Characterizing quantum detectors by Wigner functions

We propose a method for characterizing a photodetector by directly reconstructing the Wigner functions of the detector's Positive-Operator-Value-Measure (POVM) elements. This method extends the works of S. Wallentowitz and Vogel [Phys. Rev. A 53, 4528 (1996)] and Banaszek and W\'odkiewicz [Phys. Rev. Lett. 76, 4344 (1996)] for quantum state tomography via weak-field homodyne technique to characterize quantum detectors. The scheme uses displaced thermal mixtures as probes to the detector and reconstructs the Wigner function of the photodetector POVM elements from its outcome statistics. In order to make the reconstruction robust to the experimental noise, we use techniques from quadratic convex optimization.