An Analysis of Completely-Positive Trace-Preserving Maps on 2x2 Matrices

Details An Analysis of Completely-Positive Trace-Preserving Maps on 2x2 Matrices An Analysis of Completely-Positive Trace-Preserving Maps on 2x2 Matrices

2000-12-31

arxiv.org/abs/quant-ph/0101003v2

Lin. Alg. Appl. 347, 159--187 (2002)

Physics

Quantum Physics

A significantly expanded version of quant-ph/0005004. 34 pages, 3 figures Final version accepted for publication in Lin. Alg.

Scientific paper

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.

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