Concrete Quantum Channels and Algebraic Structure of Abstract Quantum Channels
M. N. N. Namboodiri
TL;DR
The paper analyzes the algebraic structure of the set $QC(M_n)$ of quantum channels and its Holevo-form subset $HQC(M_n)$, revealing semigroup properties and an idempotent structure. It ties these channels to preconditioner maps in numerical linear algebra and to resource-destroying (idempotent) channels, with implications for coding-encoding problems in quantum information. Idempotents are characterized via associated stochastic matrices, and generalized invertibility within $HQC(M_n)$ is examined under composition. The work extends to infinite dimensions, constructing infinite-dimensional analogues of the preconditioner maps and exploring asymptotic behavior and capacity implications.
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.