Rigorous calibration method for photon-number statistics

Characterization of photon statistics of a light source is one of the most basic tools in quantum optics. Although the outcome from existing methods is believed to be a good approximation when the measured light is sufficiently weak, there is no rigorous quantitative bounds on the degree of the approximation. As a result, they fail to fulfill the demand arising from emerging applications of quantum information such as quantum cryptography. Here, we propose a calibration method to produce rigorous bounds for a photon-number probability distribution by using a conventional Hanbury-Brown-Twiss setup with threshold photon detectors. We present a general framework to treat any number of detectors and non-uniformity of their efficiencies. The bounds are conveniently given as closed-form expressions of the observed coincidence rates and the detector efficiencies. We also show optimality of the bounds for light with a small mean photon number. As an application, we show that our calibration method can be used for the light source in a decoy-state quantum key distribution protocol. It replaces the a priori assumption on the distribution that has been commonly used, and achieves almost the same secure key rate when four detectors are used for the calibration.