On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras (Q1318204)

Let \(M_ n\) denote the algebra of \(n \times n\) complex matrices. The positive and completely positive maps of \(M_ n\) have been studied extensively. Let \(I\) denote the identity of \(M_ n\). A positive map \(\varphi\) is stochastic if \(\varphi (I)=I\). Further, \(\varphi\) is called doubly stochastic if \(\varphi\) is stochastic and is trace preserving. The authors identify the extremal doubly stochastic completely positive maps of \(M_ 2\) and illustrate some of the classification problems for maps of \(M_ n\) with \(n\geq 3\). They also consider the extreme points of the doubly stochastic completely positive maps of \(M_ n\) for \(n=2,3,4\).