Simple protocol for secure decoy-state quantum key distribution with a loosely controlled source

The method of decoy-state quantum key distribution (QKD) requests different intensities of light pulses. Existing theory has assumed exact control of intensities. Here we propose a simple protocol which is secure and efficient even there are errors in intensity control. In our protocol, decoy pulses and signal pulses are generated from the same father pulses with a two-value attenuation. Given the upper bound of fluctuation of the father pulses, our protocol is secure provided that the two-value attenuation is done exactly. We propose to use unbalanced beam-splitters for a stable attenuation. Given that the intensity error is bounded by $\pm5%$, with the same key rate, our method can achieve a secure distance only 1 km shorter than that of an ideal protocol with exactly controlled source.