The concept of the half density matrix is proposed. It unifies the quantum states that are described by density matrices and physical processes that are described by completely positive maps. With the help of the half-density-matrix representation of the Hermitian linear map, we show that every positive map that is not completely positive is a difference of two completely positive maps. A necessary and sufficient condition for a positive map that is not completely positive is also presented, which is illustrated by some examples.
Valid transformations between quantum states are necessarily described by completely positive maps, instead of just positive maps. Positive but not completely positive maps such as the transposition map cannot be implemented due to the existence of entanglement in composite quantum systems, but there are classes of states for which the positivity is guaranteed, e.g., states not correlated to other systems. this paper, we report on the concept of
N-copy extension of maps to quantitatively analyze the difference between positive maps and completely positive maps. We consider implementations of the action of positive but not completely positive maps on uncorrelated states by allowing an extra resource of consuming multiple copies of the input state and characterize the positive maps in terms of implementability with multiple copies. We show that by consuming multiple copies, the set of implementable positive maps becomes larger, and almost all positive maps are implementable with finite copies of an input state. The number of copies of the input state required to implement a positive map quantifies the degree by which a positive map violates complete positivity. We then analyze the optimal
N-copy implementability of a noisy version of the transposition map.