Entanglement increase from local interactions which lead to non-positive local reduced dynamics

Consider a bipartite quantum system S=AB such that each part interacts only with its local environment. Under such circumstances, one expects that the entanglement between parts A and B does not exceed its initial value during the time evolution. In fact, this is the case if the reduced dynamics of the system is given by ${\mathcal{E}}_A\otimes {\mathcal{E}}_B$, where ${\mathcal{E}}_A$ and ${\mathcal{E}}_B$ are quantum channels, i.e., completely positive trace-preserving maps. But, the reduced dynamics of the system may be given by a map as ${\mathrm{\Psi }}_A\otimes {\mathrm{\Psi }}_B$, where ${\mathrm{\Psi }}_A$ and ${\mathrm{\Psi }}_B$ are local non-positive maps. Then, the entanglement between A and B can exceed its initial value, as was shown in the case studied by Jordan et al. [Phys. Rev. A 76, 022102 (2007)]. In this paper, we first explore the general circumstances under which one can find such cases as they found. Next, we introduce another general procedure which leads to local non-positive maps that cause entanglement exceeding.