New examples of entangled states on $\mathbb{C}^3 \otimes \mathbb{C}^3$

We build apon our previous work, the Buckley-\vSivic method for simultaneous construction of families of positive maps on $3 \times 3$ self-adjoint matrices by prescribing a set of complex zeros to the associated forms. Positive maps that are not completely positive can be used to prove (witness) that certain mixed states are entangled. We obtain entanglement witnesses that are indecomposable and belong to extreme rays of the cone of positive maps. Consequently our semidefinite program returns new examples of entangled states whose entanglement cannot be certified by the transposition map nor by other well-known positive maps. The constructed states as well as the method of their construction offer some valuable insights for quantum information theory, in particular into the geometry of positive cones.