Noisy Preprocessing and the Distillation of Private States

We provide a simple security proof for prepare & measure quantum key distribution protocols employing noisy processing and one-way postprocessing of the key. This is achieved by showing that the security of such a protocol is equivalent to that of an associated key distribution protocol in which, instead of the usual maximally-entangled states, a more general {\em private state} is distilled. Besides a more general target state, the usual entanglement distillation tools are employed (in particular, Calderbank-Shor-Steane (CSS)-like codes), with the crucial difference that noisy processing allows some phase errors to be left uncorrected without compromising the privacy of the key.