Greenberger-Horne-Zeilinger (GHZ) states are quantum states involving at least three entangled particles. They are of fundamental interest in quantum information theory and have several applications in quantum communication and cryptography. Motivated by this, physicists have been designing various experiments to create high-dimensional GHZ states using multiple entangled particles. In 2017, Krenn, Gu and Zeilinger discovered a bridge between experimental quantum optics and graph theory. A large class of experiments to create a new GHZ state are associated with an edge-coloured edge-weighted graph having certain properties. Using this framework, Cervera-Lierta, Krenn, and Aspuru-Guzik proved using SAT solvers that through these experiments, the maximum dimension achieved is less than $3,4$ using $6,8$ particles, respectively. They further conjectured that using $n$ particles, the maximum dimension achievable is less than $\dfrac{n}{{2}}$ [Quantum 2022]. We make progress towards proving their conjecture by showing that the maximum dimension achieved is less than $\dfrac{n}{\sqrt{2}}$.

翻译：Greenberger-Horne-Zeilinger(GHZ)态是涉及至少三个纠缠粒子的量子态。这类量子态在量子信息论中具有基础性意义，并在量子通信与密码学领域有诸多应用。受此推动，物理学家们正在设计各类实验，利用多个纠缠粒子构建高维GHZ态。2017年，Krenn、Gu和Zeilinger发现了实验量子光学与图论之间的桥梁。大量用于生成新型GHZ态的实验与一类具备特定性质的边着色边加权图相关联。基于这一框架，Cervera-Lierta、Krenn和Aspuru-Guzik利用SAT求解器证明，通过这类实验，在分别使用6个和8个粒子时，所能达到的最大维数分别小于3和4。他们进一步猜想，在使用n个粒子时，所能达到的最大维数小于$\dfrac{n}{{2}}$[Quantum 2022]。我们通过证明最大能达到的维数小于$\dfrac{n}{\sqrt{2}}$，在证明该猜想的道路上取得了进展。