Entanglement in quantum graph states is intrinsically linked to rank-width, a graph complexity measure introduced by Oum and Seymour. In this work, we enable the preparation of maximally entangled deterministic graph states in constant depth by developing a general method to derive lower bounds on the rank-width of regular graphs from their edge expansion. By bridging edge-isoperimetric inequalities with the strong chromatic index and Jelínek's approach for lower bounding cut-rank, we systematically establish lower bounds for the rank-width of Cartesian products, including hypercubes, Hamming graphs, and grids. Extending this framework via Boolean function analysis, using a generalization of the Kahn-Kalai-Linial's Theorem, we strengthen the bounds for all Cartesian products by a non-trivial logarithmic factor. These methods result in the discovery of deterministic families of graphs on $n$ vertices with a provably maximum rank-width $Θ(n)$. Our results fill the previous gap in the literature for deterministic graph families of rank-width greater than $Θ(\sqrt{n})$.

翻译：量子图态中的纠缠本质上与秩宽相关，秩宽是由Oum和Seymour引入的一种图复杂度度量。本文通过发展一种从正则图的边扩展导出其秩宽下界的通用方法，实现了在恒定深度下制备最大纠缠确定性图态。通过将边等周不等式与强色指数及Jelínek的割秩下界方法相衔接，我们系统地建立了笛卡尔积（包括超立方体、汉明图和网格）的秩宽下界。利用布尔函数分析扩展这一框架，通过推广Kahn-Kalai-Linial定理，我们以非平凡的对数因子强化了所有笛卡尔积的下界。这些方法导致发现了$n$个顶点上具有可证明最大秩宽$Θ(n)$的确定性图族。我们的结果填补了文献中秩宽大于$Θ(\sqrt{n})$的确定性图族领域此前存在的空白。