Decoy-State Quantum Key Distribution with Arbitrary Phase Mixtures and Phase Correlations

We formulate a general method to find bounds on the statistics of states passing through an unknown channel from the statistics of another set of states. We pay special attention to the application of this method to decoy-state quantum key distribution (QKD) where the states that can be practically prepared are not always the most secure states. In contrast to standard decoy-state analysis, we do not assume that our states are phase-randomised and can consider a fairly general laser source. We also develop a method to accommodate phase correlations with minimal characterisation of the source. Thus, we develop general techniques to deal with phase imperfections in a large class of QKD protocols. We apply these methods to a simple implementation of the three-state protocol and discuss the effects of partial phase-randomisation on the key rate of the protocol