# Quantum Calibration of Photon-Number-Resolving Detectors Based on Multi-Pixel Photon Counters
## 4 Discussion and Conclusions  
We experimentally chose two certain wavelengths of light in the near infrared and the visible bands to calibrate MPPC respectively. On the premise of positional optimization, the initial values of the probability distribution can be reliably measured. Since the initial values have already contained information about unexpected parameters, QDT could calibrate the detector more accurately and completely. In QDT, the reconstruction is completed based on the direct output signals of the detector and MLE. With the method of reconstructing POVM elements, continuous curve of the probability distribution can be predicted from discrete points of experimental data. This kind of calibration maximized the amount of retained information and could be applied to any detector with photon-number-resolving capability, for example in the field of quantum information and metrological applications. Considering different wavelengths, the material of the sensitive surface demonstrates different characteristics. The different quantum efficiencies at the near-infrared and the visible bands result in different signal-to-noise ratios. The decrease of the signal-to-noise ratio leads to the reduction of the photon-number-resolving capability. The larger the average number of photons incident on MPPC is, the more indistinguishable discrete boundaries of the voltage distribution regions are.  
In conclusion, we use the method of QDT to calibrate MPPC, which is more accurate and integrated than the previous methods. In the near infrared and visible bands (791 nm and 523 nm), the POVM elements and associated Wigner functions were calculated and faithful quantum characteristics of MPPC were acquired. According to the calibration, the different characteristics of MPPC at different wavelengths are described and discussed. We believe that the quantum calibration of MPPC as a photon-number resolving detector will inform in-depth researches and applications of MPPC and other single photon detectors.  
Conceptualization, Y.C. ([PERSON]) and E.W.; methodology, Y.C. ([PERSON]), X.C. and E.W.; software, Y.C. ([PERSON]), X.C. and J.M.; validation, Y.C. ([PERSON]); formal analysis, Y.C. ([PERSON]), Y.C. ([PERSON]), X.C., J.M., G.X. and A.X.; investigation, Y.C. ([PERSON]), Y.C. ([PERSON]) and Y.W.; resources, E.W.; data curation, Y.C. ([PERSON]) and Y.C. ([PERSON]); writing--original draft preparation, Y.C. ([PERSON]) and Y.C. ([PERSON]); writing--review and editing, Y.C. ([PERSON]), Y.C. ([PERSON]), and E.W.; visualization, Y.C. ([PERSON]); supervision, E.W.; project administration, Y.C. ([PERSON]) and E.W.; funding acquisition, E.W.  
Figure 6: Cross section of the Wigner function for MPPC for 0 to \(\geq\)3 clicks outcomes at (**a**) 791 nm (red); (**b**) 523 nm (green). The solid lines provide the experimental Wigner function.  
**Funding:** This work is funded by National Natural Science Foundation of China (11722431, 11674099, 11621404); Program of Introducing Talents of Discipline to Universities (B12024); Shanghai International Cooperation Project (16520710600); Natural Science Foundation of Shanghai (16 ZR1409400); Shuguang Program (15 SG22) by Shanghai Education Development Foundation and Shanghai Municipal Education Commission.  
**Acknowledgments:** The authors thank C.D. for the discussion with MPPC settings.  
**Conflicts of Interest:** The authors declare no conflict of interest.