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

TL;DR

This paper provides a new characterization and explicit description of all completely positive, trace-preserving maps on 2x2 matrices, including their extreme points and geometric structure, facilitating easier analysis of such quantum channels.

Contribution

It introduces a new characterization, explicit extreme points, and a canonical form for stochastic maps on 2x2 matrices, advancing understanding of quantum channels.

Findings

Derived a new characterization for complete positivity of 2x2 stochastic maps.

Explicitly determined all extreme points of the set of these maps.

Showed any such map can be expressed as a convex combination of two generalized extreme points.

Abstract

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.