Entanglement Witnesses for a Class of Bipartite States of n × n Qubits

Entangled quantum states are among the most important physical resources in the manifold applications of quantum information theory [1]; from a mathematical point of view, entanglement is closely related to positive and completely positive linear maps on operator algebras. In the following, we shall consider finite dimensional bipartite quantum systems described by Hilbert spaces C n ⊗ C2n : in this case, identifying entangled states is equivalent to sorting out positive maps Λ : M2n(C) 7→ M2n(C), which are not completely positive, such that these states do not remain positive under the action of the map id ⊗ Λ. When n = 1, all entangled states are detected by acting on just one of the two parties with the transposition [2, 3]. In low dimension, all states which remain positive under partial transposition (PPT) are automatically separable for positive maps always involve the transposition [4, 5]. In higher dimension, this is not the case and there can be PPT entangled states [6]. Since a general characterization of positive maps is not available, a better control on both the entanglement of states and the positivity of maps can only come by devising new positive maps that may detect the entanglement of at least particular classes of bipartite states [7–15]. In this spirit, we consider in the following bipartite states that are diagonal with respect to the orthonormal basis generated by the action of tensor products of the form 12n⊗σ~ μ, σ~ μ = ⊗i=1σμi , on the totally symmetric state |Ψ2 n + 〉 ∈ C n⊗C2n . We first characterize the structure of positive maps detecting the entangled ones among them; then, we illustrate the result by examining some entanglement witnesses, already present in the literature [12–14] for the case n = 2, that is when the states correspond to normalized projections onto subspaces generated by orthogonal vectors of the form 14 ⊗ σμ1μ2 |Ψ+〉 ∈ C. Finally, we show how, for this class of states being separable, entangled or PPT entangled are properties related to the geometric patterns of the subsets of 16 square lattice points which identify them.