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：完美匹配和量子物理学：限制 GHZ 态的维度
翻译后的摘要：
Greenberger-Horne-Zeilinger （GHZ）态是涉及至少三个纠缠的粒子的量子态。它们在量子信息理论中具有基础性的重要性，并在量子通信和密码学中有多种应用。受此启发，物理学家一直在设计各种实验，使用多个纠缠粒子来创建高维 GHZ 态。2017 年，Krenn、Gu 和 Zeilinger 发现量子光学实验和图论之间的桥梁。创建新的 GHZ 态的大部分实验都与具有一定属性的边染色边权重图相关联。使用这个框架，Cervera-Lierta、Krenn 和 Aspuru-Guzik 利用 SAT 求解器证明，通过这些实验，使用 $6$、$8$ 粒子，最大维度分别小于 $3,4$。他们进一步猜想，使用 $n$ 个粒子，最大可达到的维度小于 $\dfrac{n}{{2}}$ [Quantum 2022]。我们通过展示实现的最大维度小于 $\dfrac{n}{\sqrt{2}}$，从而在证明他们的猜想方面取得了进展。
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