On the temperature dependence of the interaction-induced entanglement

Both direct and indirect weak nonresonant interactions are shown to produce entanglement between two initially disentangled systems prepared as a tensor product of thermal states, provided the initial temperature is sufficiently low. Entanglement is determined by the Peres-Horodeckii criterion, which establishes that a composite state is entangled if its partial transpose is not positive. If the initial temperature of the thermal states is higher than an upper critical value $T_{uc}$ the minimal eigenvalue of the partially transposed density matrix of the composite state remains positive in the course of the evolution. If the initial temperature of the thermal states is lower than a lower critical value $T_{lc}\leq T_{uc}$ the minimal eigenvalue of the partially transposed density matrix of the composite state becomes negative which means that entanglement develops. We calculate the lower bound $T_{lb}$ for $T_{lc}$ and show that the negativity of the composite state is negligibly small in the interval $T_{lb}<T<T_{uc}$. Therefore the lower bound temperature $T_{lb}$ can be considered as \textit{the} critical temperature for the generation of entanglement.