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Linear Algebra and its Applications

Characterizations of completely positive trace preserving majorants

Mohsen Kian, M. Rostamian Delavar, Mario Krnić

Published Mar 3, 2026• 0 citations
Publisher
DOI
10.1016/j.laa.2026.02.035

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We give a useful new characterization of the set of all completely positive, trace-preserving (i.e., stochastic) maps from 2x2 matrices to 2x2 matrices. These conditions allow one to easily check any trace-preserving map for complete positivity. We also determine explicitly all extreme points of this set, and give a useful parameterization after reduction to a certain canonical form. This allows a detailed examination of an important class of non-unital extreme points which can be characterized as having exactly two images on the Bloch sphere. We also discuss a number of related issues about the images and the geometry of the set of stochastic maps, and show that any stochastic map on 2x2 matrices can be written as a convex combination of two "generalized" extreme points.