A new description of orthogonal bases
A new description of orthogonal bases
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DOI：
10.1017/s0960129512000047
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发表时间：
2008-10
影响因子：
0.5
通讯作者：
B. Coecke;Dusko Pavlovic;J. Vicary
中科院分区：
文献类型：
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作者：
B. Coecke;Dusko Pavlovic;J. Vicary
我们证明了有限维希尔伯特空间的正交基可以等价地刻画为FdHilb范畴中的交换的<$-Frobenius幺半群，它以有限维希尔伯特空间为对象，以连续线性映射为态射，并对幺半群结构张量积。当相应的交换的<$-Frobenius幺半群是特殊的时，基被严格正规化。因此，正交基和标准正交基的特征都没有提到向量，而只是根据范畴结构：运算的合成，张量积和†-函子。此外，这种表征可以在操作上解释，因为f-Frobenius结构允许基向量的克隆和删除。也就是说，我们通过依赖于它们被克隆和删除的能力来捕获基向量。由于这种能力将经典数据与量子数据区分开来，我们的结果对范畴量子力学具有重要意义。
We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative †-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 †-Frobenius monoid is special. Hence, both orthogonal and orthonormal bases are characterised without mentioning vectors, but just in terms of the categorical structure: composition of operations, tensor product and the †-functor. Moreover, this characterisation can be interpreted operationally, since the †-Frobenius structure allows the cloning and deletion of basis vectors. That is, we capture the basis vectors by relying on their ability to be cloned and deleted. Since this ability distinguishes classical data from quantum data, our result has important implications for categorical quantum mechanics.