  We generally investigate necessary conditions for the generation of the
long-distance entanglement. We consider a quantum system in which a system
mediates the indirect interaction between two spins, which we refer to as probe
spins. First, we weaken the coupling between each probe spin and the mediator
system to the infinitesimal strength in order to generate the long-distance
entanglement. We give two necessary conditions for the mediator system to
generate the long-distance entanglement. We prove that the indirect interaction
cannot generate the entanglement if it is `classical.' We also give a necessary
condition for the effective fields on the probe spins to satisfy. Second, we
generate the long-distance entanglement by the use of only external fields. We
show that external fields on the adjacent spins to the probes are necessary in
addition to external fields on the probe spins. Finally, we consider the cases
where the coupling strength between each probe spin and the mediator system is
finite. In particular, we show two examples where the external fields on the
mediator system highly enhance the long-distance entanglement.
