概念，并证明这些矩阵对应于某些C* 代数上的迹。特别是，同步相关阵出现在他们的研究各种版本的量子色数的图形和其他量子版本的图论参数。在本文中，我们进一步发展这些想法，专注于同步相关阵和微观状态之间的关系。我们证明了Connes嵌入猜想等价于两族同步量子相关阵相等。证明了如果Connes的嵌入猜想有肯定的答案，则对任何图，迹秩和投射秩都是相等的。然后，我们将这些结果应用到更一般的非本地游戏。
英文摘要：
In a recent paper, the concept of synchronous quantum correlation matrices was introduced and these were shown to correspond to traces on certain C*-algebras. In particular, synchronous correlation matrices arose in their study of various versions of quantum chromatic numbers of graphs and other quantum versions of graph theoretic parameters. In this paper we develop these ideas further, focusing on the relations between synchronous correlation matrices and microstates. We prove that Connes' embedding conjecture is equivalent to the equality of two families of synchronous quantum correlation matrices. We prove that if Connes' embedding conjecture has a positive answer, then the tracial rank and projective rank are equal for every graph. We then apply these results to more general non-local games.
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