Special classes of completely positive linear maps

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O'Meara, Corey Patrick
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University of Guelph

Completely positive linear maps between 'C'*-algebras were originally developed in the 1950's as a special case of positive linear operators between matrix algebras. Within the last two decades, mathematical physicists have determined that completely positive maps play a crucial role in quantum information theory as structures which model information transfer between quantum systems. In this thesis we analyze two main classes of completely positive linear maps: the Schur maps which arise from the Schur matrix product and maps which are equal to their adjoint. In the analysis of Schur maps, we prove that many of the geometric properties of the convex set of correlation matrices may be derived from analysis of the set of Schur maps We then give several necessary and sufficient conditions on characterizing self-dual completely positive linear maps. In doing so, we completely characterize the extreme points of the convex set of 2 * 2 unital self-dual completely positive trace-preserving linear maps. Finally, we use the concept of a conditional expectation to provide a general framework for some of the special classes of completely positive linear maps.

completely positive linear maps, matrix algebra, quantum information theory, Schur maps, self-dual