Extended reduction criterion and lattice states

Abstract

We study a particular class of states of a bipartite system consisting of two 4-level parties. By means of an adapted extended reduction criterion we associate their entanglement properties to the geometric patterns of a planar lattice consisting of 16 points.

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

“…A particular sub-class of states of two pairs of two qubits, n = 2 and σ µ = σ α ⊗ σ β = σ αβ ∈ M 4 (C), were considered in [12][13][14] and called lattice states. When they are PPT, the entanglement properties of such states cannot be ascertained by standard methods, like for instance the reshuffling criterion [16,22], and new methods had to be devised.…”

“…Indeed, the only candidate point to fulfil the criterion in Proposition 5 is (2, 2); however, it belongs to I. Luckily, a refined criterion [14] based on [8] shows it to be entangled.…”

“…Example 4 Let us now consider the lattice state in Example 21: in [14], it has been shown to be entangled by using the following positive map…”

“…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][8][9][10][11][12][13][14][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 ½ 2 n ⊗σ µ , σ µ = ⊗ n i=1 σ µi , on the totally symmetric state |Ψ 2 n + ∈ C 2 n ⊗C 2 n .…”

“…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 ½ 2 n ⊗σ µ , σ µ = ⊗ n i=1 σ µi , on the totally symmetric state |Ψ 2 n + ∈ C 2 n ⊗C 2 n . 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][13][14] for the case n = 2, that is when the states correspond to normalized projections onto subspaces generated by orthogonal vectors of the form ½ 4 ⊗ σ µ1µ2 |Ψ 4 + ∈ C 16 . 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.…”