Criteria and new classes of k-positive maps
  We study k-positive maps on operators. Proofs are given to different positivity criteria. Special attention is on positive maps arising in the study of quantum information science. Results of other researchers are extended and improved. New classes of positive maps are constructed. Some open questions are answered. 
Jinchuan Hou, Chi-Kwong Li, Yiu-Tung Poon, Xiaofei Qi, and Nung-Sing   Sze
quant-ph math.FA