An analysis of completely positive trace-preserving maps on M2

Mary Beth Ruskai, Stanislaw Szarek, Elisabeth Werner

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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.

Original languageEnglish (US)
Pages (from-to)159-187
Number of pages29
JournalLinear Algebra and Its Applications
Issue number1-3
StatePublished - May 15 2002

Bibliographical note

Funding Information:
∗ Corresponding author. E-mail addresses: (M.B. Ruskai),; (S. Szarek), (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
  • States
  • Stochastic maps


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