ENTANGLEMENT WITNESSES CONSTRUCTED BY PERMUTATION PAIRS

For n >= 3, we construct a class {W-n,W-pi 1,W-pi 2} of n(2) x n(2) hermitian matrices by the permutation pairs and show that, for a pair {pi(1), pi(2)} of permutations on (1, 2,..., n), W-n,W-pi 1,W-pi 2 is an entanglement witness of the n circle times n system if {pi(1), pi(2)} has the property (C). Recall that a pair {pi(1), pi(2)} of permutations of (1, 2,..., n) has the property (C) if, for each i, one can obtain a permutation of (1,..., i - 1, i+1,..., n) from (pi(1) (1),..., pi(1)(i - 1), pi(1)(i+1),..., pi(1)(n)) and (pi(2) (1),..., pi(2) (i - 1), pi(2)(i+1),..., pi(2) (n)). We further prove that W-n,W-pi 1,W-pi 2 is not comparable with W-n,W-pi,W- which is the entanglement witness constructed from a single permutation pi; W-n,W-pi 1,W-pi 2 is decomposable if pi(1)pi(2) = id or pi(2)(1) = pi(2)(2) = id. For the low dimensional cases n 2 {3, 4}, we give a sufficient and necessary condition on pi(1), pi(2) for W-n,W-pi 1,W-pi 2 to be an entanglement witness. We also show that, for n is an element of {3, 4}, W-n,W-pi 1,W-pi 2 is decomposable if and only if pi(1)pi(2) = id or pi(2)(1) = pi(2)(2) = id; W-3,W-pi 1,W-pi 2 is optimal if and only if (pi(1), pi(2)) = (pi, pi(2)), where pi = (2, 3, 1). As applications, some entanglement criteria for states and some decomposability criteria for positive maps are established.