Positive maps that are not completely positive
Sixia Yu · Physical Review A · 2000
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.