Entanglement Witnesses Constructed By Permutation Pairs

For n ≥ 3, we construct a class $$\left\{{{W_{n,{\pi _1},{\pi _2}}}} \right\}$$ of n2 × n2 hermitian matrices by the permutation pairs and show that, for a pair {π1, π2} of permutations on (1, 2, …, n), $${{W_{n,{\pi _1},{\pi _2}}}}$$ is an entanglement witness of the n ⊗ n system if {π1, π2} has the property (C). Recall that a pair {π1, π2} 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 (π1 (1), …, π1 (i − 1), π1(i + 1), …, π1(n)) and (π2(1), …, π2(i − 1), π2(i + 1), …, π2(n)). We further prove that $${{W_{n,{\pi _1},{\pi _2}}}}$$ is not comparable with Wn,π, which is the entanglement witness constructed from a single permutation π; $${{W_{n,{\pi _1},{\pi _2}}}}$$ is decomposable if π1π2 = id or π 1 2 = π 2 2 = id. For the low dimensional cases n ∈ {3, 4}, we give a sufficient and necessary condition on π1, π2 for $${{W_{n,{\pi _1},{\pi _2}}}}$$ to be an entanglement witness. We also show that, for n ∈ {3, 4}, $${{W_{n,{\pi _1},{\pi _2}}}}$$ is decomposable if and only if π1π2 = id or π 1 2 = π 2 2 ; = id; $${{W_{3,{\pi _1},{\pi _2}}}}$$ is optimal if and only if (π1, π2) = (π, π2), where π = (2, 3,1). As applications, some entanglement criteria for states and some decomposability criteria for positive maps are established.