Perfect matchings and Quantum physics: Bounding the dimension of GHZ states

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 $\dfrac{n}{\sqrt{2}}$.