Bożejko, Marek (ed.) et al., Quantum probability. Papers presented at the 25th QP conference on quantum probability and related topics, Będlewo, Poland, June 20–26, 2004. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 73, 347-356 (2006).
Let be the algebra of matrices with complex entries. For in , we have
Thus has the structure of a -algebra. A matrix is positive iff
A linear map is called positive if is a positive matrix for every positive matrix . If , is called -positive map (resp., -copositive map) whenever (resp., is a positive element in for every positive in . If is -positive (resp., -copositive) for every , then is called completely positive (resp., completely copositive).
A positive map which is a sum of completely positive and completely copositive maps is called decomposable. It is known that if , then every positive map is decomposable. Given an extremal unital positive map , the authors construct concrete maps and which give a decomposition of . They also show that in most cases this decomposition is unique.