# Quantum Calibration of Photon-Number-Resolving Detectors Based on Multi-Pixel Photon Counters
## 3 Results
### Povm  
To fully characterize MPPC, QDT was employed to obtain the POVM. In previous theoretical models [36; 40; 41], the loss, the cross talk and the dark counts of the quantum detector have been  
Figure 3: Detector voltage amplitude histogram at 791 nm (**a**,**b**) and 523 nm (**c**,**d**). The inset shows the corresponding photon number distribution histogram. Error bars in the insets were obtained by the standard divisions on five independent measurements. considered separately. However, in the real applications, all these factors affect together on the performance of the detector. By contrast, using QDT, we do not need to consider any information of the under-test detector [38]. All these properties such as the dark counts, the after pulsing, the crosstalk and so forth are involved in QDT process [42]. The reconstruction of POVM has already represented directly the overall relationship between the real known input state and the real detector's output, and this relationship has included the effect of all non-ideal factors. Furthermore, QDT is based on the method called MLE, where no prior assumptions are required [39]. The core of MLE was a kind of iterative algorithm that satisfies Equation (2) as the constraints. The incident light passed through a set of filters served as input quantum state at two different wavelengths. Incident photons ranging from 0.5 to 250 per pulse were collected by adjusting the filters. The digital oscilloscope was connected to the output of MPPC to acquire the information of the output pulses. For simplicity, we chose 0, 1, 2 and \(\geq\)3 photons to analyze. Dividing the counts of different photons in the detector voltage amplitude histogram mentioned above by the total number of pulses, we could get the probability of detected different photons and the experimental probability distribution P. In theory, POVM elements of MPPC can be directly deduced with inversion formula of Equation (8). However, in practice, the inverse operation of formula takes a large amount of computation and may not satisfy Equation (2) as the constraints. We set the POVM matrix as a diagonal matrix since MPPC is phase-independent. The initial value of the diagonal element was the reciprocal of clicks number \(j\). The zero matrix was used to initialize the photon-number distribution matrix of probe coherent states. Then the initial probability distribution P\({}^{\prime}\) was obtained according to Equations (6) and (8). We calculated the square sum of the difference between the experimental probability distribution P and the corresponding initial value P\({}^{\prime}\) of each element and kept iterating to make the difference between P and P\({}^{\prime}\) close to zero. The reconstructed POVM elements are represented in Figure 4.  
According to the reconstructed POVM elements and given quantum states of a input state, the probability distribution can be directly obtained. At different wavelengths, reconstructed POVM elements were used to calculate the output results of MPPC under different average number of incident photons. The calculated results were compared and analyzed with the output results obtained in the experiment, to verify the feasibility and superiority of using QDT method to calibrate spatial multiplexing detectors. As shown in Figure 5, discrete points represent the probability distribution obtained from the experiment, while solid lines represent the probability distribution calculated by the reconstructed POVM elements. The coincidence degree between experimental results and calculation results is high. The fidelity can be used to quantitatively express the degree of coincidence between the reconstructed POVM elements and the actual quantum characteristics of the detector [38]:  
\[F=\sum_{n=0}^{M}\sqrt{P_{exp}\times P_{n}}. \tag{10}\]  
Figure 4: POVM elements of MPPC at (**a**) 791 nm; (**b**) 523 nm. The POVM are presented in three dimensions and truncated at twenty for clarity.