Criteria Of Positivity For Linear Maps Constructed From Permutation Pairs

In this paper,we introduce a notion of property(C)for permutation pairs and show that a D-type map ΦD:Mn→Mn with D=(n-2)In+Pπ1+Pπ2induced by a pair{π1,π2) of permutations of(1,2,...,n)is positive if{π1,π2)has property(C).The property(C)is characterized for any{π1,π2),and an easy criterion is given for the case that π1=πP and π2=πq,where π is the permutation defined by π(i)=i+1mod n and1≤p<q≤n. We studied the case n=4intensively,and give a simple necessary and sufficient condition for{71,π2}to have the property(C).Furthermore,a necessary and sufficient condition for a, D-type map Φπ11,π2on4×4matrices constructed from a pair of arbitrary permutations {π1,π2] to be positive is obtained.