Positive Maps Constructed from Permutation Pairs

Abstract

A property (C) for permutation pairs is introduced. It is shown that if a pair {π1, π2} of permutations of (1, 2, · · ·, n) has property (C), then the D-type map $${\Phi _{{\pi _{1,}}{\pi _2}}}$$ on n × n complex matrices constructed from {π1, π2} is positive. A necessary and sufficient condition is obtained for a pair {π1, π2} to have property (C), and an easily checked necessary and sufficient condition for the pairs of the form {πp, πq} to have property (C) is given, where π is the permutation defined by π(i) = i + 1 mod n and 1 ≤ p < q ≤ n.