  Being able to implement effective entanglement distribution in noisy
environments is a key step towards practical quantum communication, and
long-term efforts have been made on the development of it. Recently, it has
been found that the null-result weak measurement (NRWM) can be used to enhance
probabilistically the entanglement of a single copy of amplitude-damped
entangled state. This paper investigates remote distributions of bipartite and
multipartite entangled states in the amplitudedamping environment by combining
NRWMs and entanglement distillation protocols (EDPs). We show that the NRWM has
no positive effect on the distribution of bipartite maximally entangled states
and multipartite Greenberger-Horne-Zeilinger states, although it is able to
increase the amount of entanglement of each source state (noisy entangled
state) of EDPs with a certain probability. However, we find that the NRWM would
contribute to remote distributions of multipartite W states. We demonstrate
that the NRWM can not only reduce the fidelity thresholds for distillability of
decohered W states, but also raise the distillation efficiencies of W states.
Our results suggest a new idea for quantifying the ability of a local filtering
operation in protecting entanglement from decoherence.
