Statistical inference from imperfect photon detection

We consider the statistical properties of photon detection with imperfect detectors that exhibit dark counts and less than unit efficiency, in the context of tomographic reconstruction. In this context, the detectors are used to implement certain positive operator-valued measures (POVMs) that would allow us to reconstruct the quantum state or quantum process under consideration. Here we look at the intermediate step of inferring outcome probabilities from measured outcome frequencies, and show how this inference can be performed in a statistically sound way in the presence of detector imperfections. Merging outcome probabilities for different sets of POVMs into a consistent quantum state picture has been treated elsewhere (Audenaert and Scheel 2009 New J. Phys. 11 023028). Single-photon pulsed measurements as well as continuous wave measurements are covered.