  Triggered single-photon sources produce the vacuum state with non-negligible
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. We show that it is impossible to increase
the probability for a single photon using linear optics and photodetection on
fewer than four modes. This impossibility is due to the incoherence of the
inputs; if the inputs were pure-state superpositions, it would be possible to
obtain a perfect single-photon output. In the more general case, a chain of
beam splitters can be used to increase the probability for a single photon, but
at the expense of adding an additional multiphoton component. This improvement
is robust against detector inefficiencies, but is degraded by dark counts or
multiphoton components in the input.
