Distance dependence of entanglement generation via a bosonic heat bath

Within a generalized Caldeira-Leggett model we analyze the conditions under which a bosonic heat bath can entangle two microscopic quantum systems at a distance $r$. We find that the attainable entanglement is extremely distance-sensitive. Significant entanglement can only be achieved if the systems are within a {\em microscopic} distance that is of order of the cut-off wavelength $\lambda$ of the system-bath interaction. At larger distances the maximal entanglement is exponentially suppressed with a decay length of order $\lambda$. We conclude that entanglement generation via a heat bath is not suitable for entangling remote objects.