Entanglement Witnesses Constructed By Permutation Pairs

Abstract

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 W_n,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 ²₁ = π ²₂ = 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 ²₁ = π ²₂ ; = 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.