When is Noisy State Information at the Encoder as Useless as No   Information or as Good as Noise-Free State?
  For any binary-input channel with perfect state information at the decoder, if the mutual information between the noisy state observation at the encoder and the true channel state is below a positive threshold determined solely by the state distribution, then the capacity is the same as that with no encoder side information. A complementary phenomenon is revealed for the generalized probing capacity. Extensions beyond binary-input channels are developed. 
Rui Xu, Jun Chen, Tsachy Weissman, Jian-Kang Zhang
cs.IT math.IT