This article analyzes the algebraic structure of the set of all quantum channels and its subset consisting of quantum channels that have Holevo representation. The regularity of these semigroups under composition of mappings are analysed. It is also known that these sets are compact convex sets and, therefore, rich in geometry. An attempt is made to identify generalized invertible channels and also the idempotent channels. When channels are of the Holevo type, these two problems are fully studied in this article. The motivation behind this study is its applicability to the reversibility of channel transformations and recent developments in resource-destroying channels, which are idempotents. This is related to the coding-encoding problem in quantum information theory. Several examples are provided, with the main examples coming from pre-conditioner maps which assigns preconditioners to matrices, in numerical linear algebra.Thus the known pre-conditioner maps are viewd as a quantum-channel in finite dimentions.

翻译：本文分析了所有量子信道构成的集合及其具有Holevo表示的子集的代数结构，研究了这些映射复合下半群的正则性。已知这些集合是紧凸集，因此具有丰富的几何性质。我们尝试识别广义可逆信道及幂等信道。对于Holevo型信道，本文对这两个问题进行了充分研究。此项研究的动机源于信道变换的可逆性及其在资源破坏信道（幂等信道）最新发展中的应用，这与量子信息理论中的编码-解码问题相关。文中提供了若干实例，主要实例源自数值线性代数中为矩阵分配预处理子的预条件子映射。因此，已知的预条件子映射可被视为有限维量子信道。