Completely positive, trace-preserving linear maps (so-called stochastic maps) are well-known to play a fundamental role in quantum information theory. Therefore, it is an important problem to find useful, easily checkable characterizations of the complete positivity of linear transformations on the space of all bounded linear operators acting on the complex Hilbert space . In this paper, the authors present such a characterization in the case when is two-dimensional. Although this case may seem rather special, it is of considerable interest because of the role it plays in quantum computation and quantum communication.
It is known that if a linear transformation on the space of complex matrices is positivity-preserving and trace-preserving, then the matrix representation of can be reduced, via a change of basis in , to a rather special form: the entry in the left upper corner is 1 and it has nonzero entries only in the first column and in the diagonal. The main result of the paper gives a rather computable criterion for a linear transformation on with a matrix representation mentioned above to be completely positive. The criterion is the existence of a contractive solution of a matrix equation of the form where are given matrices with entries which are algebraic expressions of the entries of the mentioned matrix representation of .
Next, the authors determine explicitly all extreme points of the set of all stochastic maps on . 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. The authors 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 can be written as a convex combination of two “generalized” extreme points. Several interesting examples are presented.