Greenberger-Horne-Zeilinger (GHZ) states are quantum states involving at least three entangled particles. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. They are of fundamental interest in quantum information theory, and the construction of such states of high dimension has various applications in quantum communication and cryptography. Krenn, Gu and Zeilinger discovered a correspondence between a large class of quantum optical experiments which produce GHZ states and edge-weighted edge-coloured multi-graphs with some special properties called the \emph{GHZ graphs}. On such GHZ graphs, a graph parameter called \emph{dimension} can be defined, which is the same as the dimension of the GHZ state produced by the corresponding experiment. Krenn and Gu conjectured that the dimension of any GHZ graph with more than $4$ vertices is at most $2$. An affirmative resolution of the Krenn-Gu conjecture has implications for quantum resource theory. On the other hand, the construction of a GHZ graph on a large number of vertices with a high dimension would lead to breakthrough results. In this paper, we study the existence of GHZ graphs from the perspective of the Krenn-Gu conjecture and show that the conjecture is true for graphs of vertex connectivity at most 2 and for cubic graphs. We also show that the minimal counterexample to the conjecture should be $4$-connected. Such information could be of great help in the search for GHZ graphs using existing tools like PyTheus. While the impact of the work is in quantum physics, the techniques in this paper are purely combinatorial, and no background in quantum physics is required to understand them.

翻译：Greenberger-Horne-Zeilinger (GHZ) 态是涉及至少三个纠缠粒子的量子态。它们在量子信息理论中具有根本性的意义，且构建此类高维态在量子通信和密码学中具有多种应用。Krenn、Gu和Zeilinger发现了一大类能产生GHZ态的量子光学实验与一类具有特殊性质的边赋权边着色多重图（称为 \emph{GHZ 图}）之间的对应关系。在此类GHZ图上，可以定义一个称为 \emph{维度} 的图参数，该维度与对应实验所产生的GHZ态的维度相同。Krenn和Gu猜想，任何顶点数超过 $4$ 的GHZ图的维度至多为 $2$。肯定地解决Krenn-Gu猜想对量子资源理论具有重要意义。另一方面，在大量顶点上构建具有高维度的GHZ图将带来突破性的成果。在本文中，我们从Krenn-Gu猜想的视角研究GHZ图的存在性，并证明该猜想对于顶点连通度至多为2的图以及三次图成立。我们还证明了该猜想的最小反例应是 $4$-连通的。这些信息可能对利用PyTheus等现有工具搜索GHZ图有极大帮助。尽管本工作的影响在于量子物理学，但本文所使用的技术是纯组合的，理解它们无需量子物理学的背景知识。