  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.
