All 2-positive linear maps from M3 to M3 are decomposable

Abstract: Following an idea of Choi, we obtain a decomposition theorem for k-positive linear maps from Mm to Mn, where 2<=k<min{m,n}. As a consequence, we give an affirmative answer to Kye's conjecture (also solved independently by Choi) that every 2-positive linear map from M3 to M3 is decomposable.