Improving single-photon sources via linear optics and photodetection

Triggered single-photon sources produce the vacuum state with nonzero probability, but produce a much smaller multiphoton component. It is therefore reasonable to approximate the output of these photon sources as a mixture of the vacuum and single-photon states with probabilities 1-p and p, respectively. Here we are concerned with increasing the efficiency p by directing multiple copies of the single-photon-vacuum mixture into a linear optical device and applying photodetection on some outputs. We prove that it is impossible, under certain conditions, to increase p via linear optics and conditional preparation based on photodetection. We also establish a class of photodetection events for which p can be improved, although with an added multiphoton component. In addition we prove that it is not possible to obtain perfect single-photon states via this method from imperfect $(p<1)$ inputs.