Upper bound on the secret key rate distillable from effective quantum   correlations with imperfect detectors
  We provide a simple method to obtain an upper bound on the secret key rate that is particularly suited to analyze practical realizations of quantum key distribution protocols with imperfect devices. We consider the so-called trusted device scenario where Eve cannot modify the actual detection devices employed by Alice and Bob. The upper bound obtained is based on the available measurements results, but it includes the effect of the noise and losses present in the detectors of the legitimate users. 
Tobias Moroder, Marcos Curty, and Norbert L\"utkenhaus