Investigating a class of $2\otimes 2\otimes d$ bound entangled density matrices via linear and nonlinear entanglement witnesses constructed by exact convex optimization

Here we consider a class of $2\otimes 2\otimes d$ density matrices which have positive partial transposes with respect to all subsystems. The entanglement witness approach is used to investigate the entanglement of these density matrices. To demonstrate the approach, the three-qubit case is considered in detail. For constructing entanglement witnesses (EWs) detecting these density matrices, we attempt to convert the problem to an exact convex optimization problem. To this aim, we map the convex set of separable states into a convex region, named feasible region, and consider the cases in which the exact geometrical shape of the feasible region can be obtained. In this way, various linear and nonlinear EWs are constructed. The optimality and decomposability of some of the introduced EWs are also considered. Furthermore, the detection of the density matrices by the introduced EWs are discussed analytically and numerically.

Figure 1

The boundaries of feasible region for pure product states (dotted curves) which form an astroid and for mixed separable states (line segments) which form a rhombus, for EWs of relation (3.8). The $P_i$’s are dimensionless.