  We study the problem of determining the photon number statistics of an
unknown quantum state by simultaneously measuring conjugate quadratures with
double homodyne detectors. Classically, the sum of the squared outputs of the
two homodyne detectors is proportional to the intensity (thus the photon
number) of the input light. Quantum mechanically, due to vacuum noise, the
above photon number measurement is intrinsically noisy. We quantify the
information gain in a single-shot measurement and discuss potential
applications of this technology in quantum key distribution. We also show that
the photon number statistics can be recovered in repeated measurements on an
ensemble of identical input states without scanning the phase of the input
state or randomizing the phase of the local oscillator used in homodyne
detection.
