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An analysis of completely-positive trace-preserving maps on ${ℳ}_{2}$. (English) Zbl 1032.47046

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 $B\left(H\right)$ of all bounded linear operators acting on the complex Hilbert space $H$. In this paper, the authors present such a characterization in the case when $H$ 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 ${\Phi }$ on the space ${M}_{2}$ of $2×2$ complex matrices is positivity-preserving and trace-preserving, then the $4×4$ matrix representation of ${\Phi }$ can be reduced, via a change of basis in ${ℂ}^{2}$, 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 ${M}_{2}$ 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 $A=BXC$ where $A,B,C$ are given $2×2$ matrices with entries which are algebraic expressions of the entries of the mentioned matrix representation of ${\Phi }$.

Next, the authors determine explicitly all extreme points of the set of all stochastic maps on ${M}_{2}$. 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 ${M}_{2}$ can be written as a convex combination of two “generalized” extreme points. Several interesting examples are presented.

##### MSC:
 47L07 Convex sets and cones of operators 81P68 Quantum computation 15A99 Miscellaneous topics in linear algebra 46L30 States of ${C}^{*}$-algebras 46L60 Applications of selfadjoint operator algebras to physics 94A40 Channel models (including quantum)