We give a useful new characterization of the set of all completely positive, trace-preserving maps φ : M2 → M2 from which one can 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 that 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 M2 can be written as a convex combination of two "generalized" extreme points.
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∗ Corresponding author. E-mail addresses: email@example.com (M.B. Ruskai), firstname.lastname@example.org; email@example.com (S. Szarek), firstname.lastname@example.org (E. Werner). 1 Partially supported by the National Security Agency (NSA) and Advanced Research and Development Activity (ARDA) under Army Research Office (ARO) contract DAAG55-98-1-0374 and by the National Science Foundation under Grant numbers DMS-9706981 and DMS-0074566. 2 Partially supported by a Grant from the National Science Foundation. 3 Partially supported by a Grant from the National Science Foundation and by a NATO Collaborative Linkage Grant. 4 Tel.: +216-368-2901; fax: +216-368-5163.
- Bloch sphere
- Completely positive maps
- Noisy channels
- Quantum communication
- Stochastic maps