Four signalling states are sufficient to achieve the Holevo capacity for qubit channels, but in many cases are not necessary. There are examples known where the capacity is achieved for two orthogonal input states, two nonorthogonal states, and examples where three states are necessary and sufficient. Many previous results were obtained for a class of channel for which three states are sufficient. In this study, a systematic theory for this class of channel was developed. Simple criteria are presented that, when satisfied, mean that two states are sufficient for the ensemble. When these criteria are satisfied, there is a simple method to determine whether the states in the ensemble should be orthogonal or nonorthogonal. When the criteria are not satisfied, it is still possible that two states are sufficient, though it is possible that three states are necessary. In the case where three states are necessary, the form of the optimal ensemble is predicted. These results provide an efficient method for calculating the Holevo capacity for all channels in this class.
Triggered single-photon sources produce the vacuum state with non-negligible probability, but produce a much smaller multiphoton component. We describe a method for increasing the probability of a single photon via a chain of beam splitters. This method has the drawbacks that it introduces a significant multiphoton component, and it can not be used to increase the probability for a single photon above 1/2. Part of the reason for these drawbacks is the incoherence in the photon sources. If the photon sources produced pure states rather than incoherent superpositions, it would be possible to obtain a perfect single photon output. The method for incoherent inputs is robust against detector inefficiencies, but sensitive to dark counts and the multiphoton component in the input.