We obtain a complete characterization of eventually entanglement breaking (EEB) unital channels in terms of some simple commutation properties of their eigenvectors. Specifically, the following properties are equivalent for a unital channel: (i) it is EEB; (ii) it is eventually positive partial transpose (PPT); (iii) it is eventually mixed twisted dephasing; and (iv) its eigenvectors corresponding to non-zero eigenvalues commute with all its peripheral eigenvectors. Similar results hold for one-parameter semigroups of unital quantum channels, where continuity forces such semigroups to become primitive. We employ a matrix integral technique inspired by a result of Watrous. We also characterize the extreme points of the convex set of unital EB channels and, using this characterization and the theory of equiangular tight frames (ETFs), identify several extreme points beyond the twisted dephasing channels.We obtain a complete characterization of eventually entanglement breaking (EEB) unital channels in terms of some simple commutation properties of their eigenvectors. Specifically, the following properties are equivalent for a unital channel: (i) it is EEB; (ii) it is eventually positive partial transpose (PPT); (iii) it is eventually mixed twisted dephasing; and (iv) its eigenvectors corresponding to non-zero eigenvalues commute with all its peripheral eigenvectors. Similar results hold for one-parameter semigroups of unital quantum channels, where continuity forces such semigroups to become primitive. We employ a matrix integral technique inspired by a result of Watrous. We also characterize the extreme points of the convex set of unital EB channels and, using this characterization and the theory of equiangular tight frames (ETFs), identify several extreme points beyond the twisted dephasing channels.

math.OACross-listedarXiv:2609.24168Sep 22, 20260 citations

On quantum channels: extreme points, topology, and categorical properties

James Silva

Jun 15, 20260 citations

On the Extremality of the Tensor Product of Quantum Channels

James Miller S. T. da Silva

Completely positive and trace preserving (CPT) maps are important for Quantum Information Theory, because they describe a broad class of of transformations of quantum states. There are also two other related classes of maps, the unital completely positive (UCP) maps and the unital completely positive and trace preserving (UCPT) maps. For these three classes, the maps from a finite dimensional Hilbert space $X$ to another one $Y$ is a compact convex set and, as such, it is the convex hull of its extreme points. The extreme points of these convex sets are yet not well understood. In this article we investigate the preservation of extremality under the tensor product. We prove that extremality is preserved for CPT or UCP maps, but for UCPT it is not always preserved.Completely positive and trace preserving (CPT) maps are important for Quantum Information Theory, because they describe a broad class of of transformations of quantum states. There are also two other related classes of maps, the unital completely positive (UCP) maps and the unital completely positive and trace preserving (UCPT) maps. For these three classes, the maps from a finite dimensional Hilbert space $X$ to another one $Y$ is a compact convex set and, as such, it is the convex hull of its extreme points. The extreme points of these convex sets are yet not well understood. In this article we investigate the preservation of extremality under the tensor product. We prove that extremality is preserved for CPT or UCP maps, but for UCPT it is not always preserved.

arXiv:2305.05795May 9, 20230 citations

Factorizations of quantum channels

Jeremy Levick

arXiv:1808.05048Linear Algebra and its ApplicationsMay 7, 20182 citations

X

to another one

Y

is a compact convex set and, as such, it is the convex hull of its extreme points. The extreme points of these convex sets are yet not well understood. In this article we investigate the preservation of extremality under the tensor product. We prove that extremality is preserved for CPT or UCP maps, but for UCPT it is not always preserved.