Construction and testing of preconditioners of Toeplitz/block Toeplitz matrices using Korovkin's classic theorems of positive linear approximations are known. Later the map implementing preconditioners was observed to be a completely positive map, and this structure led to an abstract formulation of Korovkin-type theorems in a non-commutative setting. Interestingly enough, these preconditioner maps' properties satisfy the properties of an abstract quantum channel in quantum information theory. In this short article, this viewpoint is discussed by computing related quantities such as Kraus representation, channel capacity, fidelity etc. Moreover, the algebraic properties of the class of quantum channels are also discussed.

翻译：基于科罗夫金正线性逼近经典定理的托普利茨/块托普利茨矩阵预条件子的构造与检验已为学界所知。后续研究发现实现预条件子的映射是完备正映射，这一结构催生了非交换框架下科罗夫金型定理的抽象化表述。有趣的是，这些预条件子映射的性质恰好符合量子信息论中抽象量子通道的特性。本文通过计算Kraus表示、信道容量、保真度等相关量值对此视角展开讨论，并进一步探讨了量子通道类别的代数性质。