正交基的一种新描述
A new description of orthogonal bases
摘要：
我们表明，有限维希尔伯特空间的正交基可以等价地表征为一个可交换的匕首Frobenius幺半群的范畴FdHilb，它有有限维希尔伯特空间作为对象和连续线性映射作为态射，和张量积的幺半群结构。当相应的交换匕首-Frobenius幺半群是特殊的时，该基础被规范化。因此，正交基和标准正交基可以仅根据运算的合成和张量积来公理化，而不需要任何对底层向量空间的显式引用。这个公理化还承认一个操作解释，因为乘法复制基向量，而计数器一致地删除它们。也就是说，与量子数据相比，我们依赖于克隆和删除经典数据的独特能力来捕获基向量。因此，我们的结果对范畴量子力学有重要的意义。
英文摘要：
We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative dagger-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative dagger-Frobenius monoid is special. Hence orthogonal and orthonormal bases can be axiomatised in terms of composition of operations and tensor product only, without any explicit reference to the underlying vector spaces. This axiomatisation moreover admits an operational interpretation, as the comultiplication copies the basis vectors and the counit uniformly deletes them. That is, we rely on the distinct ability to clone and delete classical data as compared to quantum data to capture basis vectors. For this reason our result has important implications for categorical quantum mechanics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
