Temperature dependence of 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-Horodecki 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⩽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 the critical temperature for the generation of entanglement. It is conjectured that above this critical temperature a composite quantum system could be simulated using classical computers.