Classical block designs are important combinatorial structures with a wide range of applications in Computer Science and Statistics. Here we give a new abstract description of block designs based on the arrow category construction. We show that models of this structure in the category of matrices and natural numbers recover the traditional classical combinatorial objects, while models in the category of completely positive maps yield a new definition of quantum designs. We show that this generalizes both a previous notion of quantum designs given by Zauner and the traditional description of block designs. Furthermore, we demonstrate that there exists a functor which relates every categorical block design to a quantum one.

翻译：经典区组设计是重要的组合结构，在计算机科学和统计学领域具有广泛的应用。本文基于箭头范畴构造给出了一种新的区组设计抽象描述。我们证明，在矩阵与自然数范畴中，该结构的模型能够恢复传统的经典组合对象，而在完全正映射范畴中的模型则导出了量子设计的新定义。研究表明，这一框架统一了Zauner提出的早期量子设计概念与经典区组设计的传统描述，并进一步证明存在一个将每个范畴区组设计关联至量子区组设计的函子。