  :Hello there, thanks for dropping a note. 2-positive just means that the map remains positive after passing to 2 by 2 matrices. Same goes for n-positivity. I have to confess, in hindsight, that whole section is probably not well-written. The POV that ought to be emphasized, IMHO, is that completely positive maps directly generalize vector states. Mentioning 2-positivity is probably not appropriate.   