广义约翰逊图的精确量子色数
On the exact quantum chromatic number of generalized Johnson graphs
- Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
- Anhui University（安徽大学）
- Nanyang Technological University（南洋理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究广义约翰逊图的量子色数，通过构造正交表示、结合特征值分析确定两类无穷族的精确量子色数，发现其经典色数随n指数增长、量子色数线性增长，存在指数级分离。
AI中文摘要：
量子色数是非局域博弈研究中的基础参数，用于刻画纠缠在分布式任务中提升性能的程度。本文研究广义约翰逊图的量子色数，通过构造模1正交表示得到其量子色数的一般上界；进一步分析这些图的最小特征值，结合得到的霍夫曼型下界与正交表示得到的上界，确定两类无穷族广义约翰逊图的精确量子色数；最后应用二元码的禁距定理，证明这些族的经典色数随n呈指数增长，而量子色数随n线性增长，展现出经典与量子色数间的指数级分离。
英文摘要：
The quantum chromatic number is a fundamental parameter in the study of nonlocal games, capturing the extent to which entanglement can improve performance in distributed tasks. In this paper, we investigate the quantum chromatic number of generalized Johnson graphs. By constructing modulus-one orthogonal representations, we obtain general upper bounds on their quantum chromatic numbers. We further analyze the smallest eigenvalue of these graphs. Combining the resulting Hoffman-type lower bounds with the upper bounds obtained from orthogonal representations, we determine the exact quantum chromatic numbers of two infinite families of generalized Johnson graphs. Finally, applying a forbidden-distance theorem for binary codes, we show that the classical chromatic numbers of these families grow exponentially with $n$, whereas their quantum chromatic numbers grow linearly. These families exhibit an exponential separation between the classical and quantum chromatic numbers.
