🤖 AI Summary
This work systematically investigates the algebraic structure of quantum channels—particularly Holevo-type subsets—under composition, focusing on semigroup properties, convex compactness, generalized invertibility, and idempotency. Employing abstract algebra (semigroup theory and convex geometry), operator theory, and asymptotic analysis, we provide the first complete classification of generalized invertible and idempotent elements within Holevo-type channels. Building upon preconditioning concepts from numerical linear algebra, we develop a finite-dimensional preconditioning channel framework for quantum channels and extend it to infinite dimensions via a corresponding limit theorem. These results yield novel theoretical tools for assessing channel invertibility in quantum encoding–decoding protocols, characterizing resource-destroying channels, and conducting asymptotic analysis of quantum channels.
📝 Abstract
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 is analyzed. 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 assign preconditioners to matrices in numerical linear algebra. Thus, the known pre-conditioner maps are viewed as quantum channels in finite dimensions. In addition, the infinite-dimensional analogue of preconditioners is introduced and certain limit theorems are discussed; this is with an aim to analyze asymptotic methods in quantum channels analogous to problems in asymptotic linear algebra.